Progress in Optics, Vol. 13

PROGRESS IN OPTICS V O L U M E XI11

EDITORIAL ADVISORY BOARD

L. ALLEN,

Brighton, England

M. FRANCON,

Paris, France

E. INGELSTAM,

Stockholm, Sweden

K. KINOSITA,

Tokyo, Japan

A. LOHMANN,

Erlangen, Germany

M. MOVSESSIAN,

Armenia, U.S.S.R.

G . SCHULZ,

Berlin, D.D.R.

W. H. STEEL,

Chippendale, N.S. W., Australia

W. T. WELFORD,

London, England

t VOLUME XIII

EDITED BY

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

Contributors

H. P. BALTES, L. M A N D E L W. M. R O S E N B L U M , J. L. C H R I S T E N S E N G. SCHULZ, J. S C H W I D E R M. S. S O D H A , A. K. G H A T A K , V. K. T R I P A T H I W. T. W E L F O R D

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C O N T E N T S O F V O L U M E 1(1961) I. I1.

1-29 THE MODERNDEVELOPMENT OF HAMILTONIAN OPTICS.R . J. PEGIS . . . . WAVE OPTICS AND GEOMERUCAL OPTICS IN OFTICALDESIGN.K . MIYAMOTO 31-66 DISTRIBUTION AND TOTAL ILLUMJNATION OF ABERRATION-FREE 111. THEINTENSITY DIFFRACTION IMAGES. R . BARAKAT 67-108 . . . . . . . . . . . . . . . . . . D . GABOR . . . . . . . . . . . . . . . . . 109-153 IV . LIGHTAND INFORMATION. DIFFERENCESBETWEEN OPTICALAND V. ON BASIC ANALOGIESAND PRINCIPAL 155-210 ELECTRONIC INFORMATION. H . WOLTER. . . . . . . . . . . . . . . . VI . INTERFERENCECOLOR.H. KUBOTA. . . . . . . . . . . . . . . . . . 21 1-251 VII . DYNAMIC CHARACTERISTICS OF VISUALPROCESSES. A . FIORENTINI . . . . . 253-288 ALIGNMENT DEVICES. A . C . S . VAN HEEL . . . . . . . . . . . 289-329 VIII . MODERN

CONT.ENTS O F V O L U M E I1 (1963) I.

RULING. TFSTING AND USE OF OPTICAL GRATINGS FORHIGH-RESOLUTION 1-72 SPECTROSCOPY. G . W . STROKE. . . . . . . . . . . . . . . . . . . . OF DIFFRACTION GRATINGS. J . M . BURCH 73-108 APPLICATIONS I1. THE METROLOGICAL . . . . . . 109-129 NON-UNIFORM MEDIA.R . G . GIOVANELLI 111. DIFFUSIONTHROUGH OF OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS AND BY IV . CORRECTION SPATIALFREQUENCY FILTERING. J . TSUJIUCHI . . . . . . . . . . . . . 131-180 OF LIGHTBEAMS. L . MANDEL . . . . . . . . . . . . . . 181-248 V . FLUCTUATIONS FOR DETERMINING OPTICALPARAMETERS OF THINFILMS. F.ABEL~S249-288 VI . METHODS

C O N T E N T S O F V O L U M E I11 (1964) THEELEMENTS OF RADIATIVE TRANSFER. F. KOTTLER. . . . . . . . . . 1-28 I. P . JACQUINOT AND B. ROIZEN-DOSSIER . . . . . . . . . . 29-186 I1 . APODISATION. OF PARTIAL COHERENCE. H . GAMO. . . . . . . . . 187-332 I11. MATRIXTREATMENT

C O N T E N T S O F V O L U M E IV (1965) I. I1. 111. IV. V. VI . VII .

HIGHERORDERABERRATION THEORY. J . FOCKE. . . . . . . . . . . . 1-36 0. BRYNGDAHL. . . . . . 37-83 APPLICATIONSOF SHEARING INTERFEROMETRY. OF OPTICAL GLASSES. K . KINOSITA . . . . . . . . 85-143 SURFACEDETERIORATION P. ROUARD AND P. BOVSqUET . . . . 145-197 OPTICAL CONSTANTS OF THIN FILMS. . . . . . . . 199-240 THE MNAMOTO-WOLF DIFFRACTION WAVE. A . RUBINOWICZ THEORY OF GRATINGS AND GRATING MOUNTINGS. W .T .WELFORD241-280 ABERRATION DIFFRACTION AT A BLACK SCREEN.PARTI: KLRCHHOFF'S THEORY.F. KOTTLER281-314

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

OFTICALPUMPING. C. COHEN-TANNOUDJI AND A . KASTLER. . . . . . . 1-81 83-144 NON-LINEAR OPTICS.P. S. PERSHAN. . . . . . . . . . . . . . . . . 111. Two-BEAMINTERFEROMETRY. W . H . STEEL. . . . . . . . . . . . . . . 145-197 IV . INSTRUMENTSFOR THE MEASURING OF OPTICALTRANSFER FUNCTIONS. K. MIJRATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199-245 V. FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE INDEX. LIGHTREFLECTION 247-286 R . JACOBSSON. . . . . . . . . . . . . . . . . . . . . . . . . . . DETERMINATION VI . X-RAY CRYSTAL-STRUCTURE AS A BRANCHOF PHYSICAL OPTICS. H . LIPSONAND C. A . TAYLOR. . . . . . . . . . . . . . . . 287-350 VII . THEWAVE OF A MOMNG CLASSICAL ELECTRON. J . PICHT . . . . . . . . 351-370

C O N T E N T S O F V O L U M E VI (1967) I. I1. 111.

N. V.

VI. VII. VIII

RECENT ADVANCES IN HOLOGRAPHY. E . N . LEITHAND J . UPATNIEKS. . . 1-52 SCATTERINGOF LIGHTBY ROUGHSURFACES. P . BECKMANN . . . . . . . . 5x49 MBASURBMENT OF THE SECOND ORDER DEGREE OF COHERENCE. M . F R A N ~ N ANDS. MALLICK. . . . . . . . . . . . . . . . . . . . . . . . . . 71-104 DESIGNOP ZOOM LENSES.K . Y m . . . . . . . . . . . . . . . . . 105-170 SOME APPLICATIONS OF LASERS TO 1NTERFEROMETRY. D . R . HERRIOTT . . . . 171-209 EXPERIMENTALSTUDIES OF I m s m FLUCTUATIONS m L m . J . A . ARMSTRONG AND A . w . SMlTH . . . . . . . . . . . . . . . . . . . . . . 21 1-257 FOURIER SPECTROSCOPY. G. A . VANASSEAND H . SAW . . . . . . . . . 259-330 DIFFRACTIONAT A BLACKSCREEN. PART11: ELECTROMAGNETIC THEORY.F. KOT ~ L E R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331-377

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

MULTIPLE-BEAM INTERFERENCE AND NATURAL MODESIN OPENRESONATORS. G . KOPPELMAN. . . . . . . . . . . . . . . . . . . . . . . . . . 1-66 I1. METHODS OF SYNTHESISFOR DIELECTRIC MULTILAYER FILTERS. E . DELANO AND R.J.PEcrs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67-137 111. ECHOFSAT OPTICAL FREQUENCIES. I . D . h E L L A . . . . . . . . . . . . 139-168 lv. IMAGEFORMATION wnm PARTULLYCOmm LIGHT.B. J . THOMPSON . . 169-230 V . QUASI-CLASSICAL THEORY OF LASERRADIATION. A . L . MIKAELIAN AND M . L . TER-MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1-297 VI . THE~HOTOGIWH~C IMAGE.S. Oom . . . . . . . . . . . . . . . . . 299-358 VII . INTERACTION OF VERYINTENSE LIGHTwm FREEELECTRONS. J . H . EBERLY 359415

C O N T E N T S O F V O L U M E VIII (1970) SYNTHETIC-APERTURE OPTICS. J . W . GOODMAN . . . . . . . . . . . . . 1-50 THEO m c a PWF~RMANCEOF THH HUMAN EYE. G. A . FRY. . . . . . . 51-131 LIGHTBEATING SPECTROSCOPY. H . z. ClJMMINS AND H . L. SWINNEY . . . 133-200 MULTILAYER ANTIREFLECTION COATINGS. A . MUSETAND A . THELEN . . . 201-237 STATJSTICALPROPERTIESOF LASERLIGHT.H . RISKEN . . . . . . . . . . . 239-294 COHERENCETHEORY OF S~URCE-SIZE COMPENSATION IN INTERFERENCE 295-341 MICROSCOPY. T. Y m o m . . . . . . . . . . . . . . . . . . . . . L . LEVI. . . . . . . . . . . . . . . . . . 343-372 VII . VISIONIN COMMUNICATION VIII . THEORYOF PHOTOELECTRON COUNTING. C. L . MEHTA . . . . . . . . . 3 7 3 4 1.

I1. 111. IV . V. VI .

.

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

GASLASERSAND m m APPLICATION TO PRECISE LENGTHMEASUREMENTS. A . L . BLCQM . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-30 11. PICOSECOND LASERPULSES. A . J . DEMARIA. . . . . . . . . . . . . 31-71 THROUGH mw TWULENT ATMOSPHERE. J . W. 111. OPTICALPROPAGATION STROHBWN. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-122 OF OFTICALBIREFRINGENT NETWORKS. E. 0. AMMANN. . . . . 123-177 IV . SYNTHBS~S V . MODELOCKINGIN GASLASERS.L . ALLENAND D . G . C . JONES . . . . . . . 179-234 VI . CRYSTAL OPTla WTH SPATIAL DISPERSION. v. M . AGRANOVICH AND v . L . GINZBURG. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235-280 OF OFTICAL METHODS IN THE DIFFRACTION THEORY OF ELASTIC VII . APPLICATIONS . . . . . . . . . . . . . . . 281-310 WAVES.K . GNIADEK AND J . ~ETYKIEW~CZ

VIII . EVALUATION. DESIGNAND EXTRAPOLATION MJZK-IODSFOR OPTICAL SIGNALS. BASEDON USEOF THE PROLATE FUNCTIONS. B. R . FRIEDEN . . . . . . . . 3 11407

C O N T E N T S O F V O L U M E X (1972) I. I1. 111. IV . V.

144 BANDWIDTH COMPRE~~ION OF OPTICAL IMAGES. T . S. HUANG. . . . . . . 45-87 THEUSEOF IMAGETUBES AS SHUTTERS. R . W .SMITH. . . . . . . . . . . TOOLSOF THEORETICAL QUANTUM OPTICS.M .0.SCULLY AND K .G .WHITNEY 89-1 35 FIELD CORRECTORS FOR ASTRONOMICAL TELESCOPES.C . G . WYNNE. . . . . 137-164 OPTICALABSQRFTION STRENGTH OF DEFECTS IN INSULATORS. D.Y . SMITHAND 165-228 D . L. DEXTER. . . . . . . . . . . . . . . . . . . . . . . . . . . LIGHTMODULATION AND DEFLECTION. E . K . SITTIG. . . . . 229-288 VI . ELASTOOPTIC DETECTION THEORY. C . W . HELSTROM. . . . . . . . . . . . 289-369 VII . QUANTUM

C O N T E N T S O F V O L U M E XI(1973) MASTER EQUATION METHODS IN QUANTUM O m ~ c sG . . S. AGARWAL . . . . 1-76 RECENTDEVELOPMENTS IN FAR INFRARED SPECTROSCOPIC TECHNIQUES. H . YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-122 OF LIGHTAND Acousnc SURFACE WAVES.E . G . LEAN . . . . 123-166 111. INTERACTION WAVFS IN OPTICALIMAGING. IV . EVANESCENT 0. BRYNGDAHL . . . . . . . . 167-221 V . F~ODUCTION OF ELECTRON PROBES USINGA FIELDEMIILSSION SOURCE. A . V. CREW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223-246 VI . HAMILTONIAN J . A . ARNAUD . . . . 247-304 THFDRYOF BEAMMODEPROPAGATION. VII . GRADIENT INDEXLENSES.E . W . MARCHAND . . . . . . . . . . . . . . 305-337

I. I1.

C O N T E N T S O F V O L U M E XI1 (1974) I. I1. 111. IV. V.

VI .

SELF.FOCUSING.SELF.TRAPPING. AND SELF-PHASE MODULATION OF LASER . SVELTO . . . . . . . . . . . . . . . . . . . . . . . . . 1-51 BEAMS.~ SELF-INDUCED TRANSPARENCY. R . E . SLUSHER. . . . . . . . . . . . . 53-100 MODULATION TECHNIQUES IN SPECTROMETRY. M . HARWIT.J . A . DECKERJR. 101-162 INTERACXION OF LIGHTWITH MONOMOLECULAR DYELAYERS. K . H .DREXHAGE163-232 THE PHASE TRANSITION CONCEPT AND COHERENCE IN ATOMICEMISSION.R . GRAHAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233-286 BEAM-FOIL SPECTROSCOPY. S. BASHKIN. . . . . . . . . . . . . . . . 287-344

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PREFACE The review articles that are being presented in this volume cover a broad range of topics and it is hoped that many optics workers will find in it something of special interest to them. Three of the articles deal with radiation theory or with the interaction of radiation and matter; more specifically these articles are concerned with the question of validity of Kirchhoff s law of heat radiation under non-equilibrium conditions, yith possible alternatives to quantum electrodynamics and with self-focusing of laser beams in plasmas and in semi-conductors. Readers interested in physiological optics will find in the present volume a review of spherical aberration measurements of the human eye. More traditional areas of optics are represented by articles that deal with interferometric testing of smooth surfaces and with aplanatism and isoplanatism. Since the publication of the previous volume of this series,\Professor A. Rubinowicz, an esteemed member of the Editorial Advisory Board of PROGRESS IN OPTICS from its inception, passed away. Professor Rubinowicz’ significant contributions, especially in the areas of selection and polarization rules for electric dipole radiation, the Zeeman effect of electric quadrupole lines and the boundary wave theory of diffraction are well known. What is generally perhaps not so well known is that Professor Rubinowicz helped to train - and did so with great devotion - many physicists in his native Poland. He formed an active research school which has made lasting contributions to theoretical physics, especially in electromagnetic theory. Some of these researches are described in the second edition of his monograph Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, which covers a much broader field than its title suggests and which is undoubtedly one of the finest textbooks available on scalar and electromagnetic diffraction theory. Professor Rubinowicz was admired not only for his considerable scientific achievements but also for his selflessness and integrity. He was held in great affection by all who knew him. EMILWOLF Department of Physics and Astronomy University of Rochester, N . Y., 14627 November 1975

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CONTENTS I . ON THE VALIDITY OF KIRCHHOFF’S LAW OF HEAT RADIATION FOR A BODY IN A NONEQUILIBRIUM ENVIRONMENT by H . P . BALTES(Zug. Switzerland) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . FUNDAMENTAL LAWSAND DEFINITIONS . . . . . . . . . . . . . . . . . . . . 2.1 Kirchhoffs law for a body in equilibrium environment. . . . . . . . . . . 2.2 Einstein’s concept of radiative energy exchange . . . . . . . . . . . . . . 2.3 Absorptivity and emissivity ambiguously defined . . . . . . . . . . . . . 3. STIMULATED EMISSION TREATED AS NEGATIVE ABSORPTION. . . . . . . . . . . . 3.1 The concept of net absorption . . . . . . . . . . . . . . . . . . . . . 3.2 Kirchhoffs law for a weakly absorbing freely radiating body . . . . . . . . 3.3 The transmission of a weakly absorbing hot body - re-interpretation of an experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Net absorption and spontaneous emission for freely radiating metals . . . . . 4. STIMULATED EMISSION NOT CONSIDERED AS NEGATIVE ABSORPTION 4.1 Induced absorption and “total” emission for a freely radiating metal . . . . . 4.2 The deviations from Kirchhoffs law predicted by Ashby and Shocken . . . . 4.3 Re-examination of the results of Ashby and Shocken . . . . . . . . . . . 4.4 The proper thermodynamic definition of absorptivity and emissivity . . . . . 5 . SOMEEXPERIMENTAL RESULTS. . . . . . . . . . . . . . . . . . . . . . . . 6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES.

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

3 4 4 6 7 9 9 10

13 15 17 17 18 20 21 22 23 24 24

I1. THE CASE FOR AND AGAINST SEMICLASSICAL RADIATION THEORY by L. MANDEL (Rochester. N.Y.) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

EFFECT. . . . . . . . . . . . . . . . . . . . . . . . 2. THE PHOTOELECTRIC 3. RELATIONBETWEEN SEMICLASSICAL THEORIES OF PHOTODETECTION AND Q.E.D. . EMISSION OF LIGHT ACCORDING 4. SPONTANEOUS 5 . RESONANCE FLUORESCENCE . . .

TO NEOCLASSICAL THEORY.

. . . . . . . . . . . 6. FLUORESCENCE EFFECTSIN MULTI-LEVEL ATOMS. . . . 7. POLARIZATION CORRELATIONS IN AN ATOMIC CASCADE . . 8. MOMENTUM TRANSFER EXPERIMENTS. . . . . . . . . .

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29 30

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39 43 50

52

54 59

CONTENTS

XI1

9 . INTERFERENCE EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . 10. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

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

66

65

111. OBJECTIVE AND SUBJECTIVE SPHERICAL ABERRATION MEASUREMENTS O F THE HUMAN EYE by W . M . ROSENBLUM and J . L. CHRISTENSEN (Birmingham, Alabama)

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . OF THE OPTICAL ELEMENTS OF THE EYE . 2. THE ANATOMY 2.1 Cornea . . . . . . . . . . . . . . . . . . . . 2.2Aqueous . . . . . . . . . . . . . . . . . . . . 2.3 Crystalline lens . . . . . . . . . . . . . . . . . 3. THEBASICCONCEPTS OF SPHERICAL ABERRATION . . .

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

INTRODUCTION TO THE MEASUREMENT OF THE SPHERICAL ABERRATION OF 4 . HISTORICAL THEEYE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. SUBJECTIVE ABERRATION MEASUREMENTS OF THE EYE. . . . . . . . . . . . . . ABERRATION MEASUREMENTS OF THE EYE . . . . . . . . . . . . . . 6. OBJECTIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. CONCLUSIONS REFERENCES .

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71 72 72 76 76 76 77 79 86 89 90

IV . INTERFEROMETRIC TESTING OF SMOOTH SURFACES

by G . SCHULZand J . SCHWIDER (Berlin) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

TESTING BY COMPARING Two SURFACES . . . . . . . . . . . . . . . 2. RELATIVE

96

2.1 Determination of the deviation sums . . . . . . . . . . . . . . . . . . . 2.2 Determination of the deviation differences . . . . . . . . . . . . . . . . . 2.3 The use of a null lens . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Enhancement of sensitivity . . . . . . . . . . . . . . . . . . . . . . . 2.5 The measurement of interference patterns . . . . . . . . . . . . . . . . . 3. ABSOLUTE TESTING BY COMPARING SEVERAL SURFACES. . . . . . . . . . . . . 3.1 Testing flats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Testing spherical surfaces . . . . . . . . . . . . . . . . . . . . . . . 3.3 Testing aspheric surfaces . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Solutions applying uniformly to the whole surface . . . . . . . . . . . . . 4. COMP~RING A SURFACE WITH ITSELF. . . . . . . . . . . . . . . . . . . . . 4.1 Shearing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Point reference methods . . . . . . . . . . . . . . . . . . . . . . . . 5. COMPARING A SURFACE WITH A HOLOGRAM . . . . . . . . . . . . . . . . . . 5.1 Comparing with a hologram produced by interference . . . . . . . . . . . 5.2 Comparing the surface with a computer-generated hologram as master . . .

99 105 106 108 115 118 119 126 131 134 140 141 144 146 146 150

6.

~ M E S Y S T E M A T I C ~ U R C E S O F E R R O R A N D L I M I T S O F M E A S U R E M E N T.

R E ~ R E N .c ~.

. . . . . . 157

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NOTESADDEDIN PROOF. . . . . . . . . . . . . . . . . . . . SUPPLEMENTARY

162 166

xm

CONTENTS

V . SELF FOCUSING O F LASER BEAMS IN PLASMAS AND SEMICONDUCTOIRS by M . S. SODHA, A . K . GHATAK and V . K . TRIPATHI (New Delhi) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.

171 175 Effective dielectric constant . . . . . . . . . . . . . . . . . . . . . . . 175 Pondermotive force . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Heating of carriers by a Gaussian EM beam in slightly and fully ionized gases . 178 Heating of carriers in parabolic and nonparabolic semiconductors . . . . . 181 Redistribution of carriers and expressions for field dependent dielectric constant 185 2.5.1 Collisionless plasma (pondermotive mechanism) . . . . . . . . . . 190 2.5.2 Strongly ionized plasma (R -z 1, thermal conduction predominant) . . 190 2.5.3 Slightly ionized plasma (R B 1, collisional loss predominant) . . . . . 190 191 2.5.4 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 2.5.5 n-type indium antimonide . . . . . . . . . . . . . . . . . . . . . 2.5.6 Indium antimonide (both types of carriers) . . . . . . . . . . . . . 193 Nonlinearity in the dielectric constant of a magnetoplasma . . . . . . . . . 194 2.6.1 Nonlinear dielectric constant of a collisionlessmagnetoplasma: pondermo197 tive mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Nonlinear dielectric constant of a collisional magnetoplasma: R B 1 . 199

PHENOMENOLOGICAL THEORY OF

2.1 2.2 2.3 2.4 2.5

2.6

FIELDDEPENDENT DIELECTRIC CONSTANT . . . .

3. KINETIC THEORY OF FIELDDEPENDENT DIELECTRIC CONSTANT . . . . . . . . . . 203 3.1 Heating and redistribution of carriers by a Gaussian EM beam in a slightly ionized plasma and a parabolic semiconductor . . . . . . . . . . . . . . . . . 203 3.2 Nonlinearity in the dielectric constant of a magnetoplasma . . . . . . . . . 209 4. STEADY STATE SELFFOCUSING OF EM BEAMS IN PLASMA. . . . . . . . . . . . 4.1 Self focusing in a nonlinear isotropic medium . . . . . . . . . . . . . . 4.1.1 Collisionless plasma . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Collisional plasma: collisional loss . . . . . . . . . . . . . . . . 4.1.3 Fully ionized plasma: conduction loss . . . . . . . . . . . . . . . 4.1.4 Parabolic semiconductors (e.g. Ge) . . . . . . . . . . . . . . . . 4.1.5 Nonparabolic semiconductors (e.g. InSb) . . . . . . . . . . . . . 4.2 Magnetoplasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Collisionless magnetoplasma . . . . . . . . . . . . . . . . . . . . 4.2.2 Weakly ionized magnetoplasma : collisional loss . . . . . . . . . . 4.2.3 Strongly ionized magnetoplasma: thermal conduction loss . . . . . .

5. NONSTEADY STATE SELFFOCUS!NG. . . . . . . . . . . 5.1 Linear part of current density . . . . . . . . . . . . 5.2 Nonlinear current density: no redistribution of carriers 5.3 Nonlinear current density: redistribution of carriers . 5.4 Nonlinear propagation: self distortion of plane waves 5.5 Nonsteady self focusing . . . . . . . . . . . . . .

. 213 . 213 217

. 220 . 223 . 225 .

221 229 232 . 233 . 235 238

. . . . . . . . . . 238 . . . . . . . . . . . 239 . . . . . . . . . . . 240 . . . . . . . . . . . 242

. . . . . . . . . .

246

6. GROWTH OF INSTABILITY . . . . . . . . . . . . . . 6.1 Growth of instability in a plane wavefront . . . . . . . . . . . . . . . . 6.2 Growth of instability in a Gaussian beam . . . . . . . . . . . . . . . . 6.3 Growth of a Gaussian perturbation over a plane uniform wavefront . . . . .

249 249 256 260

7. EXPERIMENTAL INVESTIGATIONS ON SELF FOCUSING . . . . . . . . . . . .

261

REFERENCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262

XIV

CONTENTS

VI . APLANATISM AND ISOPLANATISM by W . T . WELFORD(London) 1. INTRODUCTION . .

. . . . . . . . . . . . . . . . . . 2. THEABBESINECQNDITION . . . . . . . . . . . . . . . 3. AXULISOPLANATISM . . . . . . . . . . . . . . . . 3.1 The Staeble-Lihotzky condition . . . . . . . . . . 3.2 Conrady's theorem . . . . . . . . . . . . . . . 3.3 Linear coma as an optical path aberration . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

269 . . . . 271 . . . . . . . . . 273 . . . . . 274 . . . . . 275

. . . . . . . . . . . 277 . . . . . . . 277

3.4 Linear coma as ray aberration or wavefront aberration . . . . 3.5 Some different definitions of axial isoplanatism . . . . . . . 3.6 Isoplanatism at varying magnification . . . . . . . . . . . 4 . ISOPLANATISM WITH NO AXISOF SYMMETRY . . . . . . . . . 4.1 The Smith optical cosine law . . . . . . . . . . . . . 4.2 The most general isoplanatism theorem . . . . . . . . 4.3 Off-axisisoplanatism in a symmetrical optical system

. . . . . . 5 . ISOPLANATISM IN HOLOGRAPHY . . . . . . . . . . . . . . . . . . REFERENCEs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

279 282 283 284 285 287 289 291

ADDENDUM

292

AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVE INDEX . VOLUMES I-XI11 . . . . . . . . . . . . . . . .

293 299 304

E. WOLF, PROGRESS IN OPTICS XI11 - NORTH-HOLLAND 1976

I ON THE VALIDITY OF KIRCHHOFF’S LAW OF HEAT RADIATION FOR A BODY IN A NONEQUILIBRIUM ENVIRONMENT * BY

H. P. BALTES Zentrale Forschung und Entwicklung, Landis & Gyr Zug AG, CH-6301 Zug, SwitzerlanP*

* This report was written under sponsorship of the National Bureau of Standards, Washington. D. C. 20234, and is therefore not subject to copyright. ** The article was written during the author’s stay at the Department of Physics, Faculty of Science, University of Waterloo, Waterloo, Ontario, Canada.

CONTENTS PAGE

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

3

6 2. FUNDAMENTAL LAWS AND DEFINITLONS . . . . .

4

Q 3 . STIMULATED EMISSION TREATED AS NEGATIVE ABSORPTION., . . . . . . . . . . . . . . . . . . . .

9

Q 4. STIMULATED EMISSION NOT CONSIDERED AS NEGATIVE ABSORPTION . . . . . . . . . . . . . . . .

17

6

22

5 . SOME EXPERIMENTAL RESULTS .

. . . . . . . . . .

Q 6 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

23

Q 7. ACKNOWLEDGEMENTS . . . . . . . . . . . . . . .

24

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

24

0

1. Introduction

Kirchhoff’s law of heat radiation states that the emissivity of radiating bodies is equal to their absorptivity. The law was derived for bodies in thermodynamic equilibrium. In particular, this condition implies thermal equilibrium between the radiating body and the radiation field in its environment: the body is supposed to be in a blackbody radiation field characterized by the body’s uniform temperature. From Einstein’s microscopic interpretation of the radiation process it was sometimes concluded that the total emission of a body is the sum of induced (or stimulated) emission and spontaneous emission (ASHBYand SHOCKEN [19651, MCCAFFREY [19661). In many experimental situations, the body is not in thermal equilibrium with the radiation field; e.g. in emission spectroscopy the sample usually shows a much higher temperature than that of the surrounding radiation field or that of the detector. For such a “freely radiating” body, the radiation field seen by the surface of a strongly absorbing, opaque sample (e.g. a metal) is different from the equilibrium field as required above. Thus the environment would lead to a different amount of stimulated emission and accordingly to a total emissivity different from the one found in thermal equilibrium. In this way, it has been concluded that Kirchhoff s law does not apply when the sample of strongly absorbing material and the radiation field surrounding it are characterized by different temperatures, and the temperature dependence of the according deviation from Kirchhoffs law has been calculated (ASHBYand SHOCKEN [1965], MCCAFFREY [1966]). On the other hand, Kirchhoff s law seems to remain valid in the case of nonequilibrium if induced emission is considered as negative absorption, as was first shown for freely radiating weakly absorbing bodies (e.g. rarified gases), WEINSTEIN p19601, BAUER[19693. Here, absorptivity is redefined as induced absorption minus induced emission, whereas emissivity is understood as being due to spontaneous emission only. Thus, in the heat radiation literature, the words absorptivity and emissivity have both been used in a different sense - a confusion that has been 3

4

K I R C H H O F F ’ S L A W O F HEAT R A D I A T I O N

[I,

92

the source of controversial conclusions. According to a recent analysis (BURKHARD,LOCHHEAD and PENCHINA [1972]), the preferable thermodynamic definition of absorptivity and emissivity is the one that treats stimulated emission as negative absorption, since this definition deals entirely with experimentally accessible quantities : for the measurable emissivity and absorptivity, Kirchhoff s law remains valid for a freely radiating body, provided that the material quantum states of the body obey the equilibrium distribution. The above controversy is reviewed in this report. The problem in question is relevant for the spectroscopy and radiometry of solids and liquids in the temperature and wavelength range where induced emission cannot be neglected, i.e. at high temperatures and/or long wavelengths such that exp (- hc/AkT)is not small compared to unity. In the case of opaque bodies, the validity of Kirchhoffs law allows us to calculate the emissivity from measured values of the reflectivity and vice versa. In section 2, the fundamental radiation laws are briefly recollected, emphasizing the underlying physical concepts and definitions. In section 3 we report some of those papers where stimulated emission is treated as negative absorption and where the validity of Kirchhoff s law is maintained. Furthermore, the interpretation of an experiment measuring the infrared transmission of weakly absorbing samples at high temperature is reexamined. In section 4,we present the results of the quantum theory of the emissivity due to electrons in metals as well as the conclusions occurring if stimulated emission is not considered as negative absorption. These conclusions are discussed together with the proper thermodynamic definitions of emission and absorption. Some experimental results concerning Kirchhoff s law are reviewed in section 5.

0

2. Fundamental Laws and Definitions

2.1. KIRCHHOFF’S LAW FOR A BODY IN EQUILIBRIUM ENVIRONMENT

Studying the spectra of flames, Kirchhoff 1859 observed that radiating bodies absorb light of just the same wavelength that they emit (KIRCHHOFF [1898] pp. 610). Restricting his interest initially to bodies emitting and absorbing only one narrow spectral band, he proved that the ratio of the emitted power and the absorptivity is the same for all bodies at the same temperature*, a result based on the second principle of thermodynamics and on geometrical optics. * “. . , dass fur Strahlen derselben Wellenliinge bei derselben Temperatur das Verhaltnis des Emissionsvermogens zum Absorptionsvermogen bei allen Korpern dasselbe ist” (KIRCHHOFF [1898] pp. 6-7).

1,

§ 21

FUNDAMENTAL LAWS A N D DEFINITIONS

5

In a more comprehensive paper (KIRCHHOFF [18981 pp. 1 1-36) the proof was generalized for arbitrary wavelengths and polarizations and the concept of the blackbody was introduced. The resulting theorem can be summarized as follows. For any body in (radiative) thermal equilibrium with its environment, the ratio between the spectral emissive power E(v, T )and the spectral absorptivity a(v, T ) for a given frequency v and a given temperature T is equal to the spectral emissive power EBB(v,T ) of the blackbody for the same frequency and temperature, E(v, T)/a(v,T ) =

EBB(v,

T).

(1)

The emissive power (“Emissionsvermogen”)E is defined as the energy flux per unit time, unit area, and unit solid angle. The absorptivity (“Absorptionsvermogen”) a is the fraction of incident radiation that the body absorbs.* Only narrow bundles in a narrow frequency band were considered by Kirchhoff. He carefully listed all the prerequisites needed, a few of which are still worth being cited. (i) The radiation emitted by the body is independent from the environment. (ii) The body radiates into the empty space. (iii) The radiating body is inside a cavity with non-transparent walls. These walls have the temperature of the body. (iv) The wavelengths occurring are infinitesimally small compared to any occurring lengths. The requirement (i) is most noteworthy. Taken literarily, it would imply that the emissive power allows exclusively for spontaneously emitted radiation. The condition (iii) implies that the body is surrounded by a blackbody radiation field characterized by the body’s temperature. This requirement is not fulfilled in many experimental situations and was the starting point of the controversy we are going to report here. We observe that the restriction (iv) eliminates any experimental arrangement where geometrical optics is invalid. This condition can be relevant for far-infrared emission spectroscopy (BALTES and K N E U B ~ H[19721) L or for samples [1968]). showing a rough surface (GEIST[1972], HUNTand VINCENT For the case where the requirement (ii) is not fulfilled, but where the space into which the body radiates is filled with a non-absorbing material

* For convenience, we denote dimensionless quantities by small, non-dimensionless quantities by capital letters. As we are concerned here with the spectral quantities,we shall drop the word “spectral” as well as the variable v from now on.

6

KIRCHHOFF’S L A W OF HEAT R A D I A T I O N

[I,

02

of refractive index ii, Kirchhoff derived the result

E = AEBB/E2,

EBB= E2EBB.

In this report, we assume ii = 1 (empty space). Introducing the emissivity

Kirchhoff s law reads e = a.

(4)

According to the energy conservation principle, the sum of absorptivity, reflexivity and transmittivity is unity, a+r+t

For the opaque body with t

= 0 we

e

=

1.

(5)

obtain

=

1-r.

(6)

This relation is sometimes referred to as Kirchhoff s law for opaque bodies and is of great practical interest for spectroscopic applications. The reader interested in the detailed derivation of Kirchhoff s law should consult the literature (PRINGSHEIM [1901], HILBERT [1912, 19131, PLANCK [1913]). For further reference let us write down the universal function EBB(v,T> established by Planck :

2.2. EINSTEIN’S CONCEPT OF RADIATIVE ENERGY EXCHANGE

Based on thermodynamics and geometrical optics, Kirchhoff s theory is phenomenological and does not describe the radiation energy transfer on an atomistic scale. A microscopic interpretation of the radiation phenomena was given more than 50 years later when Einstein introduced the following [19171) : assumptions (EINSTEIN (i) By emission of radiation energy, molecules can go over to a lower quantum state independently from whether they are excited by some surrounding radiation field or not, i.e. spontaneously. This process is originally called “Ausstrahlung” in Einstein’s paper. (ii) The energy of a molecule exposed to a radiation field can be changed by an energy transfer from the surrounding radiation field to the molecule. The energy transfer can be positive or negative, depending on the phase of the molecular resonator and the oscillating field. This process, considered

I,§

21

7

FUNDAMENTAL LAWS A N D DEFINITIONS

by Einstein as one and the same physical phenomenon for both signs, is originally called “Einstrahlung”. However, the above two types of stimulated processes contributing to the “Einstrahlung” are often considered as if physically different. Thus the interaction of light and matter is usually discussed in terms of three, not two, processes : (i) Spontuneous emission : Molecules spontaneously emit photons. This process is identical with Einstein’s “Ausstrahlung”. It is the only process compatible with the independence from environment of the emissivity as required by Kirchhoff. (ii) Induced or stimulated absorption: The radiation field induces the molecule to increase its energy by removing a photon of the appropriate frequency from the field. This is the positive sign part of Einstein’s “Einstrahlung” . (iii) Induced or stimulated emission : Photons already existing in the radiation field stimulate the molecule to decrease its energy by emission of a photon of the same frequency into the field. This is the negative sign part of Einstein’s “Einstrahlung”. We emphasize that Einstein considers induced emission and induced absorption as one and the same process, where just the sign of the energy transferred is different. Hence it is not unlikely to consider induced emission as negative absorption. It is only in the wellknown final energy balance used for the derivation of Planck’s radiation law that spontaneous and induced emission appear on the same side of the equation, because they have the same sign :

Pn exp (- E,/kT)B;p

=

P, exp (- ~ , / k T ) ( B k p+ A:)

(8)

where E, and E, > E, denote the energies ofthe higher and lower quantum B, and states with €,-en = hv, pn and p , the respective degeneracies, A: the celebrate Einstein coefficients, and where p denotes the energy density of the radiation field, which comes out to be Planck‘s spectral density. The above equation, however, does not necessarily imply that the physically measurable emissivity has to be defined as the sum of spontaneous and stimulated emission.

x,

2.3. ABSORPTIVITY A N D EMISSIVITY AMBIGUOUSLY DEFINED

We have learned above that the absorptivity a by definition is the fraction of absorbed incident radiation intensity, and that the emissivity e by definition is the ratio of the body’s emission rate compared to that of a black-

8

K I R C H H O F F ' S LAW OF H E A T R A D I A T I O N

[I,

§2

body at the same temperature. As pointed out recently (BURKHARD, LOCHPENCHINA [1972]), these definitions can be used ambiguously. As we shall expound below (see sections 3 and 4), two different uses of the words absorption and emission are popular in the literature. Definition Z: total emission is the sum of induced emission and spontaneous emission. Total absorption is just the induced absorption. Definition ZZ: total emission is just spontaneous emission. Total absorption or net absorption is induced absorption minus induced emission ; i.e. induced emission is negative absorption. The question now is whether Kirchhoff s law postulates

HEAD,

el : = es+ei

=

a, = : a'

(9)

or e11 : = es = a,--. 1

1

=

:a".

(10)

We would expect that both relations equivalently hold in the case of equilibrium, but what about the freely radiating body, where e, depends on the environmental radiation temperature? And which definition describes the experimental situation properly? We have seen above that the definition I1 seems to be favoured by the physical concepts backing both Kirchhoffs' and Einstein's results. A more detailed discussion is given in section 4. Following the relation (6), the above two options can be carried over to the definition of the (specular) reflectivity of an opaque body (t = 0). Then the first principle of thermodynamics leads to r

=

(1 1)

1-a.

According to the definitions (9) and (10) we find

+ = I . - a-

-

I-a,

=

I-(es+ei)

=

1-e'

(12)

1 - e 11,

(13)

and +I

=

1-a"

=

l + e .I - a . 1 = l - e ,

=

respectively. We might call r1 the net rejlexion and rI1 = rl+ei the gross rejlexion including net reflexion plus induced emission. We point out that the definition 11 is consistent with the requirement rlI = 1 + e , - a , 5 1 because a, 2 e, is always granted by equation (8) as exp ( - E,/kT) >= exp ( - E,/ICT) for E, 2 E, in thermal equilibrium. We finally mention the notions of the absorption coefficient c1 and emission coefficient & useful in the case of transmitting homogeneous materials. They are specific quantities defined as follows: aZdx is the amount of

I , § 31

EMISSION TREATED AS NEGATIVE A B S O R P T I O N

9

absorbed incident radiation Z after moving a distance dx, and ddvdt d VdQ is the amount of energy emitted in the frequency range v . . . v + dv by matter in the volume element d V in time dt into the solid angle dQ. In these terms, Kirchhoffs law reads B

=

aE,,

(14)

or, defining & with respect to the radiation density of the blackbody, E = &/EBB, the law reads E = a. We observe that a is the quantity well known from Bouguer’s law I(d) = I, exp (-ad)

(15)

describing the transmission of a slab of thickness d. The above options I and I1 can of course be transferred to a and E . fj 3. Stimulated Emission Treated as Negative Absorption 3.1. THE CONCEPT OF NET ABSORPTION

In the introduction of this chapter we described the conclusion that Kirchhoffs law cannot be valid for freely radiating bodies because of a lack of induced emission due to the absence of the equilibrium radiation [19601 claimed that such a conclusion field. Already 15 years ago WEINSTEIN is not correct. By considering what happens in a simple system, Weinstein demonstrates that Kirchhoff s law is valid as long as the distribution of the material states of the system is the equilibrium distribution, and that Kirchhoffs law is, in this sense, independent of the state of the radiation field. Weinstein finds that the behaviour of the induced emission is irrelevant to the validity of Kirchhoffs law, provided that the “proper” account of the effect of induced emission is taken in calculating the absorptivity : one must regard induced emission as negative absorption. The same result was independently obtained by BAUER[19691. Weinstein considers only the case of a nonscattering, nonreflecting, freely radiating body which is large enough and whose absorption coefficient is small enough that elementary geometrical optics is adequate to describe the behaviour of the radiation field. For a pencil of radiation moving through the body by a distance dx, gains and losses in specific intensity Z (radiation energy per unit volume) are balanced by the equation of transfer dl

=

-aIdx+bdx

(16)

10

K I R C H H O F F ’ S L A W OF H E A T R A D I A T I O N

[I,

53

where 01 and d denote the absorption and emission coefficient, respectively, as defined at the end of section 2.3. Calculating 01 and 8, one must decide where the induced emission (proportional to I> goes. Weinstein proposes to regard stimulated emission as negative absorption because “the induced emission is coherent with the inducing radiation and is emitted in the same direction as the inducing radiation, so that when we measure the decrease in the intensity of a beam passing through a given thickness of matter, what we find is the difference between the decrease in intensity arising from transitions in which a quantum is removed and the increase in intensity arising from transitions in which a quantum is induced to be radiated”. “Thus”, Weinstein continues, “the increase in the induced emission in thermodynamic equilibrium over that for a body radiating freely has nothing to do with the emissivity of the body, but only reflects the fact that in thermodynamlc equilibrium we have a greater amomt of radiation incident on the body so that, of course, a greater amount is transmitted”. Weinstein concludes that the true emission coefficient arises entirely from spontaneous emission processes and consequently is completely determined by the distribution over the material states and by the spontaneous transition probabilities connecting them. For Kirchhoff s law to hold it is hence sufficient that the material states of the body obey the equilibrium distribution (going along with uniform temperature of the body), whereas the temperature characterizing the radiation field is immaterial. We notice that the physical concept reported above is almost identical to Einstein’s way of describing the decrease and increase of intensity due to removing photons from and inducing photons to the field, i.e. the process called “Einstrahlung” described above (see section 2.2). We believe that the above argument is substantially correct and can as well be transferred to the case of opaque bodies with non-zero reflectivity : measuring the reflected intensity, we measure the non-absorbed intensity (defined e.g. as the amount of power not converted into heat), i.e. we measure the induced emission along with the reflexion. The argument can be extended to the general case where we have to allow for both reflexion and transmission. 3.2. KIRCHHOFF‘S LAW FOR A WEAKLY ABSORBING FREELY RADIATING BODY

Let us consider the simple system of a uniform slab of a dilute gas and

§ 31

EMISSION TREATBD AS NEGATIVE ABSORPTION

11

two non-degenerate molecular or atomic states of energies and E~ > with the respective occupation numbers n , and n2 obeying the Boltzmann coefficients [1917] B: = B: = B and A = distribution, with the EINSTEIN ~ ~ v ~ c and - ~ with B , transitions involving the frequency v = ( E ~ EJ~. Collecting together absorption and induced emission, one finds the netabsorption coefficient

a

=

const B(n, - n 2 )

=

ao(l -n2/n1)

(17)

instead of a. = const Bn,

(18)

allowing for induced absorption only. The emission coefficient arising from spontaneous emission is

8

=

(19)

const An,.

Following EINSTEIN’S paper [19171, the constant is easily evaluated and one obtains

d

=

a2hv3c-’[exp (hv/kT)- 1]-’

=

“EBB.

(20)

This is Kirchhoff s law in terms of the emission and absorption coefficient. This result is easily re-written in terms of emissivity and absorptivity by solving the equation of transfer (16) for a slab of thickness d and an angle of incidence 8 (WEINSTEIN [1960]). One obtains

e

=

1- exp (- ad/cos 6)

(21)

as required by Kirchhoffs law since

a

=

1- exp (- crd/cos e)

(22)

is the absorptivity compatible with Bouguer’s law (15). In terms of the definitions introduced in section 2.3 we thus have shown that ,I1

=

all,

(23)

i.e. spontaneous emissivity equals net-absorptivity. In this notation, we have to identify 1 - exp (- a. d ) with a’. We notice that the simple calculation indicated above simultaneously yields (i) Planck‘s formula (this was Einstein’s way to derive it); (ii) Bouguer’s law I/Io = J -a = exp ( - ad) (for normal incidence, e = 0); (24) (iii) the temperature dependence of the net-absorption coefficient,

12

KIRCHHOFF’S L A W OF H E A T R A D I A T I O N

a =

with a.

K

go[ 1 - exp

(- hv/kT)]

[I, §

3

(25)

n, , hence a a [I1 -exp (-hv/kT)]’

for a two-level system. The result (i) means that the blackbody radiation field is met with inside the body. The result (iii) means that CI decreases with temperature not only because there are fewer atoms in the ground state E, ,but also because there are more atoms in the excited state E, . Apparently unaware of Weinstein’s paper, BAUER [19691 reproduced the main features of the above calculation. He emphasizes that the emission coefficient should be a quantity characterizing the radiating material and hence must be defined in a manner independent from environmental radiation. This seems to be in the spirit of Kirchhoff s concepts (see section 2.1). Bauer suggests to accept Kirchhofl’s law in the form of equation (20). He discards the formula that is valid only for black radiation or if stimulated emission can be neglected. Bauer recommends to no longer use the notation of “Kirchhoff s law” for the formula d = a. EBB,because the underlying physical theorem is already contained in Milne’s principle of detailed balance. Bauer points out the role of the net-absorption in the extinction laws due to Bouguer, Lambert, and Beer. He discusses the difference between a and a. and stresses that both are equal only in the limit of low temperatures and for short wavelengths. If exp (- hv/kT) is not small compared to unity, we have to use the exact Kirchhoff and extinction laws € = aEBB and Z/Zo = exp (- old), respectively, including the net-absorption a instead of go, in the very same manner that we have to use Planck‘s formula instead of Wien’s approximation valid for hv z+ kT. The exact laws can be replaced by the laws using a0 for the wavelength and temperature range described by

if an error up to 1 % is accepted. Thus, the difference between u and a. has ,to be accounted for in the infrared and/or at high temperatures. Bauer mentions that the concept of the net-absorption coefficient can as well be applied to situatons where the material system itself is not in thermal equilibrium and is compatible with laser theory, where the amplification factor can be defined as just ( - a), a being negative for n, > n , .

1,

§ 31

EMISSION T R E A T E D AS N E G A T I V E A B S O R P T I O N

13

3.3. THE TRANSMISSION OF A WEAKLY ABSORBING HOT BODY - REINTERPRETATION OF AN EXPERIMENT

An experiment meeting the requirements of the calculations reviewed above was carried through by LELESand NEFF[1968]. They measured the infrared transmission of the molecular crystal copper phtalocyanine, highly diluted in a transparent pellet, as a function ofthe sample temperature The wavelength and temperature range of their investigations was

3 300Kpm 5 TI1 5 16OOOKpm,

(28)

i.e. in a region where according to Bauer's result (27) induced emission can no longer be neglected and the net-absorption coefficient a is different from the induced absorption coefficient a*. In order to interpret their measurements, Leles and Neff solve the equation of transfer in the spirit of Weinstein and Bauer introducing the coefficients u = hvc-'@(ni-nj) (29) & = hvc-1Ai.n. J J

(30)

where i denotes the ith quantum level of the molecule, which later is identified with the ith level of a harmonic oscillator. For the transmission experiment, Leles and Neff calculate the following emerging intensity : I = (&/a)[1 -exp ( - old)]

+ Zo exp ( -ad)

(31)

where I , denotes the (normally) incident radiation intensity and where d denotes the thickness of the sample. For zero spontaneous emission, the first term vanishes and Bouguer's law is obtained. For zero incident radiation, only the first term is retained : it represents the spontaneous emission of the sample, reduced by self-absorption. In order to compare the net-absorptivity a = l-exp(-ud)

(32)

with the measured apparent reduction of intensity aexp= 1 -z/zo

(33)

including both absorption and spontaneous emission, Leles and Neff rewrite the result (31) as (a- a,,,)/a

=

(&/a)/Io

(34)

where the finite band width is accounted for by integration, replacing a by Sa(v)dv.

14

KIRCHHOFF'S L A W OF H E A T R A D I A T I O N

[I,

Strange enough, the authors are not aware of Kirchhoff s law €/M leading to

I

=

EBB[l-exp (-ad)] + I , exp (-ad)

§3

= EBB

(35)

and (a - a,,,)/a

=

EB$Z,

cc [exp (hv/kT)- 11-

',

(36)

but introduce the following assumptions :, (i) As only fundamental transitions are measured, they argue, M (and a fortiori a) is temperature independent. (ii) b, however, is temperature dependent and can be approximately calculated in terms of the corresponding matrix element of the harmonic oscillator. (iii) a or the corresponding integral over the absorption band can be evaluated once for all at some very low temperature. Averaging over the oscillator levels, Leles and Neff calculate & cc [exp ( h v / k T ) - 11-l

(37)

and actually arrive at the relation (36) which they find to agree fairly well with their experimental data. Hence they claim to have shown experimentally that the harmonic approximation is reasonable even at very high temperatures, and that the absorption coefficient M is reasonably constant over the entire temperature range considered (300 K 5 T 5 800K). We think that these conclusions are not correct. The assumption (i) is not necessarily justified because this would mean to neglect induced emission, although hv B kT is not fulfilled (see relations (25), (27) and (28)). Furthermore, the occupation number of the ground state is not independent from temperature, as 3c

n1 cc 1 -

C exp [ ( E ~ - E ~ ) / ~ T ] . i=2

Even if we would admit that M does not vary appreciably with temperature, according to Kirchhoffs law this would just mean that €/EBBdoes not vary with temperature and that in this way the dependence of the right side of (34) and (36) would be reproduced. The authors overlooked that their prediction of the temperature dependence of the right side of (36) is nothing but a consequence of Kirchhoffs law and is henceforth independent of the level structure of the material system. By their very nature, the experimental results of Leles and Neff cannot prove or disprove the validity of the harmonic approximation

1.

§ 31

EM I SSI ON T R E A T E D AS N E G A T I V E A B S O R P T I O N

15

at high temperatures, but could at most be considered as a demonstration of Kirchhoff's law for a system of the type which EINSTEIN [1917], WEINSTEIN [19601, and BAUER[1969] had in mind, provided that a(T) z a(1ow T) can be taken for granted. We still have to check the authors procedure of replacing a ( r ) by a(1ow T ) on the left side of equation (34). Let us consider only the error implied by neglecting induced emission. Froln Weinstein's result (25) we know that ./ao z 1 - exp (- 14 400K pm/TA).

(39)

For TA = 10 OOOK pm, typically in the range of the measurements of Leles and Neff, this would mean ct/ct, 0.86. This result has to be inserted into

-a_ - -a l l a,

a'

-

1- [exp (- ct, d)]"'"~ l-exp(-a,d) '

where its effect can still be small provided that a, is not too far from unity, i.e. provided that sufficiently thick samples are measured. E.g. for a, = 0.8, we have exp (-sod) = 0.2 and therefore a/ao = (1 -0.2"/"0)/0.8= 0.94. Thus the procedure is feasible in principle, but has to be carefully checked in detail. It is possibly a good approximation for the 900 cm-' band measured by Leles and Neff, but is doubtful for e.g. their 728 cm-' band showing (1 -Z/l,)max z 0.6 at room temperature. As a always decreases with increasing temperature, one should start at the lowest temperature with an almost opaque sample. We mention that in the far infrared, ct is appreciably different from a, already at room temperature. 3.4. NET ABSORPTION A N D SPONTANEOUS EMISSION FOR FREELY RADIATING METALS

Developing an interpretation of his normal spectral emissivity data for metals in terms of the electronic states of the material, THOMAS [1970] used the Weinstein theory of radiative energy transfer in gases as one of his starting points. He considers absolutely freely radiating bodies, i.e. bodies of uniform temperature without any kind of incident radiation. Whereas reflexion of the body's surface can be neglected in the case of dilute gases, it plays a fundamental role in the case of metals. Einstein, Weinstein and Bauer study thermal equilibrium in the interior of the body, where field oscillators or molecules or atoms interact with the interior radiation field, which is shown to be a blackbody radiation field. The emission is controlled only by self-absorption : the normal emissivity

16

KIRCHHOFF’S L A W OF HEAT R A D I A T I O N

[I,

03

reads e = l-exp(-Nd)

(41)

with a = net-absorption coefficient and d = thickness of the slab. If the body is sufficiently thick, ad % 1, it looks black. The radiative properties are completely described by the absorption coefficient and the size. One way of visualizing a body with non-zero reflectivity r = r(v, T ) is to accept a blackbody radiation field prevailing in the interior, but admit that this field is not idealy impedance-matched to the outside vacuum by means of the reflecting surface. Hence the body does no longer look black. The interior blackbody field would then be the sum of the radiation produced and the radiation reflected back from the surface, hence e + Y = 1. Thomas, however, takes a different approach. He solves the equation of transfer (16), assuming that in equilibrium the radiation emitted into the outside vacuum is equal to the radiative power produced inside. In the same manner as Weinstein, he obtains (42)

d I = -ctIdx+aLdx

where a denotes the net-absorption coefficient and where 0rL is the emission coefficient due to spontaneous transitions. L denotes an emission power, L

=

2hv3c-2(n,/n,- i)-i

(43)

where n , and n , are the occupation numbers of electronic states. However, n l / n , is not necessarily equal to exp (hv/kT) as it was for the field oscillator occupation numbers, but this ratio depends on the electronic level structure of the solid. Assuming the boundary condition Z = 0 for x = 0 (no incident radiation), the integration of equation (42) yields I ( x ) = L[ 1- exp (- ax)].

(44)

For x + co,deep in the interior, I = L. Coming from inside to the surface, I decreases because of the radiation lost by emission. In thermal equilibrium we expect that the same amount of radiation is produced deep in the interior that is emitted outside : (45)

Emat= L.

On the other hand, by definition,

Emat= eE,,

=

e2h~’c-~[exp(hv/kT)-

11-l.

(46)

1,

P 41

17

E M I S S I O N N O T C O N S I D E R E D AS N E G A T I V E A B S O R P T I O N

From (43), (45), and (46) one concludes e

=

[exp (hv/kT)- l](n,/n, -

(47)

For a metal, we expect e < 1 because the ratio of the electronic state OCcupation numbers is in general different from that of the field oscillators in a blackbody. For a metal, nJn2 is connected with the density of states and the Fermi statistics. Thus the relation (47) is fundamental for the microscopic calculation of the emissivity of a metal. The above result does not change if we assume that the sum of the intensity in the material and the reflected intensity yields the intensity of the blackbody : from Zrefl Zmat =,Z we conclude I,,, = (1 -Zrefl/ZBB)ZBB. According to the definition of the emissivity we have e = 1 -Zren/ZBB = 1- (IBB - Zmat)/ZBB= Zma,/ZBB, which is in agreement with (47). Thus, Thomas’ approach is compatible with the concept of the non-ideal impedance match or partial transmittance of the surface. In terms of the relation (47), Thomas successfully compares his data with known results of the theory of electronic bands. The measured emissivity e reflects the deviation p = (n,/n,) exp (hv/kT),0 < p < 1, of the electronic level occupation from the one of the field oscillators. For our problem of Kirchhoffs law we learn that the concept of netabsorption can well be extended to strongly absorbing materials like metals.

+

6

4. Stimulated Emission not Considered as Negative Absorption

4.1. INDUCED ABSORPTION AND “TOTAL” EMISSION FOR A FREELY

RADIATING METAL

In papers on the many-body theory of the electrons in solids aiming at the calculation of the electronic contribution to the dielectric constant and to the absorptivity, stimulated emission is usually not considered as a negative contribution to the absorption (some references are given in BURKHARD,LOCHHEAD and PENCHINA [I19723). In quantum optics the time dependence of the electromagnetic field appears in the form exp ( fiot) where the term proportional to exp (- iot) gives rise to absorption in the pertubation theory. Then the term proportional to exp (iot), which is usually omitted, would describe the corresponding induced emission. The above separation of induced absorption and induced emission can, however, not be achieved experimentally: the real field should have the time dependence cos o t a exp( - iot) + exp (iot) where both terms are considered simultaneously. As in these calculations one usually wants to

18

KIRCHHOFF'S LAW OF HEAT RADIATION

[I,

9: 4

determine the low-temperature optical properties (ho>) kT), one is allowed to completely neglect the stimulated emission and the above distinction is pointless. It plays a role, however, if the high temperature optical properties, including emissivity, are calculated. Thus, induced emission is accounted for in the investigation of THOMAS 119701 reported above (section 3.4). As we have seen, (Thomas considers stimulated emission as negative absorption. A different approach was made by ASHBYand SHOCKEN [1965] and MCCAFFREY [1966] when developing the quantum theory of the emissivity of metals based on the appropriate electron wave functions. They considered a metal sample of temperature T, in an environment of blackbody radiation characterized by the temperature T, with T, not necessarily equal to T, . They defined total emissivity as the sum of the contributions due to spontaneous and induced emission and assumed that the absorptivity is due to induced absorption only, i.e. they adopted the definition I given in section 2.3. For T, # T, they predicted a deviation from unity of the ratio of absorptivity and emissivity, i.e. of a'/e*in the sense of section 2.3. Well aware of the fact that their result was at variance with the one of WEINSTEIN [19601, Ashby and Shocken tried to explain this disagreement by pointing out that a weakly absorbing body may be stimulated by the radiation emitted deep within the body itself, whereas the photons causing induced emission in a metal come from outside. 4.2. THE DEVIATIONS FROM KIRCHHOFFS LAW PREDICTED BY ASHBY

A N D SHOCKEN

For the rate of absorption A of photons of frequency v from the environment, per unit surface area, per unit frequency interval and unit solid angle, Ashby and Shocken obtain A

=

A' = nhvD(IM12)

(48)

where IZ = n(v, 7') = [exp(hv/kT)- 11-' is the number of photons per field oscillator of frequency v, D = D(v) is the density of states, i.e. the number of field oscillator per unit frequency range, per unit solid angle, with the polarization, direction of propagation, and frequency of the incident beam, and where ( l M I 2 ) is the average over thermal fluctuations of the squared matrix element between the initial state and the final state of the sample. For details on the perturbation Hamiltonian the reader should consult ASHBY and SHOCKEN [1965] or the more comprehensive article by MCCAFFREY [19663. Assuming incident blackbody radiation of the en-

1,

o 41

EMISSION N O T C O N S I D E R E D AS N E G A T I V E A B S O R P T I O N

19

vironmental temperature T I , the occupation number n in equation (48) reads n

=

n(T) = [exp(h~/kT)-l]-~.

(49)

From (48) the absorptivity

a

=

a'

= A'/chvDn(T,) = (IMIZ)/C

(50)

is calculated. For the rate of emission defined in the analogous manner and including both induced and spontaneous emission, Ashby and Shocken deduce from their perturbation theory the result E = E' = (n+l)hvD(JMIZ)exp (-hv/kTJ,

(51)

where the factor exp (- hv/kT,) accounts for the fact that for a solid at temperature T, there are exp (-hv/kT,) fewer electrons in the upper state capable of emitting photons, as compared with these electrons in lower states capable of absorbing photons. The term proportional to n in (51) describes the emission induced by environmental photons, hence (49) applies here again, n = n(Tr). From (51) the emissivity is calculated by comparison with a blackbody showing the sample temperature T, : e

=

e' = E'/chvDn(T,) = (n(T,)+l)exp(-hv/kT,)(IMJZ)/cn(T,).(52)

From (50) and (52) one obtains

+

e'/a' = (n(TJ 1) exp ( - hv/kT,)/n (T,)

(53)

e' - n(T,)+1 l-exp(-hv/kq) a' n( T,) 1 1-exp ( - h v / k ~ , )'

(54)

or -

+

Thus e'ja' is equal to unity for T, = T, only. For a few typical values of TJT, the "correction factor" of Kirchhoffs law was plotted as a function of wavelength by MCCAFFREY[1966]. For example, T, = 2000 K and TI = 10K with 1= 8 pm would yield el/a' = 0.63 according to (54). The respective correction for the ratio of the total emissivity and absorptivity was calculated and plotted ASHBY BY and SHOCKEN [1965]. Apparently, e'/a' -+ 1 for both hv B kTr and hv > kT,, i.e. in the limit of Wien's radiation law where induced emission is completely negligible. If meant as a prediction for measurable quantities, the above result is clearly at variance with the papers discussed in section 3.

20

KIRCHHOFF'S L A W O F H E A T R A D I A T I O N

CI,

§4

4.3. RE-EXAMINATION OF THE RESULTS OF ASHBY AND SHOCKEN

GRIMM[1970] and later BURKHARD, LOCHHEADand PENCHINA [1972] re-examined the above results. Grimm noticed that the separation of induced absorption and induced emission is the crucial step in the calculation of Ashby and Shocken. Leaving everything unchanged except replacing the definition I by the definition I1 where stimulated emission is considered as negative absorption, G r i m recovers eI1/a1' = 1 from the aboveequations. The absorption rate becomes A"

(55)

n(T,)hvD(IM(2)[1-exp (-hv/kT,)]

=

rather than (48), and the emission rate is now

E" = hvD( ]MI2)exp (- hv/kT,)

(56)

instead of (51). This yields uII

=

c-l

(IM12>[1 -exp (-hv/kT,)I,

(57)

where the factor n( TI) has cancelled : incident and net absorbed radiation are both characterized by the same temperature TI. The emissivity reads el1 = c - ~ < I M exp ~ ~ (-hv/kT,)/n(T,) )

(58)

and now depends only on T,, because it allows for spontaneous emission only. From (57) and (58) one easily concludes eI1/u1' = [exp (-hv/kT,)/n(T,)]/[l =

-exp (-hv/kT,)]

exp (-hv/kT,)[exp (hv/kT,)- 1]/[1 -exp (-hv/kT,)]

=

1,

(59)

as required by Kirchhoffs law. Thus we learn that the electron theory of Ashby and Shocken is compatible with the theories of WEINSTEIN [1960], BAUER[1969], and THOMAS [1970]. On the other hand, Weinstein pointed out that he would not have obtained the result (23) if he had not considered stimulated emission as negative absorption. The above considerations were reproduced by Burkhard, Lochhead and Penchina. They noticed that ,I1 and el are related by el' = el/[l+n(T,)] = [l-exp(-hv/kT,)]e'

(60)

and that ul'

= .I/[

1 + n(

x)]

=

[1 - exp ( - hv/kT,)]u'.

(61)

We have learned that Ashby's and Shocken's result (54) is entirely due

1 . 8 41

EMISSION N O T C O N S I D E R E D A S NEGATIVE ABSORPTION

21

to their definition of e and a. Furthermore, the calculation of the total photon flux received from an opaque sample by a detector tuned in spectral response to see only the mode under consideration, namely both the radiation emitted and the background radiation reflected, yields the same prediction for either type of definition, as was shown by BURKHARD,LOCHHEAD and PENCHINA [1972]. 4.4. THE PROPER THERMODYNAMICS DEFINITION OF ABSORPTIVITY A N D EMISSIVITY

Although both definitions I and I1 lead to the same prediction concerning the total energy received by a photodetector, the definition I1 is the preferred thermodynamic definition, as pointed out by Bauer, Burkhard, Lochhead and Penchina. The reason is that experimentally we cannot separate induced emission from the measured absorption : measured absorption is always “true” (i.e. induced) absorption less induced emission. On the other hand, either definition I or definition I1 may be adopted, provided that it is used consistently. Using the definition I, the environment dependent ratio d/a’ results as a function of T, and T, , which is equal to unity only if T, = T, . This does, however, not mean that Kirchhoff s law is valid for the case T, = T, only, but rather indicates that d and d are not the same quantities as those a p pearing in Kirchhoffs proposition. As we have seen in section 2.1, already Kirchhoff seems to have been aware of the appropriate operational definition by requiring that the radiation emittedby the body is independentfrom environment. In other words, those quantities that always give the ratio 1 and those quantities that can be measured, are d’ and a” : treating stimulated emission separately from induced absorption would involve separating cos ot into exp (- i o r ) and exp (ior) - a process which physically cannot be achieved, as Burkhard, Lochhead and Penchina point out, although it is mathematically simple. The theory of the ratio of emissivity and absorptivity developed by Ashby and Shocken was based on definition I. The operational definition occurring in an experiment measuring absorption or emission corresponds to definition 11. It is therefore inconsistent to use definition I in the calculation of e/a and to use definition I1 in the prediction of the experimental result implied by the calculation. This inconsistency would lead to the erroneous conclusion that the measured emissivity of radiating bodies depends on the background radiation as implied in the results of ASHBY and SHOCKEN [19651 and MCCAFFREY [19661.

22

K I R C H H O F F ' S L A W OF H E A T R A D I A T I O N

0

[I,

05

5. Some Experimental Results

As we have shown above, the result (54), when consistently interpreted, does not predict experimental results at variance with Kirchhoffs law. Experiments set up in order to prove or disprove the relation (52) valid for $/a' are doomed to fail, because they will always end up with $'/a" = 1. We briefly report some newer experiments. (i) KETELAAR and HAAS[19561measured the polarized infrared emission and reflexion spectra of opaque calcite slabs at 300°C for various crystal orientations in the region of the 880 cm-' and 1500 cm- reflexion bands. They observed that in the anisotropic material orientation and polarization effects strongly influence the emission spectrum, but that Kirchhoff s law in terms of e + r = 1 remains valid for any specified orientation and polarization. The values of exp (- hv/kT) corresponding to the above measurements are 0.1 1 and 0.024, hence the negative induced emission was 11 % and 2.4% of the induced absorption a' (see eq. (61)). (ii) LELESand NEW[19681measured the infrared absorptivity of copper phtalocyanine in order to demonstrate the validity of the harmonic approximation at high temperatures. As we have shown in section 3.3, their experiment cannot provide the wanted information, but can be considered as a demonstration of Kirchhoff s law in nonequilibrium environment. In the temperature and wavelength range studied by Leles and Neff, the negative induced emission is appreciable. (iii) HISANO[1968] measured the infrared emission spectra of LiF single crystals and thin slabs for various temperatures, emission angles, and slab diameters in order to verify the theory of optical modes in samples of finite thickens due to Fucm and KLIEWER [1966]. Furthermore, Hisano compares these normal emissivity data measured at 420 K with the normal reflectivity data obtained for the same material at the same temperature by JASPERSE, WAN, PLENDLand MITRA [1966]. Hisano observes that Kirchhoff s law in terms of e + r = 1 holds well in the region of the reststrahlen reflexion of the LiF crystal, but a deficiency e + r % 0.9 seems to occur near 800 cm-' where r is almost equal to zero, i.e. near the frequency of the longitudinal optical mode. We believe that the reported deviation is not larger than the experimental error involved in the measurements, in particular as results obtained by two different groups with different techniques are compared. The infrared emission experiment is particularly subject to stray radiation, as was carefully analysed by ULIN [1970], and furthermore a good black reference source is not easy to achieve in the inand STETTLER [1972]). frared spectral region (BALTES

'

1,

0 61

CONCLUSIONS

23

(iv) GRIMM[19701 recently measured the hemispherical total emittance of various metallic test surfaces in different environments. Within the accuracy of the experimental technique ( f2 %), he observed the same emitted power for the freely radiating samples and for the same samples in thermal equilibrium. Grimm therefore re-examined the theory of Ashby and Shocken by using the concept of net absorption and found that Kirchhoff s law for e*'/al' is reproduced from their equations [see section 4.3).

5

6. Conclusions

Already from studying the original concepts of KIRCHHOFF[1898] and EINSTEIN [1917] it can be inferred that the quantities compared in Kirchh o p s law are the spontaneous emission and the net absorption (defined as induced absorption minus induced emission). The emissivity is exclusively due to spontaneous emission and is consequently independent of the environmental radiation field. Therefore we expect Kirchhoff s law to hold as well for freely radiating bodies provided that the distribution of the material states of the sample is the equilibrium distribution. This result was explicitly verified for diluted systems showing no reflectivity (WEINSTEIN [19601, BAUER[1969]), and the underlying concept of net absorption was successfully extended to metals (THOMAS [1970]). GRIMM[1970] and BURKHARD, LOCHHEADand PENCHINA [1972] have shown that the devia[I9651 and tions from KirchhofPs law predicted by ASHBYand SHOCKEN MCCAFFREY [1966] for the nonequilibrium case are entirely due to the inconsistent use of a non-operational, though mathematically acceptable, definition of the emissivity (including both spontaneous and induced emission). The quantities thus defined are, however, not accessible by experiment. Therefore it has to be borne in mind that measured absorptivity (for opaque bodies often in terms of one minus reflectivity) is net absorptivity which is related by anet= [1 - exp (- hv/kT)]ain, to the induced absorptivity in the case of opaque bodies. In the case of transparent bodies, the according net-absorption coefficient unet= [1- exp (- hv/k7')]aj,, has to be inserted in order to obtain the correct version of Bouguer's law. The 50% point, where anet= +aind,is at about TA = 21000Kpm. Thereisof course no difference between the two types of definitions and accordingly between anetand aind in the case of high frequencies and low temperatures (hv B kT), where induced emission can be neglected. Thus the theory of the optical properties of metals at low temperatures will not lead to wrong predictions by accounting for induced absorption only. This procedure,

24

KIRCHHOFF’S LAW

OF H E A T R A D I A T I O N

[I

however, can lead to errors at high temperatures if induced absorption is inconsistently identified with the operational net absorption.

9

7. Acknowledgements

It is a pleasure to thank Professor R. K. Pathria for the hospitality extended to the author at the Department of Physics of the University of Waterloo, Waterloo, Ontario, Canada. Much of the material for this report was compiled while the author was a Visiting Professor at the I. Mathematisches Institut, Freie Universitat Berlin, West-Berlin, Germany. The collaboration of B. Sengebusch, then student at the Freie Universitat Berlin, is greatfully acknowledged. The author would like to thank Jon Geist, Heat Division, Institute of Basic Standards, National Bureau of Standards, Washington, D.C., who suggested the compilation of this chapter, for his hospitality and for many helpful discussions. References ASHBY,N. and K. SHOCKEN, 1965, Symposium on Thermal Radiation of Solids, ed. S. Katzoff (NASA SP-55, U.S. Government Printing Office, Washington, D.C.) p. 63. 1972, Helv. Phys. Acta 45,481. BALTES, H. P. and F. K. KNEUBOHL, BALTES. H. P. and P. STEITLER,1972. Proc. 5th ESLAB/ESRIN Symposium, eds. V. Manno and J. Ring, Astrophysics and Space Science Library 30, 160 (D. Reidel h b l . Corp., Dordrecht, The Netherlands). BAUER.A., 1969, Optik 29, 179. D. G.. J. V. S. LOCHHEAD and C. M. PENCHINA. 1972, American J. Phys. 40.1794. BURKHARD, EINSTEIN, A., 1917, Physik. Zeitschr. 18,121 ;first published in:MitteilungenderPhysikalischen Gesellschaft Zurich Nr. 18 (1916). FUCHS, R. and R. L. KLIEWER, 1966, Phys. Rev. 150, 537. GEIST.J., 1972, J. Opt. SOC. Am. 62, 602. HILBERT, D., 1912, Physik. Zeitschr. 13, 1056. HILBERT, D.,1913, Physik. Zeitschr. 14,592. T. C., 1970, AIAA Paper No. 70-858, AIAA 5th Thermophysical Conf., Los Angeles. GRIMM, HISANO,K., 1968, J. Phys. SOC.Japan 25, 1091. 1968, J. Geophys. Res. 73,6039. HUNT,G. R. and R. K. VINCENT, JASPERSE, J. R., A. KAHAN, L. N. PLENDL and S. S. MITRA,1966, Phys. Rev. 146, 146. KXLIN,R., 1970, Diplomarbeit, ETH Zurich. KETELAAR, J. J. A. and C. HAAS,1956. Physica 22, 1283. KIRCHHOFF, G., 1898, Abhandlungen iiber Emission and Absorption, ed. M. Planck (Verlag von Wilhelm Engelmann, Leipzig). The original papers appeared in : Monatsberichte der Academie der Wissenschaften zu Berlin, Dec. 1859, and: Poggendorfs Annalen 109 ( 1 860) 275. LELES,B. K. and V. D. N~m,.1968,J. Chem. Phys. 48.3557. MCCAFFREY, J. W., 1966, Theoretical Study of the Radiative Emissivity of Metals. Part I, Technical Summary Report, Contract No. NAS8 - 5210, prepared by: P.E.C. Research Associates Inc., Boulder, Colorado, for: George C. Marshall Space Flight Center, Huntsville, Alabama..

11

REFERENCES

25

PLANCK, M.,1913, Theorie der Warmestrahlung (2nd ed., Joh. Ambrosius Barth, Leipzig; reprint: Dover Publishers. New York. 1959). PRINGSHEIM. E.. 1901, Verh. d. D. Phys. Ges. 3. 81. THOMAS, L. K., 1970, Phys. Stat. Sol. 41, 681. WEINSTEIN, M. A,, 1960, American J. Phys. 28, 123.

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E. WOLF, PROGRESS IN OFTICS XI11 0 NORTH-HOLLAND 1976

I1

THE CASE FOR AND AGAINST SEMICLASSICAL RADIATION THEORY * BY

L. MANDEL Department of Physics and Astronomy, University of Rochester, Rochester. N . Y . 14627, U S A .

* This work was supported in part by the National Science Foundation and in part by the Air Force Oftice of Scientific Research.

CONTENTS PAGE

1 . INTRODUCTION

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

29

Q 2 . THE PHOTOELECTRIC EFFECT . . . . . . . . . . .

30

Q

5

3 . RELATION BETWEEN SEMICLASSICAL THEORIES OF PHOTODETECTION AND Q.E.D. . . . . . . . . . . .

39

Q 4 . SPONTANEOUS EMISSION OF LIGHT ACCORDING TO NEOCLASSICAL THEORY . . . . . . . . . . . . . .

43

Q 5 . RESONANCE FLUORESCENCE . . . . . . . . . . .

50

0

6. FLUORESCENCE EFFECTS IN MULTI-LEVEL ATOMS .

52

Q 7. POLARIZATION CORRELATIONS IN AN ATOMIC CASCADE . . . . . . . . . . . . . . . . . . . . . .

54

9 8 . MOMENTUM TRANSFER EXPERIMENTS . . . . . .

59

9

61

4

9. INTERFERENCE EXPERIMENTS . . . . . . . . . . .

10. CONCLUSION . . . . . . . . . . . . . . . . . . . .

65

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

66

0

1. Introduction

Although it is now 70 years since EINSTEIN[1905] introduced the idea that light consists of photons, and nearly 50 years since DIRAC[1927] published the first paper on the quantization of the electromagnetic field, several questions have recently been raised regarding the validity of quantum electrodynamics (Q.E.D.) in the domain of optics. While the impressive successes of Q.E.D. at higher energies are generally acknowledged by all, both the evidence and the need for a quantum theory of radiation in the optical domain have been questioned by some. The critique stems partly from some dissatisfaction with the mathematical framework of Q.E.D., the appearance of divergences and the fact that the electromagnetic field cannot strictly be treated within the confines of Hilbert space, and partly from a sense of dissatisfaction with the concep tual framework underlying the theory. Some people feel that Q.E.D. is less amenable than most theories to intuitive arguments, and that, at least within the domain of optics, photons, despite their stability, are more transcendent than other particles. Since photons cannot be localized in position precisely, because there is no dynamical variable corresponding to a position operator, they are often treated entirely within momentumspin space, and this contributes a certain elusivenessto the description of the electromagnetic field in terms of photons. But another, altogether different reason for the recent scepticism about Q.E.D. is the rediscovery of the power of semiclassical methods. For several experimental results that had long been accepted as evidence for the quantum nature of radiation, turn out, on closer examination, to be explicable in terms of a classical electromagnetic field. As a result, several alternatives to conventional Q.E.D. have recently been explored. They include the so-called neoclassical theory of Jaynes and his co-workers (JAYNES and CUMMINGS [1963], CRISPand JAYNES [1969], S m o m a n d J A Y N E S [ ~ ~J ~A~Y] N , E S [ ~ ~seealsoL~NVr[1973])in ~~]; which the electromagneticfield is described classically and expressed directly in terms of quantum expectations of the dynamical variables of the source, 29

30

SEMICLASSICAL R A D I A T I O N T H E O R Y

[II? §

2

although they are interpreted not as expectation values, but as actual values in the theory. By allowing the radiation to react back on the source, neoclassical theory is able to account for the process of spontaneous emission of an excited atom, and for certain related phenomena. Other alternative theories are the zero-point field theories (MARSHALL [1963, 19653, BOYEM [1969, 1970, 1973, 19741; see also DE LA €'ERA-AUERBACH and CETTO [1974], which contains some references to earlier work) in which the electromagnetic field is again treated classically, but is assumed to be supplemented throughout space by a universal, randomly fluctuating zero-point field, like the vacuum field of Q.E.D., which is taken to be the boundary value of the free field solution of Maxwell's equations. Finally, there are the socalled source-field theories (SERIES[1969], NESBET [1971], LAMAand MANDEL [1972, 19731, which are semiclassicalin a different sense, in which the free field is discarded, and the electromagneticfield is expressed directly in terms of the dynamical variables of the source. These source-field theories have not been developed very far, and we shall concentrate on the better known semiclassical theories in this review. Of the various semiclassical approaches, neoclassical theory is the most internally self-consistent and divergence-free theory. Moreover, its predictions frequently differ from those of Q.E.D., and can therefore be put to the test, whereas much of the work on zero-point field theory has been aimed at reproducing the results derived by Q.E.D. In the following we shall therefore tend to emphasize the evidence relating to neoclassical theory, although some of this evidence relates equally to all semiclassical theories.

0

2. The Photoelectric Effect

Perhaps the most familiar piece of alleged historical evidence for the quantum nature of electromagnetic radiation is the photoelectric effect, which was cited by EINSTEIN [1905] in his paper leading to the introduction of the photon concept. It would seem then that a full treatment of the photoelectric effect requires the apparatus of quantum electrodynamics. Actually, the photoelectric effect was already analyzed in some detail by WENTZEL [1926] before the development of the quantum theory of radiation, and the fact that it can be analyzed without quantization of the field has been repeatedly rediscovered in recent years (MANDEL,SUDARSHAN and WOLF[19643, LAMBand SCULLY [19691). Thus, if we allow an electron in some bound state of negative energy E, < 0 at time t to interact with a classical electromagnetic field at r, and

11,

4 23

31

THE PHOTOELECTRIC EFFECT

we calculate the probability P,(r, t)At that, under the influence of a perturbation in the form of a quasimonochromatic wave of midfrequency oo and complex* amplitude V(r, t ) , the electron is inducedto make a transition in a short time At (At B l/w, but At -sz coherence time of the light) to the continuum of free states, we may readily show that (MANDEL,SUDARSHAN and WOLF[1964], LAMBand SCULLY[1969]) Pl(r, t)At = BV*(r, t ) . V(r,t)At, =

0,

if hw, > - E , if hw, < - E o .

]

(1)

Here B is a constant depending on the matrix element of the electron transition and the density of states, that is characteristic of the detector. We see from eq. (1) that, without quantizing the electromagnetic field, we can account for the well-known Einstein photoelectric condition hw, > - E, ,for the fact that the photoemission probability is proportional to the instantaneous light intensity V*(r, t ) . V(r, t ) = Z(r, t), and for the appearance of a non-vanishing photoemission probability as soon as the photodetector is exposed to the light, no matter how weak the light may be. We shall return to this point shortly. As a consequence of the result embodied in eq. (l), a large number of phenomena depending on photoelectric detection of light are explicable without field quantization. In the following we shall take it for granted that ho, exceeds the binding energy of the electrons in the detector. We can generalize eq. (1) for several photodetections by one or more detectors, by noting that, if the photoelectric emissions are microscopically independent, then the joint probability P 2 ( r , , t , ; r2, t2)At1At2 for two photoemissions to occur at times t , within A t , , and t , within A t 2 , at points r1 and r2 in the field (not necessarily separate) where the light intensities are Z(rl , t , ) and I ( r z , t z )respectively, is given by

with an obvious generalization to any other number of emissions. Finally, we suppose that we are dealing with a randomly fluctuating electromagneticfield, as is generally the case in practice. In that case the field vectors have no definite values, but we have to deal with an ensemble of realizations, and subsequently average over the ensemble. Equation (2)

* The complex representation is introduced most naturally with the help of the so-called analytic signal associated with the real field. See for example GABOR [1946]; BORNand WOLF [1970]. For a quasimonochromatic field we need not be concerned about which vector yr, t ) represents. as all the complex field vectors are very simply related.

32

52

[It,

S E M I C L A S S I C A L R A D I A T I O N THEORY

then has to be replaced by Pz(r19 t1;

=

r2, t 2 ) 4 &

BlBZ(4*1,

(3)

tl)4*2, t 2 ) ) 4 4 ,

where ( ) denotes the average over the ensemble. As a consequence of the correlation between the fluctuations of the light intensity, expressed by the inequality W

l

Y h)Z(*29 t 2 ) )

#

(1(*17

(4)

W ( ~ ( r 2 , t2))

(although the equality may hold in special circumstances), it follows that P2(*19 t1; *2

9

tz)

-P P I b . 1 ,

(5)

tl) P A * 2 9 t z )

in general, so that the photoelectric emissions are coupled through the fluctuations of the electromagnetic field. This is the experimental basis of the Hanbury Brown-Twiss effect (HANBURY BROWNand TWISS[1956, 19571). Equation (3) and its higher order generalizations are applicable to a large number of experimental situations. For example, from them we may readily derive (MANDEL [1958,1959,1963], GLAUBER [1964], MANDEL and WOLF[19651) the probability p(n, t, t T ) of detecting n photoemissions from an illuminated photodetector in a finite time interval t to t T.

+

p(n, t, t + T ) =

U"e-' ( 7 , with ) U =B

+

I(*,t')dt'.

(6)

In other words, we can account for the results of so-called photon counting experiments without quantizing the electromagnetic field or introducing photons. Let us consider another example. As is well known, bosons in thermal equilibrium obey Bose-Einstein statistics. As a result, they tend to be distributed in bunches as shown in Fig. 1b rather than strictly at random as in Fig. la. The tendency towards bunching can generally be attributed to the indistinguishability of the particles, or to the symmetrization of the overlapping wave functions, and is manifest in the density fluctuations of a [1943]). When the boson gas (UHLENBECK and GROPPER [1932], LONDON light emitted by a thermal source falls on a photodetector, the bunching should be reflected in the arrival times of the photoelectric pulses, and this has indeed been observed many times (TWIS, LITTLEand HANFKJRY FERGUSON and BROWN[1957]; REBKAand POUND[1957]; BRANNEN, [19581; MORGAN and MANDEL [19661; S c m [19661; ARECCHI, WEHLAN GATTIand SONA[1966]; PHILLIPS, KLEIMAN and DAVIS[1967]). At first sight, therefore, it might seem that we have here experimental evidence not

11,

0 23

33

T H E P H O T O E L E C T R I C EFFECT

only for the quantum nature of the radiation, but also for the boson character of the photons. A moment’s reflection will show, however, that we can account for exactly the same results in terms of ordinary classical electromagnetic waves that fluctuate because of the thermal nature of the source, with the help of eq. (3). II

I I I I I I ~ I I I I I I I I I I I

I1

1 1 1 1 1 0 I I

I

I

I I

II

I I I I

II

I

I

I1

I

+.+ Mean interval

Classical distribution (a)

I

$1 f

llln

1111

-

-Mean

I 1

I1

II

1111

I

1111

I

I

Ill

II

interval

Coherence time

Quantum distribution

Degeneracy 6 = 1 (b)

Fig. 1. Illustration of typical distributions of arrival times for (a) classical particles; (b) quantum particles (bosom). (Reproduced from L. Mandel, Progress in Optics, Vol. 2, ed. E. Wolf (North-Holland Publishing Co.. Amsterdam, 1963) p. 237.)

For, as is well known, for a stationary optical field of thermal origin, that obeys Gaussian statistics, the intensity autocorrelation function has the form (HANBURY BROWNand TWISS[1956, 1957); MANDEL[1958, R MANDEL and WOLF[1965])* 1959, 19631; G L A ~ E[1964];

(I(r, t i ) Z(r, t2)> = (I(r)>2C1+ Mr, t2 - tl)121,

(7)

where ~ ( rt,, - - t , )

= (v*(r,t l ) W ,t 2 ) ) / ( W ) ,

(8)

and from eqs. (3) and (7), pZ(v7 t i ; r, tz) At1 At2 = B2(z(r)>2[1 + ly(r, t 2 -t1)l2] At1 At2.

(9) The autocorrelation function y(r, t , - t , ) has its greatest value unity when t , = t , , and falls close to zero once It, - t , 1 appreciably exceeds the coherence time T, of the light. P2(r, t , ; r, t z ) is therefore greater for It, - t , 1 << T, than for It, - t , 1 % T,, and has the general form shown in * For simplicity we have assumed that the field is uniformly polarized and may be described by a scalar function.

34

CIL

SEMICLASSICAL R A D I A T I O N THE ORY

I

I

I

I

2

pi 2

I

I

3

4

t2-t, in units of the coherence time

Fig. 2. The joint probability density P2(r,f , , r, t 2 ) for photoelectric counting as a function of the time difference t2 - t i , for polarized thermal light of Lorentzian spectral distribution.

Fig. 2 for light of Lorentzian spectral distribution. We see that the tendency of the counts to arrive preferentially in bunches of duration of order T, is well accounted for by eq. (9), both qualitatively and quantitatively. On the other hand, in the absence of intensity fluctuations, eq. (3) reduces to Pz(r1, t , ;

r2, t 2 ) = Plb-1, tl)pI(rz,

GI,

(10)

and no correlations in the arrival times of photoelectric pulses are to be expected. In particular, if we split the light beam from a single-mode laser that is almost free from intensity fluctuations with the help of a 45" half-silvered mirror, as shown in Fig. 3, and allow the two beams to fall on two photodetectors, then, by virtue of eq. (lo), no coincident arrival times of photoelectric pulses at the two detectors are to be expected, other than those that overlap by chance alone. This also has been confirmed experiand MANDEL mentally (ADAM;JANOSSY and VARGA[19551; DAVIDSON [1967]). More recently, CLAUSER [1974] has demonstrated the absence of counting coincidences in a beam splitting experiment in which the decay of single excited atoms was studied. All these results, which were obtained on the basis of semiclassical arguments and have been repeatedly confirmed by experiment, may tempt one to conclude that quantization of the electromagnetic field is indeed

11,

0 21

v

T H E P H O T O E L E C T R I C EFFECT

Amp.8 Dscr. Pbotodetector

,Beum

35

Splitter

F-f-c ----

Photodetector Amp.8 Discr.

Coincidence Counter

Fig. 3. Outline of a coincidence counting experiment in which a single light beam is split into two with the aid of a 45" half-silvered mirror.

superfluous for the treatment of photoelectric measurements. We must therefore draw attention to a fundamental weakness of the semiclassical argument, that becomes apparent as soon as we examine the problem of energy balance. The result embodied in eq. (1) was obtained by solving the Schrodinger equation of motion for the electron under the influence of a classical electromagnetic field. But, because the field is treated as an external perturbation in the form of an oscillating potential which can do work, the equation of motion does not automatically ensure that energy is conserved, as it would in Q.E.D., where the electron and the field are treated as a single, internally coupled quantum system. As a result, the equation of motion leads to a non-vanishing probability for photoelectric emission as soon as the electron encounters the field, and long before the electromagnetic wave, according to Maxwell's equation, has delivered enough energy to release the bound electron. To see this, let us consider a beam of light in the form of a long wave train of limited cross-section and of duration T, for which the total energy carried by the wave train is given by

where S is the Poynting vector, t is the time when the front of the wave train impinges on a surface normal to the beam, and the surface integral is taken over this surface. If the surface contains some bound electron with binding energy E, < 0, the combined energy of this electron and of the

36

SEMICLASSICAL RADIATION THEORY

CII,

§2

electromagnetic wave immediately before the electron is exposed to the field is evidently E, E, . Now if, under the influence of the field, the electron is thrown out of its potential well, i.e., if it makes a transition to some positive energy state, then, following the passage of the wave, the combined energy of the electron and the electromagnetic wave is clearly positive*. This must be so even if the wave is attenuated in the course of the interaction. But if energy is to be conserved, the combined energy following the interaction must equal the combined energy preceding the interaction. It follows that, if photoelectric emission takes place, we must have**

+

E, 2 -Eo.

(12)

For a light beam of given amplitude or given Poynting vector S , there should therefore exist some minimum time T,, , which we may call the classical energy accumulation time, given by

which must elapse before photoemission is energetically allowed. This condition for photoemission, which is required by energy conservation, may at times come into serious conflict with eq. (1) in which the electromagnetic energy plays no role, which predicts a non-vanishing emission probability even for times much shorter than T,,. The question whether time delays are observed in photoelectric emission following the exposure of the detector to the light, and whether they have a lower bound of T,, , is therefore an exceedingly important one. While several historical experiments dealing with this question were carried out in the first two [1913]; MAYERand decades of this century (MARXand LICHTENECKER GERLACH [1914, 19151; LAWRENCE and BEAMS[1928]), which set progressively smaller bounds-to the time delays occurring in the process of photoelectric emission, they gave no direct information on whether these delays could be smaller than T,, . The reason is that the measurement of some key parameter necessary for a determination of T,, seems to have been invariably omitted in these experiments. However, TYLER[1969] reported an experiment in which time delays in photoemission following the turn-on of a light beam were found that * We allow the wave to pass before evaluating the energy, so as to ensure that no interaction energy between the electron and the field contributes to the total energy. ** We should emphasize that there is no significant uncertaintyin the energy of the emitted electron, so long as we restrict ourselves to interaction times that are many optical periods long.

11,

8 21

T H E P H O T O E L E C T R I C EFFECT

31

were much shorter than T,, . Unfortunately, the experiment did not provide conclusive evidence for the contradiction of the classical condition (1 2), because of a weakness in the method of turning the light beam on and off. The experiment relied on the use of an electro-optic shutter, which' does not extinguish a light beam completely, but attenuates it one or two hundred times. As a result, the possibility that the photodetector was accumulating energy during the time when the light beam was nominally off, could not be entirely ruled out. Because of the crucial nature of the problem, another version of the experiment was carried out by DAVIS and MANDEL[19731, with the apparatus shown in Fig. 4. The light beam from an amplitude stabilized laser was turned on and off repeatedly with the help of a mechanical rotating shutter, and the probability distribution p(n, z) of the number of photoelectrons n emitted at the detector in a short time interval, at various times z following the turn-on of the light was measured. Although the switching time (a few psec) of the mechanical shutter was very long compared with possible electronic switching times, it was still short compared with the classical energy accumulation time Tcl, which was chosen to be 20 psec in the experiment. However, unlike electronic switches, the shutter permitted the light beam to be extinguished with very high attenuation. If the classical energy condition (1 2) holds, no photoemissions at all should be observed before 20 psec, no more than one before 40 p sec, no more than two before 60 psec, etc. Instead, as the results of Fig. 4 show, the observed probability distributions p(n, z) were virtually independent of the time delay z, in contradiction with eq. (12), but in agreement with the predictions of Q.E.D. Our picture of a classical light beam as a continuous wave described by Maxwell's equations is therefore contradicted by experiment, even though it correctly predicts the results of many other photoelectric counting experiments, including those in which beam splitters are used. Is there any way to save the situation and yet preserve the classical description of the electromagnetic field? There is actually no reason to associate a classical field with a continuous wave. According to neoclassical theory, as we shall see, each excited atom of energy hw radiates a wave packet of energy hw, whose duration is of order of the reciprocal linewidth. If we think of the light beam not as a continuous wave, but as a collection of short wave packets of energy ho,we can at once account for the arrival of photoelectric pulses before the classical energy accumulation time T c l ,because this time is defined via an average energy current S in eq. (13). However, we then encounter difficulties in accounting for the action of a 45" half-silvered mirror on the light beam. For each

38

S E M l C L A S S l C A L R A D I A T I O N THEORY

Analyzer

(a)

Fig. 4. (a) Outline of the experiment for measurement of photoelectric time delay statistics. (b) Results of the measured probability distributions p(n, T) 4s a function of n and the time delay 7. (Reproduced from W. Davis and L. Mandel, Coherence and Quantum Optics, eds. L. Mandel and E. Wolf (Plenum Press, New York, 1973) pp. 115 and 118.)

11,

§ 31

THEORIES OF PHOTODETECTION A N D Q.E.D.

39

classical wave packet of energy ho is split into two similar wave packets of reduced amplitude by the beam splitter. If these attenuated wave packets are allowed to fall on two photodetectors placed in the path of each beam, they should sometimes give rise to coincident photoemissions at the two detectors, provided the energy i h o is sufficient to cause photoemission. On the other hand, if $hho is not sufficient energy for photoemission, no photoelectric counts should be registered at all. An experiment to test these predictions is very easy to carry out*. It is readily shown that no coincident pulses (other than accidental ones) are observed at the two detectors (ADAM,JANOSSY and VARGA[1955]; DAVIDSON and MANDEL [1967]; CLAUSER [1974]). Moreover, the sum of the counting rates registered by the two detectors is very nearly equal to the counting rate registered by one in the absence of the beam splitter, even if the frequency w of the light only just exceeds the threshold frequency for photoemission. The first result clearly contradicts the classical picture of the beam as a succession of wave packets acting as perturbations on the photodetectors, while the second again violates the energy balance in the photoemission process. We see that the classical picture, either of a continuous wave or of a collection of wave packets, leads to difficulty when we invoke the requirement of energy conservation. Although it is of course possible to argue that these problems arise because of an inadequate accounting of the energy, perhaps because of some hidden internal source of energy in the detector, no evidence exists for any such energy source. The large amount of experimental data relating to photoelectric counting that is correctly explained without such ad hoc assumptions makes this an unconvincing argument.

5

3. Relation between Semiclassical Theories of Photodetection and Q.E.D.

The difficulties we have outlined all stem from the imposition of the requirement of energy conservation. How is it that, so long as this requirement is not introduced explicitly, the semiclassical theory of photodetection is able to account for most of the experimental results? This question has been discussed by SENITZKY [1965, 19681 among others, who has emphasized the possible contradictions inherent in quantizing only one of two coupled physical systems. A partial answer rests on a theorem of Q.E.D. that has been called the optical equivalence theorem (SUDARSHAN *

It makes quite an effective class demonstration.

40

SEMICLASSICAL R A D I A T I O N T HE ORY

"I,§

3

[19631; KLAUDER,MCKENNAand CURRIE[19651; KLAUDER [19661; KLAUDER and SUDARSHAN [1968]), which points to a close correspondence between the results of Q.E.D. and of semiclassical treatments of the detection problem. In Q.E.D. the electromagnetic field vectors, such as the electric field I?@, t), behave as Hilbert space operators* acting on the state of the system. For a free field, the positive and negative frequency parts v ( r , t) and @(r, t ) into which the real field may be decomposed play the role of annihilation and creation operators in Fock space, and may be given plane wave expansions in the usual form

(14)

4

V(r, t ) =

1 ~

C I*(W)

"k , s

exp { -i(k . r - ot)},

where L 3 is the normalization volume, cks is a unit polarization vector, /(a) depends on the choice of field vector and is given by ,/(hw/2~,)for the electric vector. As was first shown by GLAUBER [1963], the probability of photoelectric detection at one or more space time points r l , t , ; r, , t , , etc., is proportional to the expectation value of products of the corresponding creation and annihilation operators written in normal order. Thus, the probability for registering a photon count at r, t, within At is given by Pl(r, t)At

=

B( Pt(r, t ) . P(r, t ) ) At

=

p tr [b Pt(r, t) . p(r,t ) ] At,

(15)

where p is a constant characteristic of tne detector, and the joint counting probability at r , , t, ; r 2 , t , within A t l , At, is given by p2(r17

tl; '27

where fi is the density operator describing the state of the field. Now it has been proved that @ may be given a particularly simple and interesting diagonal representation in terms of coherent states (GLAUBER[19631) I ( u ) ) labelled by a set of complex amplitudes (Y)., in the form (SUDARSHAN [19631; KLAUDER,MCKENNAand CURRIE[1965) ; KLAUDER [19661; * We label all Hilbert space operators by the caret in order to distinguish them from cA

numbers.

11,

0 31

41

THEORIES OF PHOTODETECTION A N D Q.E.D.

KLAUDER and SUDARSHAN [19681)

B = ~ 4 ( { u l ) l { u ) , C { u l ld{ul.

(17)

Here 4 ( { u } ) is a weighting functional that plays the role of a phase space probability density. Although there exist states of the electromagnetic field, such as Fock states, for which the weighting functional 4 ( { u } ) has a highly singular form, for most fields encountered in practice, such as those produced by continuous sources, + ( { u } ) has the character of an ordinary probability density. The singular forms of 4 ( { u ) ) are generally examples of fields for which no corresponding classical descriptions exist. By substituting the expansion (17) for p in eqs. (15) or (16), and making and ( ( u } ] are right and left use of the fact that the coherent states I{}.) eigenstates of P(r, t) and Pt(r, t), respectively, with eigenvalues V(r, t) and V*(r, t) given by

P(r, t)l{u>>

1

=

5 1 l(@)&ksuks exp {i(k r - w t ) } l { u } ) ’

V(r, t)l{’}),

k,s

(18)

1

<{u}IP+(r,t )

= ({u}~

C l * ( w ) ~ i ~exp u i ~{ -i(rc

. r-wt)}

E

V*(r, t ) < { u } ~ ,

k, s

we may readily evaluate the probabilities. We find that (SUDARSHAN [1963]; KLAUDER,MCKENNAand CURRIE119651; KLAUDER[1966]; KLAUDER and SUDARSHAN [I19681; GLAUBER [1963]) PZb-1, tl ; r z

9

tz)

s s

4 ( { u } ) K*(r1, tl) Jy
=

constant x

=

constant x

=

constant x (Z(rl, tl)Z(rz, t z ) ) ,

4({u})Z(rl, tl)I(rz, t,)d(u}

(19)

where the angular brackets now denote the average over the ensemble represented by the phase space density $ ( { u } ) . This is precisely of the form given by eq. (3), provided 4 ( { u } ) can be identified with the corresponding classical probability density. We can show by a similar argument that a similar relationship between Q.E.D. and semiclassical treatments exists for all measurements corre-

42

SEMICLASSICAL R A D I A T I O N T H E O R Y

“1,

83

sponding to normally ordered field operators, which means for all measurements corresponding to the absorption of photons. It is perhaps worth emphasizing that, in practice, this applies to nearly all measurements in optics, as nearly all light detectors, whether they be photocells, photographic plates, or human eyes, depend on the photoelectric effect. Finally, let us briefly examine the question how Q.E.D. avoids the paradox posed by the half-silvered mirror for semiclassical theory. Let us suppose that, in the situation illustrated in Fig. 3, the field at r in front of the beam splitter is to be related to the fields at r l , r2 at the two detectors. As Maxwell’s equations hold in Q.E.D., we have for the amplitudes

where z,, z2 are transit times for the light from r to r1 and from r to r z , respectively, and cR, cT are the reflectance and transmittance coefficients = 1) of the beam splitter. From eq. (15) the average (with l~,1~+Ic,l’ number of photoelectric counts registered by the two detectors together, at time l + z , within At for the detector at r1 and at time t + z z within At for the detector at rz ,is given by ( n ) = B[(Pt(rl, t + z l ) . V(rl, t+Zl))+(Vt(r2, t + z , ) . V(r2, t+z,))]At, (21) where /3 is a parameter measuring the detection efficiency of each detector (assumed to be the same for both). From eqs. (20) and (21) it then follows that
=

B[IcT12(Pt(r, t ) . %, t)>+ IcR12(Pt(r, t ) . P(r, t)>]

=

fi(Pt(r, t ) . P(r, t)>,

(22)

which is just the mean number of counts that would have been recorded by either detector placed at position r at time t. On the other hand, from eq. (16) the joint probability for registering counts at both detectors, at time t f z , within At, at r , , and at time t + z 2 within Atz at r 2 , is given by Pz(r,, t,; r2t2)At,At2 =

/12(qt(rl, t + z , ) qt(r2, t+z,) f‘)(r2, t + z z ) c ( r l , t+z,))At, Atz (23)

=

$’ IcT(’IcR12(ct(r, t ) qt(r, t ) c(r, t ) c(r, t ) ) Atl At2,

which is proportional to the joint counting probability for two events at

1 1 , s 41

43

S P O N T A N E O U S EMISSION OF L I G H T

if the state of the field happens to be a one-photon state this probability vanishes. In other words, in Q.E.D., by virtue of the fact that the electromagnetic field operators and the state of the field are treated as entirely different entities, the integrity of the number of excitations, and therefore of the number of photoelectric counts, is preserved. The beam splitter splits the field, but does not change the number of photons. r , t , and

ij 4. Spontaneous Emission of Light according to Neoclassical Theory After considering problems in the absorption of light, and the difficulties inherent in a consistent accounting of all the experimental results by semiclassical approaches, we turn to the problem of light emission. The process of spontaneous emission of light by an excited atom was long considered as something of an embarrassment to semiclassical radiation theory, and its explanation (DIRAC[1927]) became one of the early triumphs of Q.E.D. However, by incorporating the reaction of the field back on the source in the formalism, we can show that neoclassical radiation theory also leads to spontaneous, but determinate, radiation emission. The first development of this idea can be traced to a little known paper by FERMI [19271 (see also FERMI [1962]). However, the approach appears to have been forgotten, until the idea was rediscovered and developed further by Jaynes and his co-workers (JAYNES and CUMMINGS [19633 ; CRISPand JAYNES[19691; STROUDand JAYNES[1970]; JAYNES[1973]; see also LANYI[1973]) in recent years. In the following we shall briefly outline the argument, because the detailed conclusions are peculiar to neoclassical theory and they permit the theory to be tested against Q.E.D. Neoclassical theory rests on an interpretation of quantum mechanics that was first advocated by SCHR~DINGER [1926]. According to this interpretation, a particle of charge e, mass M , momentum p and of wave function + ( r , t ) has an associated charge density a(r, t ) given by 4 r , 4 = ell/*@,t)ll/(r, 4,

(24)

and an associated electric current density given by

These densities have no statistical interpretation, but are to be regarded as densities of actual, classical charges and currents distributed throughout space. According to this point of view, an electron is generally to be looked ori as a moving charge cloud, that evolves in a manner determined by the

44

SEMICLASSICAL R A D I A T I O N THEORY

[I],

§4

Schrodinger equation for $(r, t). The densities given by eqs. (24) and (25) coincide with the usual probability densities of quantum mechanics, so that the electromagnetic fields produced by the particle according to Maxwell’s equations, which involve integrals over space and time, are expressed in terms of what we would normally interpret as quantum mechanical expectation values of the dynamical variables of the source. Because these fields are given by quantum expectations, they are free from all fluctuations or uncertainties, so that the theory is fully determinate so long as the initial conditions are definite, although governed by quantum mechanical equations of motion. In the following,we shall find it convenient to continue to use the Dirac notation of quantum mechanics, although the state functions are to be understood as Schodinger wave functions. We consider a two-level atom with unperturbed energy eigenstates 11) and 12), separated by an energy h a , . We suppose that the atom has an electric transition dipole moment j3, with non-vanishing matrix elements (llirl2)

= PlZ = (2lfill>,

(26)

which we take to be real by proper choice of the phases of the states. Then, in the presence of a classical electric field E(r, t), the atom experiences an electric dipole interaction, and the state I$(t)) of the atom in the interaction picture satisfies the Schrodinger equation of motion - icw . Eft)

IW>= i h

a

I$(W

(27)

The field E may have contributions from a possible external field EtE),and from the field produced by the atom itself, the so-called reaction field. has been evaluated by STROUD and JAY= [1970] and shown to be of the form

where K is a frequency whose order of magnitude is the reciprocal of the transit time of a light signal across the atom. A convenient method of solving eq. (27), in the absence of an external field, is to introduce the so-called Bloch vector representation of the atomic state, according to which any pure state is characterized by the direction of a unit vector with polar coordinates 8 , 4 in a symbolic Bloch space. An arbitrary state of the atom can then be expressed in the form I$(t)) = exp (+iyt)[sin (9)exp (iiq5)ll) +cos (#)exp (-$i4)12)],

(29)

11,

$ 41

S P O N T A N E O U S EMISSION OF L I G H T

45

where the phase factor exp(4iyt) is inserted for later convenience. From this we see that the fully excited and de-excited states of the atom correspond to Bloch vectors with 8 = 0 and 8 = n, respectively. The expectation value of the dipole moment fi in the state I+(t)) is given by

(+lfil+)

= P,, sin

8 cos (4 + wo t),

(30)

which is obviously greatest for 8 = $, and zero for 8 = 0 or n. If we now make use of eqs. (28)-(30) in the Schrodinger eq. (27) with EE) = 0, multiply by (1 I on the left, and discard terms oscillating at twice the optical frequency, we obtain the coupled -equations of motion

8

=

fi sin 8

(31)

$ = YCOS~,

(32)

in which we have substituted

The equations are readily integrated to yield e(t) = 2 tan-' [exp +(t) = +(O)-

-

P

In

P(t-to)],

cash B(t - to) cosh Pto

[

1'

(34)

where to is a parameter determined by the initial state of the atom from the relation tan [$O(O)] = exp (-

fito).

(35)

The trajectory of the Bloch vector corresponding to these solutions is shown in Fig. 5 for the special case /? = y. It will be seen that the Bloch vector falls from its initial position so as to end up pointing straight down, while precessing first in one direction and then in the other. The initial state I+(O)) = 12) or 8(0) = 0 is singular in the theory, in that the Bloch vector takes an infinite time to fall from the upright position. But, except for this singular state, the theory predicts spontaneous decay of an excited atom. However, the manner in which the atom decays is quite different from that predicted by Q.E.D. For example, the energy W(t)relative to a level midway between 11) and 12) is given by W(t) = ~ h m cos , 8 = -+hao tanh P(t-to),

(36)

46

SEMICLASSICAL R A D I A T I O N THEORY

CII,

54

I Fig. 5 . The trajectory on the unit sphere of the Bloch vector of a two-level atom undergoing spontaneous emission according to neoclassical radiation theory. (Reproduced from C. R. Stroud Jr., J. H. Eberly, W. L. Lama and L. Mandel, Phys. Rev. A5 (1972) 1098.)

and its time evolution is illustrated in Fig. 6. The intensity Z ( t ) of the electromagnetic field radiated is expressible in the form I(t) = -

d W ( t )- $ h o o p sin’ 8 dt ~

=

$ h o o p sech’

P(t-

to),

(37)

and is also illustrated in Fig. 6. It will be seen that the light intensity is greatest when the atom is in a state of half - rather than full - excitation, and that its form is not exponential in general. The same parameter fi (half the Einstein A-coefficient) determines the time scale of the decay as in Q.E.D., but, unlike Q.E.D., neoclassical theory predicts that the actual time development of the decay depends on the initial state, through the parameter t o . The significance of the parameter y becomes apparent when we calculate the instantaneous frequency of the radiated field. We find that it is shifted from the natural atomic frequency wo by an amount given by &t)

= -y

tanh

P(t-to),

(38)

11,

P 41

41

S P O N T A N E O U S EMISSION OF L I G H T

I

I

hw (Energy)

W(t)

& ( t ) / Y (Frequency Shift)

I

2

3

Pt

Fig. 6. The time development of the atomic energy, the radiated light intensity and the frequency shift according to neoclassical radiation theory.

which has the same form as the curve for W ( t ) shown in Fig. 6. The frequency shift of the field is analogous to the Lamb shift found in Q.E.D., but it varies with time in a characteristic manner, and has been called the dynamic Lamb shift (JAYNESand CUMMINGS [1963]; CRISPand JAYNES [1969]; STROUD and JAYNES [1970]; JAYNES [1973]; see also LANYI[1973]). As a result of the solutions embodied in eqs. (36)-(38), the spectrum of the radiated light depends on the degree of excitation of the atom, in contrast to the predictions of Q.E.D. Moreover, the shape of the^ power spectrum radiated by an almost fully excited atom is not Lorentzian, but has the form of a sech’ [(.n/2fi)(o - oO)] function when y -ez fi, as can be seen by taking the Fourier transform of the expression for the field. The finite spectral

48

SEMICLASSICAL R A D I A T I O N T HE ORY

[II,

54

width results from the finite life-time of the radiation process, and not from any fluctuations. At first sight it might appear that the evidence for the exponential decay of atomic fluorescence, and for the constancy of the Lamb shift, would immediately demolish neoclassical radiation theory. But closer examination reveals that the issue is not nearly so clear cut. For we note from Fig. 6 that the tails of the curves for W ( t )and I ( t ) are very nearly exponential. As a result, neoclassical theory also predicts exponential decay for an atom that is only fractionally excited. Moreover, for such an atom the instantaneous frequency shift $ ( t ) is also nearly constant in time. Any experimental evidence for exponential decay and a constant Lamb shift is therefore not decisive without additional evidence that the atoms were highly excited, and this is unfortunately lacking in most experiments. Indeed, CRISPand JAYNES [1969] have calculated the ls-2p frequency shift in atomic hydrogen from the neoclassical equations of motion (with the neglect of contributions from other levels), and find a value -0.285 an-', which is in quite good agreement with the experimental value -0.262k0.038 cm-' of HERZBERC [1956], although the coincidence is less good in some other cases. The agreement for the ls,-2p, shift may however be fortuitous, as has been suggested by VAN DEN DOELand KOKKEDEE [1974], who have repeated the neoclassical calculation when electron spin currents are included, and found strong disagreement with experiment. Systematic attempts to measure the decay law for excited atoms as a function of the degree of excitation were recently made by WESSNER, ANDERSON and ROBISCOE [1972]*, and by GIBBS[1972, 19731. The former used an atomic beam of hydrogen of 60 keV energy and studied the decay rate of the 2P, level. They observed exponential decay and found no evidence for any dependence on the relative population of the 2P, and 2S, levels. GIBBS[1972, 19731 investigated the resonance fluorescence between the 2P, and 2S, levels of an atomic beam of Rb that was optically excited by a pulsed laser beam, for various degrees of atomic excitation. Corrections for the role of neighboring energy levels were included. Figure 7 shows some experimental results for the measured fluorescence as a function of the degree of excitation, which is expressed by the so-called area of the excitation pulse (MCCALLand HAHNC1967, 1969]), which may be taken as the polar angle A 6 through which the atomic Bloch vector is rotated by the pulse. It will be seen that the fluorescence is greatest for excitations of n: and 3n: that leave the atomic system almost fully inverted, in agreement

-

* In this connection, see also the much older work of GAVIOLA [1928].

11,

6 41

S P O N T A N E O U S EMISSION OF LIGHT

49

Fig. 7. Results of measurementsof atomic fluorescence as a funtion of the degree of excitation of the atom, as expressed by the area of the excitation pulse. (Reproduced from H. M. Gibbs, Phys. Rev. AS (1973) 458.)

with Q.E.D., but in disagreement with eq. (37) of neoclassical theory, which predicts zero - or at least - minimum fluorescence when the atom is in the excited state, and maximum fluorescence when A0 is near &c, $, etc. The absolute power measurement of the exciting pulse obviously played a key role in this experiment. Because of the difficulty of measuring absolute power with any accuracy, Gibbs relied mainly on the phenomenon o f selfinduced transparency to calibrate his detector. According to the theory of MCCALLand HAHN [1967, 19693, a n-pulse is transmitted through a resonant medium without absorption, and this was used to provide a reference power level. Because of this calibration, the results would be less decisive for the neoclassical theory if an appreciable dynamic Lamb shift were present, for, according to neoclassical theory, the transparency pulse would then have an area much less than 71. However, later experiments on the hyperfine structure of atomic sodium (SCHUDA,HERCHER and STROUD [1973]) give no indication of any such dynamic shift. As the spectra of Rb and Na have similarities, we conclude that the predictions of neoclassical theory are contradicted by the results of this experiment. Gibbs also studied the time development of the fluorescence and showed that it was approximately exponential even for strongly excited atoms. The fact that, according to neoclassical theory, the electromagnetic field is expressible in terms of expectation values of the atomic variables,

50

[II,

SEMICLASSICAL R A D I A T I O N THEORY

05

implies that the field is determinate and free from all fluctuations, provided the atom is in a definite state*. It is possible that this prediction could be tested directly in photoelectric counting experiments with fluorescence radiation, but the difficulties of preparing single atoms in a definite state seem to have discouraged such experiments so far.

5

5. Resonance Fluorescence

The equations of motion (31) and (32) of neoclassical theory for a twolevel atom may readily be generalized to the situation in which an external field E(t) = E , cos ot

(39)

is present, whose amplitude we take to be constant and whose frequency o is close to the atomic frequency 0,. We then obtain, instead of eqs. (31) and (32), the equations for resonance fluorescence

4 = ycose+- E O -h

cot 8,cos [(o-o,)t-4].

4

These equations admit steady state solutions for which d = 0 and = o-o, so long as the field is not too strong. To see this let us examine the somewhat simpler problem of a resonant applied field with w = 0,. Then, if there is a steady state solution, we obtain from eqs. (40) and (41), psin8

EO.Pl2 .

= -

y sin 8 = -

~

h

sin 4

Eo * P12 cos h

from which it follows that tan

4

=

Ply

and

* In this connection see also the discussion of fluctuations by and the discussion of radiation reaction by BULLOUGH [1973].

GORDON

and NASH[1973]

11,

§

51

RESONANCE FLUORESCENCE

51

Evidently the solution for 0 exists only if the Rabi frequency IE,. plz/hl 5 which implies that the field is not too strong. In that case the equations predict that the excitation of the atomic dipole becomes constant in time (in the interaction picture), so that the atom radiates a sharp spectral line at frequency o.The reason is that any finite spectral width is invariably associated with transients in neoclassical theory, and these transients have, of course, disappeared in the steady state. The limitation on the field strength arises partly because of the exact resonance condition o = oo. It has been shown by STROUDand JAYNES [1970] that for large E,,, at least when p = 0, the smallest amount of detuning will cause the system to reach a steady state. It is perhaps worth mentioning that the same conclusion also follows from the semiclassical Bloch equations for the atom (see e.g. ALLEN and EBERLY [1974]), in which the damping is introduced phenomenologically by the addition of extra terms, rather than through the radiation reaction. When y # 0 and the applied field is large, the long term solutions of the neoclassical equations of motion (40) and (41) are periodic in general (JAYNES and Cummings [1963]; CRISPand JAYNES [1969]; STROUD and JAYNES[19701 ;JAYNES[1973] ; see also the related work of LANYI[19731). As a result, the Fourier spectrum of the resonance fluorescence has the form of a group of sharp lines rather than just one line. In no case is the spectrum continuous, however. These results are to be contrasted with the corresponding results of Q.E.D. for the two-level atom in an electromagnetic field, which have been derived by many workers (NEWSTEIN [1968]; MOLLOW C1969, 19723; CHANCand STEHLE [1971]; STROUD [1971]; SENITZKY [1972]; AGARWAL [19741 ; MILONNI[19741 ; ACKERHALT and EBERLY [19741 ; CARMICHAEL and WALLS[1975]; KIMBLE and MANDEL[1975]). Although some discrepancies between some of the calculated results exist, the linewidth of the scattered field is always finite for an incident field of finite amplitude, according to Q.E.D. While the amount of quantitative experimental information on the BRUNNER, problem of resonance fluorescence is small (HERTZ,HOFFMAN, and STEUDEL [19681 ;HANSCH, KEOL,SCHABERT, SCHMELZER PAUL,RICHTER and TOSCHEK [19691 ;SHAHIN and HANSCH[1973]), the recent development of tunable dye lasers has made it possible to study the emission spectrum in resonance fluorescence as a function of the detuning between the exciting light and the atoms. Such an experiment has recently been carried out by SCHUDA, STROUD and HERCHER [19741, who excited a hyperfine component of the D, line of an atomic beam of Na atoms by a perpendicular laser beam, and then examined the spectrum of the fluorescent light with the

,/-

52

CII,0 6

SEMICLASSICAL RADIATION THEORY

-200

-ioo

-0

loo

200

Scattered spe$trum,MHz (centered on excitation frequency)

Fig. 8. Results of measurements of the hyperfine spectrum of resonance fluorescence from sodium atoms as a function of the detuning of the excitation beam. (Reproduced from F. Schuda, C . R. Stroud Jr. and M. Hercher, J. Phys. B7 (1974) L200.)

help of a scanning Fabry-Perot interferometer. Some of their results are shown in Fig. 8. Although the interpretation of the experiment is complicated by contributions from more than two atomic levels, the results again decisively contradict the predictions of neoclassical theory. Some other recently reported experiments on resonance fluorescence (RASMUSSEN, SCHIEDER and WALTHER [1974] are also in agreement with Q.E.D. It has recently been pointed out (AGARWAL [1974], KIMBLE and MANDEL [19751) that the quantum properties of the free electromagnetic field actually play only a very minor role in the time development of the light intensity in spontaneous emission, or even in resonance fluorescence. However, measurements of the spectrum in resonance fluorescence, or of two-time correlation functions, reflect non-commuting properties of the free field and therefore test Q.E.D.

0

6. Fluorescence Effects in Multi-Level Atoms

Although the analysis of the two-level atom permits considerable simplifications to be made, few experiments involve only two atomic levels,

11,

0 61

F L U O R E N C E EFFECTS I N M U L T I - L E V E L ATOMS

53

even if the presence of the other levels is manifest only indirectly. But it is interesting to note that there are features of neoclassical theory that should show up explicitly in multi-level atoms. If the coefficients of the states 11) and 12) in eq. (29) are denoted by cl(t) and c,(t), respectively, so that

I+(W

= Cl(t)ll)

+ C2(012),

(44)

eq. (31) can be re-written in the form

which emphasizes that the rate of atomic decay from the upper state is proportional to the product of occupations of the upper and lower states. This neoclassical equation may readily be generalized to the decay of a multi-level atom, for which the rate of change of the occupation of the ith level is given by

In some recent experiments on multi-level Pb atoms, GIBBS,CHURCHILL and SALAMO [1973] showed that this relation is contradicted even when the system is weakly excited, in contrast to the situation represented by eq. (45) for the two-level atom, which becomes identical with the corresponding equation of Q.E.D.in the limit of weak excitation. As we have seen, in neoclassical theory the electromagnetic field is a cnumber which is expressed directly in terms of expectation values of the dipole moment of the source. This feature leads to some interesting differences between the predictions of neoclassical theory and Q.E.D. in the fluorescence of multi-level atoms. The argument given below follows the work of HERMAN, GROTCH, KORNBLITH and EBERLY [1975]*. The natural generalization of the dipole moment operator for a multilevel atom is of the form =

1pn,ln)(ml +h.c. = 1fin, ci,, +h-c. n. m

n,

(47)

m

We may suppose that the indices n, rn label the sublevels of a lower and of an upper group of levels, respectively. We now use Maxwell’s equations to express the electric and magnetic fields E and B produced by the atom in * Such effects have also been discussed by BREIT[19331 ; GROTCH and HERMANN [I9741 ; ACKERHALT and EBERLY [1974]. See also the discussion of a related problem by OLIVER and ATABEK[1973].

54

SEMICLASSICAL RADIATION THEORY

“1,

§7

terms of j2, and then determine the rate of atomic fluorescence from the Poynting vector S(r, t ) at any position r in the far-field at time t. But whereas E , B and S are c- numbers inneoclassicalradiation theory, expressed in terms of ( j 2 ) and its derivatives, they are Hermitian operators in Q.E.D. As a result, we are led to the following two expressions for the rate of energy flow from an atom at the origin in neoclassical theory and in Q.E.D., respectively :

We have written r1 for the unit vector from the atom to the field point Y and omm,, unn, for the frequency differences between the atomic levels Im), Im’)and In), In’), respectively. Now

which vanishes for n # n‘, whereas (cf,,,,) (S,,,.) does not vanish, in general. As a result certain Fourier components in the spectrum of the fluorescence radiation, which result from beats between different energy levels, are forbidden according to Q.E.D., although they are allowed according to neoclassical radiation theory. Recent attempts (STONER [19741) to search for these beat frequencies in fluorescence indicate that they are indeed missing, but more experimental evidence is desirable.

,

0 7. Polarization Correlations in an Atomic Cascade In a classic paper, EINSTEIN, POWLSKYand ROSENEl9351 discussed a paradox in the quantum theory of measurement, involving the appearance of correlations between separated, but previously interacting, particles. The paper has come to be generally regarded as a critique of the conventional interpretation of quantum mechanics, for they argued that the predicted correlations were contrary to experience. The argument was answered by BOHR[1935], but discussion of the problem has continued over the years [1936]; Born c1951-J; BOHM and AHARANOV to some extent (FURRY [1957]). The existence of the correlations discussed by Einstein, Podolsky

11,

5 71

POtARIZATION CORRELATIONS

55

and Rosen was finally demonstrated experimentally in correlation experiments on electron-positron annihilation radiation (BLEULERand BRADT[1948] ;Wu and SHAKNOV [1950]). More recently, optical versions of the correlation experiments were reported by KOCHERand COMMINS [19671, and FREEDMAN and CLAUSER [1972], who made use of the 6S-4PAS cascade decay of Ca atoms. An outline of the apparatus used by Kocher and Commins and the energy level diagram for calcium are shown in Fig. 9. In both experiments, an atomic beam of calcium was excited by the light of a hydrogen or deuterium arc, so as to produce some population of the 6 s level of calcium. This was the initial level for the 6 W P - 4 S cascade decay, that gave rise to two visible light pulses at wavelengths of 5513 A and 4227 A. The light pulses were detected by photomultipliers placed on opposite sides of the interaction region, that were used to search for coincidences in photoelectric emission. Interference filters tuned to the respective wavelengths, and linear polarizers whose orientations could be changed, were used to filter the light before detection. It was found in both experiments that the rate of coincidence Hg ARC

*

2275 8,

Ca beam oven

Fig. 9. (a) Outline of the apparatus used for the investigation of polarization correlations in the cascade decay of Ca atoms; (b) The energy level scheme for calcium. (Reproduced from C. A. Kocher and E. D. Commins, Phys. Rev. Letters 18 (1967) 575.)

56

S E M I C L A S S I C A L R A D I A T I O N THEORY

“1,

§1

pdarizer I

polorizer 2

pobrizatlon of light

hkX polorizer 2

pdarizer I

Fig. 10. The geometry of the polarization correlation experiment.

counting was greatest when the linear polarizers were aligned, and close to zero when the polarizers were orthogonal. The implications of the experiment have been carefully analyzed by CLAUSER [19721. Although the emphasis in some of the discussions has been a general disproof of hidden variable theories (CLAUSER, HORNE,SHIMONY and HOLT[1969]; BELL[1965,1966]), we shall briefly examine the implications for semiclassical radiation theory, for it has been argued by JAYNES[19731 that the results of this experiment contradict neoclassical theory more directly than other results. Consider the situation illustrated in Fig. 10, in which we take the interaction region to be the origin of coordinates and the two detectors to be located along the positive and negative z-axes. Because the initial and final atomic states in the cascade decay have zero angular momentum and the same parity, the two-photon state I$) produced, according to Q.E.D., must be a linear superposition of the general form (CLAUSER [19721) 1

where I f, o,4 ) is a linearly polarized one-photon state, corresponding to propagation m the direction of the z-axis, frequency o and linear polarization in a directiQn making an angle 4 with the x-axis. Despite the fact that we have w&en I$) as a pure quantum state, it may readily be shown that each photon of the pair is completely unpolarized. To see this we form the density operator a = l$>(t,hl for the combined system, and then trace over the Hilbert space associated with one of the photons. We then find

a(+)= Tr(-)D = $[I+,

wl, O>
+ I +, ol,%71)ol, (hn +,13,

(52)

which will be recognized as the density operator of an equally weighted

11,

0 71

P O L A R I Z A T I O N CORRELATIONS

57

mixture of orthogonal polarization states, and corresponds to an unpolarized photon. If we think of the measurement with the help of the optical filters and polarizers as a projection onto the state 1 +, o1, 41) I - , 02,42),then the joint probability for detection is given by Joint Detection Probability

which vanishes for orthogonal placement of the linear polarizers. Let us see how to model this situation in terms of a semiclassical theory in which the light is not quantized. Evidently we have to suppose that the atom emits two dipole wave packets, which in general are elliptically polarized. As the orientations of the polarization ellipses are likely to be different for different atoms, we take them to be similar for each pair of wave packets, but distributed at random over all directions. The calculation is simplified if we restrict ourselves to linear polarizations in some direction making an angle 43with the x-axis. The wave packets then pass through the two linear polarizers and emerge with amplitudes cos ( 4 1~ 41)and C O S ( ~ ~ -respectively, - ~ ~ ) , as can be seen by reference to Fig. 10. The probability amplitude for the joint detection of the two pulses is therefore given by Probability Amplitude cc cos ( 4 1~ 41)cos (43- 42),

(54)

and this result applies according to both Q.E.D. and the semiclassical point of view, except that the concept of probability amplitude is replaced by the actual field amplitude in neoclassical theory. The joint detection probability then follows from the square of the probability amplitude, but, because of the unknown or random orientation of the polarization, we also have to average over all values of 43. There are, however, two ways in which we can proceed. We can first average the probability amplitude over all orientations 43and then square, or we can first square the probability amplitude and then average over 43. Which of these procedures is appropriate depends on the underlying physics of the problem. As is well known, quantum mechanics requires that we form linear superpositions of states if, and only if, the states are intrinsically indistinguishable, and statistical mixtures when information about the states is otherwise lacking. It follows that if we are dealing with two photons, or two unit excitations of the field, each of which can be absorbed only once, then we can make no distinction in principle between different states of polarization before the photons have been detected. Accordingly, we

58

SEMICLASSICAL R A D I A T I O N THEORY

“I, §

7

must sum the probability amplitude given by eq. (54) over all polarization orientations 43 before squaring, and we then arrive at the result Joint detection probability for two photons OC

2(c0s

(43-41)

4

cos (43-42))2 = cos2 (42-4119

(55)

in agreement with eq. (53). We have used the symbol ( ) to denote the average over all 43.On the other hand, if we suppose that the excited Ca atom emits two classical wave packets, say in accordance with neoclassical radiation theory, then we can, in principle, determine the polarization of each wave packet before it falls on the detector by diverting a small portion of the wave with the help of a beam splitter. The joint photoelectric detection probability, which is related to the average product of the corresponding classical light intensities according to eq. (3), is therefore proportional to the square of the amplitude in eq. (54), averaged over c$3. We then obtain Joint detection probability for two classical wave packets

(43-41) C O S ~(43-42))

K ~ ( C O S ~

The average over

c # ~refers ~

=

1 ++C O S ~(42-41).

(56)

to a succession of measurements over many

1 22% 90

OO

45

67%

Angle 0 (deg)

Fig. 1 1. The results of measurements of the coincidence counting rate divided by the rate with both polarizers removed as a function of the angle I$between the two linear polarizers. (Reproduced from S . J. Freedman and J. F. Clauser, Phys. Rev. Letters 28 (1972) 941.)

11,

§ 81

MOMENTUM TRANSFER EXPERIMENTS

59

atoms, and arises here not for intrinsic reasons, but as a reflection of our ignorance of the state of polarization in any one case. It follows from both arguments that the joint detection probability should be greatest for parallel alignment of the polarizers, and least for orthogonal alignment. But whereas Q.E.D. predicts zero coincidences for orthogonal polarizers, semiclassical theory does not. It should be pointed out that both eqs. (55) and (56) are ideal relations, that need to be modified in practice to take account of the finite efficiencies and solid angles of the apparatus (CLAUSER [19721). When these modifications are made, the experimental results are clearly in agreement with Q.E.D. and in disagreement with the semiclassical theory, as the data of FREEDMAN and CLAUSER [I9721 reproduced in Fig. 11 show. We have then another piece of evidence contradicting semiclassicalradiation theory, that specificak ly involves the emission process, in contrast to the corresponding evidence discussed in 5 2 relating to the absorption of light.

0

8. Momentum Transfer Experiments

Some interesting evidence for the quantum nature of radiation is provided by an experiment by BURNHAM and WEINBERG [1970] on the parametric frequency splitting of light. As is well known, when an intense coherent beam of light falls on a non-linear dielectric, new lower frequencies can be produced by parametric down conversion (see, e.g. BLOEMBERGEN [19651). Burnham and Weinberg allowed the 3250 A light from a He-Cd laser to fall on a crystal of ADP, as shown in Fig. 12, and by careful phase matching they were able to isolate two new light beams at wavelengths of 6330 A and 6680 A, propagating at angles of 50 mrad to the axis of the incident beam. These visible beams were allowed to fall on two photodetectors. The process of phase-matched parametric frequency conversion requires that the frequencies o and wave vectors k satisfy the conditions (see e.g. BLOEMBERGEN [1965)) wp = 0 1 +o, (57) kp = k , + k , , where the suffix p refers to the incident light beam or pump beam and suffices 1 and 2 refer to the newly produced beams. Although these relations have a natural quantum interpretation, the non-linear interaction between the light and the medium can be described in completely classical terms. A coherent incident light beam should produce two parametrically generated, coherent beams. Now coherent beams give rise to random photo-

SEMICLASSICAL R A D I A T I O N THEORY

[I[,

38

X

trap

Channel 1

I I I I

I

I

Lens

I

Fig. 12. Outline of the apparatus used to study coincidences in the parametric frequency splitting of light. (Reproduced from D. C. Burnham and D. L. Weinberg, Phys. Rev. Letters 25 (1470) SS.)

Atom Detectors

Light Detectors

To Deloyed Coincidence Motrix

Fig. 13. Outline of proposed experiment to test radiation theories by measurement of correlations in the momentum given to the atom and to the light.

11,

8 91

INTERFERENCE EXPERIMENTS

61

emissions at the detector, as has been demonstrated (HANBURY BROWN and TWISS[1956, 19571; TWISS,LITTLEand HANBURY BROWN[1957]; REBKAand POUND [1957]; BRANNEN, FERGUSON and WEHLAN[1938]; MORGAN and MANDEL [1966]; SCARL[1966]; PHILLIPS, KLEIMAN and DAVIS[1967]; ARECCHI,GATTIand SONA[1966]; ADAM,JANOSSY and VARGA[1955]; DAVIDSON and MANDEL [1967] ; CLAUSER [1974]). Therefore, according to semiclassical theory, we would expect to find no coincidences between the pulses registered by the two detectors shown in Fig. 12, other than chance coincidences. Instead, it was found that the photoelectric pulses appear in coincidence with high probability, and this again demonstrates the failure of the semiclassical description. The parametric process has been analyzed quantum mechanically by MOLLOW [19731. The possibility of testing neoclassical theory through measurement of the momentum transfer in spontaneous emission is suggested by recent observations of the recoil deflection of an atomic beam (SCHIEDER, WALTHER and VIALLE [1972]). Let us consider the situation and WOsm [1972] ;RCQL@ illustrated in Fig. 13 in which a beam of excited atoms propagating into the plane of the diagram is ringed by a set of photodetectors on one side and by a set of atom detectors, such as Langmuir probes, on the other side. According to neoclassical theory, an atomic dipole that is brought to a definite state of excitation by an external field radiates an electromagnetic field in a definite dipole radiation pattern, and therefore recoils in a definite direction (actually in the forward direction in a plane wave field). If all atoms are excited identically, one particular atomic detector should register all the recoiling atoms, whereas any one of the photodetectors can register the emitted light, so long as the radiation pattern does not vanish in the direction of that photodetector. On the other hand, according to Q.E.D., the direction of the atomic recoil is indefinite, but it must at all times be opposite to the direction defined by the photodetector that registers the absorbed photon, so as to ensure microscopic momentum conservation. If the pulses registered by pairs of atomic detectors and photodetectors are analyzed for directional correlation, the results should provide a rather decisive test of the theories, that is perhaps more intuitively convincing than some of the previous experiments. However, the experiment remains to be done.

9 9. Interference Experiments Despite some fairly substantial evidence for the inadequacy of semiclassical theories, efforts to describe experiments in semiclassicalterms have

62

S E M I C L A S S I C A L R A D I A T I O N THEORY

[II,

09

persisted. The reason is partly attributable to the conceptual difficulties associated with Q.E.D., for the formalism of Q.E.D. tends to discourage interpretation in terms of a physical model of light. We shall consider one example of an experiment that is naturally explicable in terms of classical waves, but rather less naturally in terms of photons, although Q.E.D. correctly predicts the outcome. Most interference experiments make use of mutually coherent light beams, i.e., beams that are derived from the splitting of a single beam. However, it has been known for some years that it is possible also to observe interference effects with light beams produced by two completely [19551; independent sources (FORRESTER, GUDMUNDSEN and JOHNSON JAVAN, BALLIK and BOND[1962]; MAGYAR and MANDEL [1963]; LIPSETT and MANDEL [1963]), provided the observation is effectively made in a time To short compared with the reciprocal frequency spread l/Av. PFLEEGOR and MANDEL[1967, 1968]* reported an experiment in which such interference effects were detected with two independently derived light beams, that had been sufficiently attenuated to ensure that one photon was received at a time. More precisely, the average interval T, between photon emissions was made very long compared with the transit time T, of light through the apparatus, so that, with high probability, each detected photon was absorbed by the detector long before the next one was emitted by the source. Since several photons are needed to define an interference pattern, To was made greater than T,, but the number of detected photons per observation was only of order 5-10. Let us consider how the appearance of interference fringes in such an experiment is to be interpreted**. Viewed in terms of continuous electromagnetic waves, the outcome of the experiment is no mystery, and the attenuation of the light beams is irrelevant to the result. For any two waves appear coherent over a time interval To short compared with the reciprocal frequency spread l/Av. The same conclusion holds even if we look on the beam as a collection of classical wave packets of duration l/Av, rather than as a continuous wave. The interference fringes are then readily explained in terms of the overlap between successivelyemitted wave packets. On the other hand, if we wish to regard the light beam as a collection of approximately localized”photons, it is much less easy to understand the

* Somewhat similar experiments were also reported by RADLOFF[1968], HAIGand SILLITO 119681; KLOBA,ULBRICH and GING[I9691 ** The interpretation of the experiment has been discussed by PAUL,BRUNNER and RICHTER [19631, MANDEL[1964], JORDAN and GHIELMETTI HAKEN KEN [1964], PAUL[1967], MANDEL [19691; DEBROGLIE, ANDRADE and SILVA [19683 ; LIEBOWITZ [19701

11,

0 91

INTERFERENCE EXPERIMENTS

63

result. Although the formalism of Q.E.D. accurately predicts the outcome and MANDEL[1967, of the experiment, as has been shown (PLEEGOR 1968]), it provides little in the way of explanation. For the lifetime between emission and absorption of each photon is only T, in the experiment, which is very short compared with the average arrival time interval T, between one photon and the next, and this prompts the question what is interfering with what? To make these considerations a little more quantitative, let us regard the light beam incident on the detector from two different directions as corresponding to the excitation of two different modes of the field, that we label 1 and 2.If the two beams are described by state vectors l$l)lOz), 10,)I$?) in the Hilbert space spanning both modes, then the state of the total field will be represented by 11)1)1$2), GLAUBER [1963]. If d is the operator corresponding to the absorption of a photon (in the same Hilbert space), and we make a mode decomposition of d in the form (58)

b = b,+b,

then ~l$l>l$Z>

= (~lI$l))l$Z)+

I$l)(~ZlI)2)).

(59)

The probability that a photon is absorbed by a detector looking at the total field is then given by the expectation value of dtd, and from eq. (59) we have Photon absorption probability cc ( $ z ~ ( $ l ~ b ~ d ~ $ l ) ~ $ ~ )

The terms on the right of this equation have an obvious interpretation. The first two represent probabilities that a photon is absorbed out of each incident beam, while the others are interference terms that are nonzero in general. But since the entire equation refers to the absorption of one photon, the interference effect is evidently to be associated with the detection of each photon, rather than with the interference of one photon with another. This conclusion is very much in the spirit of a well known statement of DXRAC [1948] that ". . . each photon interferes only with itself. Interference between two different photons never occurs". Nevertheless, there remains the problem of understanding how the superposition state leading to the interference term arises. In a conventional interferometer, beam splitters are used to divide one light beam into two or more beams, and the superposition states are then associated with the divided beam. But in this experiment, in which the light beams are derived from completely separate sources, and are superposed only at the detector,

64

SEMICLASSICAL RADIATION THEORY

[II,

99

there appears to be no way, at first sight, in which a photon from one beam can get into the other. Rather, it is suggested by eq. (59) that the superposition state is produced in the process of absorption. In order to understand the physical reason for this, we consider the situation illustrated in Fig. 14, where the two light beams of wavelength A, inclined at some small angle 8,are brought together on a receiving screen,

i

p,

;

t

A/ 8

h&X

I

Fig. 14. Illustration of the principle of the interference experiment with light beams derived from two separate sources.

where, according to electromagnetic theory, they produce interference fringes of spacing A/O. If the photodetector that is used to explore the interference pattern is to resolve the fringes, it must have a spatial resolution Ax in the detector plane better than Ale. It then follows from the uncertainty principle that if each photon is localized to an accuracy Ax, it has a momentum uncertainty Ap, in the same direction such that lAP,l > he/A

=

21PA

(61)

wherep, is the component of the photon momentum parallel to the receiving plane. As a consequenceof eq. (61) it is impossible, in principle, to determine from which source the photon was emitted, which means that it should be regarded as coming not from one source or the other, but, in a sense, from both. The measurement forces the photon into a superposition state in which it behaves as if it was associated with both light beams, and these two states of each photon interfere. This explanation follows the lines of the usual interpretation of the formation of interference fringes in an interferometer, except for the fact that the superposition state is here produced a posteriori, so to speak, in the process of detection. This may be hard to accept, but the conceptual problems do not invalidate the argument. This experiment does however admit one possibility that has no counter-

11,

5 101

CONCLUSION

65

part in conventional interferometry. Because the light is produced by two separate sources, we could, in principle, determine from which source the photon came from an examination of each source after the measurement. Let us suppose that each source has 10 excited atoms prior to the interference measurement, and that, following the measurement, one source is found still to have 10 excited atoms, while the other one has only 7. Does this not tell us from which source the photons came? The answer is that it does, but that the information precludes the formation of interference fringes, even though the sources are left undisturbed during the interference measurement. To see this we merely note that, if the atoms start and end in a state of definite excitation, the field radiated by them will also be in a state of definite excitation, i.e., in a Fock state. But, as is well known, (a,) and (a2) vanish in a Fock state, so that the interference terms in eq. (60) vanish also. No interference fringes should therefore be observed in that case, for interference is ruled out whenever the path of the photon can be established. This provides the possibility of another interesting test of Q.E.D., but the experiment remains to be carried out. $10. Conclusion

Although semiclassical theories have had considerable success in accounting for many observed effects, including photoelectric correlations, pulse propagation through a resonant medium, photon echos and optical free induction, they fail completely in other cases, and no evidence exists that should cause us to think of giving up Q.E.D. in favor of a semiclassical theory. Despite the internal difliculties of Q.E.D. that remain to be resolved, all existing experimental evidence points rather to the validity of Q.E.D., whose predictions have never actually failed, and to the inadequacy of the semiclassical approach in certain situations. It is perhaps tempting to take the view that this issue is now settled and should not become the subject of further debate and investigation. But there are no final answers in physics, and no matter how great our faith in Q.E.D. may be, we have to continue to examine its predictions critically and search for inadequacies. And in the course of this search it should become apparent that the efforts that have gone into the construction of alternative theories have not really been wasted. For these efforts have stimulated a great deal of new activity, both in the tackling of unsolved problems of Q.E.D. and in the performance of new experimental tests of the theory, and that is the basis for progress in this as in other fields.

66

SEMICLASSICAL RADIATION THEORY

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E. WOLF. PROGRESS IN OPTICS XI11 0 NORTH-HOLLAND 1976

111

OBJECTIVE AND SUBJECTIVE SPHERICAL ABERRATION MEASUREMENTS OF THE HUMAN EYE BY

W. M. ROSENBLUM and J. L. CHRISTENSEN School of OptometrylThe Medical Center, University of Alabama in Birmingham, Birmingham, Alabama, U.S.A.

CONTENTS PAGE

Q 1 . INTRODUCTION . . . . .

. . . . . . . . .

. . .

.

71

$ 2 . THE ANATOMY OF THE OPTICAL ELEMENTS OF THE EYE . . . . . . . . . . . . . . . . . . . . . . . . .

72

9 3. THE BASIC CONCEPTS OF SPHERICAL ABERRATION.

76

Q 4. HISTORICAL INTRODUCTION TO THE MEASUREMENT OF THE SPHERICAL ABERRATION OF THE EYE . . . .

77

.

9 5. SUBJECTIVE ABERRATION MEASUREMENTS OF THE EYE . . . . . . . . . . . . . . . . . . . . . . . . . .

79

9 6. OBJECTIVE ABERRATION MEASUREMENTS OF THE EYE . . . . . . . . . . . . . . . . . . . . . . . . . .

0 7 . CONCLUSIONS . . . . .

. . .

. .

. .

86 .

89

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

90

.

. . .

. . .

“Ofall the external senses, the eye is generally supposed to be by far the best understood; yet so complicated and so diversified are its powers, that many of them have been hitherto uninvestigated; and on others much laborious research has been spent in vain”. Thomas Young, M.D., F.R.S. November 27, 1800

0

1. Introduction

The concept of the human eye as an array of symmetric, centered, homogeneous optical elements, has promoted the study of its basic optical characteristics. Using this type of modeling, the human eye is considered as a two element optical system composed of a corneal lens, a crystalline lens, and an image plane coinciding with the retina. These elements are seen in Fig. 1.1. This array can then be analyzed by use of paraxial ray trace techniques in order to locate the principal planes, nodal points and the anterior and posterior focal planes of the eye. The limitations of this type of analysis is clearly brought to light when the aberrations of the eye are considered. The anatomy of the optics of the visual system can no longer be neglected in modeling the imaging process and the application of optical ray trace techniques developed for arrays of lenses which possess an axis of revolution leads to confusing results. The need for considering the actual structure of the eye is immediately recognized when one attempts to measure, in vivo, the geometrical aberrations, such as spherical aberration. In order to understand the aberration measurements of the human eye, the following discussion will be divided into three parts : in the first part the components of the visual system will be briefly analyzed, in the second part, a historical outline of the relevent spherical aberration measurements will be presented, and in the third part, an examination of some of the most recent attempts to measure the aberrations of the human eye will be given. 71

12

S P H E R I C A L A B E R R A T I O N MEASUREMENTS

1

I

I I-visuaf oxis

I v i t r m s humor

Fig. 1 .l. Horizontal section of human eye. (From Walls, The vertebrate eye. Bloomfield Hills, Mich., Cranbrook Institute of Science, 1942.)

0

2. The Anatomy of the Optical Elements of the Eye

In the human eye (Fig. 1.1) light from an external object is focused by the cornea and lens on the retina, assuming there is no refractive error. At the retina some of the light is absorbed by the photopigments located in the outer segments of the rods and cones. Electrical activity is generated which leads to the production of nerve impulses which are transmitted via the optic nerve and tract to the portions of the brain concerned with vision. The eye’s optical array participates in the information gathering role of the visual sense by preserving at the retina the spatial relationships between the individual rays arising from separateGxterna1 object points. The optics of the eye can thus be treated as an independent, sensory subsystem. 2.1. CORNEA

The cornea is the first refracting element of the eye. It is about 11.7 mm

“1,

9 21

T H E O P T I C A L ELEMENTS OF T H E EYE

13

in horizontal extent and 10.6 mm along its vertical dimension. At its center the cornea is 0.52 mm thick and expands to 0.67 mm in the periphery. The front surface separates air, with an index of 1, from the corneal substance which has a refractive index of 1.376. With the development of the ophthalmometer*, investigators realized that the cornea was not a surface of revolution. GULLSTRAND [I19621 writing in 1896, stated:

“. . . the form of the normal cornea can be described by saying that there is a central optical zone where the curvature is approximately spherical and which extenh horizontally about 4 mm, and somewhat less than this vertically, and is decentered outwarh and also a little downwar&; and that the peripheral parts are considerably flattened, decidedly more so on the nasal side than on the temporal, and usually more so upward than downwards”. The optical zone or cap is generally regarded as being spherical with a radius of curvature of 7.8 mm. Most “schematic” eyes employ this value or one very close to it for the curvature of the anterior corneal surface. The curvature of the posterior corneal surface, which separates the cornea, of refractive index 1.376, from the aqueous humor, of refractive index 1.336,has also been measured by reflection techniques and determined to be about 6.8 mm in radius. The exact configuration of this surface has not been determined. There seems to be a great amount of variation from eye to eye in the position of the optic cap on the cornea. HELMHOLTZ [1962] believed the optic cap to be decentered nasally and superiorly. Gullstrand, as indicated in the quote above, disagreed. BIER [I9561 stated that the optic cap is surrounded by a band of concavity leading to peripheral regions which are convex but much flatter than the central zone. He also indicated that the nasal half of the cornea is flatter than the temporal half. Subsequent investigators have not been able to confirm the existence of an annulus of negative curvature as described by Bier. KNOLL[I 9621 provided the first qualitative measurements of corneal

* The ophthalmometer is a clinical instrument used to measure corneal curvature. A luminous object is placed at a specific distance before the cornea and the size of its image, formed by the cornea acting as a convex mirror, is optically determined. From the relationship between the object and image size the corneal curvature can be calculated. This device is limited in accuracy as it is designed to measure the mean curvature over approximately a 3 mm region of the cornea and to measure only the curvature of spherical surfaces.

74

SPHERICAL ABERRATION MEASUREMENTS

CIIL

§2

curvature since Gullstrand. Using the technique of photokeratoscopy* he measured the radii of curvature of 3 zones of the horizontal meridians of 67 corneas. The mean radius of curvature of a central zone having a 2 mm diameter, a zone (1) having a 4.5 mm central diameter and a 1 mm annular width and a zone (2) having an 8.5 mm central diameter and an annular width of 1 mm were determined. Zones (1) and (2) were measured for both the nasal and temporal regions of the horizontal corneal meridian. All three zones were centered on the subject’s line of sight; a line operationally determined by connecting the point of fixation to the center of the entrance pupil of the eye. The corneas were placed in one of four categories on the basis of their symmetry (less than 0.2 mm variation) or asymmetry (over 0.2 mm of variation) of zone (1) and with regard to the flattening detected in zone (2) : Number of corneas Type 19 A - zone (1) is symmetrical, zone (2) flattening is less than 2 mm 22 B - zone (1) is symmetrical,zone (2) flattens over 2 mm 11 C - zone (1) is asymmetrical, zone (2) flattening is less than 2 mm 15 D - zone (1) is asymmetrical, zone (2) flattening is over 2 mm Invariably, nasal readings of zone (2> were flatter than the central value, the zone (1) nasal value, or the temporal zone (1) or (2) readings. The nasal regions between zone (2) and the limbus were markedly flattened in many corneas. A pronounced convexity just prior to the limbus was seen in several corneas. Corneas were also found which were spherical centrally but toroidal in the periphery. Ten type A corneas, symmetrical in zone (l), were asymmetrical in zone (2). The existence of a corneal axis of symmetry is doubtful. MANDELL [19651modified a keratometer so that curvature measurements of regions of the cornea only a mm in extent could be made. Standard keratometers average the curvatures over 2.6-3.0 mm regions. By means of a series of fixation targets, peripheral curvatures could also be determined very accurately. * The technique of photokeratoscopy involves taking a photograph of the cornea imaging an object. The object can be measured as can the object distance and the image size is determined from the photograph once the magnitfication is known. Just as in keratometry, the corneal curvature can be determined from the relationship of these entities.

THE O P T I C A L ELEMENTS OF THE E Y E

Fig. 2.1. Corneal contours as measured by small-mire keratometry.

Twenty-six corneas were examined ; a great deal of individual variation was found (Fig. 2.1). Only 5 corneas were found to have corneal caps. Symmetry of flattening was found for 5 of the corneas ; 16 corneas flattened more rapidly on their nasal sides than temporally. The point of maximum corneal curvature was centered on the line of sight for 4 subjects and in the remainder was decentered about equally temporally or nasally. Mandell concludes that the classical concept of an optic cap does not accurately depict most corneas; rather, the central zone of the cornea is elliptical and the peripheral regions flatten more rapidly than would be predicted by an elliptical function. MANDELL [1971] measured the corneal contours of eight subjects using photokeratoscopy. Various mathematical curves were evaluated and the ellipse was chosen to model the central cornea. Mandell feels that schematic eyes should include elliptical corneas which would better explain the spherical aberration shown by human eyes.

16

SPHERICAL ABERRATION MEASUREMENTS

[rrr,

53

This discussion is limited to a consideration of the form of the cornea along one meridian. Most corneas exhibit various curvatures in different meridians ; a condition known as astigmatism. This adds an additional complication to the task of specifyingthe corneal contour. 2.2. AQUEOUS

The aqueous humor, located in the anterior and posterior chambers of the eye (Fig. 1.l), has a refractive index of 1.336. The aqueous is a true fluid having no internal structure. 2.3. CRYSTALLINE LENS

The crystalline lens is the most complicated optical element in the eye. The front surface of the lens in Gullstrand’s schematic eye has a radius of curvature of 10 mm and the posterior of 6 mm. In reality, the front surface of the lens seems to be elliptical in shape (FISHER[1969]). The thickness of the lens varies throughout life as additional lens fibers are formed at the equator. The adult lens is 3.U.O mm thick. By far, the most complicated aspect of the lens is its refractive index. The lens is not homogeneous with regard to refractive index; rather, the center of the lens, the nucleus, has the highest value and the periphery, the cortex, the lowest. The index is thought to decrease parabolically with distance from the lens center. There are also abrupt changes in the refractive index producing, so called, isoindicial surfaces. There is an embryological, a foetal, an infantile and an adult nucleus which probably vary from one another and from the cortex in index. The curvature of these isoindicial surfaces would have to be known to completely specify the action of the lens on light.

5

3. The Basic Concepts of Spherical Aberration

Any one attempt to classify the aberrations of the eye in terms of an optical system which is homogeneous, spherical and has an axis of revolution immediately gives rise to difficulties. For the eye, the existence of an axis of revolution is only an approximation, since the presence of astigmatism in the cornea and crystalline lens necessitates an evaluation of the aberrations in each meridian. When the astigmatism of the cornea-lens system is slight, one can speak of approximately measuring the spherical aberration on the fovea when the axis of revolution and the visual axis are

111,

§

41

77

HISTORICAL I N T R O D U C T I O N

CIRCLE OF LEAST CONFUSION

FOCUS OF PARAXIAL RAYS

LATERAL SPHERICAL ABERRATION

i

FOCUS OF MARGINAL RAYS

LONGITUDINAL SPHERICAL ABERRATION

Fig. 3.1. Positive longitudinal and lateral spherical aberration. (From J. R. Meyer-Arendt,Introduction to Classical and Modem Optics, Prentice-Hall, 1972.)

close. The measure of the spherical aberration for a symmetric system is then reduced to an analysis of the relation between the height of the entering ray, [19671).When the ray height, and the position it crosses the axis (LEGRAND measured from the optical axis is increased and the axial crossings occur in front of the paraxial focal point the system is “under corrected”. This case is commonly referred to as positive spherical aberration. In the case when the ray height is increased and the axial crossings occur beyond the paraxial focus the system is “over corrected”. The case of positive spherical aberration is depicted in Fig. 3.1 in which the marginal rays focus prior to the paraxial focal point. The longitudinal spherical aberration is defined as the distance between the paraxial and the marginal focii; the lateral spherical aberration is defined as the distance between the marginal ray and the paraxial focus, measured in the paraxial focal plane. When dealing with the eye, the longitudinal spherical aberration is given in terms of the dioptic difference between the two focii and the dioptic value of the circle of least confusion is defined as one third the sum of the marginal and paraxial diopter values. Thus, to evaluate the spherical aberration of an eye in vivo, one must measure the position of the ent‘ering rays of light with respect to an axis of symmetry and determine either objectively or subjectively the position at which they cross the axis.

0

4. Historical Introduction to the Measurement of the Spherical Aberration of theEye

Thomas YOUNG’S [18001 experiments in determining the focal distance of the human eye, made use of an instrument called an optometer. Based upon

78

i P H E R I C A L A B E R R A T I O N MEASUREMENTS

"11,

§

4

1

0"

Fig. 4.1. Various views of Thomas Young's optometer are shown. Uppermost is the insert containing a series of slit openings which are placed in front of the eye. Below, is a side and bottom view and on the left, the scale. used for measuring the dioptric power of the eye.

111,

0 51

SUBJECTIVE ABERRATION MEASUREMENTS

79

experiments performed by Porterfield and Sheiner, Young showed that if an object is at a point of perfect vision, the image on the retina will be single while objects placed in front or behind will appear doubled. The reader can construct a Young’s optometer (Fig. 4.1) by drawing a line along the center, lengthwise, of a piece of cardboard 8”-1” and observing with one eye the line through pairs of slits ranging from 1/32” to 1/16” in breadth, placed directly in front of the pupil. By noting the point where the double images of the line convergesto a single point, one can ascertain the corrective spectacles required for myopic or hyperopic eyes. The optometer served as the basic instrument for measuring the spherical aberration of the eye since each pair of slits allowed the determination of the dioptric power of a region of the pupil. By employing a series of slits of varying separation, the change in the ocular power across the pupil can be determined. The difficulty with this type of experiment lies in locating an optical axis for the eye and accurately placing the slit about this axis. Even the determination of the portion of the pupil each slit covers is difficult. Yet Young’s optometer serves as the basis for modern experiments as we shall see. VOLKMANN [18461observed a pin through four small holes placed in front of his pupil. He measured the degree of spherical aberration by noting the position of the pin first through the inner two holes, then through the outer ones. Nine observers were used, of which, five showed positive spherical aberration and four showed negative spherical aberration. Helmholtz in 1866 noted “the experiments of Young and Volkmann would undoubtly have indicated the nature and magnitude of the spherical aberration of the eye but in most meridians of the eye the points of intersection of the refracted with the central ray do not form a continuous series at all, so that the conception of spherical aberration does not apply”. The non-coincident method mentioned above was the method used by AMESand PROCTOR [1921], and VONBAHR [1945] to measure the aberrations in the vertical and horizontal meridians.

0

5. Subjective Aberration Measurements of the Eye

A direct method of examining the spherical aberration of the eye would be to pass two beams of light, one traveling centrally and the other peripherally through the pupil. No spherical aberration would be evident if both beams coincided on the retina, whereas positive aberration would cause the peripheral beam to come to a focus closer to the cornea, than the central beam, and for negative aberration, focusing would occur further from the

80

SPHERICAL ABERRATION MEASUREMENTS

[m,$ 5

(b) Fig. 5.1. Schematic representation (a) of a subjective method for viewing the upper and low portions of the lens L, upon which vertical threads are placed. The upper vertical thread is separately imaged through different portions of the eye by moving the source S 2 . The image (b) appears as an illuminated field with the vertical threads in alignment when there is no spherical aberration. In the presence of a variation of dioptric power the images will not coincide as seen in (c).

Fig. 5.2. Calculation of the value of ocular chromatic aberration as a function of the apparent displacement of the threads f l and f 2 .

111,

§ 51

SUBJECTIVE ABERRATION MEASUREMENTS

81

cornea than the central beam. The amount of difference in the two focii could be determined, yielding a measure of the spherical aberration in the human eye. IVANOF-F[19531has employed this parallax method to measure the spherical aberration of the human eye. His apparatus, is schematically shown in Fig. 5.l(a). Two point sources of light S, , S, are imaged by the lower and upper halves of a lens in the pupil of the eye. In front of the lens two reticules are placed, one reticule is fixed in the center of the lower lens, and the other is moveable across the upper half of the lens. Due to the arrangement of light sources the observer sees the lens completely filled with light and the images of the upper and lower reticules. If the images are coincident, Fig. 5.2(b), the threads coincide and no aberration would exist. When the source Sz is moved laterally across the pupil, the image forming rays enter the pupil a known distance from the centrally located source S, . The image thus formed is displaced, appearing as Fig. 5.2(b). By displacing the movable reticule until alignment is obtained, the amount of spherical aberration can be determined. Using Fig. 5.2(a), we can see that when the reticules (fl , f,) are in alignment on the lens but imaged through different portions of the ocular system, the retinal images (f; , fi) are displaced. The amount of displacement on the retina will be a function of the distance, h, of the moveable source S2 imaged in the pupil measured from the centrally located source S, imaged in the pupil. By moving the reticule fz, downward, by an amount 6 , the image is moved into alignment, Fig. 5.2(b). The +2.0 i

-2.04 Fig. 5.3. Spherical aberration of the eye.

82

SPHERICAL ABERRATION MEASUREMENTS

“11,

05

object vergence for f, is l/p,, and the object vergence for f2 is l / p2 . Since the images are coaxilly located, the difference 1/ p l - 1/p2 is the dioptric value of spherical aberration the eye exhibits. Using the small angle approximation lipl = -a,/h, lip2 = - a2/h.Thus the spherical aberration is equal to - S/pl h diopters. Let us return to the problem of defining an optical axis for the eye. As Ivanoff points out, the eye is not a centered optical system, hence the notion of an optical axis of the eye is fictitious. He replaces this concept with what he calls the “achromatic axis of the eye”, a ray directed to the center of the fovea entering the eye at a point such that there is no chromatic dispersion. From this reference, the spherical aberration of the eye is studied as a function of the points of entry and the states of accommodation. The measurements shown in Fig. 5.3 were taken along a horizontal meridian across the pupil. The curve designated by squares corresponds to the unaccommodated eye, the one designated by circles corresponds to the eye with 1.5 D of accommodation and the curve marked by triangles corresponds to an eye with 3.OD of accommodation. Ivanoff noted the asymmetry with respect to achromatic axis and that the value of the aberration is generally larger on the temporal side of the cornea than on the nasal side. The mean spherical aberration of all the individuals examined is shown in Fig. 5.4. In general, the eye is undercorrected for spherical aberration when accommodation is relaxed. The eye is approximately corrected when accommodating between 1.5 and 2.0 diopters. Over correction results when accomodation reaches the level of 3.0 diopters.

Fig. 5.4. Mean spherical aberration of the investigated ten eyes.

111,

§ 51

SUBJECTIVE ABERRATION MEASUREMENTS

83

The eye when examined on a point to point basis is seen to be irregular in its refractive power, therefore in order to obtain a spherical aberration curve for the entire eye, the experimenter would have to examine many meridians and average the results. KOOMEN[1949] has dealt with this problem by placing a series of centered annular apertures over the pupil of the eye and determining the spectacle correction for each aperture; in this way, all meridians are measured simultaneously. In order to perform this type of measurement, Koomen considered the fully dilated pupil to be divided into a central circular area of two millimeters diameter and five to seven annular areas of increasing diameter extending to the edge of the pupil. The spectaclecorrection for each annular zone considered with respect to the correction for the central zone yielded the magnitude and sign of the aberration. (The experimental apparatus is depicted in Fig. 5.5.) The observer, by placing his right eye behind the beam splitter viewed two scenes. FIXATION

OBJECT

PHOROPTOR

20 F T

Fig. 5.5. Apparatus for measurement of the spherical aberration of the eye

One scene consisted of four transparent letters illuminated from behind and acted as a fixation object. By placing a spectacle lens of proper power into this optical path, the apparent position of the four letters could be changed, thus stimulating accommodation for this distance. The other scene which was viewed through the annular apertures consisted of a distant pair of point sources whose separation was adjusted to be just above the resolution threshold for each annulus used. Measurements were made by allowing the observer to manipulate the phoropter, while viewing the fixation target until the best resolution of the double star image was obtained. This procedure was repeated for a series of annular apertures, starting with the smallest, up to the largest allowed by the pupil. A result of Koomen’s experimentation on his own eye is shown in Fig. 5.6 which shows the variation in spherical aberration with accommodation.

84

Y0N

SPHERICAL ABERRATION MEASUREMENTS

4 1

- POWER

OF I ACCWMODATION-FIXING LENS, IN DlOPlERS--4.6

,

, -3.7

-2.9

-2.1

-1.1

0

BEST SPECTACLE CORRECTION FOR ZONE ( DIOPTERS) Fig. 5.6. Spherical aberration curves of the right eye of subject M.K. (emmetrope).

Slit ight path

Registration of the position of the objective (gauged in diopters)

0,

< - -

Objective diaphragm

-turns when objective moves U

Eye position indicator

/ocular

Half mirror

I

Observer

Fig. 5.7. Arrangement of Van den Brink's apparatus.

111,

§ 51

SUBJECTIVE ABERRATION MEASURE ME NT S

85

Koomen noted that positive aberration was found for the relaxed eye; the spherical aberration was reduced as the eye accommodated as seen in the case of his own eye and negative aberration was reached at high levels of accommodation. VAN DEN BRINK[1962] extended the methods of accurately measuring the aberrations of the eye so as to consider the entire ocular surface in turns of a dioptric power mapping. The method depicted in Fig. 5.7 determines the average spherical power of a portion of the eye-lens, by use of four light paths which provide the test target, an annular comparison field around the test target, a fixation point for direction, which allows for control of the accommodation state, and an optical system for positioning the eye. The test object viewed through an off set diaphragm consisted of an array of stripes which was surrounded by a comparison field of fixed brightness. The subjective brightness of the target could thus be changed to match the comparison field. In order to determine the position of the observer’s eye a small black disk was imaged within the pupil of the eye, then enlarged to equal the size of the pupil. Any movement of the eye caused the image of the disk to move aside and a white cross to appear. A small red circle with a fixation point in its center was seen in the middle of the test target and used to determine state of accommodation.

Fig. 5.8. Dioptric power as a function of the place at several states of accommodation (observer E, left eye). Points of the same power are connected by iso-diopter lines (vertical object). State of accommodation: 0.0 diopter, vertical object.

Fig. 5.9. State of accommodation: 1.0 d i o p ter, vertical object.

86

S P H E R I C A L A B E R R A T I O N MEASUREMENTS

Cm § 6

By positioning the diaphragm in front of the objective lens and changing its diameter, a selecr part of the eye-lens was used to create an image on the retina. The objective lens was moved to focus the target for a given off axis position. The dioptric power equivalent of this movement was employed to map the dioptric power variations of the eye-lens system. The results are shown in Figs. 5.8, 5.9 which are mappings of the iso-dioptric lines of an observer with zero accommodative power and for an accommodative state of 1.Odiopters. It can be clearly seen that the dioptric power of the eye is not homogeneous and that with accommodation even more complex variations result,

0

6. Objective Aberration Measurements of the Eye

In the previous examples, the spherical aberration of the eye was studied in terms of the subjective image quality on the retina. BERNYand SLANSKY [I9701 have examined the problem in an objective manner by using the retina and the optical system of the eye to produce a real image which is examined in a Foucault’s test arrangement. In Fig. 6.1, one can see the optical system used to create an image of an illuminated sIit, M, upon the

Fig. 6.1. Optical apparatus: W, light source; W‘, electronic flash; W’, subject pupil; W”’, photographic plate; a, water cell; f, filter; M, wide slit; p, fixation test; r, retina; e, Foucault’s knife-edge; 0, photographic lens; L, beam-splitter ; LIL,, focusing lens system for the variation of the subject’s accommodation.

111.

0 61

OBJECTIVE ABERRATION MEASUREMENTS O F T H E EYE

87

Fig. 6.2. Principle of the wide slit Foucault’s method: B’D’ = R I R z = 20 = slit width; ET = transverse aberration; E = intersection of the knife-edge e and the optical axis; W” = optical system to be studied; W”’ = image given by lens 0.

I =

retina. The image on the retina acts as a secondary source, forming an image at E. A Foucault’s knife edge is placed at E intersecting the image. A photographic lens is then positioned behind the knife edge so that an image of the subject’s pupil will be formed on a photographic emulsion. A photometric evaluation of pupil thus photographed will allow the determination of the normalized wave-aberration referred to a sphere centered a t E. In order to understand this, let us look at Fig. 6.2 where the slit B D is imaged along the axis of symmetry, to form the image B’D’. If one then considers the image formation through the small segment dW“; the image will be translated downward. This shift due to the presence of aberration can be described as a local distortion of the emerging wave front and the degree of distortion determined by comparison with a reference sphere centered at E. Thus the degree of image shift or transverse aberration can be recorded in terms of a change of intensity on a photographic emulsion placed at W”‘. By carefully scanning the photographs, Fig. 6.3, the authors have determined the transverse aberration for each point in the pupil as shown in Fig. 6.4. This is a diagram of the wavefront

88

[m,0 6

S P H E R I C A L ABERRATION MEASUREMENTS

distortion in the pupil were the wavefront is represented by equal phase curves referred to as a sphere centered on a chosen focal point. The Figs. 6.4A and B are for the same eye with a 1 diopter accommodation at different times, and the use of 1257 and 6400 experimental points, respectively. Only one subject has been examined by Berny and Slansky because the analysis of the photographs require several months.

I

Y

i

I

Y

i

Fig. 6.3. Foucault’s shadows in the pupil, showing deviation components which are perpendicular to the knife-edge. Accommodation, 1 dioptre; A, A,, vertical knife-edge; BIB,, horizontal knife-edge.

89

CO NCL USI O NS

;

M s1.00

A -0.750 4- r 0.500 0 -.0.230 0 Y0.125 0

---

0 +0.125 8 +0.250 X +0.500

A +0.750

OO * +++L1.250 1.500 Dl

0

Fig. 6.4. Wavefront A of the optical system of the eye. Level curves in terms of wave number ( d / l ) : , W,-l.OOO; A, -0.750; +, -0.500; 0,-0.250; 0,-0.125; -----, 0.000; 0 , f0.125; W, +0.250; x , f0.500; A, +0.750; H,f l . O O O ; +1.250; 0 , +1.500. Reference point E. Pupil diameter 7 mm. Accommodation 1 6 . A, 1257 experimental points; B, 6400 experimental points.

*,

Q 7. Conclusions We have seen up to now how the dioptric power of the eye varies extraaxially. This variation of optic power would be expected to affect the performance of the eye. By using the modern method of measuring the modulation transfer function for various pupil sizes this prediction can be and GUBISCH [19661 have produced such a series confirmed. CAMPBELL of MTF curves for the human eye with pupil sizes 1.5,2.0,2.4, 3.0,3.8,4.9, 5.8, 6.6 mm (Fig. 7.1). It is clearly seen that the ocular contrast decreases progressively with increasing pupillary diameters. The experiments of KRAUSKOPF [1964] indicate that the line spread function is broadened for annular pupils as compared to circular pupils having the same outside diameter, indicating the absence of simple spherical aberration in the human eye. Krauskopf attributes the increase in the blur of the retinal image due to large pupillary apertures to variations in the dioptric power over the plane of the pupil. Experimental evidence shows that the concept of spherical aberration as generally applied to the human eye is a gross simplification. As modern

90

SPHERICAL ABERRATION MEASUREMENTS

“I

I.a

0.a 2 0

VI

sg Ob 4 K

c c

2

VI

0.4

c

t 0

0

0.2

0 NORMALIZED SPATIAL FREQUENCY

Fig. 7.1. Normalized modulation transfer functions of the human eye, averaged among three subjects. The results are normalized to the highest spatial frequency transmitted by an ideal optical system with light of 570 nm wave-length. Symbols are: 0 , 1.5 mm pupil; x , 2.4 mm pupil. Dotted curve gives performance of ideal system.

techniques are brought to bear on the problem of off-axis variations in ocular refracting power a more realistic model of the optical system of the eye will evolve. Such a model wodd lead to a better prediction of visual performance and its limitations. A clearer understanding of light paths in the eye, especially peripherally as in the case of laser retinal coagulation, would be another benefit of such analysis.

-References AMES, A. and C. A. PROCTOR, 1921, Dioptrics of the eye, J.O.S.A. 5, 22. BERNY,F. and S. SLANSKY, 1970, Wavefront determination resulting from Foucault test as applied to the human eye and visual instruments, in: Optical Instruments and Techniques, ed. J. Houe Dickson (Oriel Press, 1969). BIER, N., 1956, A study of the cornea in relation to contact lens practice, American J. Optom. 33,291-304. CAMPBELL, F. W. and R. W. GUBISH, 1966, Optical Quality of th6Human Eye, J. Physiol. 186, 558-578. FJSI~ER, R. F., 1969, The significance of the shape of the lens and capsular energy changes in accommodation, J. Physiol. 201, 21-47.

1111

REFERENCES

91

GULLSTRAND, A., 1962, in: Helmholtz’s Treatise on Physiological Optics, Appendix I1 (Dover Press, N.Y.). H., 1962, Helmholtz’s Treatise on Physiological Optics (Dover Press, N.Y.). HELMHOLTZ, IVANOFF, A,, 1953, Les Aberration de l’oeil. KNOLL,H., 1962, Corneal contours in the general population as revealed by the photokeratoscope, Amer. J. Optom. 38,389-397, 1961. KOOMEN, M., R. TOUSEY and R. SCOLUIK, 1949, The spherical aberration of the eye, J.O.S.A. 39,270-376. KRAUSKOPF, J., 1963, Further Measurements of Human Retinal Images, J.O.S.A. 54,715-716. LEGRAND, Y., 1967, Form and Space Vision, Rev. ed. transl. by M. Millodot and G. Heath (Indiana University Press). MANDELL, R. B., 1965, Contact Lens Practice: Basic and Advanced (Charles C . n o m a s , Springfield, IL) pp. 3 5 4 7 . R. B., 1971, Mathematical model of the corneal contour, Brit. J. Physiol. Optics MANDELL, 26, 183-197. VANDEN BRINK,G., 1962, Measurements ofthe geometrical aberrations of the eye, Vis. Res. 2, 233-244. VOLKMANN, A,, 1846, Wagners Handworterb. Physiol. VON BAHR,G., 1945, Investigations in the spherical and chromatic aberrations of the eye and their influence on its refraction, Acta Ophthalmologica 23, 1 . YOUNG, T., 1801, On the mechanism of the eye, Phil. Trans. 91, 23-88.

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E. WOLF, PROGRESS IN OPTICS XI11 0 NORTH-HOLLAND 1976

IV

INTERFEROMETRIC TESTING OF SMOOTH SURFACES BY

G . SCHULZ and J. SCHWIDER Zentralinstitut fur Optik und Spektroskopie, Akadernie der Wissenschuften der DDR, Berlin-Adlershof, DDR

CONTENTS PAGE

§ 1. INTRODUCTION.

9 9

9

. .

. . . . . . .

.

.

. .

.

. . .

95

2. RELATIVE TESTING BY COMPARING TWO SURFACES

96

3. ABSOLUTE TESTING BY COMPARING SEVERAL SURFACES . . . . . . . . . . . . . . . . . . . . . . . .

118

4. COMPARING A SURFACE WITH ITSELF . .

140

.

. . .

.

5 5. COMPARING A SURFACE WITH A HOLOGRAM

146

§ 6. SOME SYSTEMATIC SOURCES OF ERROR AND LIMITS OF MEASUREMENT . . . . . . . . . . . . . . . . .

157

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

162

SUPPLEMENTARY NOTES ADDED IN PROOF. . . . . .

166

0

1. Introduction

The testing of smooth surfaces by interferometry has recently gained in importance. This article is intended as a survey of the progress made in this field. It deals with the quantitative examination of smooth surfaces of components used in optical systems of all kinds. These surfaces are usually boundaries between a solid, e.g., glass, and the surrounding air. In special cases the solid surface is replaced by a liquid one. The exact geometric shape of these surfaces is determined by optical interference methods, using especially reflected light. Methods are covered where the accuracy with which the depth is measured reaches an order of magnitude of 1/10 or less of the wavelength. This chapter will concentrate on the description of methods involving comparatively high accuracy of depth measurement. On account of the optical uncertainty relation (Q 6), this leads to a lower degree of accuracy in the lateral measurement. Microscopic interference methods, on the other hand, (see, e.g., TOLANSKY [1960], KRUG, RIENITZand SCHULZ [1964], YAMAMOTO [1970]) are not discussed helz, and the surfaces to be measured are assumed to be polished and smooth. Interferometry makes possible the measurement of differences in the path of light and thereby of variations in the shape of surfaces from which this light is reflected (BORNand WOLF[I9641 p. 256). Measuring such variation; amounts to a comparison of surface contour. It includes several possibilities: the comparison of different surfaces (this is dealt with in # 2 and 3); comparing a surface with itself (§4), e.g., when this surface reflects a well defined wavefront, which is then doubled by a shearing method and compared to itself; comparing a surface with a diffracting component, i.e., with a hologram (0 5), by comparing the wavefronts emerging from the surface and the hologram. Finally, § 6 deals with some general problems of systematic errors and limits-of accuracy. When comparing different surfaces, one distinguishes between relative and absolute methods. The former are dealt with in 9 2, the latter in Q 3. 95

96

I N T E R F E R O M E T R I C T E S T I N G OF SMOOTH S U R F A C E S

CIV,

P2

Using only relative methods, one can only obtain relations between different surfaces, e.g., the sum of the deviations of the two surfaces from ideal planes. Absolute methods, on the other hand, provide knowledge of the surface itself, e.g. of its deviations from a mathematical plane.

5

2. RelativeTesting by Comparing Two Surfaces

By interferometric comparison of two surfaces A, B, one can ascertain their differences in shape. These differences in shape first appear as local variations of thickness of a layer of air. The layer of air is situated between the two surfaces either really or virtually, and is bounded by them.

Fig. 2.1. Basic arrangement for determining the deviation sums a(<,q ) = x(<, q ) + y ( c , q ) of the surfaces A, B by interference. The light entering from above is partially reflected by A and B, and the two beams interfere.

For example, Fig. 2.1 shows a real layer of air, formed between two almost flat surfaces A, B placed on top of each other. From above, an almost plane wave of ligh-t impinges at right angles. Through reflection at the boundaries of the layer of air, waves are formed which interfere. The figure shows the case of two-beam interference in reflected, light (BORNand WOLF[1964] p. 286). One can measure the optical path difference of the interfering beams (at least its local changes; for details of the measurements see 9 2.4 and § 2.5). At the point l, q the difference between the paths is :

45, rl) = 2{t(l, rl)- C4l,rl) + At?rl,l>.o;

(2.1)

here, no is the index of refraction of the layer of air (for errors and corrections to (2.1) see 4 6). The linear function t(5, v )

=

t(O,O)+&

t+P,rl

(2.2)

is defined by the position of the reference planes PL, , PL, . The reference plane should fit the surface as closely as possible. The reference plane can be defined by the condition that it passes through 3 fixed points of the

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surface. The function t(5, q) is then determined by the path differences at these points. Taking into account (2.1) and (2.2), one canthen measure the sum

4549 +fie,

?) =

4 5 , rl)

(2.3)

of the surface deviations. Generally, we consider a surface deviation to be positive when it increases the thickness of the test piece. It therefore increases in the direction of the normal to the surface which points into the air. The normals nA, ng of the two surfaces to be tested point in opposite directions in Fig. 2.1. Testing the surfaces yields relative results when only a relation between the two surface deviations can be determined, but not the deviations themselves. In Fig. 2.1, this relation is the sum (2.3). BEAM S P L m

Fig. 2.2. Basic arrangement for determining the deviation differencesd(& q ) = x(5, q ) - y ( < , q ) of the surfaces A, B by interference. Left: Michelson interferometer. Right: Image of the surfaces A, B in the same optical space after reflection at the beam splitter BS.

Fig. 2.2, on the other hand, shows a virtual layer of air, such as one finds, for example, when comparing the surfaces A, B in a Michelson interferometer. The two normals nA, nB to the surfaces here virtually have the same direction, as shown on the right of Fig. 2.2. In this case the path difference of the interfering waves is given by:

At the virtual layer of air one, therefore, measures the difference of the surface deviations :

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The distinction between the two basic types, i.e., the difference in direction of the norm?ls to the surfaces and, therefore, the distinction between sum and difference, shall serve as systematic characteristic for description, the reason being its value in testing surfaces. For measurement purposes the two sides of the test piece are not equivalent; for the most accurate measurements are usually only possible on thin layers of air. On the other hand, when I (c, q) depends on the refractive index of the test piece, disturbances may arise through the inhomogeneity of the refractive index. If one wants to be independent of this, then measurement of the sum requires optical arrangements, basically different from those for measuring the difference. Thus, when determining the sum, it is not necessary to have an optical element (usually required to be free from defects) in the path of the interfering beams between the two surfaces A, B (see Fig. 2.1). On the other hand, such an element is necessary for determining the difference; in Fig. 2.2 it is the beam splitter. Unlike Fig. 2.2, Fig. 2.1 allows for multiplebeam interference. Determination of the sum is dealt with in § 2.1, of the difference in 0 2.2. In principle, comparison of different surfaces is also dealt with in 5 2.3. For this comparison, however, a “null lens” is used as a further optical component, which adds some special problems. The difference in shape of the smooth surfaces to be tested must not be such that in some areas the interference fringes are too close together and cannot be sufficiently resolved. This means that there are restrictive requirements regarding the shape of the surfaces, particularly regarding differencesof inclination. Thus, Fig. 2.1 would not, in general, allow for the comparison of a sphere to a plane, but to a second sphere, in which latter case the reference surfaces are ideal spheres. One must, therefore, distinguish between tests on planes and spheres, etc. The layer of air between the test surfaces is usually chosen to be in the shape of a wedge (wedge adjustment, PI or P2 # 0). One thereby obtains an interferogram with almost straight interference fringes of equal thickness. The lateral deviations of these fringes from the ideal fringe system, which would be attained in the case

are then proportional to the sums or differences of the surface deviations to be measured. In the simplest case, a deviation equal to the distance between two fringes corresponds to a surface deviation of half a wavelength. With parallel adjustment (fil = fi2 = 0), fringes of equal inclination (Haidinger rings) have been used in place of fringes of equal thickness for the measurement of deviation sums varying slowly with position.

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2.1. DETERMINATION OF THE DEVIATION SUMS

For testing the deviation of a surface from a true plane the Fizeau interferometer is used in most cases. As this interferometer is sufficiently known, only a short description referring to Fig. 2.3 will be given here. Light from a spectrum lamp falls on the collimator by way of a mirror at a small angle to the optical axis. A fine pinhole, set in the focal plane of the collimator, serves to restrict the aperture. The surfaces to be tested, A, B, are placed together into the path of the parallel beam. A and B each reflect a wave which is focused in the focal plane of the collimator and appears as image of the pinhole. To observe the test surfaces, the eye is brought to the spot where these images of the pinhole are formed. At the position of the layer of air between the plates interference fringes of equal thickness are formed. For photographic recording, the eye is replaced by a camera, which is also focused onto the layer of air.

L Fig. 2.3. Fizeau interferometer for testing optical flats. A, B : combination of the test surfaces.

The collimator should be free from spherical aberration to avoid distortion of the interference pattern (TAYLOR [1957]). A well corrected objective with an aperturef/6 (wherefis the focal length) is sufficiently free from such faults. To examine a Fizeau interferometer for such distortions, the following procedure is suitable (DEW[19671) : 2 interference photographs of a combination of flats AB with the same number of fringes per diameter are taken with wedge adjustment (92.0). In both photographs the plane surfaces have the same lateral position with respect to the collimator, only the edge of the wedge lies once to the left and once to the right

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of the combination of surfaces. After reducing the deviations to parallel adjustment and subtracting the deviation sums of the two photographs, the non-linear distortions of the collimator remain, a distortion amounting to the distance between adjacent fringes causing an error of 112 in the deviation sums. A number of methods have been developed to evaluate the wedge interferograms. In the end they are all based on subtracting the deviations of an ideal wedge interference pattern from those of the real wedge interference pattern. To measure along a line, the values of the wedge thickness of a known ideal wedge are subtracted from the real thickness values (see 0 2.0 and Fig. 2.1, SCH~NROCK [1939], COSYNS [1953], BUNNAGEL [1956b], DEW[1966a]). The deviation sums related to an ideal pair of planes can be measured by a method combining graphical representation and calculation (SCHWIDER, SCHULZ,RIEKHER and MINKWITZ [1966]). To determine the positions of the reference planes relative to one another, the deviation sums at 3 points spaced as far apart as possible are equated to zero. For present purposes, this is sufficient for determining the pair of reference planes. (The remaining degrees of freedom of the pair of reference planes, which determine its absolute position and orientation, are unimportant for the resulting picture of contour lines of the layer of air.) Determination of the pair of reference planes which give the best fit can be made by the method of least squares (compare JONES and KADAKIA [1968]). When, in relation to an arbitrarily chosen and fixed pair of reference planes, the deviation sums of the surfaces at the points (lpr qv) are labeled upvand the deviation sums of the variable pair of reference planes are denoted by upv,then

has to be made a minimum. Hereby the Gpv form a linear position function : -,v

=

Yo+Y15p+Yz~".

(2.7)

S is made a minimum by varying the constants y o , y l , y z ; the values of the constants are determined from the 3 equations :

(2.8) In this way the pair of reference planes is obtained with maximum reliability, since it is defined by a combination of all the values measured. Equally important for optics is the examination of spherical surfaces.

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Here the deviation sums are also obtained with a Fizeau arrangement. The entering wavefront is nearly perpendicular to the test surfaces at every point. This is achieved by making the entering wave emerge from or run into the common center of curvature of the surfaces. Optical arrangements as in Figs. 2.4 and 2.5 are used. Concave surfaces can be tested with an arrangement as shown in Fig. 2.4 (KOLOMITZOV and DUCHOPEL [1956]). The use of a microobjective enables one to have,large numerical apertures and still fulfill the since condition. By using an auxiliary lens, convex surfaces can be examined, whereby the order in which test and reference surfaces are arranged is changed around. With an auxiliary lens the test surface can be fitted to a divergent wavefront (BIDDLES [1969]). By using a special condenser system convex surfaces can be examined without additional auxiliary lenses (KOLOMITZOV and DUCHOPEL [1956] ; GATES[1958] ; BERNARDand HABERMANN [1962] ; GUBEL,DUCHOPEL, MJASNIKOV and URNIS[1973]). The condensor of RIDE and ZIECLER[1968] includes aplanatic menisci, that of HARRIS [I9711 consists of achromatic doublets. Fig. 2.5 shows the schema of the interferometer. Either the reference surface is concentric with the second boundary surface of the master, or the latter forms an aplanatic meniscus /SPECTRUM LAMP

--MICROOBJECTIVE

Fig. 2.4. Fizeau interferometer for testing spherical surfaces. When testing convex surfaces, the order in which the reference surface and the surface to be tested are inserted is reversed. In this case the surface to be tested must be fitted to the entering spherical wave surface by means of an auxiliary lens.

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[IV,

d2

X/SPECTRUM LAMP

* I

\

I

+CONDENSOR

REFERENCE SURFACE

SUWACE TO

BE TESTED

‘d’

Fig. 2.5. Universal Fizeau interferometerfor testing spherical surfaces. Convex (top) as well as concave (bottom) surfaces can be examined. In the latter case one must imagine the two upper plates as having been removed. In this case the striatedpath of the rays applies.

together with the reference surface (HOPKINS [19711). This guarantees the fulfillment of the sine condition for the whole system, consisting of reference surface and condensor, for different reference surfaces. (For the effect of the aberrations and their influence on the accuracy of measurement see 9 6.) On account of the large number of lenses necessary for the condensor the interference pattern is degraded. When using a laser, therefore, disturbing interference systems appear. By using thermal light sources (spectrum lamps) the coherence can be so strongly reduced that parasitic interference fringes no longer occur, while only the contrast of the interference pattern is reduced by the disturbing light. By using aspheric condensors [1967], HOPKINS [1972]) the number of surfaces can be sub(HERRIOTT stantially reduced. A variant of the spherical Fizeau interferometer of Fig. 2.5 is made [1966]; HEINTZE, possible by the use of a laser light source (HERRIOTT POLSTER and VRABEL[1967]). Here the interferometer surfaces are both concave and arranged in such a way that the almost coinciding centers of curvature lie between these surfaces. Even with this configuration, multiple-

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beam interference patterns can be produced. On account of the large path differences walk-off (Q 6) can occur, which, however, can be removed by a field lens at the center of curvature. The arrangement mentioned in the last paragraph can also be used to test flatness with the Common-test. For this the flat to be examined is placed obliquely between the second of the two spheres and its center of curvature. By reflection at the flat the optical axis is bent, the light rays being reflected from both spheres just as before. Therefore the flat can be compared interferometrically with the second sphere, when the other parts of the interferometer (especially the other sphere) are ideal. As POLSTER [1970] has proved, fwo interferograms are necessary for testing the flat (while the shape of the sphere is known), the second being taken after turning the flat through 90" about its normal. This is because stigmatic imaging between a real point F and a virtual point F can be obtained not only with a plane, but also with an arbitrary hyperboloid of rotation with focal points F and F', the plane being only a special case. With a single interferogram one cannot distinguish between the plane and such a hyperboloid, the centers of curvature of the two spheres being at F and F . As mentioned above, this uncertainty can be removed by rotation through 90" (the hyperboloid has different Lurvatures along the meridional and transverse directions). - The sphere can be determined without a flat, by comparison with a known sphere. The deviations from true spherical shape which have been found are deducted in the two interferograms previously mentioned, from which the deviations from the plane of the reflecting flat can be determined with a computer (ROBERTS [1972]). - Methods for the KINGand evaluation of the Common-test are also described by RIMMER, Fox [19721. For testing surfaces it is essential that the interferograms should be easy to interpret, so that information about the deviations can also be gained by visual observation. When the surfaces have an ideal shape, they should be adjusted so that the path difference is the same in the whole field of view. An interference test which makes this possible for ideal surfaces is called a null test. In 5 2.0 the concepts of parallel and wedge adjustment for planes were introduced. For spheres parallel adjustment means that the centers of curvature coincide. In this connection wedge adjustment means that the centers of curvature are laterally displaced with respect to one another. Only with planes is adjustment for the null test easy. With spheres concentric rings may appear with a combination of two ideal surfaces on account of an axial displacement. To make these connections clearer, one may consider a system of rectangular coordinates 5, q, in the space of the two

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ideal spheres, with the two coinciding centers of curvature on the optical axis ( = 9 = 0. Let an arbitrary point P on one of the spheres (radius R) be denoted by the coordinates 5, q. If one displaces this sphere by an infinitesimal vector with the components 6,, 6,, 6,, then the thickness of the ideal layer of air between the two spheres in P is given by (HOPKINS [1971]) :

This relation, which is valid for suitable signs of the variables and for apertures which are not too large, takes the place of (2.2) for testing spheres. t(5, q) can be thought of as being measured in the direction of the normal to the surface. For 6, = 0, (2.9) takes the form (2.2). In this case straight and equidistant interference fringes are formed in the plane of observation if the image is formed while obeying the sine condition (for further details, see 5 6). The interpretation and evaluation of the course of the fringes as observed can then take place in an analogous manner to that used when testing flats. For 6, # 0 the course of the fringes determined experimentally can be reduced to the case of a given wedge adjustment or parallel adjustment. - In this way the sums of the deviations of two real spherical surfaces from two ideal reference spheres can be determined. To determine the position of a reference sphere relative to the real surface, one candeal with the 4 parameters of the sphere in a manner analogous to that used for the 3 parameters of the plane. Thus one can make the condition that one of the reference spheres must pass through 4 given points of the surface. Or one [ 19701;HARRIS [19711;cOmpare can use methods of least squares (HOPKINS also SAUNDERS [1961]). The quantities u, in (2.6) then no longer form a linear position function. Their constants depend omthe radius and position of the reference spheres. By varying the constants, one can find their optimum values. For aspheric surfaces the relation between the position of, and distance t ( ( , q ) between, the reference surfaces is considerably more complicated. For the position, 3 further degrees of freedom have to be considered for arbitrary aspheric surfaces and 2 for rotationally symmetrical aspheric surfaces. Tilting about the vertex of the surfaces 5 = q = 0 leads to approximately straight and equidistant fringes, as the distance t((, q ) increases or decreases proportionally to ( or ?. With steeply curved aspheric surfaces, at least, the forming of the image of the fringe system in d plane is accompanied by errors of projection. With rotationally symmetric surfaces a displacement 6, leads to concentric rings, only in contrast to (2.9) the radius of curvature R depends on the zone examined, i.e., R = R ( t 2+ q 2 ) .

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The distance t(l,q ) is approximately proportional to the displacement 6, or 6,, respectively, and to the first partial derivative of the surface with and BENNETT [19721have investigated respect to 5 or q, respectively. WYANT the effect of some errors with synthetic holograms, which are also of interest in this connection. When using the method of “least squares”, the different values of t(5, q) depending on all the parameters must be used, and the essential constants determined in a manner analogous to (2.7-2.9). 2.2. DETERMINATION O F THE DEVIATION DIFFERENCES

As already discussed in 4 2.0, the Michelson (Twyman-Green) interferometer (Fig. 2.2) is suitable for determining the deviation differences. The distance between adjacent fringes corresponds to a difference of ;1/2 in the thickness of the layer of air. Optical arrangements using transmitted light are also suitable for determining the deviation differences. Thus the Mach-Zehnder interferometer (BORNand WOLF [1964] p. 313) can be used for examining nearly polished or aspheric surfaces. The distance between adjacent fringes corresponds to a difference in thickness of A/(n- l)(n is the refractive index).The sensitivity is about 4 times smaller than that obtained with reflected light. In addition, the structure of the interference pattern is more complex with the Mach-Zehnder interferometer (SCHULZ[1964], SCHULZand SCHWIDER [1964]) and the adjustment therefore more complicated. Generally speaking, the Michelson or TwymanGreen interferometer is used more frequently. Each Twyman-Green interferometer contains optical elements, which must in the first instance be assumed to be free from faults. WEINSTEIN [1951] provided the beam splitter cube with spherical entrance and exit surfaces, in order to eliminate its spherical aberration. In other types of systems, the beam splitter is penetrated by plane waves. In the case of sphericity control, one must then insert a beam diverger into the measuring arm of the interferometer. VANHEELand SIMONS [1967] used a sphere in combination with a plane parallel plate as beam diverger, the plane parallel plate serving as a correcting member for spherical aberration. Both with the “surface measuring interferometer” (SMI), (TROPEL Inc. [19701) and BUCCINI and the “laser unequal path interferometer” (LUPI) of HOUSTON, OWEILL[19671 specially corrected systems are inserted as beam diverger. As laser light sources are used, the differences of dispersion in the interferometer are of no importance. An interferometer with a special beam diverger for white light was described by BALDWIN [1968]. The Twyman-Green interferometer is particularly useful for examining

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52

aspheric surfaces; for it is possible to compare concave surfaces to other concave surfaces and convex surfaces to other convex ones. This avoids the necessity of producing a complimentary aspheric surface, which would be difficult in most cases. BIRCHand GREEN [1972a] as well as SCHWIDER [19733 have used the Twyman-Green interferometer for relative comparison of aspheric surfaces. One of the advantages over other optical arrangements is the possibility of using white light interference fringes (see 0 5.1) for adjustment purposes. As previously mentioned, the faults of the components of the interferometer show in the interferograms, and a null test is made difficult by this. Therefore, JONESand KADAKIA[1968] have, for a Fizeau interferometer, subtracted the data of the empty interferometer when testing for homogeneity. This also removes errors due to the interferometer plates. The interferometer errors can also be eliminated in a purely analogous manner (SCHWIDER [1973]). This is described in 0 4 together with a thetashearing technique. 2.3. THE USE OF A NULL LENS

To test aspheric surfaces, a special group of optical interference arrangements is used. A generally applicable absolute method, e.g., comparison of 3 suitable surfaces (see 0 3), for arbitrary aspheric surfaces is not known. Mechanical measuring processes are not sufficiently accurate for determining the errors of a master surface.

w

'FIELO

nskfERlC LENS

s'RFAC€

Fig. 2.6. Testing with the aid of a null lens. The null lens compensates the wavefront in such a way that the wave leaving it is everywhere perpendicular to the aspheric surface. After reflection at the aspheric surface, the null lens reshapes the wave into a plane wave. Deviations of the aspheric surface from the ideal shape appear as deviations of the plane wave.

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The testing of aspheric surfaces can be reduced to an examination of spherical or plane waves by insertion of a special optical system - a “null lens”. For such an examination an arrangement, as shown in Fig. 2.6, is used. With this arrangement the null lens compensates the aspheric wave in such a way that a spherical or plane wave is formed. The null lens described by OFFNER[I9631 is relatively simple. A combination of two lenses is used to compensate the spherical aberration, which is introduced by a parabolic mirror with 1 : 1 image formation. Similarly, JAMESand WATERWORTH [1965] have used a null lens for examining spherical single lenses with transmitted light. HOUSTON, BUCCINI and O’NEILL[19673 have proposed for the LUPI (§ 2.2) to measure aspheric mirrors by insertion of a null lens. The method of HOLLERAN [1963] is an exceptional case allowing, as it does, for conic sections to be examined in reflected light by dipping them into a liquid of known refractive index. The liquid lens, which is bounded by the plane surface of the liquid and the aspheric surface, produces, for a matched refractive index, a spherical wave in reflection. This spherical wave surface can then be tested by interference. The Cartesian surfaces are a special kind of aspheric surfaces (KLEIN[197Q]). They make stigmatic point to point imaging possible. MARIOGE,METZ, MILLETand MALIE[19701 have interferometrically measured a condensor which converts a plane into a spherical wave. The examination was carried out in a Twyman-Green interferometer. The reference mirror was flat, and the aspheric lens was examined with double passage. Parabolic mirrors can be tested by reflection (HERRIOTT [1967]), if a plane mirror is used in the autocollimating system. In this case the reflecting flat takes over, as it were, the function of the null lens. After reflection at the flat and the paraboloid a spherical wave is formed, which can be tested with a Michelson interferometer. WOHLER[1970] has dealt extensively with the problem of the null lens and proposed systems which fit a plane or spherical wavefront to the aspheric surface to be tested. In order to caIculate the null lens, the reflected wavefront has to be determined. Mathematical procedures and solutions for this are given for surfaces to the 10th degree. HILBERT and RIMMER[1970] have proposed a null lens consisting of aspheric plates, which are inserted into the path of the divergent beams and which correct the aberrations of a paraboloid or other aspheric surface,

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2

2.4. ENHANCEMENT OF SENSITIVITY

Mostly, the deviations of the surfaces from their desired shape are stored as deviations of the interference fringes from equidistance, parallellity, and straightness. According to the measuring situation, the interference patterns are either “evaluated visually”, or one measures the position of the extremes. To make visual evaluation possible, it is necessary to increase the sensitivity.of the interference patterns with respect to the surface deviations. In this sense an increase can already be obtained with a highly reflecting coating on the interferometer plates, whereby multiple-beam interference patterns are formed (BORNand WOLF [1964] p. 323). This involves a change from sinusoidal two-beam interference to the Airy distribution. In transmitted light, bright and sharp maxima appear on a dark background, and in reflected light sharp minima are formed on a bright background. The position of the extremes and the fringe deviations can then be determined more accurately. The sharpening of the fringes is described by the finesse or by the effective number of interfering bundles. The finesse is the ratio of the half width of the Airy distribution to the distance between successive orders. Since the deviation sums are to be determined from the position of the extremes, all that is known about the surface in the intermediate region is that the deviation sums differ by less than 4 2 from those of the adjacent fringes. Transmission interferometry permits us to fill in the region between successive orders, on which information is lacking. For this, procedures with stepwise variation of the wavelength or simultaneous irradiation with several wavelengths are suitable. When the refractive index between the plates of the interferometer is varied stepwise, then the distance between and SHISHIW successive orders can be subdivided equidistantly (SAKURAI [1948], SAUNDERS [ 19511). The variation of the refractive index is achieved by evacuating the space between the interferometer plates. The positions of the separate fringes are frozen in a photographic plate by multiple exposure. If several wavelengths, e.g., from a helium spectrum lamp, enter simultaneously, then with small order numbers subdivision of the distance [1951]), which can be calibrated between the orders results (SAUNDERS with respect to the thickness of the layer of air. When irradiating simultaneously with several wavelengths, one must make sure that the scale of wavelengths is chosen to fii the distance between the interferometer plates. With transmitted light, HERRIOTT [1961] (see

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also HERRIOTT [1967]) uses a monochromator in combination with a Fizeau interferometer for transmitted light. The entrance slit of the monochromator is replaced by a grating of variable frequency. The set of waves emerging from the exit slit are used to illuminate the Fizeau interferometer. The grating frequency is varied by imaging a fixed grating by means of a zoom system. Further information can be obtained from the last reference. The subdivision is nearly equidistant, if the order m + 1 of the wavelength 1, coincides with the order m of the wavelength I,+, . The range of wavelengths A1 = 1, - &+, depends on rn as follows :

A1 = (AM+1,+,)/2m (BORNand WOLF[1964] p. 335). In place of a monochromator, one can insert a Fabry-Perot filter (FP) (SCHWIDER C1966, 19683). This greatly increases light intensity (GIRARD and JACQUINOT[1966]). At the same time this type of multiple-wavelength interferometry produces a criterion for the equidistance of the subdivision into N parts. Fig. 2.7 shows a diagram of the arrangement. The white spectrum of a white light source is filtered through a Fabry-Perot filter. By selecting a small range of angles near the axis, the axial eigen-wavelengths of the FP are separated. The axial eigen-wavelengths A,+, fulfill the condition =

AM/(l-dM)>

where M is the order number of the Fabry-Perot interferometer (M= 2T/A,). When the distance T between the plates of the Fabry-Perot interferometer is N times the distance t between the test plates in their central region, then M = Nm (rn is the order belonging to t). Thus AM+,, x &[I +p/Nm+(p/Nm)’+

. . .].

BEAM SPLITTER INTERFERENCE

WHITE UWT SOURCE

UNDER TEST

WAVELENGTH SELECTING

WRY-PEROT-

FIZEAUINTERFEROMETER

FILTER

Fig. 2.7. Multiple-wavelength method. For purposes of measurement, an interference filter is inserted into the path of the rays. In order to count the subdivisions, onemayadd the lightofa spectrum lamp.

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For equidistant subdivision into N parts of the order distance, the wavelengths must fulfill the condition = &[l + p / N r n ] . This condition is fulfilled to a good approximation by the axial eigen-wavelengths,ifthe order number m of the Fizeau interferometer is sufficiently large. Take the extreme case ,u = N and rn = 100, then the relative maximum error of the wavelength is about lop4. Fringes of superposition in white light, which presuppose a rational number for the ratio of the thicknesses T/t, are used as criterion for adjustment. In order to count the subdivisions, it is useful simultaneously to radiate a line from a spectrum lamp onto the center of the selected wavelength range. The fringes at distance ;1/2 are then marked

Fig. 2.8. Interference patterns obtained with the multiple-wavelength method. Top left: Pattern showing the subdivision of the 1/2 step into 25 parts. Bright fringes correspond to a change in thickness of the layer of air by 4 2 . Between 2 fringes following each other closely the layer of air changes by 1/50. Top right to bottom right: Photographs of 2 Fabry-Perot plates with changes of the thickness of the layer of air by A/lO, 1/30 and 1/50 from fringe to fringe.

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(Fig. 2.8 top left). Measurement of the actual deviation sums is then carried out with an interference fiIter in the path of the rays in combination with a white light source, when the patterns of Fig. 2.8 are formed. - A similar solution (HESSE,Ross and Z~LLNER [1967] and Z~LLNER [1968]) uses a Michelson interferometer, instead of an FP, as wavelength analyzer. In place of simultaneous irradiation by several wavelengths, one can also use a number of discrete directions of irradiation, as the path difference also depends on the angle of incidence (MURTY[1962]). A successive interpolation of the order distance can also be attained by varying the wavelength by means of a tunable single mode laser (FILSTON and STEINBERG [19671). Since all these methods presuppose transmitted light distributions, many authors have tried to convert the interference distributions obtained with reflected light into distributions that would be obtained with transmitted light. For fairly large wedge angles this can be done in the Fizeau interferometer, since it is then possible to intervene in the focal plane of the collimator of the Fizeau interferometer. In this focal plane images of the light source are formed which correspond to waves which have traversed the layer of air between the test surfaces 0, 2, 4, . . . 2N times. Different forms of intervention are used to convert the distribudon obtained with reflected light into one as obtained with transmitted light (KIMMEL [1955], BLANKEand LOHMANN [1957], FARRANDS [1959], COWNIE [1963], PASTOR and LEE[19683). So far these methods have mostly been used for examination in the micro region. This is also true for the method described by KOCH [19601, who removed the first reflected beam with a special arrangement of absorbing and dielectric layers. LANGENBECK [1967, 19691 has developed multipass interferometry by separating from the rest the wave which has passed N times through the Fizeau interferometer and superposing onto it a suitable reference wave. As with multiple-wavelength interferometry (see above), for 2N passages through the layer of air one fringe spacing corresponds to a difference in thickness of A2/2N. To carry out the method described above, a system as shown in Fig. 2.9 is used. A plane monochromatic wave is divided by the beam splitter and illuminates on the one hand the Fizeau interferometer, and on the other the reference mirror. The wave which has traversed the Fizeau interferometer 2N times is allowed to pass the beam separator. The other waves of the Fizeau interferometer are not allowed to pass through this. The reference mirror is tilted in such a way that the reference wave also passes through the beam separator. By superposing both waves in the plane of the camera, an interferogram at low frequencies is formed with the fringe

112

I N T E R F E R O M E T R I C T E S T I N G OF S M O O T H S U R F A C E S

CAMERA

Fig. 2.9. Multipass method. The beam which is reflected several times (here twice) between the surfaces A and B, interferes with the beam of the reference mirror, which is inclined so that it can pass the beam separator.

spacing mentioned above. If x and y are the deviations of the surfaces A and B, u the deviations of the reference mirror, and u the errors of the beam splitter, then

-

N(x+y)-u-v

= a.

If the reference, splitter, and Fizeau plates are of equal quality, then the deviations (u u), which would appear as errors, can be ignored in comparison with N ( x + y ) , when N is chosen sufficiently large. - LANGENBECK [1967] has suggested a similar method for spheres, combining a concave and a convex sphere. To match the magnitudes of the amplitudes of the two interfering waves, optical polarizing methods are suitable. Whereas with the multipass Twyman-Green interferometer an interferogram at low frequencies can be obtained by matching the direction of the reference wave, a multipass interferogram at high frequencies can be transformed back via Moire when a Fizeau interferometer is used. Until now only methods with fringes of equal thickness have been discussed. Methods will now be described for which fringes of equal inclination are used for measuring the deviations. To observe the Haidinger rings a telescope is used, as these interference patterns are formed at infinity. This means that for constant thickness of a layer bounded by parallel planes the same direction results in the same phase shift. If the thickness of the layer of air changes, the fringes of equal inclination swell or shrink. The criterion is used, for example, for adjusting a Fabry-Perot interferometer. SCH~NROCK[19391 has measured the change in diameter of the

+

IV, §

21

RELATIVE T E S T I N G B Y C O M P A R I N G T W O S U R F A C E S

I13

Haidinger rings with a telescope with small aperture while scanning the aperture of the plates to be tested. Here the local variations of the thickness of the layer of air are calculated from the changes in inclination. RIEKHER [19581 has converted the quadratic relation between the thickness and the radius of the ring into a linear relation by means of a parabolic linkage, and thereby simplified reading off and reducing the values. In a modification of the method (KOPPELMANN and KREBS[1961], ROESLER[1962]) the whole aperture of the Fabry-Perot interferometer is covered at the same time with a telescope. The intensity of the central spot of the Haidinger system is measured photoelectrically. By moving a small diaphragm over the plates, the photoelectric current is changed. Here the relation between the change in intensity and the change in the thickness of the layer of air is approximately linear for small changes, if the average distance of the interferometer plates is chosen so that the slope of the Airy or sine distribution falls on the photoelectric cell. A forerunner of this method was the photographic recording of the intensity distribution. The brightness is and SHISHIDO [1948], SAUNDERS then a measure of the deviations (SAKURAI [19513). Change of the radii of the rings has also been connected with an electronic display system (BENEDETTI-MICHELANGELI [I19681). For this, periodic variation of the angle of incidence is necessary, so that the distribution of the angle is transferred to the time axis. The change of angle is produced by a swinging mirror which alters the direction of a laser beam. The intensity transmitted by the FP interferometer is detected by a photo-receiver. Calibration of the angular distribution is achieved by means of a coarse grating positioned in the focal plane of the image forming telescope objective. Thus it is possible to determine the change of angle by counting. The FP plates are measured point by point in this way. Until now methods for increasing the sensitivity by special interferometric arrangements or adjustments partially linked with electronic measuring techniques have been dealt with. It is, however, also possible, a posterioi to subject a given interference pattern to several methods of evaluation, with the aim of improving the localization of the extremes of the fringes, especially with two-beam interference. Thus, the sine shaped two-beam interference distribution can be converted with a hard limiter into a pattern with alternating opaque and transparent fringes (ZORLL [1952]). The position of the fringes can be determined even better by Sabattier equidensities (LAU and KRUG [19681) converting the opaquetransparent transitions into sharp lines. To determine the position of the extremes, one can use the geometricmean of the lines of equal density. At the

114

INTERFEROMETRIC T E ST I NG OF SMOOTH SU R F A C ES

“v, 9 2

same time, illumination errors are compensated (SCHWIDER, SCHULZ, and MINKWITZ [19661). RIEHKER Non-linear photographic effects can also be used in a different way (BRYNGDAHL and LOHMANN[1968] ; BRYNGDAHL [1969] ; SCHWIDER [1970] ; MATSUMOTO and TAKASHIMA[19703; MUSTAFIN and SELESNEV [1972]). On the photographic plate, the distribution of intensity in a twobeam interferometer with visibility 1 is given by

45, v ) = i o ( l + cos [ 2 w ?- 9(5,v)l>, where g is the spatial frequency of the fringes and i, the average intensity, while ( ~ ( 5v, ) describes the position of the fringes in the interferogram at the point q). The photographic plate, exposed in this way and developed, is copied with a hard limiter. Here multiples of the basic frequency g are also generated. This involves multiplication of the deviations, which are stored in the interferogram through the fringe position ( ~ ( 5q). , By means of optical Fourier analysis in a double-diffracting apparatus the multiples of the basic frequency g can be separated. By superposing the waves of the +Nth and -Nth order of diffraction multiplication factors 2N can be reached. Multiplication of the carrier frequency from g to 2Ng can be transformed back via Moire. Thus, proceeding from the interferograms of Fig. 2.8 with a 25 times subdivision of the order spacing, Fig. 2.10 is obtained. Here one fringe spacing corresponds to a level difference of 1/200 or A/400, respectively.

(r,

Fig. 2.10. Enhancement of the bending of the fringes with the aid of non-linear photographic effects. Left: Fringe spacing corresponds to 4200. Right: Fringe spacing corresponds to L/400.An interferogram with a fringe spacing corresponding to Lj50 (Fig. 2.8) was treated (the method includes linear transformation of the pattern of contour lines).

IV,

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RELATIVE TESTING B Y C O M P A R I N G TWO S U R F A C E S

115

2.5. THE MEASUREMENT OF INTERFERENCE PATTERNS

Interference patterns are obtained either as a “live” system of interference fringes, or as a photographic recording. With increasing requirements for accuracy methods for measuring two-beam interference patterns are especially in demand. First of all the measurement of interferograms will be described. Here the extremes of the interference distribution have to be brought into coincidence with a mark, and the positions of a r-7 co-ordinate stage have to be stored.The accuracy of setting depends on the shape of the mark used for measuring. It increases from using a single stroke to a double stroke by a factor of x 3 (SCHULZE [19671). In this way it should be possible to obtain an accuracy of evaluation of *l/50 of the fringe spacing. There are several special methods for increasing the sensitivity of detection, e.g., photometric balancing of the intensity of surface regions or use of three-beam interference (BOITEMA[19571). These, however, have not been used testing surfaces. For further information cf. e.g., KARTASHEV and EZIN[1972]. The eye is sensitive to color differences (SCHOBER [19581). When testing surfaces, interference colors are used for judging the surface by means of a test glass. The layer of air between the surfaces must then be very small (e.g.. x 1 pm), so that one is able to measure or judge the quality of the surfaces. The coincidence method can be improved by bringing an interference fringe to coincide with itself after turning it by 180”. For this the interference pattern is cut along a line and sections along this line are turned by 180”.For this rotation a dove prism can be used. KONTIEVSKY, KOTSHKOVA and PERESHOGIN [1968] have inserted a set of dove prisms between the collimator and the pair of test surfaces which perform the rotation through 180”. The dove prisms are inserted along a diameter and parallel to the average direction of the fringes. By shifting the set of dove prisms at right angles to the interference fringes, coincidence of the extreme for one fringe can be obtained. The shift serves as measure of the deviation of the fringe. GATES [1954] uses 2 intcrference patterns of the same pair of surfaces. The number and the average direction of the interference fringes are the same for both patterns. The order number of the fringes, however, increases in the two pictures in opposite directions. The two patterns are brought into optical contact along a line in a special double-microscopeand measured point by point with coincidence adjustment. The tatter is effected by mirror rotation which acts in opposite directions on the two interference patterns. A difference between the number of fringes in the two interferograms is equivalent to the addition of a linear function to the deviation sums.

116

INTERFEROMETRIC TESTING OF SMOOTH S U R F A C E S

[IV,

§2

The setting with an oppositely oriented partner can also be carried out with electronic aids. TOMKINS and FRED[1951] have developed a method for determining the position of spectrum lines, which was applied by PRIMAK [1967] to the evaluation of interferograms. Here an image of the interference fringe is formed on a slit and the intensity converted into potential differences by a photoelectric receiver. A rotating octagonal transmission prism is inserted into the path of the beam, which periodically moves the interference fringe over the slit. By covering every second prism surface, the interference fringe is made to pass the slit with frequency 2w, when the octagonal prism rotates with frequency w. For display, a voltage proportional to the transparency of the interference photograph is fed to the vertical plates, and a synchronuous triangular voltage of the basic frequency w is fed to the horizontal plates of an oscilloscope. In general, 2 oscilloscope pictures of the interference fringe are visible, which can be made to coincide by shifting the interferogram. The displacement is a measure of the fringe deviation. BIRCH[19721 compares the electronic picture of the interference fringe on the screen of the oscilloscope with a reference pulse. This reference pulse is produced by a rotating glass plate. In this way a fixed reference mark for the whole interferogram is formed, and adjustment to the maximum is facilitated. The spatial position of the extremes of a sine distribution can be determined with great accuracy by measurements at the slopes, since these contain the region where the gradient is greatest. Here equality of intensity in the two slope regions is used as criterion for adjustment ;if the intensities are equal, then the extreme lies mid-way between the points of equal intensity. DYSON [1963a] has used a calcite crystal to produce 2 images of the slit on the interferogram, which are polarized at right angles to one another. An image of the interferogram irradiated in this way is formed on a photoelectric receiver. A rotating polarizer is inserted into the beam path, which modulates the intensities in the pattern with frequency w. The intensities of the two slope regions are in opposite phase. The photo current is proportional to the sum of these intensities i, cos (wz +). and i, cos m. When, by shifting the interferogram relative to the double-slit pattern, i, = i,, then the ac component of the photo current becomes zero. This provides a sensitive criterion for fringe setting. DEW[1967] has inserted 2 photoelectric detectors into the slope regions of the sine-shaped density distributions. By feeding the two photo-currents in opposite directions to a galvanometer a zero reading was obtained for

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117

equal intensities,i.e., for symmetrical setting with respect to the extremes. JONES and KADAKIA [19681 used a recording photometer to measure the interference fringes. Here the deviations from an ideal wedge were determined with a computer, the ideal wedge being determined by a least squares fitting procedure using all the experimental values; compare equations (2.6) to (2.8). - A scanning arrangement described by RANCOURT and SHANNON [1966) operates in a similar way, using a computer to transform a given interferogram into a map of contour lines. To measure the interference pattern produced in the interferometer, “real-time” procedures have also been used with success, i.e., the interference pattern is processed electronically, without the need to record it first. Already in the method of KOPPELMANN and KREBS [1961] the local variations of the intensity were used to measure the deviation sums. Modulation methods (CRANE [19671) are especially resistant to fluctuations of stability in electronic processing. This modulation can be attained, for instance, by varying the path difference in one arm of the Michelson interferometer with time (Fig. 2.1 1). Optical polarizing methods with REFERENCE MIRROR

~

BEAM SPLITTER

SHlFTER

DETECTORS=

4-k &FIER

Fig. 2.11. Interferometer with phase modulation. Through the phase shifter, the interference pattern is vaned periodically in parallel adjustment. The variable intensity is fed to a fixed and a movable detector.

rotating 1/4 plates are used for this purpose. The variation with time of the path difference must take place in the same manner at all points of the interference pattern. To measure the variations of path difference with location, one uses the phase difference q(5, q) between the sine-shaped photo currents of two surface regions. One of the positions used for measurement serves as reference position, while the whole field is surveyed with

118

INTERFEROMETRIC TESTING O F SMOOTH S U R F A C E S

ClV, §

3

the other detector. The signals are fed to a differential amplifier ;the difference is rectified and recorded. The sinusoidally modulated photo currents are proportional to sin oz and sin [oz+cp(t, q)]. If the amplitudes are equal, the difference is given by : 2 sin [Mt, q)] cos [oz +id<, ?)IFor small values of cp the amplitude of the difference of the photo currents is proportional to the phase difference q({, q ) and thereby to the variation of the path difference. Similarly, a method using the Twyman-Green interferometer employs a piezoelectric modulator, but a matrix of receiversof 32 x 32 photodetectors (GALLAGHER and HERRIOTT [1971], BRUNING, GALLAGHER, HERRIOTT and NENNINGER [1972], BRUNING [1973]). The modulated signals are fed to a computer, whereby the position of the interference fringes can be determined with an accuracy of about 1/50 of the fringe spacing. Faults of the interferometer can be eliminated by measuring a master surface with the same arrangement and subtracting these data. The display takes place on an storage oscillograph, which allows representation of interpolated deviation curves as well as perspective views of the deviations.

5

3. Absolute Testing by Comparing Several Surfaces

$ 2 dealt with the comparison of two surfaces by interferometry. Such a comparison enables the shapes of these surfaces to be determined relative to one another. This type of method can become part of an absolute method, if one can obtain the information on one of the surfaces in a different way, e.g., whenit takes the form of a liquid mirror. Comparison with this known surface then provides information about the other surface. Basically, this is the simplest and most obvious possibility of an absolute test. It is described in the first part of $ 3.1. Other methods do not start off with a known surface, but only compare surfaces which at the beginning are all unknown ; by suitable combination such comparisons furnish absolute information about these surfaces. This is dealt with in subsequent parts of $3. Absolute tests give the shape of the surfaces as deviation from a mathematically ideal reference surface, e.g., from a plane (6 3.1) or from a sphere (§ 3.2). In principle, a few special asphe& surfaces ($ 3.3) can also be included. The real surfaces must not deviate too far from these ideal shapes, or else the accuracy of measurement is too low. Thus one must distinguish between tests on planes, spheres, and aspheric surfaces. But it is also possible to make distinctions based on other char-

IV,

0 31

ABSOLUTE TESTING

I19

acteristic differences. For instance one can find the required surface deviations at a group of points, along lines, or on the whole surface. This is connected with the loci of the unique solvability of systems of equations whose linear dependence varies with the position on the surface. This will be explained in more detail later. Absolute tests furnish more information than mere relative ones. This increase in iRformation also demands an increased effort. Therefore absolute tests are often suitable for the creation of standards or master surfaces of accurately known shape, to which other surfaces can then be compared by interferometry. This latter comparison can be carried out with the simpler methods described in Q 2. 3.1. TESTING FLATS

Tests for flatness are of fundamental importance, also as a basis for other optical tests. The plane has a basic quality which distinguishes it from all other surface shapes. In physics the definition of the plane is connected with the validity of certain physical laws or assumptions. Thus one bases it on the laws of the propagation of light and defines a plane as a surface on which the waves of a far distant point source of light are in phase. But this does not yet furnish a standard with which a test surface can be compared interferometrically. - But one can make the plane materialize with the help of other physical laws, in particular the law of gravity and its effect on the surface of a liquid. If, with a rotationally symmetrical distribution of masses, the center of gravity is far removed and there are no other substantial disturbing influences, then the surface of the liquid may be regarded as plane. Lord RAYLEIGH [I8931 already proposed the use of a liquid mirror as a standard for the interferometric testing of flats. Schonrock, too, has been concerned with this problem in the nineties, but without success (EINSFWRN [1954]). Later BARRELL and MARRINER[1949], EINSPORN [1954, 1955, 19611, B~~NNAGEL [1956a, 1965, 19683 and DEW [1966a, 19673 have investigated the problems involved in a more thorough manner and have tested such mirrors and used them experimentally. The following were recommended as suitable liquids : medicinal liquid parafin (Barrel1 and Marriner), water (Einsporn), mercury (Bunnagel), silicone oil (Dew). The optical arrangement consists of a Fizeau interferometer with horizontally placed surfaces; compare Fig. 2.3. In general, the lower surface (A) is the liquid mirror and the upper surface (B) the test surface. Under these conditions the test sample must be transparent, e.g., a glass

I20

I N T E R F E R O M E T R I C T E S T I N G OF S M O O T H S U R F A C E S

CIV,

53

plate. If, however, the test surface is opaque, it can be used as lower surface (A) ; the liquid immediately above must, of course, then be transparent. The choice of a suitable liquid is limited by the requirement that disturbing influences must be avoided. Some of the disturbing influences that have to be taken into account are : mechanical vibration, temperature variations, evaporation effects, electric charges, capillary effects at the boundary, dust particles on the liquid, disturbances caused by unevenness of the bottom of the vessel when the layer of liquid is too thin, the effect of the curvature of the earth, sagging of the test surface by gravitation on account of the horizontal position required. Sagging is a fault of a general nature, which will be dealt with in 0 6 . If the liquid mirrors are not too large, the effect of the earth's curvature is negligibly small; in any case this effect is known, and can be taken into account when necessary. The other effects entail certain requirements with regard to the properties of the liquid and the thickness of its layer. The reflectivity of the liquid mirror must also be considered, as it affects the contrast and definition of the interference fringes. When using a mirror made of mercury, the test surface with its partially transparent aluminium coating is connected electrically with the mercury vessel, in order to allow charges formed during the cleaning of the test surface to drain away. Producing such a liquid mirror requires experience. The liquid mirror enables one to make an absolute test of flatness by making the ideal reference surface materialize. Other methods differing from this latter method are described below. These make it possible to carry out an absolute test of flatness by combining relative measurements between unknown surfaces in a suitable way. Usually these are the surfaces of round glass or fused quartz optical flats. Of the two polished surfaces of each flat only one is examined, and this is done in reflected light. - The basic principle of such a method was already known to Lord RAYLEIGH [1893], and SCHCJNROCK[1908, 19391 has used and extended it. Three unknown plate surfaces A, B, C are combined in pairs in an arrangement similar to Fig. 2.3 and compared interferometrically.Thus one can measure the sum x + y of the deviations from planeness x from A and y from B in the combination AB (Fig. 2.1, equation (2.3)); similar relations are valid for the combinations BC and CA: x + y = a,

y + z = b, z + x = c.

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121

The quantities a, b, c are the deviation sums which have been measured according to 5 2.1 and are, therefore, known. Thus the unknown deviations from planeness can be determined absolutely from 3 equations with 3 unknowns. This, however, is not valid for all points on the surface, but only along one plate diameter of each surface (more about this later). According [I9391 one can extend this method to apply to 3 diameters to SCH~NROCK of each surface which form angles of 60" and 120"with one another. For this the reference plane of each surface to which its deviations from planeness are related must be placed through 3 points on this surface which form the corners of an equilateral triangle. The center of this triangle coincides with the center of the plate (i.e., the center of gravity of the triangle and the center of the plate are superposed on one another, if considered in the direction of observation). In each surface combination, it must be possible to determine the position of the two reference planes with respect to one another. This is the case, if the surface points determining these planes coincide in pairs. Since these points form the corners of equilateral triangles, this coincidence can be achieved in three different positions. Thus deviations from planeness along three diameters of each plate have been determined (SCHWIDER, SCHULZ,RIEHKER and MINKWITZ [19663, compare also SCHULZand SCHWIDER [1967]). In addition one can determine the deviations from planeness along each bisector of angle, and this is also done with the aid of equations (3.1). For this, according to SCH~NROCK [19391 ( N + 2) different combinations of positions (interference patterns) are required, if one wants to determine the deviations from an ideal plane along N diameters of each plate. - By adding combinations of positions with non-coinciding plate centers, additional deviations from planeness along [19671). The determination results chords can be determined (SCHWIDER partly from (3.1) (chords take the place of the diameters), partly from a comparison of a chord with a known diameter. All this is valid, if one can regard the plates as rigid bodies whose shape is independent of time and position (i.e., of all those positions in which the plates are tested and later used). This physical assumption of a "rigid body" is generally a basic requirement in the use of optical surfaces of solids. But it may not completely apply if the plates are tested in the horizontal position and sag under the influence of gravity. In order to simplify the solution of these problems one can, according to DEW[1966a, 19671, first of all limit absolute testing to the more easily manageable one-dimensional case, i.e., to the testing of linearity. This limitation suggests itself for the single reason that the system of equations (3.1) is, for 3 combinations of position, only valid and uniquely solvable along one line, as mentioned earlier. Ac-

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INTERFEROMETRIC TESTING O F SMOOTH SURFACES

“v. § 3

cordingly, in place of standards for flatness, one can first of all produce standards for straightness, i.e., glass “beams” with known deflection, see § 6. These are then used for testing the flatness of extended surfaces. Here a beam is brought to coincide, one by one, with a series of equidistant parallel chords of the test surface. The Fizeau interferogram along each chord is photographed. The deviation from straightness of the surface along this chord can be determined from this, apart from a linear function of position. The two constants of this function characterize the reference line with respect to which the deviations from straightness are ascertained (these reference lines are fixed later). A secondseriesof chords, at right angles to the first, is dealt with in the same way, and so is a pair of diameters at right angles which intersects the two series at 45”. From now on all reference lines are fitted to one another and fixed so that they lie in one plane. This plane is the reference plane of the test surface, with respect to which the deviations from planeness along the previously described chords have been found. The methods mentioned so far require a greater number of combinations of positions (interference photographs), if the deviation of a surface from an ideal plane is to be measured along a larger number of diameters or chords. By contrast, with a method of SCHULZ [1967], the smallest possible number, i.e., 4, of combinations of positions are required for an arbitrary number of diameters (central sections) suitable for testing. These combinations of positions can be chosen in such a way that a system of equations with an arbitrarily chosen number of unknowns can be formulated and solved. These 4 combinations of the 3 surfaces A, B, C (Fig. 3.1) are shown

SURFAE A

9JRFAC-E C

Fig. 3.1. Three surfaces A, B and C, each of which is to be tested absolutely along N diameters (the example shows N = 5). x,: deviation from planeness of A along the vth diameter,i.e., central section ; y. : corresponding deviation from planeness of B ; z,: corresponding deviation from planeness of C . Top: view from above onto the surfaces. Bottom: Side view (central section through the surfaces).

IV,

8 31

123

ABSOLUTE TESTING

-20

A-

1- COMBINATION

ZK’COMBINATION

3Q COMBINATION

4”’COMBINATION

Fig. 3.2. Four combinations of positionsofthesurfaces A, B and C of Fig. 3.1. They enable one to test along N diameters (central sections). In the 4th combination the surface B is rotated through the angle = 271x M / N , compared to the 1st combination ( M and N are natural numbers which are prime to each other, the example shows M = 1, N = 5). @J

in Fig. 3.2. The 4th is distinguished from the 1 st by the fact that the surface B is rotated by an angle @ = 2n x MIN

(3.2)

round the optical axis. Here M and N are natural numbers which are prime to each other. In the lst, 2nd, and 3rd combinations the vth diameter coincides with the (- v)th diameter of the other surface; in the 4th combination, however, the vth diameter of the surface A coincides with the (2- v)th diameter of the surface B (v = 0, & 1, . ..). This means that the following deviation sums can be measured (such sums are written in Fig. 3.2 against the corresponding diameters) : (1st comb.):

xv+y-,

= a,,

(2nd comb.):

y,+z-,

=

(3rd comb.):

z,+x-,

= c,,

(4th comb.):

x,+y,-,

=

b,,

(3.3)

a:.

The right-hand sides are measured according to 52, and are therefore known. The quantities on the left are the unknown deviations from an ideal plane which are to be determined. These are 3N unknowns. Their determination follows from the system of equations (3.3). All quantities in (3.3) refer to the same p-value (p is the coordinate of the position along

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INTERFEROMETRIC TESTING O F SMOOTH SURFACES

[IV,

D3

each diameter of the length 2r, IpJ is the distance from the center, --I 5 p 5 r). The first 3 lines of (3.3) are identical for v = 0 with the equations (3.1). They alone allow the testing of one diameter on each surface. Adding the 4th combination of positions allows one to test all the other 3N diameters as well (Fig. 3.1). This becomes clear if one looks at them successively (Fig. 3.2) : The diameter of A (deviation from planeness xo)which has been tested using the first 3 combinations of positions is made to coincide with a still unknown diameter of B (j2) and compared interferometrically in the 4th combination, so that the latter diameter also becomes known. This again, coincides in the 1st combination with a still unknown diameter of A (x- 2 ) , whereby this also becomes known, etc. The reference plane PL,, with respect to which the deviations from planeness of the surface A are determined, must have the same position with respect to A for all combinations of positions (a corresponding condition holds for B and C ) . It is also necessary to ensure that the position of the two reference planes with respect to each other can be determined in every combination. This is achieved, for example, if for each combination 3 known points on one surface (i.e., 3 points with known deviations from the reference plane) coincide with 3 known points on the other surface. In the first 3 combinations of Fig. 3.2, these are in each case the reference points indicated by 3 full circles, whose deviations from the reference plane are zero by definition (this is how the reference plane is defined). In the 4th combination, these are 2 each of the points mentioned and the center (empty circle). The deviations of this point from the reference plane (i.e., the deviations x,, and y o for p = 0) can be determined from the first 3 combinations and, therefore, be regarded as known. In each case the

Fig. 3.3. Three combinations of 3 surfaces.Top: The surfaces (A, B, C) with their co-ordinate systems ([, q). Bottom: The 3 combinations (AB, BC, CA). The surface mentioned second in each case has been brought from the position shown above to that shown below by rotation through 180" round the striated line (<= 0). Thereby the [-axis of this surface is reversed (the reverse position has not been lettered with

r).

IV,§

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A B S O L U T E TESTING

125

coinciding points are surrounded by an additional circle. They form the corners of an isosceles triangle. - 3 N equations can be chosen from the equations (3.3), which have a unique solution along the N diameters for determining the 3 N unknowns. The remaining equations can be used for controlling and compensating measuring errors (SCHULZ [19671, SCHULZ, SCHWIDER, HILLER and KICKER [1971]). With only 3 combinations of positions, testing for flatness is only possible along one diameter of each of the 3 plates; if lateral shifting of the plates SCHULZ, RIEKHER and MINKis permitted only along a chord (SCHWIDER, WITZ [1966], DEW [1966a], POLSTER [1968]). Furthermore, with an arbitrary number of combinations of an arbitrary number of plates, testing for flatness is still only possible along straight lines (SCHWDER [1967]). This is certainly true, if one measures the deviation sums on pairs of these plates, which is what the authors have done. Let us clarify the mathematical background of the last paragraph for 3 combinations of positions of 3 plates covering one another in pairs. The systems of co-ordinates of the plate surfaces cannot be brought into coincidence in all 3 combinations, the systems can at best be brought to a position where their mirror inverted images coincide, see Fig. 3.3. Only along the line of reversion or mirroring (here the q axis) do the equations which can be formulated for the deviations from an ideal plane have a unique solution. According to Fig. 3.3, the following equations correspond to the equations (3.1):

4 5 , rl) + Y( - 4 = rX rl) + 4 - 5 9 4 9 = q), z(t, r l ) + x( - t, a) = 4 5 , I.r). 5 7

v(57

4 5 7

b(57

(3.4)

Here x(<, q) represents the deviation from the ideal plane of A at the point

(5, q); correspondinglyy refers to B and z to C . The first line corresponds to the measurement of the deviation sum a(5, q) at the point (5, q) of the combination AB; similarly, the second line applies to BC, and the third to CA. Since in Fig. 3.3 only points which have mirror symmetry coincide, only the deviation sums of such points can be measured. These points are, for the chosen systems of coordinates, points with opposing values of 5 and equal values of q. Accordingly, (3.4) has a unique solution as a system of 3 equations with 3 unknowns only along the diameter t = 0. For 5 # 0 one cannot obtain a unique solution from (3.4). One can consider the point (5, q) and its mirror image (- 5, q) in all 3 combinations; this results in 6 equations with 6 unknowns: They consist of the 3 equations (3.4) and

I26

[IV, § 3

INTERFEROMETRIC TESTING OF SMOOTH SURFACES

those which follow from(3.4), if 5 is replaced by - 5. But these 6 equations with 6 unknowns have no unique solution; for the right-hand sides of the equations have to satisfy the identity

4 5 , v)+&

r)+c(t, 49-4-5,

?)-b(-5, d-c(-t7

11)

=0

(3.5)

B B c P s l Fig. 3.4. Three combinations of 3 plates which are not plane, but still show no deviations from planeness in the interference patterns. The diameter 5 = 0 is perpendicular to the plane of the drawing. Top: The surfaces (A, B, C). Bottom: The 3 combinations (AB, BC, CA). The figure demonstrates that the equations which can be set up here do not have a unique solution for the deviations from planeness for the whole surface. - The figure shows the side views (the vertical views are shown in Fig. 3.3).

which can be easily seen. This means that the locus of unique solvability is the diameter 5 = 0. There is, however, no unique solution for the whole surface in this case. Accordingly, surfaces can be specified which are not plane and still do not show any deviation from planeness in the interference pattern of any of the 3 combinations (SCHULZand SCHWIDER[1967]), see Fig. 3.4. Such surfaces have the following property:

4-5,

?)

=

-x(57 'I)= - y e , ?)

= -45

?).

3.2. TESTING SPHERICAL SURFACES

The plane has 3 parameters; these are necessary to determine its position in space. By contrast, the sphere has 4 parameters, one of which is necessary to define its curvature. We are concerned with surfaces which are meant to be spherical and whose small deviations from a sphere are to be determined with as much accuracy as possible. This problem cannot be solved by comparison with a known plane, since the interference fringes would generally be much too closely spaced. Rather, comparisons between several (approximately) spherical surfaces are suitable. At first these surfaces are all supposed to be unknown. After testing them absolutely, they can be used as master surfaces of high precision for the testing of lens and mirror surfaces, as described in 9 2. The above-mentioned absolute testing of the master surfaces is related to the interferometric determination of their small deviations from an ideal

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reference sphere, it does not include the exact determination of the radius of curvature R of that reference sphere. R can be determined by other methods. For, in practice, it is not necessary to determine R with the same accuracy (HERRIOTT [19671, HOPKINS [19701). Absolute testing of spherical surfaces is carried out as follows according to HOPKINS[I19701 and HARRIS[1971]: Measurements in a spherical Fizeau interferometer (Fig. 2.5) are combined according to the method described by SCHULZ[1967] for flatness tests (Fig. 3.2, equations (3.3)). This is possible if, in the combinations of positions of two concave surfaces, the latter are on opposite sides of the approximately coinciding centers of curvature (compare HERRIOTT [1967]). The reference spheres are determined by least squares methods as the best fitting spheres. Fig. 3.5 shows a side view of the 3 surfaces to be tested; namely, 2 concave surfaces A, C and 1 convex surface B. The view from above is identical with Fig. 3.1 (top). The surfaces are compared interferometrically in 4 combinations of positions; their side view is represented in Fig. 3.6, the vertical view is identical with Fig. 3.2 (top). In each combination of positions, the sums of the deviations from sphericity are measured at coinciding points (the measurement itself is carried out according to 0 2). In the 3rd combination

Fig. 3.5. Three sphericalsurfaces A, B, C, whose deviations from a true spherex,, y,, z,along N central sections are to be determined absolutely. The figure shows the side view in the central section v = 0. The vertical view is identical with the top part of Fig. 3. I , where the other central sections are also shown.

Fig. 3.6. Four combinationsofpositions of the surfaces A, B,C of Fig. 3.5 in side view. The view from above is identical with Fig. 3.2, top. In the 4th combination the surface B has been rotated in comparison with the 1st combination by the an& @ = 27t x M / N round the optical axis. The 4 combinations enable one to test spherical surfaces along N central sections.

128

INTERFEROMETRIC T E ST I NG O F SMOOTH S U R F A C E S

“v,

63

points coincide between which a beam of light passes really (not virtually) through the approximately common center of curvature*. The points are made to coincide in the 4 combinations by orienting the surfaces in a suitable way. The coincidences agree with those represented in Fig. 3.2 (top). Therefore, the system of equations (3.3) can also be used here, and its solution yields the absolute deviations from sphericity along the chosen number N of central sections. In place of two concave and one convex surfaces, one can also use 3 concave surfaces. Then all 4 combinations of positions are of the type of the 3rd combination. Otherwise, the method is the same as described above. The reference surface (broken lines) with respect to which the deviations of the surface A are determined must have the same position with respect to A in all combinations of positions (a corresponding condition holds for B and C). For an explanation of how this is achieved by using reference points when testing for flatness compare 0 3.1. When testing spheres, as just described, the least squares criterion has been used (HOPKINS [1970], HARRIS[1971] ;compare also 0 2.1). This demands that the reference sphere of a test surface is the sphere whose mean-square deviation from this surface is a minimum. This uniquely defines the reference sphere, and the minimum mentioned is a measure of the asphericity of the surface**. The method described above permits the determination of deviations from an ideal reference sphere along central sections, i.e., along the reference surface along lines. The following method (SCHULZ[1973a, 1973b1) permits the determination in a group ofpoints. These cover the surface of the sphere in both dimensions; the distance between neighbouring points can be chosen at will. The method operates with two (A, B) rather than with three (A, B, C) test surfaces. - The principle on which this is based is as follows: If, with respect to a sphere, a second concentric sphere is turned arbitrarily, while all the time keeping its concentric position, then interference of equal thickness between the two spheres always shows a locally constant path difference (referred to the center of the spheres), when the spheres are ideal. This has previously been used to show whether two surfaces are spheres within the accuracy of measurement. One should also * These “points” are not, of course, points in the mathematical sense. This is not only because of the lack of definition on account of diffraction inherent in every optical image, but also because of the directional dispersion by diffraction at the test surface structures(compare optical uncertainty relation, 6 6). ** Correspondingly, when the image of a point is formed by an optical system suffering from aberrations, the mean-squaredeformation of the wavefront with respect to its best fitting reference sphere can be taken as a measure of the quality of the image (see e.g., BORN and WOLF[I9641 pp. 463 and 468).

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be able, however, to use such rotations when determining quantitatively the deviations of a surface from sphericity. It can be shown that one can specify 3 positions of the two surfaces relative to one another and evaluate the 3 interference patterns in such a way, that one obtains quantitative values for the deviations from an ideal sphere. If the curvatures are fitted to one another, then in all three positions the two surfaces are really (SCHULZ [1973a]) or virtually (SCHULZ [1973b])incloseproximitywithout, however, touching one another. In this case, therefore, only small optical path differences are used. To demonstrate this principle, let us consider the case where the two surfaces are close to one another really and the radius of curvature of the reference sphere is large compared to the diameter of the test surface. In this case the system of co-ordinates shown in Fig. 3.7 and the 3 positions shown in Fig. 3.8 can be used. At the points P,, (some of these are marked in the figures) the deviations from sphericity xPvof the surface A are to be determined; the same holds at the points QlrV foryllYof B. For this purpose the surfacesare compared by interferometry in a basic position and in positions of rotational and lateral displacement. Fig. 3.8(a) shows the basic position. Here the pairs of points P,,, QPv (p, v any whole number) coincide. In Fig. 3.8(b) the position of rotational displacement is shown. Here the surface B has been rotated round its point Qooby 90" with respect to the basic position, so that the point pairs PPv,Q,-,, coincide (this coincidence holds :,ere to a

Y-20

Fig. 3.7. Two surfacesA, B, which are to be tested absolutely,with co-ordinatesystems. Top left: co-ordinatesystem with points P,,. of the reference sphere of the surface A (viewed from above). Bottom left: A with reference sphere in side view. Right: the same for the surface B with points Q,". The reference spheres are represented in broken lines; p, v = 0, I , 52, . . . . The points p = const. lie on meridians, the points v = const. on parallels of latitude, the equator is the line v = 0.

130

;

[IV, §

INTERFEROMETRIC T E S T I N G OF SMOOTH S U R F A C E S

(a)

0

0

Lq;Q;*p

0

0

0

0

0

0

0

0

.

0

0

0

%Qoo

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3

P,..(x,

0 I‘

P

%o, 4 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

4

(C) 0 0

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0

0

0

Fig. 3.8. Relative positions of the test surfaces A, B (Fig. 3.7) and thereby of the coordinate systems of their reference spheres. Top of (a), (b), (c): the 3 positions viewed from above. Lower part of (a), (b),(c): the same in side view.

sufficient approximation). Fig. 3.8(c) shows the position of lateral displacement. Here the surface B has been shifted with respect to the basic position in such a way that the pairs of points P,,,, Q(,,+ coincide. The dotted arrows show the movement of the B co-ordinate system which has taken place ‘with respect to the basic position. These movements always mean rotations of the B reference sphere round its centre 0. For the “position of rotational displacements” one rotates by 90” round the line OQ,, ; for the “position of lateral displacement” one rotates about a line at right angles to OQ,, by a smaller angle. The interference fringes of equal thickness

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131

formed between A and B can be measured in a spherical Fizeau interferometer (§ 2). In each of the 3 positions the reference sphere of a surface must maintain the same position in relation to the latter. This can also be achieved by the use of reference points, in which the deviations from an ideal sphere are by definition equated to zero. These reference points are shown in Figs. 3.7 and 3.8 by fun circles. They enable one to ascertain the exact relative position of both reference spheres with respect to one another, for instance by coincidence of 3 reference points of either surface. The latter is the case in the basic position and in the position of rotational displacement (the coinciding reference points have an additional circle round them). - In the side view the reference spheres are shown concentrically. In practice, a small lateral shift of their centers with respect to one another is useful. The course of the interference fringes thereby observed can be reduced to the exact concentric position. Supposing this has been done, then the following deviation sums can be measured :

+Y p v x p v +Y v - p

-

apv,

=

bpv,

xpv+Y(p+l)"

=

cpv'

xpv

(3.6)

The first line is valid for the basic position, the second for the position of rotational and the third for that of lateral displacement. - One can consider the right-hand side of the equations as known, and the deviations from sphericity on the left-hand side can be determined one after the other in suitable order from the equations (3.6). Since there are more equations than unknowns, there is again a possibility here for controlling and compensating measuring errors. 3.3.

TESTING ASPHERIC SURFACES

Tests on aspheric surfaces by methods analogous to those described in

0 3.1 and 9 3.2 have not, so far, been carried out; different methods have been used. Optical elements (e.g., flats and objectives) are inserted, which have been tested in a different way and found to be sufficiently free from faults. This is justified, since tests on aspheric surfaces are in general more difficult to carry out and, therefore, only a lower accuracy can be expected, than with, say, tests on plane surfaces. The aspheric surface itself is then tested by, for instance, methods which have been described or were referred to in 0 2.3. Or the aspheric surface is compared by methods mentioned in 0 4; or it is compared to a synthetic hologram, see 9 5 .

132

INTERFEROMETRIC TESTING OF SMOOTH SURFACES

[IV.

§3

Apart from such methods, there are, in principle, methods of testing certain aspheric surfaces without insertion of other optical elements; i.e., for certain aspheric surfaces there are combinations of positions which enable an absolute determination of the surface deviations from the corresponding ideal aspheric surface, if for the present one neglects problems of adjustment and image formation. The ideal aspheric surface is one which, suitably adjusted, produces locally constant path differences, i.e., disappearing interference fringes. Such aspheric surfaces are, for example, paraboloids and cylinders in the following combinations of [1973b]): positions (SCHULZ

$45 1 5 ~

COMBINATION

ZND AN0 3" COMBINATION

C ~ " A N O sTH COMBINATION

Fig. 3.9. Basic arrangement for testingabsolutely two paraboloids ofrotation A, B and a third surface C with the aid of 6 combinations of positions (figure shows the first 5, in side view). C must be thought of as having been rotated by 180"round the line Iz3 during transition from the 2nd to the 3rd combination, round the line Iz4 during transition from the 2nd to the 4th combination, and round 145 during transition from the 4th to the 5th combination.

Rotational paraboloids can be combined according to Fig. 3.9. In the first combination the course of the beams of light is as shown. Ideally they are here no longer directly reflected within themselves. Rather, one has a finite angle of reflection and a beam only meets itself again after twice passing each surface. This means, on the one hand, that each surface deviation to be determined is affected by a factor dependent on the angle of reflection. On the other hand, it means that 4, rather than 2, points coincide (e.g, the four points joined in the figure by dotted lines). The interference fringes formed in this manner furnish in each case the sum of 4 surface deviations (the first of the following equations). From the 2nd to the 5th combination, however, it is possible to measure for each case the sum of 2 surface deviations (2nd to 5th equation):

ABSOLUTE T E S T I N G

x;+x;+y;+y;

133

=

a,,

+ z ; = b,,

x6

+zb = c,

X;

yb

(3.7)

+z; = e,,

&+z;

=

h,.

The above-mentioned factor (the result of a finite angle of reflection) may here be considered as having been taken into account. These equations have a unique solution. One obtains for instance :

+

xb = %uo 3b, - c, - e,

- h,).

8 s-= I

I-

Fig. 3.10. Comparing one combination of positions each for testing flats, cylindrical and spherical surfaces. Left : testing flats ; middle : testing cylindrical surfaces; right : testing spherical surfaces. The 5-v co-ordinate systems of the surfaces I and I1 are chosen in such a way that the system of each surface together with the normal to that surface pointing in the direction of the air forms a right-handed system. In the top part of the figure, the co-ordinate systems are shown as they are projected onto one another through the courses of rays shown below. Top: view from above; middle: side view along the section q = 0; bottom: side view along the section = 0.

134

INTERFEROMETRIC TESTING OF SMOOTH S U R F A C E S

[IV,

03

This solution is valid for the cross-section shown in the plane of the figure. The addition of a further combination of positions (here the 6th) furnishes the solution for a further (N-1) sections, in a manner analogous to the 4th combination in Fig. 3.2. - A and B are concave paraboloids. The convex surface C is a. paraboloid (approximately) only for a very small distance from A and B ;in more general terms, the shape is such that a wave leaving A or B at right angles will impinge perpendicularly on C as well. Three circular cylinders A, B, C can be combined in 3 combinations of positions AB, BC, CA; the middle column of Fig. 3.10 shows any one of these 3 combinations for the case of three concave cylinders (I denotes for each case the first and I1 the second surface of a combination). The 5-q systems of co-ordinates coincide in all 3 combinations of positions. The coincidence is brought about by the light rays (which cut the axis of the cylinder at right angles). At the coinciding points (e.g.,at those joined by dotted lines) the following deviation sums can thus be measured : x(5, 17) + At, rl) =

45, d,

Y e , r)+ 4 5 , q) = b(t,rl),

(3.8)

4 5 , V ) + X ( t , q ) = 45, q). Here x(5, q) is the looked-for surface deviation of A at the point 5, q, and correspondingly y relates to B, and z to C. Each line follows from one of the 3 combinations of positions. Solving for x, y and z is uniformly possible over the whole surface: X(5, V ) =

3[ a(5, V ) -

b(5, rl)+

C(5,

111,

r) = C t 4 5 , $4b(5, v ) -45, v)], z(5, r) = 3c -45, ?) + b(5, 49 + 45, ?)I.

Y(5,

(3.9)

This is based on the possibility of making all the systems of co-ordinates coincide; this possibility is provided by the cylindrical shape (also in the case of two concave and one convex cylinder, which is not dealt with here). The surface of the cylinder occupies a special position in this connection. This becomes especially clear, when one compares methods of testing cylindrical surfaces with the corresponding tests on planes and spheres (see § 3.4). 3.4. SOLUTIONS APPLYING UNIFORMLY TO THE WHOLE SURFACE

The last-mentioned method for testing cylindrical surfaces furnishes a

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uniform solution for the whole surface. Now we will deal more generally with some other cases of this type. It becomes apparent that there are additional possibilities for materializing an ideal standard. When the expression for the surface deviation is valid for all points, a solution holding uniformly for the whole surface is obtained. That is to say, the solution of the system of equations for the deviations of the surfaces from their desired shape can be formulated independently of the exact place on the surface, if the co-ordinates are suitably chosen. For instance, in (3.8), unlike (3.4), one can leave out the parameters (5, q ) which are the same for all values and formulate the solution uniformly according to (3.9). Each solution is, of course, required to be unique if the number of equations is equal to the number of unknowns. The formally simplest example of a uniform solution for a whole surface is the comparison of a surface with a liquid mirror. The other tests for planes and spheres discussed above, however, furnish unique solutions purely mathematically only along lines or at points. Here the solutions along different lines - or at different points - generally have different mathematical expressions. In this respect the surface of the cylinder occupies a special place as compared to the plane and the sphere, see Fig. 3.10 (SCHULZ[1973b]). Through the combination of the surfaces the perpendiculars to the surfaces pointing in the direction of the air are adjusted so that they point in opposite directions. Thus, when testing flats, the 5 axes point in opposite directions, if one assumes q axes pointing in the same direction. When testing cylindrical surfaces, however, the rays parallel to one plane are crossed. This reverses the image of one of the two 5 axes; i.e., not only the q but also the 5 axes coincide in position and direction. When testing spherical surfaces, however, the rays are crossed in each plane going through the optical axis. Compared to the testing flats, this reverses the image of the 5 and the q axes of one surface. That is to say, the 5 axes coincide pointing in the same direction, but the q axes coincide in opposite directions. The following can therefore be pointed out with regard to Fig. 3.10: while testing cylindrical surfaces allows for all the systems of co-ordinates to coincide, such coincidence is impossible with corresponding tests on flats (cf. Fig. 3.3). Testing spherical surfaces is analogous in this respect to testing flats. For both, the image of one of the two co-ordinates of the surface is reversed, which amounts to mirroring. This mirroring can be compensated (especially when testing spherical surfaces), if one inserts optically a suitable mirror (SCHWIDER [1973]). A totally plane mirror PL can be used for this purpose (e.g., a liquid mirror),

136

INTERFEROMETRIC T E ST I NG OF SMOOTH SU R F A C ES

[IVY §

3

Fig. 3.1 1. Absolute sphericity determination on the whole surface with the aid of a plane mirror PL. (When PL is at the focus, 0x11~the micro region needed for reflection need be assumed to be ideally plane.)

see Fig. 3.11. The required 3 combinations of positions of the 3 spherical surfaces A, B, C are shown in side-view. The 5-q co-ordinate system of each concave surface (A, C) is placed so that it forms a right-handed system with the perpendicular to the surface pointing into the air, while with the convex surface (B) the corresponding system is left-handed. The q axes are everywhere perpendicular to the plane of the figure; in the first two combinations they are parallel to each other, in the 3rd antiparallel. The first two combinations are identical with the first two in Fig. 3.6; the 3rd differs from the 3rd in Fig. 3.6 in that the mirror PL is inserted. This latter reserves the direction of one of the two co-ordinate axes. Thereby all <-q systems of co-ordinates can be brought to coincide. For the required surface deviations the equations (3.8) with the solution (3.9) then hold uniformly for the whole surface.

F'AND 2"O COMBINATION

3RDCOMBINATION

Fig. 3.12. Testing of two spherical surfaces A, B in 3 combinations of position. During transition from the 1st to the 2nd combination, A is rotated by 180" round the optical axis. In the 3rd combination, one half of the field of view is screened, which is possible on account of 180" symmetry. Even without this screen disturbing waves can be eliminated by inclining A and filtering the entering wave in the plane of the image of the convergence point.

IV:

5 31

ABSOLUTE TESTING

137

The reflecting flat in Fig. 3.6 need not be completely plane, if its deviations have been previously determined by a different method. These deviations are then taken into account as correction in the last line of (3.8). - On the other hand, the surface PL, when it is infocus, need only be plane over the micro-region necessary for reflection. When testing smooth surfaces, one assumes in any case smoothness within the micro-region. This can be put to use for obtaining solutions from even and odd functions. Only 2 spherical surfaces are used and tested, one concave (A) and one convex (B). Their deviations from an ideal sphere which are to be determined are x(5, q) andy(t, q). The odd function can be deduced from two interference patterns according to Fig. 3.12, left, by turning the surface A for the second combination by 180" round the optical axis:

4 5 , rl) +fit, v ) = 4 5 , rl), 4- 5, v ) = b(5, v). -v)+fi57

From this follows the odd function :

d- 4 -5, -q) = a(<, v ) - b(5, 49. (3.10) The even function [x(<, q ) +x( - 5, - q)] follows from the arrangement 4 5 7

shown on the right of Fig. 3.12. This presupposes that the surface B is so smooth at the point (0,O)that it may be assumed to be idea!!y fat within the required micro-region. It follows that

and therefore :

Uniform solutions over the whole surface open up an additional possibility (SCHWIDER [19721). First of all, this possibility refers to the evaluation of interference patterns. This evaluation can be carried out here by a (non-numerical) optical analogue process. The result of this process is a compensating hologram. The surface deviations of a surface A which has been examined can be read from this hologram. In addition this non-ideal surface A, combined with the hologram just mentioned, can be used as an ideal master surface for further tests. Here the compensating hologram diffracts the light reflected at the real surface A in such a way that the result is the same as if it had been reflected by an ideal surface. In other words: the real surface A and the compensating hologram form a unit which

138

I N T E R F E R O M E T R I C TESTING OF SMOOTH S U R F A C E S

[IV,

93

functions like an ideal surface suitable as standard. Its function is similar to that of a liquid mirror for testing flats. Using a test on spheres as example (Fig. 3.12), we will explain the production of the compensating hologram by an analogue process. The absolute deviations x(5, q ) of the surface A are deduced, as described above, from three interferograms according to equation (3.12). - In interferometry it is possible, in particular, to carry out subtractions, additions, and multiplications with small whole numbers in an analogous manner. That is why we start with equation (3.12) in the form 24t-7

?d =

4

5

9

49 - Ht, q)+ 4 5 , q).

In the first place one carries out one subtraction and addition each. The hologram so obtained contains the double deviations 2x(<, q ) from the ideal shape, which is why it is necessary, when inserting the hologram, to use an arrangement in which the deviations x of the master surface A are also doubled. Addition and subtraction of interferometric data are made possible by superposition via MoirC. Generally known are the low frequency and subtractive Moire’s. But an interferogram does not contain the sign; only when it is known in which direction the order number of the fringes increases or decreases, can the deviations be determined with the right sign. - For the hologram we define a spatial frequency vector k,* in such a way that its positive direction points to the edge of the wedge, i.e., in the direction of decreasing order number, and that its magnitude is proportional to the number of interference fringes per unit length. Both the magnitude and direction are fixed by the adjustment procedure. With a sensible selection of k vectors both subtraction and addition are possible. In order to carry out the analogue operations a double diffraction apparatus is used (CUTRONA, LEITH, PALERMO and PORCELLO [1960]), in whose Fourier plane only the suitable orders of diffraction of the combined interferograms are allowed to pass. Fig. 3.13 illustrates the procedure of producing the compensating hologram. The three steps are depicted in succession. At first subtraction is carried out by superposingand filtering the patterns with deviation sums a, b. The fringes, or the k vectors belonging to them, are then placed as shown in Fig. 3.13 (top). The pattern with the deviation sum c and the vector * The spatial frequency vectors only represent the carrier frequency vectors. The deviation sums a, b, c, etc., lead to a change of frequency A&, which depends on the location. It is assumed that IAkl is small compared to Ikl, so that the veqtors shown in Fig. 3.13 are representative of all the vectors of a hologram.

IV,

§ 31

A B S O L U T E TESTING

I39

ADDlTlOy 1 (a-b) * c

E E E F z

kk-b)

k &+&I k2x

k,

-

Fig. 3.1 3. Diagram for choosing the spatial frequency vectors for producing the compensating hologram without calculation. The k-vectors are marked in the index by the deviation sums or the absolute deviations, respectively.

k, is then combined with the resulting difference hologram with the carrier frequency vector which, by properly choosing the direction of k,, leads to the resulting vector k,, and thereby yields the compensating hologram. This compensating hologram is then placed in the position of the image of the pair of plates A, D (master, surface to be tested) in an arrangement corresponding to Fig. 3.14, i.e., the hologram is used as an image hologram. With a laser light source and a condensor light is directed onto the combination of plates AD. Part of the reflected light is directed through a beam splitter and onto a second optical system, which images the layer of air between the two surfaces A, D on the hologram. Since, however, twice the value of the absolute deviation - i.e., 2 4 5 , q ) - is contained in the hologram, 2 x + 2 w ( w being the absolute deviations of the surface to be tested) must be produced. For this purpose A and D are coated with re[19673 flecting films. The multipass method described by LANGENBECK makes multiplication of the deviations with small whole numbers possible. - If the surfaces A and D are inclined relative to one another, bundles with different directions are generated in the space of the parallel beam which have passed the interferometer a varying number of times. These are separated in the back focal plane F, (Fig. 3.14) of a lens. Thus the wave reflected at A only and the one which has passed twide back and forth through the interferometer can be separated from all others. A double-diffracting apparatus is added to the hologram. In the filtering plane F, only the diffracted master wave and the wave which has twice passed back and forth without diffraction can pass. They form the final interferogram in the plane

140

INTERFEROMETRIC TESTING OF SMOOTH SURFACES

“v,

04

BEAM SPLITTER

x: R

--IMAGING SYSTEM

--

-mna

I,\

FILTER 5

it

COMPENSATING HOLOGRAM

~

--sF#TIAL

FILTER F2

-CAMERA

Fig. 3.14. Absolute interferometer with a real master and compensating hologram. The surfaces are inclined relative to one another. The reflected light is screened in the plane F, except for 2 waves. These illuminate the compensating hologram. In the first order of diffraction, a wave with the deviations 2x(5, q ) and one with the deviations 2x(<, q)+2w(5, q ) are then superposed so that one is subtracted from the other.

of the camera (Fig. 3.13, bottom). In the case of testing spherical surfaces, the final interferogram is only determined to a quadratic function of the surface co-ordinates t, q (see $2.1).

5

4. Comparing a Surface with Itself

So far we have considered comparisons between different surfaces. But one can also compare a surface with itself. One can, for example, compare every point on a surface with a different point on it (shearing methods,

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Q 4.1). But it is also possible to compare every point on a surface with a single spot on that same surface (point reference methods, 5 4.2). Such interferometers, which have also been called common-path interferometers, have already been reviewed several times (BRYNGDAHL [19651, STEEL[1966], FRANCON and MALLICK[1971]). What follows is therefore largely limited to those aspects important for testing a surface which have not yet been reported. Such testing is generally carried out in reflected light, for tests in transmitted light give less accurate information about the shapes of the surfaces. Only for tests on aspheric surfaces, where very high accuracy can generally not be obtained, does one use transmitted light (compare Q 5). 4.1. SHEARING METHODS

Wavefront shearing interferometers were introduced under this name by BATFS [1947] (BORNand WOLF[1964] p. 312). Related methods have been developed by WAETZMANN [1912] and RONCHI[1926]. A shearing interferometer produces two non-coinciding wavefronts from one wavefront in such a way that different points of the test object are compared interferometrically. It is often typical that the entering wave surface first passes the object and only then enters the shearing interferometer, where it is divided, but not re-united. This produces corresponding requirements for coherence ; also, the wavefront must not have unknown phase structures before passing the object, since these would be superposed onto the object structures searched for. - The shearing interferometer itself is assumed to be acting ideally. The two wavefronts which carry the object structures and are to be compared are usually congruent, but spatially displaced with respect to one another (lateral shearing, rotational shearing). Or the wavefronts are mirror images of one another (reversion). Doubling the original wavefront is usually carried out by splitting the amplitudes at partially transmitting surfaces (also by double diffraction). Diffraction gratings, however, are also used to split the wavefront. A lateral shearing method was used by SAUNDERS and BRUENING [19681 for testing a 2m mirror telescope, see Fig. 4.1. The shearing interferometer used (SAUNDERS [1964]) consists of 2 prisms cemented together; it splits the entering wavefront into two which are inclined at a small angle to one another. The corresponding images of the aperture of the telescope produce interference fringes in the region where they overlap. Evaluation of these interferograms furnishes one with contour maps of the wavefront relative

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A

FILTER

m -*

EYE

Fig. 4.1. Testing of a reflecting telescope (mirrors A, B) with a lateral shearing interferometer SH. SH (shown on a larger scale below) is in a tube which has been put in the place of the eyepiece of the telescope.

to a best fitting reference sphere. A star is the source of light. The influence of atmospheric disturbances has to be kept inoperative. This is done by using a sufficientlysmall shear and is controlled experimentally. -A method for the workshop testing of optics with similar interferometers is described by SAUNDERS [19701. BIRCHand GREEN [1972a1have described a shearing method for aspheric surfaces. The entering wave penetrates as plane wave an obliquely placed plane parallel plate and reaches the aspheric test surface via a lens as closely at right angles to it as possible. Here it is reflected and, after again passing the lens, falls once more onto the plane parallel plate. This produces, after reflection, two interferingwavefronts displaced with respect to one another. The interference pattern is evaluated with the aid of lateral shearingmethods. The lens must be well corrected. WYANT[1972] has pointed out that optical elements such as a diverger combined with aspheric wavefronts may produce large additional aberrations. Using special optical systems it is also possible to produce and compare two images of a wavefront, which are rotated with respect to one another round the optical axis by an arbitrary angle. Such shearing methods are called theta or rotary shearing (ARMITAGE and Lomm [1964]). Rotation is effected with, for instance, a Michelson arrangement, whose two reflecting flats are replaced by 90" double mirrors (roof prisms). By turning one mirror round the optical axis any arbitrary angle of rotation can be arranged. Or, with a different arrangement, the rotation is achieved by means of a Dove prism (compare also MURTYand HAGEROTT [19661).

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The system of fringes produced with the optical arrangements mentioned above, can also be obtained in a different way (SCHWIDER [1973]). But here the wavefront in front of the object and the interferometer need not be ideally free from faults. One of the arms of a Michelson arrangement contains the surface to be tested and the other a comparison surface (this need not be free from faults either). Two interference photographs are taken with wedge adjustment. The second differs from the first in that the surface to be tested has been rotated by a known angle round the optical axis. The two patterns are superposed to form a MoirC picture. The Moirk fringes represent theta shearing interference fringes. By this difference technique, errors due to the incident wave, the interferometer, and the comparison surface, are eliminated. - If for the second photograph one replaces the surface by a different one, then the deviations of these two surfaces from each other can be determined in the same way. In the case of plane or spherical surfaces one can do without the second photograph, and use a live fringe technique instead. Reversing interferometers are counted by some authors amongst the shearing interferometers. Some of these interferometers (GATES[19551, SAUNDERS [1958]) are designed especially for testing surfaces. An example from the last reference is shown in Fig. 4.2. One obtains the difference of the surface deviations of mirrored points of the test surface A :

4 5 , ?)-X(-

t,?)= a?).

Such an arrangement differs from typical shearing interferometers in that the optical element to be tested lies within the actual interference path, and faults of the wavefront entering the interferometer from outside do not in

Fig. 4.2. Wavefront reversing interferometer.

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general affect the result of the measurement, because the waves are reunited. Looking at it from the point of view of the type of interferometer used, such arrangements belong to those dealt with in 9 2. For if one assumes the surface to be tested to be divided in the middle, then a Michelson arrangement of the Kosters type is obtained. A Ronchi grating (RONCHI[1926]) may be regarded as a shearing interferometer (BRIERS [19721, for further references cf. there). A wavefront is split into several wavefronts by diffraction at a grating. In this way several virtual images of the original wavefront are formed which partially overlap. WYANT[1973] has replaced the Ronchi grating by a holographic twofrequency grating. In this way a shear image formed only by the first order of diffraction can be observed. With lateral displacement of the grating, the intensities in the shear image change sinusoidally. The time frequency of this change is proportional to the difference of the two spatial frequencies of the grating, and the position of the phases is determined by the spatial position of the fringes. In this way a modulated light method can be used for evaluation (5 2.5). - If a lattice grating is used, then it is possible to use simultaneously two shear directions at right angles. LANGENBECK [1971] has developed a method with a neutral reference beam from a shearing interferometer. A shearing interferometer, together with the object to be tested, produces two separated real images of the point source of light. Both images of the light source contain information about the object. But if one inserts a very small pinhole at the place of one of the images of the light source, then this image (in the limiting case of the pinhole diameter tending to zero) no longer contains information about the object. Rather, it furnishes a neutral reference wave, which interferes with the wave of the other image of the light source. The interference pattern furnishes information about the object. The amplitudes of the interfering waves are matched to one another by the requirement that the shearing interferometer produces the image of the light source which is to be filtered with correspondingly stronger amplitude. 4.2. POINT REFERENCE METHODS

Here every point on a test surface is compared to one particular point on that surface. One therefore does not have a reference surface, but, as it were, a point. One can, therefore, speak of “point reference methods” (compare also NOMARSKY [1971, 19731). Such methods were first described by BURCH[1953] with a scatter-plate and by DYSON[1957] with

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a calcite lens as beam divider. Since reviews of such methods already exist (e.g., by STEEL[1969]), we will here limit ourselves to supplementing them. MURTY[1963] replaced the two identical scatter-plates used by Burch (one to divide the beams, the other to unite them) by two identical Fresnel zone plates.

I

Fig. 4.3. Point reference method with a scatter-plate with a center of symmetry.

Scorn [1967] used and analyzed an arrangement with only one scatterplate to divide the beams and unite them, see Fig. 4.3. The ideal structure of this scatter-plate is statistical, but it has point symmetry with respect to its centre 0, which is at or near the centre of curvature of the concave mirror A which is to be tested. An image of the light source, i.e., of a pinhole P I , is formed at the centre Po of A. Part of the light, however, is scattered while first passing through the scatter-plate (path P,P,PQ), a second part during the second passage (path P,PoP2Q). The latter bundle is the reference bundle. The two bundles interfere in the image plane. In this way one compares every point P of A interferometrically to the reference point Po. SHOEMAKER and MURTY[1966] have replaced one of Burch’s two

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scatter-plates by a mirror ;illumination and observation take place through the remaining scatter-plate (via a beam divider placed before it). Reflection at the mirror just mentioned has the effect that the two interfering bundles meet the test surface twice, the reference bundle always at the same spot P,(t = 9 = 0), the other bundle at two spots which are point symmetrical to Po. In this way one measures the sum

of the surface deviations of point symmetrical spots. Because of the way the beams are guided, the interferometer is particularly insensitive to [1963b1). vibrations (compare also DYSON

0

5. Comparing a Surface with a Hologram

So far we have considered comparisons of a surface with other surfaces or with itself involving mutual comparison of the wavefronts reflected by the surfaces. But one can also compare the wavefront reflected by a surface with that diffracted by a hologram, i.e., carry out a comparison of a surface with a hologram. This hologram may have been produced by interference (9 5.1) or calculated with a computer and produced synthetically (8 5.2). 5.1. COMPARING WITH A HOLOGRAM PRODUCED BY INTERFERENCE

In holography one has the possibility of storing a wavefront on a photographic plate, if certain limitations regarding the spatial frequency apply (LEITHand UPATNIEKS [19671). When testing smooth surfaces, these conditions are generally fulfilled. The optical test glass (Fig. 5.1) can, therefore, be replaced by a hologram which has stored the wave of the master surface (SNOWand VAN DEWARKER [1970]). In a hologram interferometer with smooth surfaces focusing and therefore repositioning of optical elements is very difficult due to high coherence. One can facilitate this by limiting the spatial coherence or by storing an auxiliary wave whose focus is on the surface to be tested (FERCHFR and TORGE[1970]). PASTOR[1967] has proposed the use of holograms for general comparisons of surfaces, particularly for aspheric surfaces. Thus one can, for instance, store a state in a hologram, which would not otherwise be available for comparison on account of changes which the object might undergo, e.g., correction of the surface or further polishing. Here real-time holography offers the possibility of comparing two states at time z1 and z2 (Fig. 5.2). If high ac-

C O M P A R I N G A S U R F A C E W I T H A HOLOGRAM

147

BEAM SPLITTER

OBSERWTlON

Fig. 5.1. Holographic test glass. The wave reflected by it is stored in a hologram. After replacing the test glass by the surface to be tested, the deviations of the latter can be measured.

ASPHERlC SURFACE

Fig. 5.2. Hologram interferometer for testing aspheric surfaces. A polished aspheric surface is illuminated with diffuse light. The reflected wave is stored in a hologram. After this the aspheric surface can be removed and the polishing continued. After repositioning, the deviations form the initial state are visible.

curacy and reproducibility of the results are required, holographic methods will be problematic, if the carrier frequency in the hologram is too high. In this case lateral shrinking of the photographic layer is particularly noticeable. Furthermore, the holographic plates are, in general, not sufficiently stable, and this also leads to disturbances of the wavefront. In some cases it is useful to reduce the carrier frequency by fitting the shape of the reference wave to the wave of the object. In 0 4.1 such a method [1973]). The method can also be was described for spheres (SCHWIDER applied to aspheric surfaces. For this, two as’phericsurfaces of similar shape are inserted into a Twyman-Green interferometer (Fig. 5.3). By fitting the shapes of the aspheric surfaces A and B (or C) to one another, a carrier frequency hologram of low frequency (here hologram 11) can be used for

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I N T E R F E R O M E T R I C T E S T I N G OF S M O O T H S U R F A C E S

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Fig. 5.3. Doubly compensated hologram interferometer for testing aspheric surfaces. The hologram I serves to illuminate the interferometer. After reflection at the aspheric surfaces A, B or C, approximately spherical waves are formed. The deviations of the surface B from A are stored in hologram 11. After replacing the surface B by C, the deviations of the surface C from B can be measured.

storing the deviations of the surface B from A. Since the interferometer is not free from faults, the use of a difference method (see 0 2.2) when comparing the aspheric surfaces B and C is necessary to compensate for errors. Since the surface A is used as a reference surface, its deviations when comparing the surface B to the surface C are also eliminated. To obtain the difference z-y of the deviations of the surfaces C and B the Moire comparison between hologram 11, which has stored the deviation difference x - y of the surfaces A and B, and the live-fringe pattern of the surfaces A and C is used. Here the contrast is at first relatively small. To improve the contrast, it is necessary to separate the wave of the 0th order of diffraction. But with strongly deformed aspheric surfaces a separation of the 0th and 1st orders of diffraction is not possible if the carrier frequencies are small (e.g., 1O/mm), since the position of strongest constriction of the beam extends further than the spacing between the 0th and 1st orders of diffraction. To compress the cross-section of the beam, the wave entering the interferometer is deformed in advance in such a way that, after reflection at the aspheric surfaces A, B, or C, almost spherical waves are produced. These then pass through an almost point shaped region. A hologram I serves to deform the wave in advance. This is obtained by superposing an aspheric and a plane wave, which enter from the same side with respect to the hologram. For reconstruction, a plane wave enters from the back and

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in reverse direction. LEITHand UPATNIEKS [1966] have shown, that in this case even the' deformation of a wave formed by a diffuser can be compensated. Here an aspheric mirror takes the place of the latter, whose effect is cancelled in this way. Fig. 5.4 shows the deviations of two aspheric surfaces from each other with parallel and wedge adjustment. The Twyman-Green arrangement and the precompensation of the wavefront furnish important aides for adjustment: on the one hand, white light interference fringes can be used for adjustment, on the other, observation of the focal plane of the precompensated wavefront makes it possible to supervise the preadjustment ofthe repositioned surfaces B or C (Fig. 5.5). In this way different aspheric surfaces of the same batch and different states of the same aspheric surface can be compared, and knowledge of the rotational symmetry of a surface can be attained.

Fig. 5.4. Deviations of 2 aspheric surfaces from one another (distortions due to the interferometer are eliminated). Left: parallel adjustment. Right: wedge adjustment.

Fig. 5.5. Section near to the place of strongest beam constriction (slightly defocussed). Left: ideal adjustment. Right: same as left but aspheric surface misaligned.

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5.2. COMPARING THE SURFACE WITH A COMPUTER-GENERATEDHOLOGRAM AS MASTER

The absolute methods described in 5 3 are not suitable for most aspheric surfaces, particularly not for those which are strongly deformed. Relative measurements by means of relative interferometric comparisons (9 2) can, however, be made. But this requires producing a master. With interferometry, one compares wavefronts. It is therefore sufficient to generate a master wave for the comparison in place of an actual master surface. BROWNand LOHMANN [1966], WATERS[1966], and LOHMANN and PARIS [1967, 19681 have introduced the computer-generated hologram into holography. PASTOR[1967] has discussed producing in this way a wavefront which can serve as master wave for testing aspheric surfaces, etc. Such a synthetic hologram is well suited for representing the master wave, since any wavefront can be calculated and it is not necessary to produce an aspheric master surface. For this purpose one must calculate, for instance, the positions of the maxima of the imaginary interference pattern between the ideal master wave and an arbitrary reference wave. Thus, at each point of the hologram, the path differences between reference and master wave must be calculated. Then the maxima are situated where the path difference 1 is an integral multiple of the wavelength A. To calculate the path differences between master and reference wave, ray-tracing methods are used. In most cases the positions of the maxima can only be determined by iterative methods, which is why a relatively large computer is required. Plane or spherical waves are particularly suitable as reference waves, since they can be produced and tested comparatively easily. But other reference waves can also be used, provided they are known. In this sense we consider waves as known, when they are produced by known single lenses or by whole systems, the properties of which are known through ray tracing or measurement. Fundamentally testing methods with computer-generated holograms (CGH) can be divided into those using reflected and those using transmitted light. Correspondingly, the types of interferometer used with these methods have been developed from the Twyman-Green or Mach-Zehnder interferometers. - There are 3 possibilities for inserting the CGH : 1. The CGH is inserted into the parallel arm of the interferometer used for testing and directly replaces a real master. 2. The CGH is used for compensating the aspheric wavefront to produce a plane, spherical, or otherwiseknown reference wave ;it is therefore inserted into the same beam as the aspheric surface and replaces a null lens (5 2.3).

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C O M P A R I N G A S U R F A C E WITH A HOLOGRAM

3. The CGH is inserted after the interferometer and produces information about the aspheric surface in Moirt. Examples of the different arrangements are given below. 1. The hologram replaces the reference mirror in, for instance, a Twyman-Green interferometer (SCHWIDER [1970, 1971]), Fig. 5.6. The diffracted wave is superposed onto the aspheric wave in a null test. By combining the CGH with an optical system, e.g., a single lens, the hologram can be relieved in terms of frequency. An example for testing a concave aH-G',BEAM

z

SPUTTER

IMAGING LENS

smp INTERFERENCE PLANE

Fig. 5.6. Michelson interferometer with CGH as reference mirror. The CGH can be of the rotationally symmetric (as shown) as well as of the off-axis type. The very intense zeroth order of diffraction, in particular, should be removed by a stop.

EAM SPUTTER

L

Fig. 5.7. Michelson interferometer with a combination of auxiliary lens and CGH. The auxiliary lens is used to produce the spherical curvature of the wavefront. The CGH only generates the aspheric deviations.

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aspheric surface is given in Fig. 5.7. For testing with transmitted light either Mach-Zehnder or Twyman-Green arrangements with double traverse are used. Here also the wavefronts can be fitted to one another by prior compensation with known optical elements, where the CGH generates, for example, only the aspheric deviations. Fig. 5.8 shows such an example for transmitted light. PLA"Mx LENS

Fig. 5.8. Mach-Zehnder interferometer with a combination of auxiliary lens and CGH. The CGH in combination with an auxiliary lens (planoconvex lens) and the aspheric lens are each inserted into one arm of the interferometer.

2. In holography, the object and reference waves complement one another, i.e., when the object wave impinges on the CGH, the reference wave used in the calculation is generated. One can make use of this relation when compensating the aspheric wavefront. The CGH then plays a part REFEMMIRROR

'e' BEAM SPLITTER

ERGER

A

CGH

-INTERFERENCE

PLANE

Fig. 5.9. CGH compensation of an aspheric surface. A diverger with a CGH and the aspheric mirror are inserted in series into one arm of the Michelson interferometer. From the entering spherical wave the CGH generates a wave which meets the mirror everywhere at right angles. After reflection and further diffraction at the CGH an approximately spherical wave is generated, which carries the deviations of the aspheric surface.

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similar to that of the “null lens” (see 9 2.3) and enables one to test aspheric surfaces by testing a plane or spherical wavefront. MACGOVERN and WYANT[19711 have described an arrangement with reflected light, where the hologram compensates the aspheric deviations of a mirror (Fig. 5.9). LARIONOV, LUKINand MUSTAHN [ 19721have proposed optical arrangements for compensating in reflected as well as in transmitted light. For the latter Fig. 5.10 can serve as example. Here the CGH follows the aspheric surface and produces a spherical wave which, in turn, can be tested interferometrically. m f f

STOP

LENS

Fig. 5.10. CGH compensation in transmitted light. The aspheric lens is compensated wlth a CGH (here RSH) to give a spherical wave. This can then be compared to an ideal spherical wave or tested with shearing methods (0 4).

3. BUINOV,LUKIN,MIRUMJAIZ and MUSTAFIN[1969, 19711 have described the first arrangements with CGH outside, the interferometer (see also BUINOV,LARIONOV, LUKIN,MUSTAFINand RAFIKOV [1971]); cf. Fig. 5.11. A plane wave falls onto the parabolic mirror to be tested and EAM SPLITTER

-SF#TIAL FILTER

INTERFERP(CE -PLANE

Fig. 5.1 1. RSH test for aspheric mirrors. An entering plane wave is transformed at a parabolic mirror into a spherical wave. The latter is diffracted at the RSH and interferes with the plane reference wave, which is reflected in the same direction by the reference mirror.

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produces a spherical wave, onto which a plane reference wave is superposed. Then a Moire between the live fringe pattern and the CGH is formed in the plane of the hologram and shows the errors of the surface to be tested. MACGOVERN and WYANT[1971] have used an arrangement as shown in Fig. 5.12. Here the hologram is illuminated by the plane reference wave and generates, in the first order of diffraction, a master object wave, which is

Fig. 5.12. Interferometer in series with carrier frequency-CGH. The CGH is illuminated by a plane wave and produces, in the first order of diffraction, a reference wave for the wave reflected at the aspheric mirror to be tested. A spatial filter separates the diffracted from the undiffracted light.

superposed onto the real object wave. - The interferometer arrangements dealt with so far can serve as examples of a number of similar arrangements. Thus FERCHER and KRIESE[1972] as well as BIRCHand GREEN[1972b] have also used CGH for testing, and have used carrier frequency holograms which were produced by a commercial plotter. The arrangements used for this purpose are essentially similar to those in Figs. 5.9 and 5.12. FERCHER [1970] and FERCHER,KRIFSE and KOHN [I9721 have shown how, with additional storing of a spherical auxiliary wave, the process of adjusting the testing interferometer can be facilitated. WYANT[19721 has adapted the system to the limited possibilities of a commercial plotter by using a relatively simple null lens in combination with a CGH. For the CGH there are two basic types - the carrier frequency hologram and the “in line” hologram of the Gabor type, which is especially important for rotationally symmetrical surfaces (see below). Three methods have been investigatedusing the carrier frequency hologram for testing optical surfaces

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(MACGOVERN and WYANT[1971], BIRCHand GREEN[1972b1, and WYANT [19721). The most widely used form is the hologram introduced by BROWNand LOHMANN[1966] with the “detour phase” for representing the phase function. When, for the sake of simplicity, the amplitude is assumed to be constant (hologram in the plane of the image) then only the phase, i.e., the position of the diffracting apertures has to be determined and recorded. On account of the limited number of resolvable points of a plotter, it is necessary to carry out phase quantization, while limiting oneself to reasonable accuracy (for example & fringe). Because of the limited number of elementary cells in the CGH one has to make a compromise between phase accuracy and carrier frequency for given maximum differences of inclination of the tangential plane of the test piece. For the CGH according to LEE [1970] (compare MACGOVERN and WYANT[1971], Fig. 5.13) the elementary cell is divided into 4 iubcells which store the positive and negative real and imaginary parts. The phase function is then stored within the subcell in the size of the spot or in the greyness of a spot which has the size of a cell. WYANT[19721 has used a special electronically controlled laserbeam recorder. Here the recording light beam follows the position of each maximum. Thus one co-ordinate is not screened. The most important group of aspheric surfaces are rotationally symmetrical surfaces, which may become even more widespread if production problems can be overcome. As far as these surfaces are concerned it makes sense to use rotationally symmetrical synthetic holograms (RSH) - more or LUKIN,MUSTAFIN and less of the Fresnel zone type. BUINOV,LARIONOV,

Fig. 5.13. Lee-type CGH (courtesy J. Wyant).

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RAFIKOV[1971] have scratched RSH onto aluminium with a circle engraving machine. Parabolic mirrors with a focal distance of up to 700 mm were examined. The maximum density of the lines was 50/mm. Likewise, SCHWIDER [1970], MATSUMOTO and SUZUKI[1970], KETTERER [1972], and ICHIOKAand LBHMANN [19721have proposed RSH for testing aspheric surfaces. The latter authors have compared the number of fringes for carrier frequency holograms and of rings for RSH. This showed that the RSH need a number of interference maxima which is, in the cases discussed by the authors, smaller by a factor of about 4 to 6. Since a parabolic mirror with a central hole was examined, conditions for separating the waves of different orders of diffraction on the axis could be given. - To free the master wave of a RSH from the disturbing waves of other orders of diffraction (especially from the undiffracted zeroth order), stops are positioned at the optical axis. Representing a master wavefront by a synthetic hologram is not fully equivalent to using a real aspheric master surface. This is because of the different effects of synthetic and material master surfaces on an entering wavefront. The synthetic hologram generates the master wave by diffraction, whereas the aspheric surface generates it by reflection or refraction. For this reason, while polychromatic interferometry is possible with real masters, it is not possible with synthetic holograms. On the other hand, a binary synthetic hologram also generates waves of higher orders (MATSUMOTO and SUZUKI[1970]), which can also be used for testing. Furthermore, computer-generated holograms can in principle be used for comparison with all smooth wave fields, including those produced by an optical system. When testing aspheric surfaces considerable problems of adjustment generally occur, since the position of the surface relative to the master or to the CGH has to be adjusted for 5 degrees of freedom for rotationally symmetrical aspheric surfaces and for 6 degrees of freedom for aspheric surfaces without rotational symmetry. Owing to errors of adjustment, evaluation of the interferograms is particularly complicated with steeply curved aspheric surfaces (WYANTand BENNETT [1972]). Furthermore, synthetic holograms are also suitable for compensating the errors of interferometers and master surfaces, provided these errors are known. In particular, there is the possibility of constructing an absolute interferometer, where the errors of the interferometer including the reference surface are compensated with a CGH, so that the absolute deviations of a test piece can be obtained directly in an interferogram (compare Fig. 3.14).

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6. Some Systematic Sources of Error and Limits of Measurement

So far errors and fundamental limits of measurement have not yet been considered. Some systematic errors of a general nature will now be dealt with, though we will not here discuss special investigations of errors, e.g., for special procedures, methods of adjustment, or receivers. - To begin with, we will consider some non-optical sources of error. The use of optical surfaces is generally based on the fiction of a “rigid body”. This assumption ignores the effect of gravity. When larger plates are in a horizontal position, perceptible bending may occur (NADAI[1925], TIMOSHENKO and WOINOWSKY-KRIEGER[ 1959]), when the flexural rigidity (proportional to the 3rd power of the plate thickness) is not large enough. This was recognized and investigated early in the course of tests on flats, and attempts were made to eliminate this bending. For absolute tests on three surfaces A, B, C (Q3.1), SCH~NROCK [1908] placed each plate in each of its three combinations of positions, once on top and once underneath : B A . C B . A C A’ B’ B’ C’ C’ A’ He then made the following assumption: When a plate lies underneath, the test surface bends towards the concave side in exactly the same way as it bends to the convex side when the plate is on top. By averaging the measurements, for instance at the combinations 2 and t, he obtained the values which would apply without gravity. Thus he obtained the shape of a plate which would apply in the absence of gravity. This shape must be realized experimentally, when the plate is used as a standard for flatness. For this purpose Schonrock‘floated the plate on mercury. A plate D, which was to be compared interferometrically with this floating plate, was placed horizontally above it. In this way one can find the shape of the surface, which D has in this position (in this position it can then be used). - Surfaces used in the vertical position have been examined in this position [19711, among others. Changing by SCHULZ, SCHWIDER, HILLER and KICKER over the plates from top to bottom is then no longer necessary. - The method [1952] is partly similar to that described by Schijnrock. used by EMERSON In addition, he has determined the sagging quantitatively by a combination of measurement and theory. For a fused quartz optical flat, supported at 3 equidistant points at its edge, and of diameter 270 mm and t6ckness 36.5 mm, he obtained sagging in the middle of the flat of about 0.1 pm. - DEW

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[rv, ci 6

[1966a] has measured the sagging of glass beams, i.e., of straightness standards (0 3.1). In a combination AB of two macroscopically similar beams (A below, B above) the beams are supported in 4 different ways: each of the two beams is supported once at the edges and once at the points of minimum deflection. In the latter case sagging is very small. Therefore its theoretical value is used which, for beams 33 cm long, 3 cm wide, and 7.6 cm high, is at most 0.009 x 1/2(2 = 0.546 pm). In the case of the beams being supported at the edges, the sagging is obtained from 2 x 2 equations obtained from interference patterns ;the latter are obtained with the abovementioned 4 different ways of support. The sagging values so determined can be inserted as corrections into equations (3.1), from which the overall shape of the beams can then be determined. In the middle region of a beam, sagging of more than 0.3 x 4 2 and sagging differences (beam below - beam above) of up to 0.013 x 1/2 are obtained. - For circular and rectangular plates DEW[1966b] has investigated various systems of support. - For large astronomical mirrors with changing load the mechanical tensions are reduced by giving them a suitable shape (honeycomb structure) and by mechanical relief systems (RIEKHER [1957]) or by placing them onto a thin layer of air (HORNE[1972]). But gravity is only one of the possible cquses of bending. Even when the plates are placed vertically, bending may occur. Larger plates are placed into padded metal straps or supported in the neighbourhood of the middle plane. KOPPELMANN and KREBS[1961b] have examined the bending of Fabry-Perot plates in the traditional adjustment with a distance spacer, and found that the deformations due to pressing-on fade away within a few mm of the spot where pressing occurs. Besides mechanical tensions temperature differences in the plates cause deformations of the surface. Therefore, good thermal insulation should be employed so that temperature variations from the exterior are attenuated as much as possible when they reach the measuring apparatus. On the other hand, after reaching the parts containing the test surfaces, they should be levelled out as quickly as possible through good thermal conductivity and, furthermore, the coefficient of thermal expansion of these parts should be as small as possible. For this reason fused quartz is often preferred to glass. Cer-Vit has an even lower thermal coefficient of expansion than fused [1967], HORNE[1972]). - DEW[1966a] was quartz (DIETZand BENNETT able to trace changes of the interference pattern to very small temperature differences by means of temperature measurements at the upper and lower surfaces of two optical glass flats of 30 cm diameter and 3.8 an thickness. Thus, for a temperature difference of *0.02", the variation was k0.2

IV,§ 61

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159

fringes. By surrounding the interferometer plates with a combination of layers of aluminium and 5 cm expanded polystyrene the resulting differences of temperature could be reduced to f0.001". - SCHWIDER [I9661 has investigated the variations with time of interferometer plates due to thermal disturbances with the aid of multiple-wavelength interferometry. In doing so he proved the exponential character of the dying out process and determined the constants of this process for fused quartz and BK7. - If glass is used, experience shows that up to 24 hours may elapse after placing the plates into the shield described above until thermal equilibrium is reached. For measurements of great accuracy and ordinary optical glass (e.g., BK7) the temperature of the surroundings must be kept highly constant - i.e., better than f0.01"C. A fundamental limitation to the measuring accuracy in determining path differences results from the statistical fluctuations* of the number of photons in the field of radiation, since the path differences are measured via the intensity in the interference field. With this measurement, intensities are compared at different points or at different times (see 9 2.4 and 0 2.5). Uncorrelated variations of the intensities therefore reduce the accuracy of measurement. Such variations AN about the average number of photons (N) are the less noticeable, the larger (N), since A N / ( N ) decreases with increasing (N). ( N ) can for example be increased by: increasing the luminous density per wavelength interval, prolonging the time of observation, increasing the quantum efficiency of the receiver. As receivers one uses either photographic emulsions or photodetectors. With these the measuring or recording time is in many cases of the order of seconds. Estimates made by HANES[1959] are used for comparison. This author found, for example, that, due to photon noise, a relative uncertainty Am/m z

i x lo-''

should be obtained in a Fabry-Perot interferometer when determining the order m ; this holds for A = 6056 A (86Kr, line width in wave numbers 1.4/m, luminous density 0.3 W/m2 steradian) with a measuring time of 1 sec and a distance of 5 cm between the interferometer plates (diameter 4 cm), when the quantum efficiency of the receiver is 0.07. Lasers strongly and increase the luminous density per wavelength interval. - KARTASHEV EZIN[I9721 have reviewed all the above-mentioned effects, which is why we refer the reader to their bibliography. Through diffraction at the structure of the test object, the light suffers * See for example MANDEL[1963] and MANLXL and WOLF[1970].

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a change of direction. The uncertainty thereby produced can be represented in the form of an optical uncertainty relation (WOLTER[1950, 19561). For multiple-beam interferometry, INGELSTAM [19531has proposed an uncertainty principle. KOPPELMANN [19691 has already reviewed papers which have since appeared on this subject. According to ROSENBERC [1958] and KOPPELMANN [19651, the uncertainty relation is :

Here A( is the smallest resolvable width and A[ the smallest resolvable depth of a small step of a surface of a multiple-beam interferometer (two plates at a distance of mI/2). m is the order of the interference, I is the wavelength, and K is a number of the order of 1. Apart from the papers and EZIN[1972] quoted here and by Koppelmann, papers by KARTASHEV and by LANGand SCOTT[I9681 should be mentioned. Kartashev and Ezin show a connection between (6.1) and the Heisenberg uncertainty relation. LANCand SCOTT[19681 have carried out measurements under different experimental conditions by working with fringes of equal chromatic order (TOLANSKY [1960]), i.e., by making use of the non-monochromasy of their light source. Thus, for m = 1 and ilx 0.6 pm, they For experimental conditions of this kind, obtained A( x 4 pm, A[ x they have formulated an uncertainty relation, in which the product A(A[ is proportional to mlZ,multiplied by a quantity dependent on the aperture angle and the reflectivity. Apart from limits to the resolution caused by diffraction, which restrict the similarity between object and image, there are other conditions to be observed with multiple-beam interference (BORNand WOLF[19641 p. 35 1). One of these is the phase condition of TOLANSKY [1948] (compare also BROSSEL[1947]). When it is not fulfilled, the interference fringes are broadened, asymmetric, and displaced compared to their normal position. KOPPELMANN [1972] has reviewed the papers on this subject, proposed confining the condition just mentioned, and calculated fringe profile curves for different parameters. Fringe shifts do not affect the testing of surfaces, as long as they are the same for all parts of the surface. With non-ideal surfaces another effect is also important: the lateral displacement of a light ray which has been reflected several times between non-parallel surfaces (TOLANSKY [1948]; BORNand WOLF[1964], 1.c.). This is also called walkoff effect. The lateral region thus covered on a test surface must be regarded as an unsharp region. It is particularly large when larger path differences and a high spatial carrier frequency (wedge adjustment) are required.

IV,§ 61

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161

The spectral properties, size, and position of the light source have an effect, via the complex degree of coherence (BORNand WOLF[19641 p. 491), on the visibility and the position of two-beam interference fringes. With incoherent, quasi-monochromatic sources the visibility (as a function of space) depends particularly critically on the size of the source (e.g., for arrangements with plane mirrors investigated by SCHULZand MINKWITZ [1961] and by SCHULZ [1964], and for arrangements with spherical mirrors by HOPKINS[1971]). Low visibility has a disadvantageous effect, since it makes the determination of the position of the fringes more difficult (9 2). Systematic errors, however, are introduced, when the change in thickness corresponding to a fringe spacing is wrongly taken into account. But the effect of the size of an incoherent source on the fringe spacing is hardly critical with the surface tests dealt with here; for the change of thickness At corresponding to a fringe spacing, if 212 in the ideal case, is given by:

At z (1 + a2/4)x 212 (IGNATOWSKY [19351). Here 2a is the angular aperture corresponding to the size of the light source, when the central light ray meets the surfaces approximately at right angles. Since the surface deviations generally extend at most over a few fringe spacings, and considering the small values of t~ involved, the term a2/4 hardly has any effect. Coherent light sources (lasers) are free from the effects of the size of the light source here discussed. If one wants to determine the deviations of a surface from its desired shape, then the course of the interference fringes for this desired shape must be known. It is further desirable that the fringe pattern should not only be known but have a shape which can be easily evaluated. In particular, it is desirable that, when the test surfaces do not deviate from their ideal form, straight and parallel interference fringes or locally constant intensity is obtained in the plane of the receiver. In flatness tests with Fizeau fringes this can be achieved comparatively easily, if the optical system imaging the fringes is free from distortions*; for, in the narrow and practically flat region between the weakly inclined wedge surfaces, the original fringes are already straight. With curved surfaces, though, things are different. If straight interference fringes in the ;lane of the receiver are to be obtained when testing spherical surfaces with Fizeau interference patterns, it is necessary for the sine condition to be fulfilled for those rays which go through the centre of the small aperture diaphragm, which approximately coincides virtually with the two centres of the test spheres (BIDDLES [1969], *

Regatding such distortions see

5 2.1.

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“v

HOPKINS[1971], HARRIS[1971]). In the ideal case, the rays mentioned above should fall as a bundle of parallel rays onto the plane of the receiver, or the sine condition has to be replaced by a sine-tangent condition where the sine in the space of the test spheres is proportional to the tangent in the space of the receiver (GATES [19583). In addition, when there are no other optical elements between the test surfaces, then aberrations of the wavefront before entry into the actual interferometer are of no great importance in practice, as long as the surfaces are sufficiently close together. For in this case such a wave aberration has practically the same effect on the waves reflected by the two surfaces, and their difference A W is zero. Otherwise, according to BIDDLES[1969], HOPKINS [19711, and HARRIS[1971], in the spherical Fizeau interferometer (Fig. 2.4 or 2.5) AW = u2t, where a (locally not constant) is the angle of incidence onto the two surfaces, which are assumed to be ideal and concentric, and t the difference of their radii of curvature. References ARMITAGE, J. D. and A. LOHMANN, 1964, Paper presented at the ICO Conference in Sydney; see also Opt. Acta 12 (1965) 185. BALDWIN,R. R.,1968, U.S. Pat. 3.512.891. 1949, Brit. Sci. News 2. 130. BARRELL, H. and R. MARRINER, BATES,W. J., 1947, Proc. Phys. SOC.59, 940. BENEDETTI-MICHELANGELI, G., 1968, Appl. Opt. 7, 712. E. and E. HABERMANN, 1960, US Pat. 3.028.782. BERNHARDT, BIDDLES,B. J., 1969, Opt. Acta 16, 137. BIRCH,K. G., 1972, Opt. Acta 19, 519. BIRCH,K. G. and F. J. GREEN,1972a, NPL Op. Met. 12. BIRCH,K. G. and F. J. GREEN,1972b, J. Phys. D : Appl. Phys. 5, 1982. R. and A. W. LOHMANN, 1957, Optik 14, 361. BLANKE, BORN, M. and E. WOLF, 1964, Principles of Optics (second ed., Pergamon Press, Oxford, London, Edinburgh, New York, Paris, Frankfurt). BOTTEMA, M., 1957, Photometric Setting Methods in Interferometry, Thesis, Groningen. BRIERS,J. D., 1972, Opt. and Laser Technol. 4, 28. BROSSEL, J., 1947, Proc. Phys. SOC.59, 224. BROWN,B. R. and A. W. LOHMANN, 1966, Appl. Opt. 5, 967. BRUNING,J. H., J. E. GALLAGHER, D. R. HERRIOTTand C. L. NENNINGER, 1972, OSA Fall Meeting, WE 16. BRUNING, J. H., 1973, private communication. BRYNGDAHL, O., 1965, Applications of Shearing Interferometry, in : Progress in Optics, Vol. IV, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) Ch. 11. BRYNGDAHL, 0. and A. W. LOHMANN, 1968, J. Opt. SOC.Am. 58, 141. and K. S. MUSTAFIN, 1969, Sov. Pat. 277269. BUINOV, G. N., A. W. LUKIN,S. 0. MIRUMJAIZ BUINOV,G. N., N. P. LARIONOV, A. W. LUKIN,K. S. MUSTAFIN and R. A. RAFIKOV, 1971, Opt. Mech. Prom. No. 4, 6.

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SCHWIDER, J., 1973, Paper presented at the 2. Internationale Tagung Laser und ihre Anwendungen in Dresden. J., G. SCHULZ,R. RIEKHER and G. MINKWITZ, 1966, Opt. Acta 13, 103. SCHWIDER, SCOTT,R. M., 1967, in: New Developments in Interferometry (Perkin-Elmer Corporation) p. 4-1; see also Appl. Opt. 8 (1969) 531. SHOEMAKER, A. H. and M. V. R. K. MURTY,1966, Appl. Opt. 5, 603. SNOW,K. and R. VAN DE WARKER,1970, Appl. Opt. 9, 822. STEEI, W. H., 1966 Two-Beam Interferometry, in: Progress in Optics, Vol. V, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) Ch. 111. TAYLOR, N. G. A., 1957, J. Sci. Instr. 34,399. TIMOSHENKO, S. and S. WorNowsKY-KRIecER, 1959, Theory of Plates and Shells (second ed., McGraw-Hill Book Co., New York, Toronto, London). TOLANSKY, S., 1948, Multiple-Beam Interferometry (Clarendon Press, Oxford). TOLANSKY, S., 1960, Surface Microtopography (Longmans, London). TOMKINS, F. S. and M. FRED,1951, J. Opt. SOC.Am. 41, 641. TROPELInc., 1970, Technical Description of SMI. 1967, Appl. Opt. 6, 803. VANHEEL,A. C. S. and C. A. J. SIMONS, WAETZMANN, E., 1912, Ann. Physik (4) 39, 1042. J. P., 1966, Appl. Phys. Lett. 9, 405. WATERS, WEINSTEIN, W., 1951, J. Sci. Instr. 28, 351. WOHLER,J. F., 1970, Priifung aspharischer optischer Flachen durch interferometrische Methoden, Dissertation, Universitat Stuttgart. WOLTER,H., 1950, Ann Physik (6) 7, 341. WOLTER, H., 1956, Schlieren, Phasenkontrast- und Lichtschnittverfahren, in: Handbuch der Physik, Vol. XXIV, ed. S. Fliigge (Springer, Berlin Gottingen, Heidelberg) p. 555. WYANT,J. C., 1972, GCH Maksutov Test, Report Itek-Corp. WYANT, J. C., 1973, Appl. Opt. 12, 2057. 1972, Appl. Opt. 11, 2833. WYANT,J. C. an&V. P. BENNETT, YAMAMOTO, T., 1970, Coherence Theory of Source-Size Compensation in Interference Microscopy, in: Progress in Optics, Vol. VIII, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) Ch. VI. ZOLLNER. F:, 1968, uber die raumliche Interferenzerscheinung im Michelson-Interferometer bei Filterung mittels Fabry-Perot-Interferometer, Dissertation, Friedrich-%hiller-Universitat, Jena. ZORLL,U., 1952, Optik 9, 449.

Supplementary notes added in proof:

92: MARIOGE and MAHE(1973, Opt. Acta 20, 413) used alternating light methods to measure the fringe position. - DYWN (1970, Interferometry as a measuring tool, Machinery Publ. Co, Brighton) proposed a fringe setting method via modulation of the path difference and phase sensitive detection techniques giving high sensitivity. - BRUNING et al. (1974, Appl. Opt, 13, 2693) gave a more detailed description of the digital wavefront measuring interferometer mentioned at the end of Q 2. The article includes solutions of detection problems, evaluation techniques, absolute testing (cf. Q 3, Fig. 3.12), and elimination of environmental disturbances by special sampling techniques. Using an on-line computer allows for real time tests of surfaces.

IVI

S U P P L E M E N T A R Y NOTES

167

Q 3: BIRCHand Cox (1973, NPL MOM 5) dealt with flatness calculations based on measurements made along surface generators (cf. the glass beams of Dew). - HOPKINSgave a paper on an absolute interferometric test for sphericity (1974, Optik 41, 360). Q 4 : Further lateral shearing interferometers were described by HARIHARAN (1975, Appl. Opt. 14, 1056), WYANT (1974, Appl. Opt. 13, 200), RIMMER and WYANT (1975, Appl. Opt. 14, 142) and EBERSOLE and WYANT (1974, Appl. Opt. 13, 1004 - accoustooptic lateral shearing). - The interB ROYCHOUDHURI (1974, Opt. Communic. 12, 29) ferometer of F o ~ R and combines lateral and radial shearing by using an off-axis zone plate (cf. also FOUIMand MALACARA, 1974, Appl. Opt. 13,2035). - LUKIN,MUSTAFIN and RAFIKOV (1975, Optika i Spektr. 38,350) used a synthetic hologram of the RSH type in a radial shear interferometer when testing a paraboloid. - Other radial shearing methods were reviewed by FouiM (1974, Opt. and Laser Technol., p. 181). - SMARTT (1974, Appl. Opt. 13, 1093) described a developed form of the Fresnel zone-plate interferometer of MURTY[1963]. 4 5: WYANTand O’NEILL(1974, Appl. Opt. 13, 2762) described the use of a CGH in combination with a Maksutov sphere in a Twyman-Green interferometer. - In a further publication, FAULDE et al. (1973, Opt. Communic. 7, 363) gave a partially compensated CGH-test for aspherical mirrors. - SCHWIDER and BUROW(1974, paper Intercamera Prague; 1975, paper ICO 10 Prague) tested aspherics by using high frequent RSH’s made on a special (interferometrically controlled) plotter. The influence of misalignments on the interference pattern was studied by comparison of computer-simulated and real interferograms. Q 6: DEW(1974, Opt. Acta 21, 609) observed the long-term stability of optical flats and found significant changes within a few years. - WALKUP and GOODMAN (1973, J. Opt. SOC. Am. 63, 399) investigated the minimum number of registered photoevents required for determination of fringe position to a given accuracy, using maximum-likelihood methods. - The investigations of KOPPELMANN [19721on multiple-beam interference fringes were continued in Optik 37 (1973) 164, 40 (1974) 89 and 43 (1975) 35. At last we refer to a survey on sphericity tests of DUCHOPEL and FEDINA (1973, Opt. Mech. Prom., No. 8, p. 50), a bibliography of MALACARA, CORNEJO and MURTYon various testing methods (1975; Appl. Opt. 14, 1065), and to papers summarized in J. Opt. SOC.Am. 64 (1974) 1361-1370 (TuG1, 2, 3, 4, 6, WC 13-19).

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E:WOLF, PROGRESS IN OPTICS XI11 0 NORTH-HOLLAND 1976

V

SELF FOCUSING OF LASER BEAMS IN PLASMAS AND SEMICONDUCTORS * BY

M. S. SODHA, A. K. GHATAK and V. K. TRIPATHI Indian Institute of Technology, Hauz Khas, N e w Delhi-110029,India

*

Work supported by National Science Foundation, U.S.A.

CONTENTS PAGE

$ 1 . INTRODUCTION

. . . . . . . .

. .

. .

. . .

. . . .

171

0 2. PHENOMENOLOGICAL THEORY OF FIELD DEPENDENT DIELECTRIC CONSTANT. . . . . . . . . . . . 175

5 3.

KINETIC THEORY OF FIELD DEPENDENT DIELECTRIC CONSTANT. . . . . . . . . . . . . . . 203

$ 4 . STEADY STATE SELF FOCUSING OF EM BEAMS IN PLASMAS. . . . . . . . . . . . . . . . . . . . . . . 213

5 5.

NONSTEADY STATE SELF FOCUSING. . . . . . . . .

5 6. GROWTH OF INSTABILITY

. . .

.

.

. .

.

.

.

. .

. .

238 249

5 7. EXPERIMENTAL INVESTIGATIONS ON SELF FOCUSI N G . . . . . . . . . . . . . . . . . . . . . . . . . . 261 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . 262

0

1. Introduction

With the availability of high power laser beams, a large number of interesting nonlinear phenomena (see for example AKHMANOV and KHOKHLOV [19721, BALDWIN [19691, BLOEMBERGEN [19651) have been studied, both theoretically and experimentally, during the last decade. The phenomenon of self focusing occupies a unique place in nonlinear optics because it considerably iefluences all other nonlinear phenomena. The self focusing of laser beams in dielectrics has been extensively studied in recent SUKHORUKOV and KHOKHLOV[1968], years; reviews* by AKHMANOV, AKHMANOV, KHOKHLOV and SUKHORUKOV [19721, SVELTO [19741 and an elementary monograph by SODHA, GHATAK and TRIPATHI [I9741 present a unified treatment of the important aspects of the field and indicate references to significant work. The self focusing of laser beams in plasmas has recently attracted a great deal of attention on account of its applicability to controlled fusion. The instabilities of nonlinear origin in the plasma are enhanced by the enhancement of the intensity of the beam due to self focusing. In turn this promotes the collisionless heating of the plasma. At larger time scales ( k zE,where tE is the energy relaxation time), the self focusing of the beam gives rise to a large enhancement in the collisional (ohmic) heating of the plasma which is again useful for fusion. The collisional heating of the plasma is also applicable to the heating of the ionosphere by a ground based HF transmitter and has been of much interest in space communications and ionospheric diagnostics. The self focusing of laser beams in semiconductors has also been a field of sizable activity in recent years; the motivation for this work appears to be a better understanding of free carrier nonlinearity and the characterization of semiconductors. In this review we have presented a unified treatment of self focusing in

* A pedagogically simple version of the review by AKHMANOV, SUKHORUKOV and KHOKHLOV [1968] has been given by SODHA[1973]. 171

i 72

SELF F O C U S I N G OF LASER BEAMS

cv, § 1

plasmas and semiconductors; the emphasis has been on the derivation and discussion of the field dependent dielectric constant on the basis of phenomenological and kinetic models of carrier transport, and on the parabolic equation approach, developed by Akhmanov and his collaborators. The steady state and transient aspects of self focusing of laser beams have been considered; the latter being important in the case of fast laser pulses which carry very high powers (BLOFMBERGEN [1973], RIENTJES,CARMAN and SHIMIZU [1973]). Phenomenologically, the nonlinear dependence of the effective dielectric constant of a plasma/semiconductor on the electric field can be understood as follows : the free carriers of the plasma absorb energy from the field and in the steady state attain a temperature higher than the equilibrium value such that the rate of power absorption from the field becomes equal to. the rate of power lost by the carriers in thermal conduction and in collisions with heavy particles (CONWELL [1967], GINZBURG [1970]). Such carriers whose temperature is modified appreciably by the field are called hot carriers and may give rise to these important effects : i) The mobility of carriers changes on account of the energy dependence of the relaxation time and the energy dependence of the carrier mass; the latter holds for nonparabolic semiconductors (see for example CONWELL [19671). ii) The carriers may be transferred from the low energy valley to higher energy valleys resulting in a large change in the mobility. This holds in many valley semiconductors (see for example CONWELL [19671, LEVINSTEIN and SHUR[19721). iii) The local concentration of carriers may change on account of (A) non-uniform heating and redistribution of carriers if the field is nonuniform and (B) the break down of the plasma (see for example SODHA, GHATAK and TRIPATHI[19741, SHKAROFSKY, JOHNSTON and BACHYNSKI [1966)). These modifications in carrier parameters lead to the nonlinear relationship between the current density and the electric vector and hence to a field dependent effective dielectric constant. A vast literature on the behaviour of hot carriers in semiconductors (see for example CONWELL [19671) and plasmas (GINZBURG [19701) is available, as such studies have found many applications in devices and semiconductor/plasma diagnostics. From the viewpoint of interaction of high amplitude electromagnetic waves with plasmas and semiconductors, the literature on hot carrier effects is, however, restricted mainly to the case of uniform beams where the local concentration of carriers is assumed to be unaffected by the presence of the radiation.

v, 8

11

INTRODUCTION

173

In the case of nonuniform beams the dominant nonlinearity in the effective dielectric constant of plasmas and compensated semiconductors arises through the heating and redistribution of carriers. The redistribution of carriers is also caused by the pondermotive force and is mainly important in collisionless plasmas. It is interesting to mention here that, owing to the space charge effects the positive and negative charge carriers are redistributed through ambipolar diffusion. The nonlinearity arising through the energy dependence of carrier mass (in the case of nonparabolic semiconductors) and the intervalley transfer of carriers is also important in causing the self focusing of the beam. The other types of nonlinearities are insignificant in self focusing. It would be useful to visualize physically the phenomenon of self focusing in a medium where the refractive index is an increasing function of the intensity E P of the beam (say n = n,+n,EEF). We consider a plane uniform wave front incident on a circular aperture (of radius ro) in a nonlinear medium. The portion of the medium illuminated by the beam has a refractive index (n = no +n, EEF) higher than that of the non-illuminated portion. Therefore the secondary wavelets diverging at an angle 0 from the wavenormal suffer total internal reflection at the boundary of the fictitious cylinder of radius Y, when

where

ec = cos-l

n0 no + n, EE*

corresponds to the critical angle. It is also known from the diffraction theory that a very large fraction of the power will be carried by rays making an angle less than 0, with the axis ; 0, = 0.6110/2r,n,

where I , is the wavelength of the radiation in free space. We may now consider three obvious possibilities. (i) When 0, > 8, the beam will diverge due to the predominance of diffraction effects. (ii) When 8, = 8, the beam should propagate in the selfmade waveguide. The corresponding power of the beam can easily be shown to be given by (1.22)21;c p = p = '' 128n,

174

S E L F F O C U S I N G OF LASER BEAMS

cv, § 1

where P,, is the critical power of the beam and c is the velocity of light in vacuum. (iii) When OD < Oc one expects the beam to focus. The above speculations are supported by rigorous analyses. In 4 2 we have given a complete derivation of the field dependent dielectric constant of the plasma/semiconductor with and without a static magnetic field by following a phenomenological approach. The cases of weakly and strongly ionized plasmas are considered separately to show the importance of the two mechanisms of energy loss viz. the collisional loss and the conduction loss ;in a strongly ionized plasma the relative predominance of conduction and collisional losses is reversed by the application of a static magnetic field. In the case of collisionless plasmas the pondermotive force is the source of nonlinearity. In both cases the redistribution of positive and negative charges takes place through the ambipolar diffusion. In the case of semiconductors containing only a single type of carriers, the redistribution of carriers does not take place. However, the nonlinearities arising through the energy dependence of carrier mass (in nonparabolic semiconductors, e.g. n-InSb) and the intervalley transfer of carriers in many valley semiconductors (e.g. GaAs) are important; the latter type of nonlinearity is not considered in the present review but a reference is made to a paper by GUHAand TRIPATHI [19723. The phenomenological treatment of 3 2 is followed by a rigorous kinetic treatment of the field dependent dielectric constant, based on the solution of Boltzmann’s equation in 6 3. The kinetic treatment gives identical results with phenomenological theory when the collision frequency of carriers is velocity independent i.e., the mobility is independent of the field. In the case of velocity dependent collisions the results by the two approaches may differ considerably. For a given dependence of the dielectric constant on the intensity of the beam, the propagation and focusing/defocusing of an electromagnetic wave (having Gaussian intensity distribution along its wavefront) has been studied in 4 4. The general treatment is based on the procedure developed by Akhmanov and his coworkers (see, for example, AKHMANOV, SUKHORUKOV and KHOKHLOV [1966,1968]). The effect of the saturation of nonlinearity has been discussed and conditions for uniform waveguide propagation and focusing/defocusingof the beam have been obtained. This is followed by considering the particular cases of self focusing/defocusingof Gaussian electromagnetic waves in (i) collisionless, collisional and fully ionized plasmas, (ii) parabolic and nonparabolic semiconductors, and (iii) in a magnetoplasma.

"3

0 21

PHENOMENOLOGICAL THEORY

175

In 6 5 the focusing/defocusing of a nonstationary electromagnetic wave in a nonlinear medium has been studied. With the present state of ultrafast ( 5 lo-'' sec) pulse technology, and with the availability of enormously high power pulses the study of self focusing under nonlinear transient conditions is very important. Assuming that the current density depends not only on the instantaneous value of the electric vector but also on its past history, expressions for the nonlinear current density have been derived when there is no redistribution of carriers and when there is redistribution of carriers. The self distortion of plane waves has then been studied and the moving foci (as a function of time) have been discussed. In fj 6 the growth rate of perturbations in the intensity of an electromagnetic beam has been studied. In particular it has been shown that a plane electromagnetic wave propagating in a plasma is unstable against modulation in the direction perpendicular to the direction of propagation. The latter part of this section discusses the growth of a small amplitude perturbation in an electromagnetic beam having Gaussian intensity distribution. In particular it has been shown that a self made waveguide in a plasma is stable for small scale filaments. In fj 7 we have discussed the experimental investigations on self focussing.

5

2. Phenomenological Theory of Field Dependent Dielectric Constant

2.1. EFFECTIVE DIELECTRIC CONSTANT

For a discussion of propagation of electromagnetic waves in conducting media it is very helpful to introduce the concept of effective dielectric constant from consideration of Maxwell's equations and other auxiliary relations, valid for conducting media. Maxwell's equations* are (BORNand WOLF[19701) V .D = 4~p, (2.la) V . B = 0,

(2.lb) (2.lc)

VxH

1 aD

47c

= --J+

c

~

-,

c at

(2.ld)

* In common with most workers in the field the CGS Gaussian system of units has been adopted except when otherwise stated.

176

SELF FOCUSING OF LASER B E A M S

cv, § 2

where E and D are the electric and displacement vectors, H and B are the magnetic and magnetic induction vectors, J is the free carrier current density and p is the free carrier charge density. The other auxiliary relations for conducting media are D = cLE

(2.le)

B

(2.10

and =

pH

where cL is the lattice dielectric constant in the case of semiconductors (e.g. cL -" 16 for Ge, eL -" 100 for Bi) and the dielectric constant of the gas (i.e. due to bound electrons) in the case of gaseous plasmas (8, -" 1 for plasmas) and p, the magnetic permeability, is approximately equal to unity for plasmas and nonmagnetic media. The free charge density p and current density J are related through the equation of continuity viz., V . J + -aP at

=

0.

If we specify the time variation of the electric vector E as e.up(iwt), the current density J , in the steady state can be expressed as J

=

a(EE*,o)E

(2.3)

where o(EE*, o)is the conductivity (complex in nature) of the plasma. In presence of a static magnetic field, the conductivity a is a tensor of second rank. Using eqs. (2.1H2.3)we obtain V . Deff= 0

(2.1 a')

and (2. Id)

where Deff = EE

and &(EE*,0) = cL-

4nio(EE*, o) 0

(2.4)

Eqs. (2.la') and (2.1d) are identical with eqs. (2.la) and (2.1d) when both p and J are put equal to zero. Thus we see that the conducting medium

"9

9 21

PHENOMENOLOGICAL T H E O R Y

177

behaves as a neutral ( p = 0) and nonconducting medium (J = 0), having a complex dielectric constant E , given by eq. (2.4). In the remaining part of this section we have considered some physical phenomena which lead to the electric field dependence of the effective dielectric constant ; explicit expressions for this field dependence corresponding to the various mechanisms have also been derived. 2.2. PONDERMOTIVE FORCE

In a collisionless plasma the drift velocity of electrons is governed by the [1970]) following equation (GINZBURG e 1 - ( v x B ) - -V(N,k,T,), C Ne

v = -eE-

(2.5)

where the right-hand side denotes the force on an electron (of charge - e and mass rn) on account of the electric and magnetic fields and the gradient of partial pressure of electrons; c, k,, N , and To denote the velocity of light in vacuum, Boltzmann's constant, electron concentration and electron temperature respectively. Eq.(2.5) may also be expressed as

au m-=

1 -V(N,k,T,), Ne

F,-eE-

at

(2.5a)

where F =-e-

vxB

- m(v . V)v,

C

is usually termed as the pondermotive force. It may be readily seen that this force arises on account of the interaction of drift velocity of electrons (which is due to the electric vector of the wave) with the magnetic vector B of the wave and from the gradient of the drift velocity. The magnitude of Fp and the last term in eq. (2.5a) is very small as compared to the force (- eE) on account of the electric field. Thus to the zeroeth order of approximation the steady state solution of eq. (2.5) can be written as eE mio

v=------.

(2.7)

The magnetic induction vector B of the wave may be evaluated using eq. (2.1~)as

B =

-

C

~ ( V X E ) . 10

178

SELF F O C U S I N G OF LASER B E A M S

[v,

02

Using eqs. (2.7) and (2.8) the time independent part of Fp(which is relevant to self focusing) can be written as

F

=-P

e2

4m02 V(EE*).

(2.9)

The same expression for the pondermotive force has been derived by many workers (BOOT,SELFand HARVIE [1958]; HORA,PFIRSCHand SCHULTER [1967]; HORA[1969, 19723) by different approaches. The effect of Fp in the evaluation of field dependence of effective dielectric constant is considered in § 2.5. 2.3. HEATING OF CARRIERS BY A GAUSSIAN EM BEAM IN SLIGHTLY A N D FULLY IONIZED GASES

In a collisional plasma the main source of field dependence of effective dielectric constant is the nonuniform redistribution of carriers on account of inhomogeneous heating of the carriers because of transverse variation of electric field along the wavefront (DYSTHE[1968]; LITVAK[1968, 19693; RASHAD[1966]; SODHA,KHANNAand TRIPATHI[1973]; PRASADand TRIPATHI [1974]; SODHA,GHATAK and TRIPATHI [1974]). In steady state this mechanism is seen to be much more effective than the pondermotive force in characterizing the field dependence of the effective dielectric constant (see Q 2.5). When an electromagnetic beam propagates through a plasma, the electrons acquire momentum and energy from the wave. In equilibrium (i.e. in the absence of an electric field), the temperature of electrons is the same as that of the heavy particles (ions and molecules) so that the net energy exchange between electrons and heavy particles is zero. When an electric field is applied, energy is gained by the electrons (from the field), which in steady state attain a temperature higher than the equilibrium value, such that the power gained by the electrons from the field is equal to that lost in collisions. Such electrons are called hot electrons. In strongly ionized plasmas the hot electrons lose much more energy by transverse thermal conduction along the wavefront than in collisions with heavy particles. An inhomogeneous temperature distribution along the wavefront occurs on account of a nonuniform transverse distribution of intensity EE*. Before deriving expressions for the electron temperature it is instructive to recapitulate the conditions for the validity of the effective electron temperature concept.

v, § 21

PHENOMENOLOGICAL THEORY

179

The concept of local temperature is valid only when the plasma is in quasithermal equilibrium i.e. the variation of temperature over a mean free path A,,, is small. Taking the characteristic length of temperature variation as a (= r o f the dimension of the beam) this condition reduces to

1, < a. In addition to this, the dimension of the beam should be greater than the Debye length to justify a macroscopic description ef the plasma. The latter condition is almost always satisfied. When the above conditions are satisfied the drift velocity u and temperature T, of electrons are governed by du mdt

=

-eE-

1 -V(NekoT,)-mvu, Ne

(2.10)

and

where v is the number of momentum randomizing collisions per second. The last term in eq. (2.10) is on account of the loss of momentum in collisions. In eq. (2.1 l), the first term on the right-hand side is the energy gained by a carrier from the field per second, the second term is on account of energy loss to scatterers in collisions and the third term is on account of energy loss in thermal conduction. The pondermotive force has been omitted on account of its small magnitude in comparison to other terms in eq. (2.10). The rate of enqrgy loss on account of elastic collisions is given by (GINZBURG [19701) (2.12a)

where To is the equilibrium temperature of the plasma and 6 = 2m/M is the mean of fraction of excess energy lost per collision with the scatterers of mass M ; inelastic collisions in which the energy transfer per collision is comparable to (2m/M) times the excess energy, can be taken into account by using a value of 6 higher than 2m/M, which can be determined from experimental data on variation of drift mobility with electric field. The conduction loss term can be calculated by considering a cylinder (coaxial with the beam) of radius r and length unity. The amount of energy

180

[v,

S E L F FO CUSI NG O F L A S E R B E A M S

D2

flowing radially outward from the curved surface per second is Q(r) = -2nrk,aT,/ar,

where k, is the electronic thermal conductivity. The amount of energy flowing out per second from the coaxial cylinder of radius r dr is

+

Q(r+dr)

=

-2nrk,

e

Hence the energy lost, per unit volume per unit time on account of thermal conduction is given by Q(r + dr) - Q(r) 2x7-dr

Dividing this by the electron density N , the conduction term is given by (2.12b) On taking the characteristic length of temperature variation as a = r o f (i.e. a / & a- ’) the ratio of the two terms (given by eqs. (2.12a) and (2.12b) respectively) is given by N

R

=

”(

width of the beam M mean free path of electroG)

(2.12c) ’

In the D region of ionosphere and other weakly ionized laboratory plasmas R B 1 for wave phenomena of interest ; therefore the excess electron energy is dissipated mainly in collisions with the neutral particles. In the F region of ionosphere and strongly ionized hot plasmas the reverse is true. In the present review we have considered the two cases of R > 1 (slightly ionized gases) and R 1 (highly ionized gases) separately. For intermediate values of R, an analysis has been presented in 9 2.6.2. In both cases, using eq. (2.10) (to the zeroeth order) the oscillatory component of u can be written as u = -

eE rn(v + io)

Using eq. (2.13) the steady state solution of eq. (2.11) for R

T,- To To

=

aEE*

(2.13) 1 is

(2.14)

v, 5 21

PHENOMENOLOGICAL THEORY

181

where a=

e2M 6k0 Too2m2

(2.14a)

and the approximation v2 ez 0 2 ,valid for laser beams, has been used. The latter approximation is essential to observe the phenomenon of self focusing as discussed in 5 5. In the other limit of R ez 1, the solution of eq. (2.11) comes out to be (SODHA, KHANNA and TRIPATHI [19731) (2.15) where

Po

= N,e2v/(mo2),

and C1 is the constant of integration; in order to study the self focusing of the beam it is not necessary to evaluate the constant of integration, hence we do not make any attempt to this end. In writing eq. (2.15) we have used the following expression for thermal conductivity at constant pressure (SHKAROFSKY, JOHNSTON and BACHYNSKJ [19661) k,

=

5Nki To 0.65m(vei) *

The average electron-ion collision frequency can be evaluated as

where

It may also be mentioned here that v = (vei)g, where go is a quantity of the order of unity as defined by SHKAROFSKY, JOHNSTON and BACHYNSKI [19661. 2.4. HEATING OF CARRIERS IN PARABOLIC A N D NONPARABOLIC SEMICONDUCTORS

The interaction of electrical and electromagnetic fields with semiconductors closely resembles that with gaseous plasmas. The behaviour of

182

SELF F O C U S I N G OF LASER BEAMS

[v. § 2

free electrons and holes in a semiconductor is similar to that of free electrons in gaseous plasmas. The Debye length and plasma frequency retain their significance as space charge screening length and natural frequency of oscillation; the macroscopic equations characterizing momentum and energy balance of electrons are also valid and useful. However, on account of the background of the lattice, carrier parameters and their relationships have to be carefully characterized. For example, the energy d of the carrier is not a parabolic function of the momentum in many semiconductors and hence the classical concept of carrier mass loses significance. In the approximation of free electron theory the electrons and holes can be considered as mobile carriers of effective mass m* defined by

In many semiconductors (e.g. elemental semiconductors as Ge, Si) the energy & is a parabolic function of the wave vector k = p / h to a good approximation, i.e. & a kZ. Thus the energy bands are parabolic and the surfaces of constant energy are spherical; in this approximation m* is scalar and independent of energy. In another class of important semiconductors viz. 111-V compounds (e.g. InSb), the surfaces of constant energy are spherical but the bands are nonparabolic; in the present review we have specifically discussed InSb, a typical representative of this class. The &-k relation for electrons in the conduction band of InSb is (KANE C19571) (2.16)

-

where m is the free space electron mass, E, ( 0.17 eV) is the energy band [1957]) is a matrix gap and p o ( " 8.5 x lo-* eV cm, cf. EHRENREICH element defined by KANE[1957]. Using eq. (2.16) the effective mass of an electron can be obtained as (2.17) where

Eq. (2.17) has been written in the approximation of m* << m. The energy dependence of carrier effective mass is an additional source of nonlinearity in nonparabolic semiconductors.

v, § 21

PHENOMENOLOGICAL THEORY

183

The holes in InSb, can be treated through a parabolic relation between I and k (CONWLLC1967-J) and the effective mass of holes is thus independent of energy. A perfect lattice does not scatter the carriers (as they can be treated as plane waves propagating through a three dimensional grating structure of infinite extent which scatters them only in the forward direction). However, owing to thermal vibrations the distortions in the lattice cause the scattering of carriers. This phenomenon depends on the mode (e.g. acoustical or optical), frequency and energy of lattice vibrations. A convenient way to study the scattering of carriers by lattice vibrations is to replace the scattering modes by equivalent number of phonons; a vibration of frequency a, having energy (N,+$))hw, is equivalent to N , phonons, where N, is an integer and q is the wave vector of phonons. For a lattice in thermal equilibrium the distribution of N, is given by the Bose Einstein law viz. N,

=

1 exp (ho,/ko To)- 1 '

For the purpose of scattering of free carriers we can treat the acoustical phonons as billiard ball type scatterers of effective mass M (SHOCKLEY [195l]), given by M = ko To/$

(2.18)

where so is the velocity of sound in the crystal. Thus the phenomenon of carrier scattering by acoustical phonons can be accounted for by introducing an electron temperature dependent effective collision frequency v and the phonon effective mass M . In germanium below the room temperature, the momentum and energy losses are mainly due to acoustical phonons. Then the results of 0 2.2 are applicable (when v2 K 0 2 viz. ) 1)=

-

eE(v - io) m*02 '

(2.19) (2.20)

where a, = e2M/(6m*202koTo).

(2.20a)

Eqs. (2.19) and (2.20) are valid for holes also when u, T,, -e, m*,and a, are replaced by u,,, Th, e, m: and ah respectively. In evaluating the carrier

184

S E L F FOCUSING OF L A S E R BEAMS

[v. § 2

temperatures we have neglected the energy loss due to thermal conduction because it is very small, compared to that for collisions. In InSb the acoustical phonons do not play an important role in the scattering process in the temperature range 40 OK < To < 300 OK. In this region the ionized impurities present in the lattice are the main source of momentum loss and an effective collision frequency for momentum transfer can be defined. For the loss of excess carrier energy, the main contribution comes from the polar optical mode; the ionized impurity scattering does not contribute to energy loss because the effective mass of an impurity ion is the same as the mass of the whole lattice (i.e. M - , a). For a Maxwellian distribution function of carriers, the rate of average [19671) energy loss per electron can be shown to be (CONWELL exp ( X , - X , ) - 1 exp W 0 ) - 1

where (a) ik, T, (in the approximation of drift energy being much smaller than the random energy, which is always true in the steady state), 0, is the Debye temperature, X , = Ro,/ko T o , X , = hw,/k, T,, KO is the Bessel function of first kind and h a , is the energy of the polar optical phonon. Epoin eq. (2.21) is the field of scattering potential and is given by (CONWLL[19671) m*e2ho, IelE,

=

h2

($

-7) ;

(2.22)

where xo and xrn are the dielectric constant at zero and infinite frequencies. In the phenomenological treatment of this section, the effective mass of electrons m* in InSb can be replaced by its average value over the Maxwellian distribution function (SODHA,TEWARI, TRIPATHI and KAMAL[19721) viz. (2.23)

In the perturbation approximation i.e. when ( T e - T,)/To < 1, eq. (2.21) can be written as =

where

-

$k,( T,- To)/zs

(2.24)

v, 0 21

185

PHENOMENOLOGICAL THEORY

hw, 2 Ko(+Xo) (2.24a) koTo 3k,T0’ exp (XO)- 1 ~

~

z6 is known as the energy relaxation time. Using eq. (2.24) in eq. (2.1 1) we obtain the following expression for the rise in electron temperature

K-

To

=

(2.25)

u,EE*

TO

where u, =

e2ng 3m, w2k, To ’

m,

=

3

+ €$ko

To

L

The drift velocity is still given by eq. (2.19). Eq. (2.25) is applicable to holes also when the corresponding parameters are used. 2.5. REDISTRIBUTION OF CARRIERS A N D EXPRESSIONS FOR FIELD DEPENDENT DIELECTRIC CONSTANT

The expressions for the rise in electron/hole temperature derived in show a nonuniform distribution of temperature along the wavefront of a Gaussian beam; the temperature is maximum on the axis of the beam and has a profile similar to that of EE*. This causes an imbalance of partial pressure and the carriers diffuse from the high pressure to low pressure regions. This effect also occurs on account of the pondermotive force. The diffusion of carriers is accompanied by the generation of the space charge field. This field drags ions (holes in the case of semiconductors) along with electrons and the ambipolar diffusion of this assembly takes place. Before deriving the expressions for the electronic concentration in the various cases mentioned above, let us investigate the role of the space charge field in detail. Following SODHA,GHATAK and TRIPATHI [1974], we consider a semiconductor having both (+ue and - ve) types of charge carriers with No, and Noh as their equilibrium concentrations, In equilibrium, the semiconductor is quasineutral, i.e. over a dimension of Debye length A,, the net charge density is zero. In the presence of an intense inhomogeneous electromagnetic beam the carriers are heated viz.

$5 2.2-2.3

T, = To(l+ a, E E * ) and

(2.26) = To(l+uhEE*)

186

[v,

S E L F FOCUSING OF LASER BEAMS

D2

and due to the nonuniform distribution of T, and T,, they are redistributed. When the changes in the local densities of + ue and - ue charges are equal (i.e. ANe N ANh) the quasineutrality of the system is maintained. If ANe # A N , a strong space charge field E , is generated. From eq. (2.10), the steady state is described by -

1 -

V ( N e k oT J - e E ,

=

0

(2.27a)

Ne

and (2.27b) The space charge field E, is related to the charge density by eq. (2.la), E ~ VE,. = 4np = 4ne(AN,-AN,),

(2.28)

where A N , = N h - Noh and ANe = N , - N o , are the changes in the local concentrations of holes and electrons respectively due to the redistribution ; N o , and No, denote equilibrium concentration of electrons and holes respectively. To solve the coupled eqs. (2.27H2.28) we consider a two dimensional problem so as to keep the mathematics simple and tractable. We take the beam axis to be along the z-axis and the intensity distribution in the xdirection as Gaussian, EE*

=

Eg exp (- x2/a2).

The variations along they-direction are ignored (i.e. d/dy = 0). The intensity varies in the z-direction also due to diffraction, self focusing and absorption effects, but we assume this variation to be much smaller than that in the x-direction. Then the rise in electron and hole temperatures (AT, and AT,) can be determined from eq. (2.26). For the changes in electron and hole concentrations (AN, and AN,) when AN, cgc No, and AN, Q: Noh, eqs. (2.27a) and (2.27b) simplify to give (2.29a) and (2.29b) Subtracting eq. (2.29b) from (2.29a) and differentiating the resulting

v>§ 21

I87

PHENOMENOLOGICAL THEORY

equation once with respect to x we obtain

where Y

=

AN,-AN,

and AD

=

[&Lk0 T O / { 4 n e 2 ( N O e

+

is the Debye length. In writing eq. (2.30) we have made use of eqs. (2.26). As the beam is symmetrical about x = 0, AN, and AN, should also be symmetrical and hence a maximum/minimum in AN,, AN, should occur at x = 0 i.e., (2.31a) Also at points x z+-u the values of ANe and A N , must be zero i.e., Y=O

as x + + _ o o .

(2.31b)

Using the boundary conditions (2.31a) and (2.31b) we obtain the following solution of eq. (2.30)

x exp ( - x2/a2)dx-

iJ0

- exp (- x/AD)

+ r e x p (-x/AD)(l-

s,I

exp (- x/AD)(l-x2/a2)exp f - x'/a2))dx}

exp (x/A,)(l- xZ/a2)exp (- x2/a2)dx xz/az)exp (- xz/uz)dx}]

,

(2.32)

To evaluate AN, and AN, explicitly we need one more equation. This can be obtained by adding eqs. (2.29a) and (2.29b) as

-+---

-

AT,+AT,

+C (2.33) TO where C is a constant independent of x ; at x > a, AN,, AN,, A T e , AT, are

188

[v, 4 2

S E L F FOCUSING O F LASER B E A M S

zero ; hence C = 0. Usingeq. (2.32), the expressions for AN, and ANh can be written as AN,

=

- (aefcrh)NOeNOh

E 2, exp ( - x 2 / a2 )+ Y ~-

(2.34a)

(NOefNOh)

and ANh= -

(aefah)NOe “Oe

E i exp (- x2/a2)- Y

+

+

(NOe

.(2.34b)

A rigorous analytical examination of eqs. (2.32), (2.34a) and (2.34b) for all values of x is very difficult. However, in the approximation of xz ez a2, Y

=

2n;

-

-Z-(~e~oe-ah~oh)~;

(2.35)

U

It can be easily concluded from eqs. (2.34a) and (2.34b) that the last term is negligibly small as compared to the first term except for the case when either No, N 0 or NohN 0; the Debye length is usually cm and hence I;/a2 is very small (this is necessary also for the validity of a macroscopic theory). Thus ANe N ANh except for the extreme case of No, or Nohbeing zero. Physically this can be stated as follows. The electrons and holes on being heated by the electric field of the wave are redistributed. Due to Coulomb attraction between electrons and holes, they move almost collectively i.e. they are dragged together. In other words, the space charge field is strong enough to keep the electrons and holes (ions in the case of plasmas) together but not strong enough to stop the diffusion of carriers. It is instructive to mention that in a semiconductor having only single type of carriers (either electrons or holes only) ANe, N 0 i.e. the redistribution of carriers is negligible. The nonlinearity appearing through the redistribution of carriers is observed only when both types of carriers are present. In n-InSb the source of nonlinearity is through the energy dependent mass (i.e. through the nonparabolicity of conduction band). In n or p type parabolic semiconductors the nonlinearity appearing through the temperature dependence of collision frequency is not enough to cause appreciable self focusing (SODHA,GHATAK and TRIPATHI [1974]). Now we evaluate the expressions for the change in carrier concentration in different cases. In the case of plasmas the time independent part of the momentum balance

-

”, § 21

189

PHENOMENOLOGICAL THEORY

equation for ions (in the steady state) gives 1

0 = eE, - - V(Nik , To).

(2.36)

Ni

Solving eqs. (2.5) and (2.36) for the case when the pondermotive force is dominant (and taking ANe N ANi as an extrapolation of the result derived above) we get ANe = - No(l -exp { -3(m/M)aEE*})

(2.37)

which is an exponentially saturating function of EE*. No in eq. (2.37) is the equilibrium concentration of electrons. In the case of slightly ionized collisional plasmas, the expression for ANe as obtained from eqs. (2.10) and (2.35) is AN, ANe =

= -

~

NoaEE* collisional loss predominant ( R > 1) 2 + aEE* ’

” r2,

-

16keT0 f 2

conduction loss predominant ( R =K 1).

(2.38a) (2.38b)

Eq. (2.38a) is written without any perturbation approximation. The expressions for the changes in electron and hole concentrations in a semiconductor can be rewritten from eqs. (2.34a) and (2.34b) as (2.39) Eq. (2.39) is valid for all sorts of intensity distributions as long as the dimension of the beam is smaller than the size of the sample. Now we may proceed to evaluate the current density (varying as exp(iwt)) in the plasma due to the electromagnetic beam. The current density in a plasma is related to electron drift velocity by the relation

J

=

(2.4)

-Neeu.

Using eq. (2.13) for u we obtain in the approximation of vz

(<

w2

N, e2 (v - iw)E = o(EE*, o ) E . mw2

J=--

(2.41)

Then the effective dielectric constant of the plasma can be written as &

==

1-

4zN, e2 mw3

___ (w

+iv).

(2.42)

190

S E L F FOCUSING OF LASER B E A M S

rv,

Q2

Using eqs. (2.37H2.38) we can explicitly write the effective nonlinear dielectric constant of a plasma in various cases. 2.5.1. Collisionless plasma (pondermotive mechanism) E = E~ =

t0+@(EE*)

1- w i / 0 2 ,

(2.43)

up= ( 4 n N O e 2 / m ) )

@ = %(l--exp

j-

3 .aEE*}) m

w2

(2.44a) (2.44b)

Eq. (2.44b) has been discussed by using a different approach by KAW, SCHMIDT and WILCOX[1973], and HORA[1969b, 19721. Eqs. (2.43H2.44b) are valid even for large value of EE*.In the limit of ( m / M )a EE* < 1, @ has a quadratic dependence on EE*,i.e. (2.44~) 2.5.2. Strongly ionized plasma ( R K I, thermal conduction predominant) E = E~

+ @(EE*)- iEi

(2.45) (2.46a) (2.46b)

and E~ is given by eq. (2.44a). The expression (2.46b) for @ is valid only in the perturbation approximation and was first derived by SODHA,&ANNA and TRIPATHI [19731. 2.5.3. Slightly ionized plasma ( R >> 1, collisional loss predominant) E =

E~+@(EE*)-~E~

(2.47)

uEE* w 2 2+uEE*

(2.47a)

w2

@(EE*)= 2

and z0 and ei are given by eqs. (2.44a) and (2.46a) respectively. In the approximation of uEE* <1, eq. (2.47) simplifies to w2 a

@(EE*)= 2

-

w2 2

EE*

(2.47b)

", P 21

PHENOMENOLOGICAL THEORY

191

A comparison of eqs. (2.44~)and (2.47b) reveals that (in the steady state) the nonlinear part of the dielectric constant due to the heating and redistribution of carriers is much larger (four orders of magnitude higher) than that on account of pondermotive force. Hence, over a time scale of energy relaxation time the pondermotive nonlinearity is unimportant. The dielectric constant of a semiconductor having electrons and holes can be written on similar lines as & = &

L

- -

4nN, e2 - 471Nhe2 m*(v + io) m:(vh io)

+

(2.48)

We simplify eq. (2.48) for the following specific cases of Ge and JnSb when vz K 02. 2.5.4. Germanium

With only single type of carriers the carriers are not appreciably redistributed and hence N , (or Nh) remains independent of the electric field of the beam. The nonlinearity can arise through the energy dependence of v only, which, as shown in 9 5 is unimportant in self focusing. Therefore the only case of interest is the one when both types of carriers are present. Using eq. (2.39) the effective dielectric constant can be written as E

= E~+@(EE*)-~E~

(2.49)

where (2.50a)

)*:

4nNo,e2 UP=(

and &.=P

(2.50~)

The nonlinear part @ of the dielectric constant tends to zero when either No, 0 or Noh-+ 0. A plot of this variation is given in Fig. 2.1 (SODHA, GHATAK and TRIPATHI [19741). This shows that for maximum nonlinearity to occur the concentration of electrons and holes should be equal. It may, --+

192

S E L F FOCUSING OF L A S E R B E A M S

& 0.0 0 . 4 0.8 I 0.8

0.4

0.0

Fig. 2.1. Variation of nonlinear part of dielectric constant of germanium with relative concentration of electrons and holes; E, = e? for No,= Noh = N o .

however, be again emphasized that eq. (2.50b) is valid only in the limit uEE* ez 1 since the expressions for TJT used in the derivation are based on this assumption. 2.5.5. n-type indium antimonide In n-InSb the nonlinearity, important in self focusing appears through the energy dependent electron mass. The redistribution of carriers is unimportant in this case, as shown earlier. Using eqs. (2.23), (2.26) and (2.48) the expression for the effective dielectric constant comes out to be (SODHA, TEWARI, TRIPATHI and KAMAL[1972]) E

=

€0

maSS

EE" - isi

(2.51)

where (2.52a) 3ko To

(2.52b)

and (2.52~)

"3

§ 21

I93

PHENOMENOLOGICAL T H E O R Y

These expressions are valid only in the perturbation approximation when a,EE* < 1. 2.5.6. Indium antimonide (both types of carriers) In this case the redistribution of carriers is also important. Using eqs. (2.23), (2.26), (2.39) and (2.48) the dielectric constant can be written as (SODHA,KHANNA and TRIPATHI [1973b]) EO+E2EE*-ki

E =

(2.53)

where, (2.54a) (2.54b) &2 = ' 2 mass +'2

(2.54~)

redistribution

and o2 &2 redistribution =

0.20-

['e

202 I

EE*

~

mo I

+

a,EE*

~

Noem:

1

-

-

0.15 -

-

.-s C

6 b

g

-

-

z o . 1 0-

-

0.05-

------------1

0

0-3

I

I

06

0.9

1.2

1

.

(2.54d)

194

S E L F FO CUSI NG OF L A S E R B E A M S

[v. 5 2

A comparison of eqs. (2.54d) and (2.52b) reveals that in a compensated ~A sample of InSb (Nee = Noh),E~~~~~~~~~~~~~~~ is much greater than E plot of these two c2 as a function of hole concentration has been given in Fig. 2.2. 2.6. NONLINEARITY IN THE DIELECTRIC CONSTANT OF A MAGNETOPLASMA

The application of a static magnetic field to a plasma makes it anisotropic, i.e. the current density and the electric vector, in general, have different directions and are related by

J=Q.E

(2.55)

where Q is a tensor of 2nd rank. As mentioned in 9 2.1 relation (2.55) is true only for electric fields having time dependence of the form exp (iot). The conductivity tensor consists of a set of nine coefficients. A generalization of eq. (2.4) can be made to define an effective dielectric tensor as E = E~

I - 4niaJo

(2.56)

where I is the unity tensor of rank two. The formal expression for the effective dielectrictensor can be obtained by solving the momentum balance equation for the time varying component [exp (imt)] of drift velocity. The equation is

dv dt

m-

=

-eE-

ev x B,

____ - mvv,

(2.57)

C

where B , is the static magnetic field. The second term on the right-hand side denotes the force on an electron due to static magnetic field. To solve eq. (2.57) we specify the direction of B , as the z axis. To obtain the solution of eq. (2.57) for u, and v,,, we multiply the y component of eq. (2.57) by i and add to the x component of eq. (2.57) then

+

21,+lU

= -

+

e(E, iE,)

m(v + i(o-0,)) ’

(2.58a)

where o,= eB,/rnc is the electron cyclotron frequency. On repeating the same operation using - i instead of + i, we get U,-lUy

= -

e(E, - iE,) m( v i(o w,)) .

+ +

(2.58b)

~

~

">

§ 21

195

PHENOMENOLOGICAL THEORY

For the z component, eq. (2.57) gives u, =

- eE, m( v + io)

(2.58~)

'

Eqs. (2.58a) and (2.58b) can be solved to evaluate ux and 0, separately. The current density can be obtained from the following relation J

-Nev.

=

It is useful to evaluate J, + iJ, and J, - iJy; thus J,+iJ,

=

a+(E,+iE,)

(2.59a)

J,-iJ,

=

o-(E,+iE,)

(2.59b)

QZZE,

(2.59~)

and J, =

where =

Ne2 m(v -ti(o - 0,))

(2.60a)

c- =

Ne2 m(v i(o w,))

(2.60b)

Q+

ozz

+ +

=

Ne2 m(v io) .

(2.60~)

+

Using eqs. (2.60) the components of conductivity tensor can be obtained in terms of CT+ and Q-. The resulting expression for the dielectric tensor has the following components Exx = E,,

E,,

Ex,

=

E,,

= -E,,

= Eyz = E,,

=

= =

(2.61a)

%(E++E-)

$(E+

0,

(2.61b)

-E-)

E,,

=

1-

47ci o

- czz

(2.6 1c)

and &*

=

4ni l--Qa,.

w

(2.61d)

In the presence of an intense electromagnetic beam the electronic concentration becomes a function of EE* and hence the dielectric tensor becomes nonlinear.

196

cv, 8 2

SELF F O C U S I N G OF L A S E R BEAMS,

Before proceeding further to evaluate the field dependent dielectric tensor explicitly, let us discuss in some detail the salient aspects of electromagnetic wave propagation in a magnetoplasma in the linear regime. We consider the propagation of a plane electromagneticwave in a magnetoplasma along the direction of the static magnetic field (i.e., in the z direction). On taking d/dx = 0, a/ay = 0, the Maxwell’s equations (2.1~)and (2.ld) combine to give the following equations for the x and y components of the electric vector, (2.62a) and a2EY ~

az2

io2

+ ~ ( - E x y E x + E , , E y ) = 0.

(2.62b)

For a detailed derivation of eqs. (2.62) reference is made to Ginzburg’s classic treatise (GINZBURG [1970]). A more general form of the above equations is derived and discussed in 95.These equations are coupledand as such no propagation vector can be defined which describe the independent propagation of Ex and E,. However, to investigate the independent modes of propagation we multiply eq. (2.62b) by u’ and add it to (2.62a). The resulting equation can be put in the following form

If u‘ is chosen such that u’ written as a2 az2

+

( E x a’E,)

-

= -l/d

+ o2( -

i.e.

c1‘

=

+ i then eq. (2.63) can be

+

E -~C L~ ’ E , ~ ) ( E d ~E , ) =

C2

0.

(2.64)

This equation on remembering the time variation of Ex and Ey represents the propagation of a function (Ex+ a’E,). On putting explicitly the values of c1’ in eq. (2.64) we obtain

a2 (E,+iE,)+k~(E,+iE,)

--

az2

=

0

(2.65a)

and (2.65b)

v, § 21

PHENOMENOLOGICAL THEORY

I97

where (2.65~) and

k-

=

w -

(ExX+iExy)*.

(2.65d)

C

Eqs. (2.65a) and (2.65b) are independent of each other. The former represents the propagation of the function A = Ex+iEyand the latter of the function A , = Ex-iEy with k , and k - as their respective wave vectors. The functions A, and A, always propagate with their specific wave vectors but Ey and Ex independently do not. In other words if the wave has to propagate with only one propagation vector k, then A, must vanish i.e. Ex = iEy. Consequently, a wave for which Ex = iEy= Ey exp (in/2) (i.e. the electric vector is right handed circularly polarized when viewed along B,) propagates with a propagation vector k,. Similarly the wave with Ex = -iEy (circularly polarized in the left handed sense) propagates with wave vector k- . These two field configurations with their respective wave vectors are known as the two independent modes of propagation, viz. the extraordinary and ordinary modes respectively. Now we analyse the field dependent dielectric tensor of a magnetoplasma for different cases of interest. The treatment can be easily generalized to the case of semiconductors.

2.6.1. Nonlinear dielectric constant of a collisionless magnetoplasma :pondermotive mechanism The pondermotive force on electrons in a plasma is given by eq. (2.6), i.e.

F

=--

eu x B

- ( u . V)u.

(2.66)

C

Following SODHA,MITTAL,VIRMANI and TRIPATHI [19741 we consider the presence of only one mode at a time so as to obtain an explicit expression for F,. For the extra-ordinary mode (propagating along the z axis) the magnetic induction vector B can be obtained from eq. (2. lc) as

B = irVxE. w

(2.67a)

198

S E L F F O C U S I N G OF L A S E R B E A M S

[v,

si 2

From eqs. (2.58a) and (2.58b), taking v K o,we obtain u,+lvy

- e(E,

=

+iE,)

mi(o - 0,)

(2.67b)

and o,-ivy

=

0.

(2.67~)

Using eqs. (2.67a) and (2.67b) the expression for the (time independent part of) pondermotive force is

(2.68) It is instructive to note from eq. (2.68) that for an em beam having a Gaussian distribution of intensity, the pondermotive force has an outward radial direction when o,/2w< 1 and an inward radial direction when oc/2w> 1 ; i.e. the plasma is rarefied in the former case and compressed in the latter. In the steady state the dc drift of electrons is zero and hence 1 -V(N,k,T,)

-eE,+F,-

= 0.

(2.69)

Ne

The equation for ions is the same as eq. (2.36). Using the procedure outlined in section 2.4, these equations can be solved for the local concentration of electrons. The result is N, = N,exp

[

3m 8M

(1 -0,/2w)

- __ c! -____

(1 -w,/o)2

(2.70)

The effective dielectric constant for the extraordinary mode can now be written as 8,

= E,+

+@+(AIAT),

(2.71a)

where (2.71b) (2.71~) and o, = (4nNOe2/m)+.

v, 0 21

199

PHENOMENOLOGICAL THE ORY

In this case when the nonlinearity is brought about by the extraordinary mode, E - and E,, also become functions of A , A : but we would not consider them. It is seen from eq. (2.17~)that @+ is always positive except when 2 > oc/w > 1. It is sensitive to cyclotron resonance also. By a similar procedure the nonlinear dielectric constant due to the ordinary mode can be evaluated; E-

=

E~-+@-(A,A~),

(2.72a)

where (2.72b) and (2.72~)

2.6.2. Nonlinear dielectric constant of a collisional magnetoplasma: R

ZB-

1

Following LITVAK [1969], SODHA,MITTAL, VIRMANI and TRIPATHI [19743 and SODHA,KHANNAand TRIPATHI [1974] the drift velocity of electrons (oscillatory component) is still given by eqs. (2.58a) and (2.58b). Further, E, may be assumed to be much smaller than E x , E, . Then we can solve the energy balance equation to evaluate the rise in electron temperature. When. the main mechanism of energy loss is due to elastic collisions between electrons and heavy particles (e.g., in a slightly ionized plasma) the energy balance equation results in the following expression for T, (when v 2 < (0-0,)2) (2.73) c( is given by eq. (2.14a). It is interesting to note from eq. (2.73) that the heating of electrons on account of the two modes is additive; the extraordinary mode is more effective in the heating of electrons. In the case when electron thermal conduction is the main source of energy dissipation, the expression for the rate of energy loss (in eq. (2.11)) has to be modified due to the presence of the static magnetic field (SODHA,KHANNA and TRIPATHI [19741). Following SHKAROSFSKY, JOHNSTONand BACHYNSKI [1966], the transverse components of the thermal current density Q can be given in terms of V T, as

(2.74a)

200

S E L F F O C U S I N G O F LASER BEAMS

cv, 9 2

and

where (2.75) The functions gk' and hkr are graphically displayed by SHKAROFSKY, JOHNSTON and BACHYNSKI [19661. The gradient of electron temperature along the z axis has been neglected. The rate of energy dissipated per second per unit volume in thermal conduction is = V .Q . Hence, (2 3k0 T)J e conduction loss - - V Ne

. Q , - -K;,

,

In writing the last step we have treated K; as the same as in the absence of the electric field (i.e. corresponding to To) which is valid as long as (Te- To)/ToK 1. It is again important to assess the relative magnitudes of the two energy dissipatingmechanisms.On taking the characteristic length of temperature variation as a( = r& the ratio of the collisional loss to the conduction loss is m width of the beam R z -M[ mean free path

1 7 v 2 +I$

(2.77)

which is different from the ratio wheno, = 0 (cf. eq. (2.12~)).Thus a plasma in which R K 1 at w, = 0 may turn out to be the one for which R >> 1 at w, >> v, i.e., at high magnetic fields the main mechanism of energy loss is due to collisions. Then the only interesting case worth discussing in a magnetoplasma in which thermal conduction plays an important role corresponds to the ca'se when v and ocare of the same order. As already pointed out o2B v 2 and hence o2B 0," in this particular case. Then the expressions (2.6%) and (2.65d) are almost equal and the propagation of extraordinary and ordinary modes need not be considered separately. Considering only the extraordinary mode, the solution of the energy balance equation for T, in the paraxial ray approximation comes out to be

where Po and C , are referred to 0 2.3. In writing the above equation we have employed the perturbation approximation (viz. T e - To K To).It may be

v, 9: 21

20 1

PHENOMENOLOGICAL THE ORY

seen from eq. (2.78) that the electron temperature is an increasing function of the magnetic field, because the electron thermal conductivity decreases with w, and hence the energy dissipation rate is slowed down. To determine the local electronic concentration, we make use of the condition, udc = 0; from eq. (2.10),

-eE,-V(N,k,T,)

=

(2.79)

0.

For ions, eq. (2.36) is still valid. A simultaneous solution of these equations results in the following expressions for electronic concentration :

Collisional loss (both modes present) N,

=

AIAT

N o /[l+&wZ (( w - 0,)Z

)].

(2.80)

-+.‘]

(2.81)

+ (w +w

y

Conduction loss (only one mode present) BoA,AT(z = 0, r = O)r2

16K;

Tof’

where perturbation approximation has been employed. The effective dielectric constants for the two modes in the two cases are &*

= so* + @ *-isi+

(2.82)

where E,+ are given by eqs. (2.71b) and (2.72b), (2.83a) and (2.83b) The explicit expressions for @+ can be written straight away by using eqs. (2.80) and (2.81) in the two cases discussed above. The above mentioned treatment of field dependence of effective dielectric constant may now be generalized to include the two mechanisms of energy loss (viz. the collisional and the conduction) simultaneously in the analysis. In this case the energy balance equation, on using eqs. (2.12a) and (2.76) can be written as

z)

I d dT, 2nl (rK;l +gNoko(T,- To)- (vei)gk, r dr M ~~

e 2

= - -Re

[ u . E*]

(2.84)

202

SELF F O C U S I N G OF L A S E R B E A M S

[v,

02

where Re denotes the real part of the quantity. Introducing a new variable T = (T,- To)/Toand using eqs. (2.58a) and (2.58b) we obtain d2T' 1 d T + -r dr + G , T dr2

__

~

=

G,

AlA?(vei)gu (0 - Oc)zhtr

+

A2'Z(vei)gu] (0 mc)2hkf

+

(2.85)

where

and G -

-

N o e2 4K;, Tom

and the condition ( v , ~ ) ~ (w-o,)~ is used. For solving eq. (2.85) we would consider the presence of only one mode at a time. Considering the presence of extraordinary mode only (i.e. A, = 0), the solution of eq. (2.85) can be obtained by the Green's function technique. When the source term (viz. the term on the right-hand side of eq. (2.85)) is set equal to zero, the resulting homogeneous equation is the Bessel equation of zeroeth order, the two solutions of which are Jo(Gtr) and Yo(Gfr).To solve the inhomogeneous equation we need the Wronskian W(Gir)of the two independent solutions of the homogeneous equation; W(Gfr) is given by (MORSEand FESHBACH C19531) W ( G t r )= J,(Gtr)Y:(Gtr)

-

Yo(Gfr)Jb(Gtr)

(2.86)

where Yo and Jk are the derivatives of Yo and Jo with respect to their arguments. Then the general solution of eq. (2.85) comes out to be

T'

=

F(r) = Jo(Gfr)

Yo(Gi r')S'(r')

dr'

where S'(r') is the source term given by

Using eq. (2.87) the expression for the local concentration of electrons can be obtained in the paraxial ray approximation as

KINETIC THEORY

203

N, - N o = - No( T,- T0)/2T0

(2.88) The perturbation approximation is implied in the above equation. The effective dielectric constant of the plasma for the extraordinary mode is again given by eq. (2.82) where @+ can be obtained as (2.89)

9

3. Kinetic Theory of Field Dependent Dielectric Constant

The phenomenological treatment of 2 is applicable when either the collision frequency of carriers is velocity independent or when only order of magnitude estimates are required in the perturbation approximation (i.e. T, - To e To).Beyond these restrictions, a rigorous derivation of the field dependent effective dielectric constant, based on the solution of Boltzmann’s equation for the velocity distribution function of carriers, is necessary. The present section is devoted to this study. 3.1. HEATING A N D REDISTRIBUTION OF CARRIERS BY A GAUSSIAN EM

BEAM IN A SLIGHTLY IONIZED PLASMA AND A PARABOLIC SEMICONDUCTOR

Following SODHA, T E W ~ KUMAR I, and TRIPATHI [1974] and TRIPATHI, SODHAand TEWARI[1973] we consider an electromagnetic beam (having Gaussian distribution of intensity in the radial direction) propagating in a plasma along the z-axis. The collisions are abundent to ensure the neglect of thermal conduction and also the pondermotive force effect. The dominant mechanism of carrier scattering is elastic electron-neutral particle collisions in a plasma and elastic scattering by acoustical phonons in a parabolic semiconductor. The carriers interact with the electric vector of the beam and the distribution function of their velocities is modified. In the presence of the electromagneticbeam, the distribution functionf,(x, u, I ) of electrons (and holes in the case of a semiconductor) is governed by the Boltzmann’s equation,

204

[v,

SELF FOCUSING OF LASER B E A M S

D3

where F , is the force on the electron due to the electric and magnetic vectors, x is the space coordinate vector and t3fe/t3tlcal, is the rate of change of

distribution function due to collisions; the motion of ions can still be treated by the phenomenological analysis of 4 2. In the absence of external magnetic fields, P, = - eE' where E' is the total electric field present in the plasma ;the effect of the magnetic field of the wave has been neglected. The total electric field E comprises of the two fields viz., the electric vector E of the wave and the space charge field E , . In the steady state it can be shown that (for fields of interest) the drift velocity of carriers is much smaller than the random velocity (GINZBURG [1970]), hence the distribution function of carriers can be expanded in spherical harmonics (in velocity space) or equivalently in Cartesian tensors (SHKAROFSKY, JOHNSTONand BACHYNSKI [19661) as fe(x, V, t) =

f&, 0,

t)+

U'

f,

~

V

(x, 0, t).

(34

Substituting eq. (3.2) in eq. (3.1) and using the orthogonality property of spherical harmonics we obtain the following equations forfo and f l ,

and -afl +vVf,--

-= (3.3b) m au call The collision terms (dfO/dt)coll and ( d f l /dt),,,, for elastic collisions of electrons with heavy scatterers of mass M are (CHAPMANN and COWELLING [19391, DESLOGE and MATTHYSSE [19601, SHKAROFSKY, JOHNSTON and BACHYNSKI [19661)

at

m m

av

(3.4a)

and ( ~ f l / a ~ ) c o l= l

-V f l .

(3.4b)

The velocity dependence of collision frequency v may be expressed as v

=

v0[u/(2k0 To/m)*]Is

(3.4c)

where vo is a constant. In many diatomic gaseous plasmas s = 2; in some

V,

o 31

205

KINETIC THEORY

other cases s = 1. For electron-ion scattering s = - 3. The electron-acoustical phonon scattering corresponds to s = 1. To obtain the steady state solution of eqs. (3.3a) and (3.3b), we expand f o andf, as

fo

=

f$+foeiof

(3.4d)

fi

=

f~+j~eiO*.

(3.44

and It may be mentioned here that f;: -szj," and fi K f:. Also f p K f because f t is the source off:. Using these inequalities in eq. (3.3b), we obtain the following expression for f:

fi'

=

eE m(v+io)

af:

x

(3.5) *

Similarly the formal solution of eq. (3.3b) for f: is

Using eqs. (3.5) and (3.6) and remembering that ReA.ReB

= $Re[A.B*+AB],

eq. (3.3a) forfl takes the form

The general solution of eq. (3.7) is difficult; however, if we ignore the first two terms on left-hand side and assume o2>> vz, the solution is

where

T, = To(l+aEE*) e2M 6m2k0Tooz

= -______

(3.9)

206

[v, § 3

S E L F F O C U S I N G O F LASER B E A M S

and N , ( x ) is the local concentration of electrons. Eq. (3.8) has been written using the normalization condition

[

f,0d3v

=

N,(x).

Using eqs. (3.8) and (3.9) in (3.7) and taking the characteristic length of temperature variation as a (the width of the beam) we can evaluate the ratio of the third term on the left-hand side of eq. (3.7) to the first term;

’

1.

m width of the beam Ratio R = (3.10) M mean free path The first and second terms on the left-hand side of eq. (3.7) are of the same order. This ratio is the same as expression (2.12~)in 9 2. Thus R B 1 implies the neglect of thermal conduction. In the present section we are limiting ourselves to this case. The isotropic part of the distribution functionf: as given by eq. (3.8) is Maxwellian at an elevated temperature T, . Thus the electrons are in quasithermal equilibrium. The expression (3.9) for temperature is the same as obtained by the elementary approach. To obtain an expression for the local electronic concentration we rewrite eq. (3.6) as

(

where we have used eq. (3.8). Now we characterize the steady state as the one in which the average dc drift velocity of carriers is zero. The drift velocity, in general is defined as

: :=

( u ) = Sufd’v /Jfd3v

[03/’

du.

(3.12)

For the time independent part, (3.13)

Hence, in the steady state (u),, = 0 and eqs. (3.1 1) and (3.13) give (3.14)

V,

I 31

KINETIC THEORY

207

This equation can be rewritten as 1

- -[k,

T,/'V(N,

Ne

q'-s/2)] = eE,.

(3.15)

A comparison of eq. (3.15) with the time independent part of eq. (2.10) reveals that the force on electrons due to the imbalance of partial pressure (or gradients in N, and T, to be more precise) is not equal to (- Ne-'V(Nek,Te)) but ( - N , - ' k o ~ ' Z V ( N T ~ ' - s ~ This 2 ) ) . is owing to the temperature dependence of mobility (or v). The equation for the ions is the same as given by eq. (2.36). A simultaneous solution of eqs. (3.15) and (2.36) results in the following expression for the local concentration of electrons N,

=

N0(2T0/(T,+

(3.16)

where N o is the equilibrium concentration (in the absence of the field) of electrons. It can be seen from eq. (3.16) that for s = 2, N, = No i.e. no redistribution of electrons (and ions) takes place. The maximum redistribution of electrons takes place when s = - 3, i.e. when electron-ion collisions predominate. Eq. (3.16) is in considerablevariance from eq. (2.38a) except for the case when s = 0 (i.e. when the collisions are velocity independent). Using eq. (3.16) for electronic concentration the oscillatory component (varying as e'"') of current density can be evaluated. From eqs. (3.12) and (3.5), J

=

-N,e(v)

=

oE

The effective dielectric constant of the plasma can now be written as E =

~~f@(EE*)-is,,

(3.18)

where Eo =

1 - m;/w2

(3.19a) (3.19b)

and (3.19~)

208

SELF F O C U S I N G OF L A S E R B E A M S

[V>

03

A comparison of eqs. (3.19b) and (2.47b) reveals that even in the perturbation approximation the field dependent real part of dielectric constant obtained by the kinetic treatment is (1 -is) times the one obtained by phenomenological treatment. The above treatment is applicable to a parabolic semiconductor also when eq. (2.36) is replaced by the following equation (similar to eq. (3.15)) for holes

(3.20) lV h

and the current density is evaluated using the formula J

=

-N,e(u),+N,e(u),.

However, the simultaneous solution of eqs. (3.15) and (3.20) in the general case is very difficult. Hence, to obtain mathematically tractable results we consider two limiting cases. (i) Compensated semiconductor (No, = Noh) In this case, the solution of eqs. (3.19) and (3.20) for local electronic (or hole) concentration is N,

= N,

=

N,,[(T,+T,)/2To]-’+”’z.

Using the above expression, the nonlinear dielectric constant can be written as (3.18) with (3.21a)

(3.21b) and

+ -mm-t

vOh (l+a,EE*)”” o (I +~a,+a,)EE*)’-”/’

]5 r (7) 3

where a, =

a(m/rn*)’

and

ah =

a(m/mz)2.

3

(3.21~)

V,

o 31

209

KINETIC THEORY

In writing eqs. (3.21a-c), no perturbation approximation has been made. (ii) Perturbation approximation (No,, Noharbitrary) In this approximation eqs. (3.15) and (3.20) may be solved to give (TRIPATHI, SODHA and TEWARI [1973])

where s = 1 has been used. For the validity of this equation, the inequalities ANe < No,, ANh < Noh must be satisfied. Using the expressions for electron and hole concentrations the field dependent dielectric constant can be again given by eq. (3.18) with the following expressions for E, , @ and E ~ , (3.22a)

and

-& w2 m

Ei =

vO+

m

5 mh ‘Oh)

4

~

5+s

3fir(T).

(3.22~)

A comparison of eqs. (2.50b) and (3.22b) shows that on account of the velocity dependent collisions, the nonlinear dielectric constant @ derived by the rigorous kinetic treatment is half of @ obtained by the phenomenological treatment. 3.2. NONLINEARITY IN THE DIELECTRIC CONSTANT OF A MAGNETOPLASMA

In this section, following TEWARI and KUMAR[1974], we generalize the kinetic treatment of 5 3.1 to include a static magnetic field B , in the z direction. The magnetic field introduces an additional interaction term in the Boltzmann’s equation. The force on an electron having velocity u in the presence of the static magnetic field and other electric fields is given by F , = - e E - e v x B,/c.

(3.23)

Using the above expression for force and expanding the distribution function in Cartesian tensors (cf. eq. (3.2)) the following equation forfi can

210

SELF F O C U S I N G OF LASER BEAMS

CV.

§3

be obtained from the Boltzmann's equation

afl

__

at

+vVf,-

eE' afo -OcXfl m av

~

~

(3.24)

= -Vf1,

where a,= - eB,/rnc is the electron cyclotron frequency. The equation for f:.is the same as eq. (3.3a). To obtain the steady state solution of eqs. (3.3a) and (3.24) we expand the distribution function in time harmonics as eqs. (3.4d) and (3.k). Forf: eq. (3.24) gives (v+ico)f/

-0,

x

f,'

=

eE af," ~. m au

(3.25)

-

To solve this equation we multiply the y component of eq. (3.25) by + i and add to the x component of eq. (3.25), to obtain (3.26) whereA, = E,+iE,. On repeating the same operation with - i instead of

+ i we get (3.27)

where A, = Ex- iE,. For the z component eq. (3.25) gives (3.28) The explicit evaluation of 8,and f l y is straightforward from eqs. (3.26), (3.27). The procedure of evaluatingf: is similar to the one adopted in Q 3.1 ; in the approximation v2 ez ( w - w , ) ~ the result is (3.29) where (3.30) CI is the same as defined in 5 3.1 ;f:again turns out to be Maxwellian at an effective electron temperature T, . The expression for T, is the same as obtained in the previous section by the phenomenological treatment. It may

V,

P 31

21 1

K I N E T I C THEORY

be again emphasized here that the heating of carriers by the two modes (described by A , and A , ) in a magnetoplasma is additive; the heating by the extraordinary mode ( A is more pronounced. In order to evaluate the normalization constant (the local concentration of electrons), we need fp. From eq. (3.24) the expressions for the x, y components off: are

In the steady state (v),,

= 0, hence,

r m

Jo

r m

fP,v3dv

=

0,

J,

fP,v3dv

=

0.

(3.32)

To evaluate the integrals explicitly we restrict ourselves to the two limiting cases (i) v B w, and (ii) v K 0,. The former case corresponds to the isotropic case discussed in the last section. In the second case, condition (3.32) gives -eE,-

1 -

V ( N e k o T , )= 0.

(3.33a)

Ne

Eq. (3.33a) is similar to the one obtained by phenomenological treatment. Thus in the limit of w, B v , the velocity dependence of collision frequency does not alter the process of carrier redistribution. Following the above analysis, the equation describing the redistribution of holes can be written as eEs-

1 -

V(Nhko q)= 0.

(3.33b)

Nh

Eq. (3.33b) is valid for ions also when q is replaced by To. The condition w,lion > vion is not necessary because the rise in the temperature of ions (or heavy holes) is negligible. The simultaneous solution of eqs. (3.33a) and (3.33b), following the procedure of 4 2, results in the following expression for the local carrier concentration when No, N Noh: Ne-No

=

N,-N0,

=

-No

[IT:; 1-

_____

(3.34)

In the case of plasmas eq. (3.34) is valid with mh and Nohreplaced by M and No( iVoe).

212

S E L F F O C U S I N G OF LASER B E A M S

cv, 0 3

Now we can evaluate the first harmonic component (oscillating as e’““)of current density. Using eqs. (3.26H3.28), we obtain

Following the treatment of 0 2, we can evaluate the conductivity tensor from eqs. (3.35) and (3.36). The effective nonlinear dielectric constant is of interest to us, which using eq. (3.34) and the conductivity tensor, can be written as Ex,

= Eyy = $ E + + E - )

EXy = Eyx =

Ezz

-

where

and

EL-

4nN, e2 m * 0 2 [l-

___

1

-

E+--E~

2i

(2) r(g5+ ] s/2

s))

r(5/2)

V,

D 41

21 3

SELF FO CUSI NG OF EM B E A M S

5

4. Steady State Self Focusing of EM Beams in Plasmas

4.1. SELF FOCUSING IN A NONLINEAR ISOTROPIC MEDIUM

In the preceding sections we have derived the expressions for the field dependent effective dielectric constant E of a plasma/semiconductor. In this section we proceed to study the propagation of a Gaussian electromagnetic beam in such a plasma. Such studies in the case of nonlinear dielectrics have been extensively made in recent years. As we have seen in 9 2.1, Maxwell’s equations in a plasma have the same form, in terms of the effective dielectric constant E , as they have in a dielectric; hence the mathematical methods to solve these equations are the same. In what follows we have adopted the approach outlined by AKHMANOV, SUKHORUK& and KHOKHLOV [19681. Using eqs. (2.1~)and (2.1d’), the following equation can be obtained for the electric vector in the steady state V ~ E - V ( VE. )

= -

oL

-&E.

(4-1)

C2

From eq. (2. la’) V. E

=

- E . VF/E,

hence, eq. (4.1) can be rewritten as V2E+

&)

o2 E . VE EE+V = 0. C

The last term on the left-hand side of eq. (4.2) is negligible if k-’V’(ln E ) ez 1 where k represents the wave vector. The above inequality is satisfied in almost all cases of practical interest. Thus o2 V2E+ - F E E = 0. (4.3) C2

We employ WKB approximation to solve this equation. On expressing the solution of eq. (4.3), for cylindrically symmetric beams, as E

=

A(r, z ) exp {i(ot- k z ) } ,

k

0 =

- E*0

C

(4.4)

and neglecting d 2 A / d z 2 which implies that the characteristic distance (in the z-direction) of the intensity variation is much greater than the wavelength, eq. (4.3) reduces to

214

CV, § 4

S E L F FOCUSING OF LASER BEAMS

In writing eq. (4.5) we have expressed E in its general form as E =

E,+@(EE*)-~E,

where @(,YE*),zi K E~ is implied. Eq. (4.5) is known as the parabolic equatiofl and has been extensively employed by various workers for propagation and radiation problems. The case, when E~ is a function of EE* is very difficult to pursue mathematically, hence we limit ourselves to the case when zi is field independent (i.e. absorption is linear); for the case of nonlinear absorption a reference may be made to SODHA, TEWARI, KUMAR and TRIPATHI [1974] who have solved the wave equation numerically. To solve eq. (4.5), we express A as A

=

Ao(r, z) exp { -ikS(r, z)},

(4.6)

where A, and S are real functions of r and z. Eq. (4.6) is true only when (i) the beam is plane polarized and (ii) its polarization does not change with its propagation (which is always the case for a slowly converging/diverging beam, at least for paraxial rays); otherwise Swill be different for A,, A, and A,. Then, we may choose without any loss of generality x or y axis of the coordinate system along the direction of the electric vector. Hence, eq. (4.5) is true for scalar A as well. For a more rigorous treatment one is referred to SODHA, GHATAK and TRIPATHI [19741. Substituting for A from eq. (4.6) in eq. (4.5) and equating the real and imaginary parts we get

as

2-aZ

+):(

2

1 = -@(A:)+ 80

-{$ 1 k2A,

a2A

+ r1 3) ar

(4.7a)

and

SUKHORUKOV and KHOKHLOV [19681, the solutions Following AKHMANOV, of eqs. (4.7a) and (4.7b) for an initially Gaussian beam can be written as S

=

$r’fl(z)+4

(4.8a) (4.8b)

and (4.8~)

V,

Q 41

SELF F O CUS I NG OF EM B E A M S

215

where ki= k ~ ~and / /?-I 2 ~may ~ be interpreted as the radius of curvature of the beam. The parameterfin eq. (4.8b) is a measure of axial intensity as well as the width of the beam. Usually it is known as the beam width parameter*. As we are interested only in initially plane wavefronts (at z = 0), dffdz = O

at z = O

(4.8d)

and without loss of generality f = 1

at z = O .

(4.8e)

To obtain an expression for the beam width parameterf, we employ the paraxial ray approximation (i.e. restrict to the case when r2 < rzfL). Then

A; = A&=o. (1 - r 2/ r 2O f2 + . . .)

(4.9a)

and

where

Substituting for A ; and Q, from eqs. (4.9a) and (4.9b) in eq. (4.7a) and equating the coefficients of r2 on both sides we get, d2f 1 _ - ~-

E; exp dz2 R i f 3 &or:f where & = k r ; is the diffraction length. In the absence of the nonlinear term (i.e. in a linear medium) the solution of eq. (4.10), using the boundary conditions (4.8~)and (4.8d), is

f

=

1+z2fRj

(4.11)

which represents the diffraction divergence of the beam. Thus propagating a distance Rd in a linear medium the width of the beam (= r , j )is enlarged by a factor of 23 and the axial amplitude is decreased by the same factor. In the presence of nonlinearity the behavior offdepends on the relative magnitude of the two terms at z = 0. When the magnitude of the second term on the right-hand side of eq. (4.10) is greater than the magnitude of the * In the geometerical optics approximation r = af(z)

represents a meridianal ray.

216

SELF F O C U S I N G OF LASER B E A M S

cv>§ 4

diffraction term (the first one), d2f /dz2 is negative and hence f decreases with z, i.e. the focusing of the beam takes place. In the opposite case, defocusing occurs. We may define a critical value of E,, viz., Eocr(when other parameters are fuced) for which the two terms on the right-hand side of eq. (4.10) cancel each other (i.e. initially the beam neither tends to converge nor to diverge); thus (4.12) The corresponding power of the beam (known as the critical power) is

P

=

8n

E$

J:Eicr

exp (- rz/ri) f2nrdr =

C E E;,, ~

8

ri .

(4.13)

In the special case when @(EE*)= cZEE* (i.e. the field dependence of dielectric constant is quadratic) @' = zz, the critical power can be rewritten explicitly as C3&$

P,,

=

~.

8w2&,

(4.13a)

P,, is thus independent of the radius of the beam. A beam having P > P,, (P being the power of the beam) tends to focus, at least initially. Later the power is attenuated due to absorption and the diffraction term in eq. (4.10) may dominate over the other to cause the divergence of the beam. Thus owing to the absorption, the extent of self focusing is reduced. To have appreciable focusing of the beam the characteristic length for self focusing should be smaller than the length for absorption k; '. In the case of quadratic field dependence, it is useful to define a parameter R,

=

rO(EO/E2Ei)t

(4.14)

in terms of which, eq. (4.10) can be rewritten as (4.15) R, may be interpreted as the self focusing length when diffraction and absorption of the beam are negligible, taking R, + cc and ki -+0, the solution of eq. (4.15) is f 2

=

1-z2/Ri.

It may be inferred from the above discussion and the numerical integration of eq. (4.15) that for appreciable focusing of the beam to occur the in-

V,

o 41

217

S E L F F O C U S I N G OF E M B E A M S

'

equalities R, < Rd and R, < k; must both be satisfied (SODHA,GHATAK [19741; TRIPATHI, TEWARI,PANDEY and AGARWAL [1973al). and TRIPATHI To discuss the, detailed behavior of eq. (4.10) we consider the following cases. 4.1.1. Collisionless plasma In this case the nonlinearity is mainly due to the pondermotive force (SHEARER and EDDLEMAN [1973]). Using eq. (2.43) for the field dependent dielectric constant and taking k' 70, eq. (4.10) can be written as d2f_-- 1- _ R:E; _ _ _ o_i m~a exp dq2 f 3 cot-; 0 2 M f 3

(-

")

M a F

(4.16)

where q

=

zfRd.

First we shall discuss the behaviour of eq. (4.16) in the case of weak nonlinearity i.e. maEi/MfZ e~ 1. In this approximation the solution of eq. (4.16) can be obtained as f2

=

i+(i-~:/~,Z)q~

(4.17a)

where

Using eq. (4.13a) the critical power (for self focusing to occur) can be written as (4.17b) In a typical case of o = l O I 4 sec-', No = 10l8 ~ m - ~To, = 105"K, P,, N 0.4 M watt/cm2. Such high powers are easily available these days. When P = P,,, Rd = R, and f = 1 for all values of q , i.e., the beam propagates uniformly without changing its curvature. This is usually referred to as the uniform waveguide propagation. For P < P,, , Rd < R, and hence the beam diverges (i.e. the diffraction effects predominate). For P > P,,, Rd > R, and f decreases with q, i.e., the focusing of the beam takes place. The analysis given above is valid only before the focus is reached because in the vicinity of the focus the axial intensity (of even a relatively weak beam) is enhanced to a large extent (maEi/MfZ is no longer much less than unity). For this reason we discuss below the general solution of eq. (4.15).

218

SELF FOCUSING O F LASER BEAMS

[v,

P4

To start with, it is important to determine the critical power of the beam in the general case (arbitrary value of (m/M)orEg). For other parameters given, E,,, is determined by the equation (4.18) It can be seen from eq. (4.18) that the maximum value of the left-hand side is e - ' ( z 0.3) and the minimum radius of the beam which can satisfy eq. (4.18) is c

ro

=

-

(4.19)

e+.

0,

Any beam of ro < rOmincannot be focused. The beam width would increase first, reach a maximum value and then decrease to yo and then increase again (GHATAK, GOYAL and SODHA119723). To obtain a general solution of eq. (4.18), the graphical method is employed. The result viz. ((m/M)clE~,, as a function of r&i/c2) is shown in Fig. 4.1. Using EOer in eq. (4.13) the critical power of the beam can be calculated.

10 8 6 4 2

-

0 0

1

2

3

4

5

6

m

7d E&r Fig. 4.1. The radius of the beam as a function of (m/M)c&.. for uniform wave guide propagation (i.e. when P = Per) in a collisionless plasma.

A beam with P > P,,,should get converged. However, with decreasingf, the diffraction term increases more rapidly than the nonlinear term and one expects a minimum infat z = zop. Beyond z = zop the beam should defocus due to the predominance of diffraction effects. To study this behaviour we solve eq. (4.16) by multiplying this equation by 2fdf/dq and integrating with respect to q once. The result is

S E L F F O C U S I N G OF EM B E A M S

219

(4.20) where we have used the initial conditions viz. f = 1 at z = 0 and df/dq = 0 at q = 0. The sign of df/dq has to be decided by the expression (4.16) for d2f/du2; dfldq should decrease when d’fldq’ is negative. For a maximum/ minimum in f( = fm), df/dq should be zero. Hence,

(-

“f)

(- ;

I-;---1 “ (4.21) jexp ?! a -exp a E i ) } = 0. f,’ R: maEi M fm One of the solutions of this equation is,f,, = 1 and the other has to be evaluated by a graphical method. This second solution fmll as a function of (rn/M)uEi is displayed in Fig. 4.2. For beams having dZf/dq2(ll=o as negative, fmll is less than],, and hence, the parameter characterising the beam dimension (in course of its propagation) oscillates between the values 1 and fm,,. Thus an oscillatory waveguide is formed. It is interesting to note that f,,, tends to 1 as the power of the beam increases i.e., the oscillatory waveguide tends to a uniform waveguide.

urn,,)

Fig. 4.2. Variation of minimum beam width parameter as a function of the dimensionless = 50 when the nonlinearity is due to pondermotive force. power of the beam for R&ow,’/ozr$so

220

SELF FOCUSING O F L A S E R B E A M S

cv, § 4

Any further integration of eq. (4.20) is difficult; hence, we have obtained the solution by numerical integration. The results have been displayed in Fig. 4.3.

725

Z/Rd

Fig. 4.3. Beam width parameter as a function of distance of propagation for (m/M)aEi = 1 and R;/R.’ = 100.

4.1.2. Collisional plasma: collisional loss In this case the expressions derived in 8 3 would be used. Two cases would be considered to justify the treatment of 0 4.1.1 for solving the wave equation. (i) Perturbation approximation (linear absorption) Assuming (aEE*) -=z1, the following equation can be obtained from eq. (4.10) by using eqs. (3.18H3.19) for the dielectric constant d2f -

-

dq2

R i exp(-2kIq) , R,2 f’

1 f’

(4.22)

where r/

= Z/Rd,

R,

=

kri,

and ki

=

ksi Rd/2&,.

It can be seen from eq. (4.22) that on account of absorption the self focusing term increases less rapidly with distance than the diffraction term. This is because the energy of the beam decreases as exp ( - 2kiq) which is equivalent to a weakening of the nonlinearity effect. We may discuss an interesting case

V, P 41

22 1

S E L F F O C U S I N G OF EM B E A M S

of a beam for which P > P,,(Rd > R, but RdIR, is not much different from unity). Such a beam converges in the beginning (fdecreases with v ] ) . After propagating some distance in the plasma, the power of the beam is reduced due to absorption and the first term on the right-hand side of eq. (4.22) dominates over the second term. After propagating some more distance (so that df/dq which was hitherto negative, becomes zero), the beam starts diverging. Thus the absorption brings about a reduction in the extent of self focusing ;f does not decrease without limit but reaches a minimum fmin (depending on the value of kf)at some optimum value of q = qOpand then increases. To have a quantitative estimate off,, and qOp,eq. (4.22) has been solved numerically and the results displayed in Fig. 4.4.It is interesting to see from the figure, thatf,,, increases with increasing kl while qOpshows an opposite trend. For a given value of v ] , the value off is higher for higher values of kf.

0.2 -

I

I

I

I

I

To have a better understanding of the behaviour of self focusing in the presence of absorption, eq. (4.22) can be solved analytically when

222

rv. 5 4

SELF FOCUSING OF LASER B E A M S

the result is (4.23) Eq. (4.23) reveals all the interesting features discussedabove. The critical power for self focusing of the beam (corresponding to R, = R, at z = 0) can be obtained (cf. eqs. (3.19) and (4.13a)) as (4.24)

A comparison of eqs. (4.17b) and (4.24) reveals that the critical power for self focusing due to pondermotive force is M / m times the critical power for self focusing due to the heating (and subsequent redistribution) of carriers; all parameters being the same, the two differ by three-four orders of magnitude. In a typical case of w = 10’’ sec-l, N o = loi4 cmP3, To = io4”K, s = - 3, P,, N 10 watt/cm2 for He plasma, P,, is inversely proportional to the mass of the heavy scatterer.

-

(ii) Arbitrary nonlinearity (no absorption) Neglecting absorption (ci -+ 0) andusingeqs. (3.18) and (3.19) in eq. (4.10) we obtain the following equation for the beam width parameter

where rl = z JR, . The critical field, E,,,,, is determined by the equation ~

+

~ L I 2

-

1

_____

(1 &f,,/2)2 -” (1 -s/2)(r,w,/c)2

(4.26) ’

The maximum value of the left-hand side for (s < 2) is (1 -s/2)/(2 -s/2)’ -si2 corresponding to aE&, = 2/(1 -s/2). Hence, the minimum value of yo satisfying eq. (4.26) is (4.26a) This value of rOminis of the same order as in the case of pondemotive force nonlinearity (cf. eq. (4.19)). A beam of radius ro < rOmincannot be at all focused irrespective of power. A numerical solution of eq. (4.26) has been obtained and the results

v, § 41

SELF F O C U S I N G O F EM B E A M S

223

(showing the variation of $a& with r,o,/c) are displayed in Fig. 4.5. Using the value of Eocrin eq. (4.13), the critical power of the beam can be calculated in a straightforward manner. 10

8

2

Fig. 4.5. The radius of the beam as a function of :aE& for uniform waveguide propagation in a plasma when the nonlinearity is on account of heating and redistribution of electrons.

As in the case of pondermotive nonlinearity (0 4.1.1), the nonlinearity in the present case too is saturating and the behaviour of self focusing is similar. Integration of eq. (4.25) gives

The sign of dfldq has to be decided by the sign of d2f/dy2 at q = 0; df/dy should decrease when d2f/dq2 is negative. For a maximum/minimum in f( =fm), df/dq should be zero. Hence,

One of the roots of eq. (4.28) is fml = 1. The other root fml, has to be calculated numerically. The results (fmIl as a function of aEg) of such calculation are shown in Fig. 4.6. For P > P,,,numerical solution of eq. (4.27) is displayed in Fig. 4.7 (SODHA, TEWARI, KUMAR and TRIPATHI [19741). The oscillatory waveguide behaviour is explicitly revealed by the figure. 4.1.3. Fully ionized plasma: conduction loss Using eqs. (2.45H2.46) (which are valid in the perturbation approximation), the equation (4.10) for the beam width parameter can be written

224

S E L F FO CUSI NG OF LASER B E A M S

I

I

I

I

I

[v, § 4 I

0.3

0.3,

frnn a2

0.24

Fig. 4.6. Variation of minimum beam width parameter as a function of the dimensionless power of the beam in a plasma for s = 1 when R&O;/W~E,,I:~ = 50. The nonlinearity is due to the heating and redistribution of electrons.

4

Fig. 4.7. Oscillatory waveguide propagation in a plasma for R & o ~ / w z t o= r ~25, aE$ = 1

explicitly as (4.29)

V,

o 41

225

SELF F O C U S I N G OF EM B E A M S

The corresponding critical power of the beam is given by (4.30) The numerical solution of eq. (4.29) is displayed in Fig. 4.8for the following sec-'; ro = 500p; No = I O I 6 ~ r n - ~To ; = 105"K; parameters o = v = 5 x 10" sec-'; Eo = lo5 esu; 2 x lo5 esu. The power of the beam is 10" watt. It isinteresting to see from Fig. 4.8 and also from eq. (4.29) that an oscillatory waveguide is formed even when the dielectric constant does not have a saturating profile (SODHA,KHANNA and TRIPATHI [1973a]).

-

0.01

I

1.0

I

2.0 = Z/Rd

-

3.0

4.

Fig. 4.8. Variation of beam width parameter with z in a strongly ionized plasma (thermal conduction predominant) form = 1015 sec-', ro = SOOk, No = 10l6 ~ r n - To ~ , = lo5 "K. The continuous curve corresponds to Eo = 2 x lo5 esu and the dotted curve corresponds to Eo = lo5 esu.

4.1.4. Parabolic semiconductors (e.g. Ge) Under the class of nonlinear solid state plasmas we limit ourselves to the perturbation approximation. Using the rigorous expression for the dielectric constant the treatment of 9 4.1.2 may be followed to study the self focusing in a parabolic semiconductor (Ge to be specific). The equation governing

226

SELF FOCUSING O F LASER B E A M S

[V>

D4

the beam width parameter (cf. eq. (4.10)) is -d2f =-- 1

dq2

f 3

Ri -exp(-2kjq) R,2

1

-

f3

(4.31)

where

and c 0 , ci and c2 are given by eqs. (3.22a)-(3.22~). The critical power for self focusing of the beam is given by (cf. eq. (4.13a)) (4.32) The above equation has been written when s = 1, 1.e. acoustical phonon scattering is dominant. For a typical set of parameters in Ge,w = lOI4sec- ', No, 2: Noh= 1015 To = 77"K, m* = 0.1 m and rn; = 0.3 m., the critical power for self focusing turns out to be 10 watt. This is much smaller than the power required to observe the self focusing of a laser beam in a nonlinear dielectric. To visualize the self focusing in Ge, the numerical solution of eq. (4.31) is displayed in Fig. 4.9 corresponding to the following parameters

f2

0.0

0.4

0.8

1.2

-g-z/Rd Fig. 4.9. Variation of beam width parameter with the distance of advancement of the beam in germanium. The curves correspond to different values of vim,v = v h .

227

SELF F O C U S I N G OF E M B E A M S

No,

=

No,

=

1015cmP3, o

=

1014sec-',

To = 77"K, ro

=

E,

250pm,

=

20esu.

An important result of the investigation is that to achieve appreciable focusing of the beam Rn must be smaller than the damping length, k; '. It is very instructive to mention here that the condition R,, < k; cannot be satisfied when the nonlinearity appears through the velocity dependence of collision-frequency or of relaxation time (TRIPATHI, TEWARI, PANDEY and ACARWAL[1973a]). Thus in the absence of redistribution of carriers there is no mechanism in a parabolic semiconductor which can cause the self focusing of the beam. 4.1.5. Nonparabolic semiconductors (e.g. InSb) Using eqs. (2.53H2.54) and (4.10), the equation for the beam width parameter comes out to be same as eq. (4.31). The numerical solution of this equation for an n-InSb sample is displayed in Fig. 4.10 for the following parameters : No,

=

1015~ m - ~o, = loi4 sec-',

To = 77"K, ro = 250pm,

E,

=

4 esu and 5.6esu.

For the above parameters the critical power for self focusing in n-InSb is w 5 watt which is about three orders less than that obtained from the

f2

0.2

a0

0.4

0.8

I .2

f -Z / R d Fig. 4.10. Variation of beam width parameter with the distance of propagation in n-InSb

228

SELF FOCUSING OF LASER BEAMS

[v. § 4

expressions derived by TZOARand GERSTEN [1971, 19721. The treatment of TZOARand GERSTEN [1971, 19721 is erroneous in the sense that they have ignored the heating of electrons and considered the nonlinearity appearing through the modulation of mass of electrons by their drift energy. As a matter of fact the drift energy of electrons is much smaller than the rise in average random energy (SODHA,TEWARI,KAMAL,PANDEY, AGARWAL and TRIPATHI [1973]), and hence the latter is mainly responsible for the nonlinearity. The proposition of DUBEY and PARANJAPE C1972, 19731 that the self focusing of a Gaussian electromagnetic beam can be achieved by the nonlinearity appearing through the energy dependenceof collisionfrequency is completely ruled out because over the whole range of parameters (v, o,T o , No etc.) the characteristic length of absorption is much smaller than the length for self focusing. To show this effect explicitly we consider the case of a parabolic semiconductor and neglect the redistribution of carriers. Then the effective dielectric constant of the semiconductor (having electrons only) in terms of effective collision frequency can be written as

Treating V~ as a function of electron temperature T, wemay expandit in the perturbation approximation as

where

and Q'

=

e2M 6m2k0T,(02+ v t ) .

In writing the above expressions we have used (Te- To)/To= a'EE* the expression obtained as a generalization of eq. (2.20). The coefficient v2 is positive for acoustical phonon scattering in which veff increases as ( Te/To)*and v2 = &'v0 ;for other scatteringmechanisms similar functional dependences are available. Using the expression for veff, the dielectric constant can be rewritten as Eeff = E~

+E~ EE* -iq

V,

P 41

229

SELF F O C U S I N G O F E M BEAMS

where

In the plane wave approximation the amplitude of the wave decreases as exp (- k , Z )where

ki

= Imaginary part of

; when ei

< g o , ki

0 Ei N

-

2c

&$.

The self focusing length (following eq. (4.14)) using the expression for e2 can be obtained as

R,

N

ro &vf 20,

+w 2 )

Jm

where ro is the initial width of the beam. Similarly the diffraction length is

For self focusing to occur the following conditions muct be satisfied simultaneously (i) k;’ > R, and (ii) R, > R,; the first inequality is to ensure the less effectiveness of absorption and the second for diffraction. The inequalities (i) and (ii) can be written as 4c and 3

0

r 0 A > 4c (v2;E*) ~

.

These two inequalities cannot be satisfied by reasonable parameters and hence self focusing of the beam cannot occur on account of the mechanism suggested by DUBEY and PARANJAPE [1972, 19731. 4.2. MAGNETOPLASMA

The dielectric constant of a magnetoplasma is a tensor. The procedure of solving the wave equation for the electric vector in such a medium is

230

S E L F F O C U S I N G OF L A S E R B E A M S

[v. 0 4

similar to the one discussed in 94.1. From (2.1~)and (2.1d'), treating a tensor, the wave equation can be obtained as

E

o2 V2E-V(V. E ) + 7E . E

=

0.

(4.33)

C

For its components, eq. (4.33) gives

and

In order to solve eqs. (4.34aH4.34~)we assume that the variations of field in the z-direction (i.e. along the static magnetic field) are more rapid than in the x-y plane. So that the waves can be treated as transverse in the zeroeth order approximation. From eq. (2.la')

On multiplying eq. (4.34b) by + i and adding it to eq. (4.34a) and using eq. (4.35) we obtain

o2 +[ E + ~ + @ + ( A ~ AA2TA, 3 ] A , = 0 c2

(4.364

and

+

Q2 -

C2

[ E - , , + @ _ ( A ~ A TA, 2 A ; ) ] A 2 = 0.

(4.36b)

In writing eqs. (4.36a, b) we have assumed that @ & < E* and the terms of the order of (i32A,/dx2)'@+have been ignored. It can be seen from eqs.

V,

8 41

23 1

SELF F O C U S I N G O F EM BEAMS

(4.36a) and (4.36b) that the two modes are coupled not only through the nonlinear term but also through the linear term involving x and y variations of A, and A,. However, as the coupling terms are relatively weak we can assume to a good approximation that one of the two modes is zero and the behaviour of the other mode can be studied. When both modes exist eqs. (4.36a) and (4.36b) have to be solved simultaneously. On assuming A , N 0, eq. (4.36a) for A , gives =

0.

EOZZ

(4.37)

A similar equation can be obtained for A, on assuming A, N 0. In the WKB approximation the solution of eq. (4.37) can be taken to be a generalized plane wave viz. A,

=

(4.38)

A exp {i(wt- k + z)}

where AA*J,=,

=

E i exp (-r2/r&

k,

w =

-

c

et+ 0 9

(4.39)

and A is the complex amplitude. Substituting (4.38) in (4.37) and omitting a 2 A / a z 2we obtain @+(AA*)A= 0.

(4.40)

The solution of the above equation can be obtained by following the treatment of 9 4.1. Taking A

=

Ao(r,z) exp { -ik+ S(r, z)}

(4.41)

the solutions for A, and S in the paraxial ray approximation are obtained to be

and S = +B(z)r'

where

+ $(z)

(4.42b)

232

S E L F F O C U S I N G O F LASER B E A M S

cv. 5 4

and f is governed by

@$ denotes the first derivative of @+ with respect to its argument, R,, = k + ri and k + i = +k+ E + J E + ~ . The initial conditions on f are

The behaviour of the beam width parameter can be discussed in the following cases of interest. 4.2.1. Collisionless magnetoplasma In the perturbation (i.e. when (m/M)aEi/? a l), eq. (4.43) reduces to the form (4.44) and its solution can be written as

where R",

=

r&+o/E+2Ei3+.

It may be remembered here that for the self focusing of the beam to occur E + ~ must have a +ue value which is possible only when either (i) w > o, or (ii) w < $),;for intermediate values of w defocusing of the beam takes place. Using eq. (4.45), one may obtain the self focusing length zlf as (4.46) The variation of zlf (for both modes of propagation) with the static magnetic field is shown in Fig. 4.11 for typicai plasma parameters. It is interesting to note that the cyclotron effects enhance the self focusing of the ordinary mode. The same behaviour may be seen even when the saturation effects in the nonlinear dielectric constant are important. The oscillatory wave guide

V,

o 41

233

SELF FO CUSI NG OF EM B E A M S

11

.1

. 21

I

I

I I I l l 1

.5

1 *C/

I

2

1

I

I

5

I I l l I

0

Fig. 4.1 1. Distance of focusing of the beam ( z ~ as ) a function of dimensionless static magnetic field for extraordinary and ordinary modes whenoi/02 = 0.2, (rn/M)aEg = O.O1,oro/c = 10’. Extraordinary mode is not focused for 0.8 < o,/o < 2.0 and formc/o > 5.0.Ordinary mode is not focused for o,/w > 4.0.

propagation of the beam in the plasma is similar to the one discussed in 84.1.

4.2.2. Weakly ionized magnetoplasma :collisional loss Using the expressions derived for the field dependent dielectric tensor in 8 3, the numerical integration of eq. (4.43) can be performed following TEWARI and KUMAR[1974]. The numerical results showing the variation of characteristic parameters of a self made oscillatory waveguide with the static magnetic field are displayed in Figs. 4.12 and 4.13. The data for these figures correspond to microwave frequencies because the plasma effects in weakly ionized plasmas at laser frequencies are weak. It may be noted from the figures that the minimum dimension of the waveguide cfmin) increases with increasing power of the beam and attains a value of unity.

234

S E L F FO CUSI NG OF L A S E R B E A M S

I

0.20

I

1

I

0.50

Fig. 4.12. Variation of minimum beam width parameterf,,, and the corresponding distance of focusing qc(= zf/Rdl) for the extraordinary mode with the applied static magnetic field for different values of aE& The other parameters are op/w = 0.5, w = IOGHz, ro = 15 cm and a = 104.

0.20

0

I

2

I

4

I

6

I

8

I

I0

.5

I2

d E,' Fig. 4.13. Variation of minimum beam width parameterf,,, and the corresponding distance of focusing qf( = zf/Rdl) with aEi for different values of magnetic field. The other parameters refer to Fig. 4.12.

V,

P 41

S E L F F O C U S I N G OF EM B E A M S

235

zf also shows a steady increase with the intensity of the beam. With w, approaching w, the focusing of the beam (extraordinary mode) occurs at large distances and the minimum spot size increases. The behaviour of the oscillatory wave guide in a collisionless plasma is qualitatively similar to that of the collisional plasma. However, it may be mentioned here that in the case when w, > w, the effective dielectric constant of the plasma for the extraordinary mode is an increasing function of electron concentration hence at such frequencies the self focusing of the beam cannot be attained by the heating type of nonlinearity. Similar results have been obtained by DEMCHENKO and HUSSEIN[1973], TANIUTIand WASHIMI[19693 and WASHIMI [19731. It may be mentioned here that the discussion of self focusing in a weakly ionized plasma is also applicableto compensated semiconductors. In the case of nonparabolic semiconductors (e.g. n-InSb) also, where the nonlinearity arises through the energy dependence of effective mass, the above mentioned treatment is applicable when appropriate modifications in E + ~are incorporated. JAIN, GERSTEN and TZOAR[19731 have recently investigated some aspects of self focusing of laser beams in n-InSb in the presence of a static magnetic field but their treatment lacks (i) in not taking the effects of heating of.carriers into account (which are orders of magnitude higher than the considered effect of drift velocity) and (ii) in not accounting properly for the anisotropy in the dielectric tensor while solving the wave equation. TEWARI, PANDEY,AGARWAL and TRIPATHI [19731 have also investigated the same problem at microwave frequencies by taking account of hot carrier effects. Their results are similar to that of a weakly ionized plasma. 4.2.3. Strongly ionized magnetoplasma : thermal conduction loss In the general case when the energy loss mechanism for the electrons is due to collisions and thermal conduction, the equation for the beam width parameter has to be solved numerically. In the two limiting cases (i) when loss of energy due to thermal conduction is dominant (at low values of magnetic field v w c )and (ii) when collisional loss of energy is predominant (at high values of magnetic field v a w , ) , eq. (4.43) can be simplified as (SODHA,KHANNA and TRIPATHI [19741)

-

6)

236

SELF F O C U S I N G OF LASER B E A M S

[v, 0 4

and

JOHNSTONand The parameters g,, gk,and hk, are given by SHKAROFSKY, BACHYNSKI [1966). The corresponding expressions for the critical power can be obtained as

(4.49a) and

(ii)

(4.49b) EOzz

Per

Fig. 4.14. The critical power for self focusing of the beam in a strongly ionized magnetoplasma and w i / d = 0.2. as a function of static magnetic field for
v, P 41

231

SELF F O C U S I N G OF EM BEAMS

( 1 1 1 1 1 1 1 1 1.6 1 1 1 1 2.4 1

0.20.0

0.4

0.8

1.2

2.0

'Ic= Z/%*, Fig. 4.15.Variation of beam width parameter with the distance of propagation in a strongly = o&' = 0.2, Ei = 100, ro = 5 an, o = loi2 sec-', ionized plasma for (vei)/o T = lo5 "K whenw,/o 5 i s . the loss of energy due to thermal conduction predominates. The two modes focus in the same manner.

4

Fig. 4.16.Variation off'with z for o,/o2 (i.e. the loss of energy due to collisional loss predominates). The other parameters are the same as in Fig. 4.15.

238

[v, 9 5

SELF F O C U S I N G O F LASER B E A M S

The variation of P,,with the static magnetic field is displayed in Fig. 4.14. The critical power shows a tremendous fall with o,/o. To have a numerical appreciation of the effect of a static magnetic field on the self focusing the results of numerical integration of eqs. (4.43) and (4.47)are shown in Figs. 4.15 and 4.16.

8

5. Nonsteady State Self Focusing

5.1. LINEAR PART OF CURRENT DENSITY

The treatment of preceding sections is applicable when the relation J ( t ) = aE(t) holds, i.e. the current density at time t is expressible in terms of the electric vector E only at time t . As pointed out in 0 2, this relation is justified only in the case of monochromatic waves. In the case when the amplitude of the electric vector of the wave is a function of time, the current density has to be re-evaluated as a function of the electric vector. In the present section we have investigated this relationship. The drift velicity of electrons in the presence of an electromagnetic wave of time varying amplitude in an isotropic plasma is governed by the equation du dt

-

eE m

-~ - vu,

where the nonlinear terms have been dropped. The solution of eq. (5.1) is (5.2a)

and the corresponding expression for current density is J ( t ) = - N,eu(t)

=

e2 I:e-".B(t-

m

t')dt'.

(5.2b)

The current density depends not only on the instantaneous value of the electric vector, but also on its past history. Eq. (5.2b) may be simplified by expressing the electric vector as E(t) = A(t)e'"', (5.3) where A is a slowly varying function of time. 'For monochromatic disturbances A is independent of time and hence eq. (5.2b) results in the instantaneous relationship between J and E viz. N,e2E J = (5.4) m(v + iw)

V,

P 51

NONSTEADY STATE S E L F F O C U S I N G

239

However, in the present case where A is a function of time, eq. (5.2b) has to be solved by expanding A(t - t’) as

The resulting expression for current density is J ( t ) = N,e2E(t) - N e e 2 m(v+iio) m(v+io)’

(5.5)

i.e., the current density in the plasma depends on the instantaneous value of the electric vector as well as on its time derivative. The appearance of this extra term (the second on the right-hand side of eq. (5.5)) in the expression for current density is a characteristic feature of a dispersive medium (viz. a plasma); in the case of dielectrics a similar relation is obtained when the dielectric is dispersive otherwise an instantaneous relationship is observed between D and E . On account of the dependence of J (in the case of plasmas and D in the case of dielectrics) on aA/at, the group velocity ug of a wave (with which the amplitude envelope propagates) in a dispersive medium is different from the phase velocity vp . Therefore whatever be the smallness of this term in eq. (5.5) it has to be retained. 5.2. NONLINEAR CURRENT DENSITY : NO REDISTRIBUTION OF CARRIERS

In this section we have studied the nonlinear current density in a collisional plasma when a high amplitude plane electromagnetic wave propagates (along the z-axis) through it. The wave is uniform in the x-y plane and has frequency higher than the effective collision frequency (0’B v2). The nonlinearity arises due to the energy dependence of electron collision frequency ; the pondermotive force and redistribution of electrons are negligible. To obtain a mathematically tractable expression for current density, the perturbation approximation has been employed. The drift velocity of electrons is still governed by eq. (5.1) where the temperature dependence of v can be taken as

s accounts for the nature of scattering cross section; s = - 3 for Coulomb collisions between electrons and ions, s = 2 for diatomic gaseous plasmas and s = 1 for other plasmas. The electron temperature is governed by the energy balance equation (eq. (2.11)). To solve for the electron temperature

240

S E L F FO CUSI NG O F L A S E R B E A M S

[v. 5 5

we need the zeroeth order drift velocity of electrons, which is obtained from eq. (5.5) (u = -J/N,e) by replacing v by vo and N , by N o . Using these expressions in the energy balance equation we obtain the following expression for electron temperature

-K - TO - a6vojrAA*(t- t') exp (- 6vot')dt' TO

(5.7)

where we have neglected the terms involving aAA*dA/at; c1 is defined in tj 2.2. Using eqs. (5.1), (5.6) and (5.7), the equation for the current density can be obtained as

aJ

-

at

Noe2 E - V O J - v1J o m

where

v1

:j

= +vo sa6vo

AA*(t - t') exp (- 6vot')dt'

and

Jo

=

N o e2(vo- iw)

mco'

A(t)e'"'.

The solution of eq. (5.8) is

J--

N o e2 E Noe2 a A eiot m(vo+ iw)' at m(vo+ iw) -

+No e2v0sc16vo m(vo+ io)2 A(t)e'..'[AA*(t-t')

exp (-6vot')dt'.

(5.9)

5.3. NONLINEAR CURRENT DENSITY : REDISTRIBUTION OF CARRIERS

When a Gaussian electromagnetic beam propagates through a plasma the main source of nonlinearity is through the heating and redistribution of electrons. The nonlinearity arising due to the temperature dependence of collision frequency is a smaller effect when v < w and hence can be neglected. We consider the propagation of an electromagnetic beam in a plasma in the z-direction. The intensity distribution of the beam in the x-direction is Gaussian and it is uniform along the y-axis; the investigations on the three dimensional beams are not reported in the literature. The amplitude varies slowly with time; the time scale of interest is the energy relaxation time

v, (i 51

241

N O N S T E A D Y STATE SELF F O C U S I N G

z,(=1/6vo). Following the treatment of electron temperature may be obtained as N , e2(vo- iw) mo2

J -

6

5.2 the current density and

+

-1

vo i o eiot aA at

(5.10)

and

:1

T,-- TO - ctdv, TO

AA*(t - t') exp (- Sv, t')dt'.

(5.11)

In these expressions N , is hitherto unknown. To obtain an expression for N , we consider the ambipolar diffusion of electrons and ions. The velocity of diffusion of electrons and ions is governed by

a

mat

-eE,-mvudc-

udc =

1 -V(N,k,T,) Ne

(5.12)

and

a

1

(5.13) V ( N ,k, To) Ni where v' is the ion-neutral particle collision frequency; the Coulomb collisions are neglected. On account of the space charge effects, the electrons and ions are diffused together, therefore udc N V,, , N , N N i. Also as the time scale for the variation of udc is very large ( k 1/6v,) as compared to the collision free time (- I/v ), the time derivatives of udc in eqs. (5.12) and (5.13) can be neglected. Then on adding eqs. (5.12) and (5.13) we obtain the following expression for the drift velocity of diffusion, m

-

V,,

=

eE, - Mv'Vdc-

-

at

(5.14)

The continuity equation, relating N , and udc can also be written as aNe

__

at

+v . (N,UdC)= 0.

(5.15)

Using eq. (5.14) in (5.15), aN, at

__-

2k,T0 V 2 N e = k o N e VZT,. mvfMv' mv + Mv'

(5.16)

To solve eq. (5.16) we assume that the beam is two dimensional (i.e. a/ay = 0) and employ the Fourier transform technique. Define

242

SELF FOCUSING OF LASER B E A M S

[v,

g(k‘) = 9 ( N e - N o ) =

D5

(5.17)

Then eq. (5.16) can be transformed as

_adk’) __

2koTo (- K 2 )g(k!) = ko No mv+ Mv’ mv+Mv’

at

d2T,

(p)

(5.18)

where the Fourier transform of d2Te/dx2using eq. (5.17) and taking the xdependence of the electric vector as e(Ei/J)exp (- x2/a2f2)can be expressed as

(2)

=-26vcrT0 e - d v t

u2

x

J27c

iexp

f

f dt‘ jm dx (1 --m

-03

[- (5

-

E )Eg(t’) U”f2

I

2

-$ikuf)

- $k u f +6vt’ . r 2 2 2

(5.19)

The solution of eq. (5.18) is

dk‘)=

mm 2Tobvua k.”j:dtlj:dt2E:(f-t1 2Jz koNo ’ ____

2k2k0TOt2

+

(mv Mv’)

- t 2 ) exp ( - 6 v t J

+ i k , 2 a2f 2 (t-tt,-tt,,

Z)

and the corresponding expression for ( N , - No) is

Using eq. (5.21) the current density can be expressed (from eq. (5.10)) explicitly in terms of No and the electric field. 5.4. NONLINEAR PROPAGATION: SELF DISTORTION OF PLANE WAVES

From eqs. ( 2 . 1 ~and ) (2. Id) the one dimensional wave equation, governing the behaviour of the electric vector of a plane uniform wave in a nonlinear plasma can be obtained as a2E az2

d2E - + -2-1 .at2

4R

aJ

c2

at

- -

(5.22)

V,

o 51

243

N O N S T E A D Y STATE S E L F F O C U S I N G

Using eq. (5.3) for the electric vector and eq. (5.9) for the nonlinear current density, eq. (5.22) for waves of slowly varying (in time) amplitudes takes the form a2A w2 = - -(Eo-iEi)A+ az2 C2

__

m

AA*(t - t’) exp ( - 6vo t’)dt’,

(5.23)

where E~

= 1-wi/02,

0;=

4nNOe2/m,

E~ =

v,o,2/u3,

E~

< E,. (5.24)

The z-variation of A can be taken as A

=

A,(z, t)e-ikZexp { -(si/2&,)kz)

(5.25)

where A , is a slowly varying function of z and k = (co/c)E$. Substituting eq. (5.25) in eq. (5.23) and neglecting d2A0/dz2we obtain

-

1 1 o2 2 sadA, exp

4

C E i 0’

(-

kz)I:A,A;(t

- t’) exp (-

6v, t’)dt’, (5.26)

where ug = dw/dk = c2/v, and is known as the group velocity of the wave; it differs considerably from phase velocity. To have physical significance of ug it can be seen from eq. (5.26)that in the case when vo -+ 0 (i.e. absorption and nonlinear effects are negligible) the solution for A , is A0 = F(t-z/v,).

(5.27)

This expression clearly shows that ug is the velocity of propagation of amplitude. In the present case when absorption and nonlinear term are finite eq. (5.26) can be solved by iteration using a new set of variables ( 2 , 4) in place of (2,t ) viz. z

=

z

and

=

t-zlv,.

(5.28)

Hence (5.29a)

244

v, 0 5

SELF F O C U S I N G OF L A S E R B E A M S

and alat

=

a/ag

(5.29b)

Expressing A , as A -A 0

-

00

e-ikS

(5.30)

(where A,, and S are real) and using eqs. (5.29), eq. (5.26)can be split into two equations (separating real and imaginary parts) aA,,

k

&.

o2

as

-+-22az ug 2 E 0 0 2 ag -

___

O,” v2sa6A,,

4csi w 2

O

exp

(-

z k z ) I:Aio({-rr) exp (-6v,t’)dt’

(5.31a) and (5.31b) The initial condition on S is S = 0 at z = 0. In the. iterative approach followed here A,, may be taken to be independent of z in the zeroeth order and hence, the solution of eq. (5.31b) is (5.32) Using the above expression in eq. (5.31a) it may be seen that the effect of S on A,, is of second order hence can be ignored. Then the solution of eq. (5.31a) is

1

x JomF(g- t’) exp ( - 6v, t’)dt’

.

(5.33)

Following SODHA,PRASAD and TRIPATHI [1974] numerical results of eq. (5.33) for the amplitude envelope A,, of an initially Gaussian (in time) pulse [F(<) = E,, exp (- 5’/?)] at some finite distance of propagation are displayed in Figs. 5.1-5.2 for typical plasma parameters. It is interesting to note from the figures that in a plasma, where Coulomb collisions are dominant the amplitude of the pulse is higher (for all values of time) in the

v, § 51

245

N O N S T E A D Y STATE S E L F FOCUSING

nonlinear case than in the linear case. The amplitudemaximum is enhanced and shifted towards higher values of time giving rise to a nonsymmetrical self distortion of the pulse. As the ratio of the pulse duration z to the energy relaxation time (zE = 1/6v,) increases, the nonlinear distortion of the pulse

Fig. 5.1. Amplitude profile of the pulse at z = 2520 (20 is the free space wavelength) for the following parameters of a laboratory plasma oi/oz= 0.6, VO/W = 0.2, s = 1, -3, aE& = 0.2, T/T. = 0,5, ( 5 , = 1/6vo).

1.2

- a

7

-

LO

0.8

0.4

v

I

I

5.-3

r_%= &.O

I

a4

I

0.8

I

I

$0

( t -Z/Vt Fig. 5.2. Amplitude profile of the pulse at z = 25L0 for the same parameters of a laboratory plasma as in Fig. 5.1 except for T/T, = 4.0.

246

[v, 0 5

S E L F F.0CUSING O F L A S E R B E A M S

is enhanced. Similar results are obtained in a weakly ionized plasma but the wave is more attenuated due to nonlinearity and amplitude maximum shifts towards the lower values of time. 5.5 NONSTEADY SELF FOCUSING

The literature on the self focusing of pulsed/modulated electromagnetic beams in plasmas is limited to two dimensional beams only. Therefore we would restrict ourselves only to this case. Using eqs. (2.la-d) and (5.10) the wave equation for the electric vector of a two dimensional beam can be written as

d’E

a’E

0%N,-No

)

E+

2iw dA . c2 at elm*.

(5.34)

In eq. (5.34) and what follows, we have not considered the effect of absorption. The electric vector E may be expressed as

E

=

A(z, x, t )

A(z, x, t ) exp {i(ot- k z ) }

=

Ao(z,x, t ) exp { -ikS(x, z, t)}

where A, and S are real. Neglecting the second derivatives of A with respect to z and t, the wave equation can be split into the real and imaginary parts to obtain (5.35) and

as 2 2-+--+ dz

as

ug at

(E) -

@ 1 PAo +.-+-c0 k2Ao ax’

=o

(5.36)

where

On transforming the variables (z, t) to (z = z and 5 = t - z / u , ) eqs. (5.35) and (5.36) assume the same forms as eqs. (4.7b) and (4.7a) respectively. Then the solutions of these equations can be written in a straightforward manner as

N O N S T E A D Y S T A T E S E L F FOCUSING

241

The equation governing the beam width parameter is given by

a’

+f ’(2, 5 - t , - t,) 4

Y

Et(5 - t , - t,) exp (-6vt,)dt, dt,

(5.37)

where Rd = ka2, and 9=

36k0 T,aao:v 168, 0 2 ( M v ’+ rnv) .

SODHA,SHARMA and TRIPATHI [1974a] have recently given a numerical solution of eq. (5.37) for amplitude modulated signals (E,(t) = E,, x

2 .o

1.6

1.2 AoCX- 0)

0.8

0.4

0.0

0.8 1.2 1.6 2 .o JLg Fig. 5. 3. Amplitude profile (at x = 0) for an amplitude modulated wave at z = 0.6& for r, = i040K, w ~ / 0 2= 0.2, aEi = 0.09, v = lo7 sec-’, 0 = 3.1 x lo3 sec-’, a = 20 cm, o = 1o”sec-’.

0.0

0.4

248

cv>5 5

SELF FO CUSI NG OF LASER B E A M S

(1 + p cos at)),the results for typical plasma parameters are displayed in Fig. 5.3. In this case the self focusing results not only in the severe overmodulation of the wave but also distorts the amplitude envelope to a considerable extent. These results are contrary to those reported in the literature for the self distortion of amplitude modulated plane uniform waves in weakly ionized plasmas. In the case of plane uniform waves the nonlinearity arising through v gives rise to the phenomenon of demodulation of waves and is a smaller order effect as compared to the one due to self focusing. Self focusing of three dimensional beams can be discussed on similar lines. For this case when the characteristic time of variation of E , is large as compared to 1/6v, the equation for the beam width parameter is of the form

(5.38)

I

0.0

-2

I 2

I

0

1

4

I

6

t/T Fig. 5.4. The location of the focus as a function of time for Rd = 10 cm, w i / o z = 0.75 and R:/R: = 10 (continuous curve) and 50 (dotted curve).

G R O W T H OF I N S T A B I L I T Y

-1.5

-1.0 -0.5

0

0.5

1.0

249

1.5

Fig. 5.5. Variation of the axial intensity of the beam as a function of time (when relaxation effects are unimportant) for R:/R: = 10.

where R, = ~ , ( E , / E ~ Eand ~ , )c2~ is given by eq. (2.54~). The solution of eq. (5.38) is

(5.39) The focus is given by (5.40)

For different values of time, zf has different values, thus a moving focus is obtained. The nature of moving focus is displayed in Fig. 5.4 for a Gaussian (in space and time) pulse; EE(t)/EG, = exp (- t 2 / z 2 ) .

The envelope of the distorted signal is shown in Fig. 5.5. The distortion of the pulse is on account of the different self focusing of the pulse in different time intervals.

9

6. Growth of Instability

6.1. GROWTH OF INSTABILITY IN A PLANE WAVEFRONT

From the analysis of preceding sections it is obvious that a nonlinear medium whose refractive index varies with the intensity of the beam, has an

250

S E L F F O C U S I N G OF L A S E R B E A M S

[v, 9: 6

inherent character of concentrating the electromagnetic energy around the points of intensity maxima. In the case of a Gaussian electromagnetic beam the intensity is maximum on the axis and hence the power is concentrated towards the axis resulting in the self focusing of the beam. By a similar reasoning one can predict the behaviour of small perturbations in the intensity distribution of a plane uniform wavefront. The effective dielectric constant of the plasma for the high intensity portion (the variation of intensity along the wavefront is caused by the perturbation) is higher than that for the lower intensity portion, as a consequence the energy should concentrate around the intensity maxima ; the intensity minima should be depleted in energy. Thus the perturbation could grow in the plasma which leads to light filamentation. The problem of stability of intensity distribution of an intense electromagnetic beam in a plasma has recently become important. KAw, SCHMIDT and WILCOX[1973] and SODHA,KUMAR, TRIPATHI and KAw [19731 have investigated the nature of perturbations in the intensity distribution of an otherwise uniform plane wavefront. They investigated the propagation of sinusoidal perturbations in the intensity distribution along the wavefront and found that small scale perturbations could grow in the course of propaga\ion of the beam. In the present section we have followed these treatments to study the growth rate of perturbations in the intensity of an otherwise uniform plane wavefront. Only those perturbations are considered which have a life time (duration) greater than the relaxation time for nonlinearity, otherwise it would not be possible for the nonlinearity in the dielectric constant of the plasma to be established according to the new intensity profile and hence the perturbation could not grow. We consider that a plane electromagnetic wave, polarized in the y direction is propagating along the z-axis in a plasma. The intensity of the wave is high but the uniform intensity distribution does not allow any redistribution of carriers; the pondermotive force and the gradient in the thermal pressure of the plasma are zero. Therefore if we ignore the attenuation of the wave (i.e. v <( o),as we have done, the effective dielectric constant of the plasma for this wave is field independent viz. E = E, . Then in the absence of any perturbation the electric vector of the wave in the plasma can be taken as

where k = (O/C)E~ and E, is a real constant. Over this electric field, we superimpose a small perturbation. Then the total electric field may be expressed as

v, 0 61

E

=

25 1

G R O W T H OF I N S T A B I L I T Y

E , exp ( i ( w & - k z ) ) + E , ( x , y, z ) exp {i(ot-kz));

\Ell << E , .

(6.2)

The nature of E l has to be investigated. In the presence of this electric field the dielectric constant of the plasma becomes field dependent. First we consider the heating and redistribution of electrons. Following the analysis of 0 2, the rise in electron temperature may be obtained as

and the corresponding expression for the electron concentration (due to the redistribution) as

Using eq. (6.4) the effective dielectric constant of the plasma can be written as E =

E,+E~E,-(E~+ET),

(6.5)

where (6.5a)

w2 2+aEg'

Other symbols have been defined in 5 2. In the case of pondermotive nonlinearity a similar treatment results in the expression (6.5) for the dielectric constant with E~ as given by (6.5b) The wave equation describing the behaviour of the electric vector can be written as O '

V 2 E - V ( V * E ) + 7EE = 0.

(6.6)

C

Using eq. (2.1a'), eq. (6.6) takes the form

( : ) : :+

V2E+V E . - V E

-&E

=

0.

(6.7)

Substituting the expression (6.2) for E in (6.7), we obtain the following equation (after linearization) for E l ,

252

[v. § 6

SELF F O C U S I N G OF LASER B E A M S

+ coz C

E~

Eo(E1,+ ET,)Eo

=

0.

(6.8)

It can be seen from eq. (6.8) that the nonlinear effects in the propagation of E l , and E l , are finite only when E l , # 0. The perturbations for which E l , = 0 are not affected by the nonlinear effects and hence are of no interest to us. Moreover when E l , # 0, the parts of E l , and E l , affected by the nonlinearity are very small and can be ignored. Thus the perturbations of interest are virtually polarized in the y-direction (i.e. El 1 I E , ) and we solve eq. (6.8) only for Ely.Expressing El, = El,+iEli, we can write eq. (6.8) for its real and imaginary parts as

a aZ

V2E,,+2k-Eli+

a2 o2 2E2 -Ei-EE,,+2-~2E~E,,

dY2

Eo

= 0,

(6.9a)

C2

and V2Eli-2k

a ~

aZ

El,

=

0.

(6.9b)

In order to solve these coupled equations for the real quantities E l , and Eli we employ the complex notation and assume the variation of the form exp (-i(ki,z+k; . r ) }

(6.10)

where r =x +y is the projection of the coordinate vector in the x-y plane, k' = k , ;+ k ; , the suffixes I I and I refer to the components parallel and normal to the z-direction. k;, and k; are to be determined below. Using eq. (6.10) in eqs. (6.9a) and (6.9b) we obtain -(k'+k;t)E1,-

2E2 E i ~

80

qzEl,+

2w2 --

CZ

~ ~ E i E ~ , - 2 k k ; ,=i E0 ~ ~

(6.11a)

and 2kk;,iEl,-(k;Z+k;t)E,i

=

(6.11b)

0.

These equations have non trivial solutions only when the determinant of the coefficients of E l , and Eli is zero, i.e., when

(K'

+k ; f )

[kf +kf +

2~ E 2 2

0 &O

k:

202 -

~

C2

e2

"$1

=

4k2k;f.

(6.12)

v, P 61

253

G R O W T H OF I N S T A B I L I T Y

On treating k;; < k;’ (i.e. the scale length of perturbation in the x-y plane is small) which is a valid assumption, the dispersion relation (6.12) simplifies to give (6.13) It is obvious from this relation that whenever, (6.14)

k;, becomes imaginary, i.e., the perturbation grows as it advances in the z-direction. The growth rate is given by

To discuss the behaviour of growth rate we consider two special cases. Let k; = 0, k; = k:. In this case the wave vector of perturbation is at right angle to the electric vector and the perturbation propagates as a TE wave. The growth rate from eq. (6.15) comes out to be

r is a function of k; and shows a maximum corresponding to dk;,/dkx = 0, i.e. kLJ,,,

=

k(E2E ~ / E ~ ) *

and the maximum growth rate is

rmax = k s , E;/2c0. We consider another special case corresponding to k; the growth rate (cf. eq. (6.15)) comes out to be

(6.15a) = 0,

k;

= k;,

then

This optimizes around

ky,,,5 k [A + 2 ] - ’ E2

E,2

with the maximum growth rate as (6.15b)

254

SELF FO CUSI NG OF LASER B E A M S

[v, 0 6

The consequence of this instability is that as the plane wave propagates in a nonlinear medium in the z-direction, it may split up in filaments in the direction normal to the z-axis. The transverse scale length for filaments (or the perturbation) is k ; ’ . It is very interesting to note from the treatment of 9.5 that the nonlinear self focusing length for a beam of dimension k;;:, in a nonlinear plasma is

Using the expressions for k;opt(k:opt or kiopt)derived above, R, can be estimated to be (6.16) This shows that the growth of perturbation may be treated as the self focusing of the beam around the intensity maxima due to the field dependent dielectric constant. This is a big coincidence that R, and r,& are of the same order. Now we examine the equilibrium intensity distribution of the beam to which it is driven by the instability. For the sake of mathematical simplicity the analysis is restricted to TE mode only, i.e., we take the electric vector in they-direction and neglect all variations in this direction. We consider a solution representing the wave propagation along the z-direction as E

= j?E,(x)

exp { i(wt - fi, z ) }

(6.17)

where fi0 is some space independent constant to be determined. Using eq. (6.17) in eq. (6.6), we obtain

:(

+ -&-fig

) E,

=

0.

(6.18)

On multiplying eq. (6.18) by 2dE,/dx and integrating once, we get (6.19) In writing (6.19) we have assumed that E, and dE,/dx both vanish at infinity, i.e. we are limiting ourselves to monotonically decreasing solutions. At the centre of the beam E, = E,, and dE,/dx = 0. Hence, eq. (6.19) gives (6.20)

"2

§ 61

255

G R O W T H OF I N S T A B I L I T Y

or (6.20a)

To solve eq. (6.19), we evaluate the integral in two cases

(A) Pondermotive nonlinearity E = E~

o2 +2 (1 - exp { - (m/iM)aE;}), o2

joE'&

dE; = e0 E; + II/(E;)

where (6.21)

(B) Heating/redistribution nonlinearity &

= E0+

o2

2 [1-(1+&E;)"'2-'], w2

joE2&dE;=

E~

Ei

+ Y(Ei)

where (6.22)

s/2

- l)]

for s # 0.

(6.23)

CIS

Using eqs. (6.21)-(6.23) in eq. (6.19) we get (6.24) Analytical integration of eq. (6.24) is difficult. However, the perpendicular scale length can be estimated as

Y

=

(m2&o/c2- P o 2)

-+.

(6.25)

256

SELF F O C U S I N G OF LASER BEAMS

Iv, 4 6

It can be seen from eqs. (6.15a), (6.21)-(6.25) that the perpendicular scale lengths of the equilibrium are typically of the same order as the optimum perpendicular scale length for the instability discussed earlier. Thus it is quite likely that these equilibria denote the final state of the instability. Similar result is expected for the other case when a / a x = 0, a/dy # 0. We might thus conclude that the final state of the intense electromagnetic wave will be cylindrical filaments with typical scale length given by eq. (6.25). 6.2. GROWTH OF INSTABILITY IN A GAUSSIAN BEAM

In this section we take up the problem of growth of a small amplitude perturbation in the intensity distribution Lf a Gaussian electromagnetic beam. The perturbations considered are of (i) small scale length (as compared to the dimension of the beam) and (ii) have mall characteristic length for their growth (as compared to the focusing length of the main beam). As a matter of fact condition (ii) is implied in (i) because the self focusing length (either of the main beam or of the perturbation) is proportional to size of the beam (or perturbation); the other factors in the expression for R, or F;: are almost the same. These restrictions are necessary because the intensity distribution of a Gaussian electromagnetic beam in a nonlinear plasma is not yet investigated beyond the paraxial region. The growth of perturbation in the present case should differ from that in the case of a uniform plane wavefront in two ways. (i) The main beam is mainly responsible for the redistribution of carriers; the perturbation causes a small modification. Therefore the saturation effects of field dependence of dielectric constant should influence the growth rate. (ii) The main beam gets focused in course of its propagation and hence the growth rate of perturbation should have an axial variation. We consider the propagation of a two dimensional Gaussian electromagnetic beam along the z-direction in a plasma. In the paraxial ray approximation the electric vector of the beam can be taken as (consistent with the analysis of 4 4) E A

=

A:

A exp ji(ot-kz)),

=

A,(x, z) exp { -ikS(x, z)}, =

E; ~

f

exp (- x”/r; f”),

s = +x’P(z)+ cp(z)

(6.26a) (6.26b) (6.26~)

"7

§ 61

G R O W T H OF I N S T A B I L I T Y

251

where f is governed by (6.27) Symbols are defined in fj 4.In writing eq. (6.27) we have expressed the field dependent dielectric constant as E =

&,+@(EE*)

where @, in the case of pondermotive nonlinearity is given by eq. (2.44b) and in the case of heating induced nonlinearity by eq. (3.19b). To the electric vector of the main beam we add a small perturbation viz.

E = A, exp (i(ot - k(z + s)))+ E' exp {i(m- k(z + s)))

(6.28)

lA,l.

IE'I

A may be taken to be polarized in the y-direction. Substituting for E in eq. (6.6) the equation governing E; (on linearization) can be written as V2Eb-2ik-

aEy aZ

+ __ (I rif2

$)

El - ik/lEb - 2ikBx

~

aEy ax

where we have expanded the dielectric constant as E(E. E*)

=

&(A;)+ @(A;)A,, . (E' + E * ) .

(6.30)

The x and z-components of E are affected by the nonlinearity to a much lesser extent and hence are of no interest to us. Writing EI = Er'-iE:, eq. (6.29) can be separated in its real and imaginary parts V2E:+2k-

dEI

1 + az r 2 f 2 (1~

6)

aE; E:+k/lE;+2kp~-

ax

and aE: V2E;-2k- aZ

1

+ __ r g f 2 (1-

6)

aE; EI-k/lE:-2Wx- ax = 0.

For small scale fluctuations (around x

G

(6.3ib)

0) the solution of eqs. (6.31a) and

258

SELF F O C U S I N G O F L A S E R BEAMS

Cv. 5 6

(6.31b) can be obtained in a similar manner as in 5 6.1. Expressing the space variations of E: and E; as

+

+

exp [ - i(kilz k; . (x y ) ) ] the dispersion relation for k;,can be obtained as

The conditions for the validity of eq. (6.32) are

and

k~'

ro f

(6.33b)

where Rneffis the distance over which the self focusing of the main beam occurs. (i.e. for TE waves), the It may be seen from eq. (6.32) that for k; = perturbation grows when 1

:k < -r t f'

+ 2w2 @'A: ~

c'

(6.34)

and the growth rate is given by

This optimizes around (6.36a) and the maximum growth rate is 1

0 '

rmax = g?+- - @'A;. 2k c2

(6.36b)

Almost similar expressions can be written for the growth rate when k; is aligned along the y-axis. We discuss the growth of perturbation in two interesting cases.

(A) Uniform selfmade wateguide This occurs when the nonlinear term in eq. (6.27) is balanced by the diffraction term i.e..

v, 0 61

G R O W T H OF I N S T A B I L I T Y

@'EO/cOri= l/Ri.

259

(6.37)

In this case d2f/dz2 and dfdz are zero and f remains unity for all values of z. Using (6.37) in (6.34) we see that only those perturbations can grow for which

k: < $ / Y O ,

(6.38)

i.e. when the perturbation is of very large dimensions. The condition (6.38) violates eq. (6.33b) hence cannot be investigated. However, it is obvious from the present analysis that a selfmade waveguide in a plasma is stable for small scale perturbations.

(B) Oscillatory waveguide For the validity of the treatment given here the condition @'E:/cOri > 1/Ri

(6.39)

must be satisfied. Eq. (6.39) corresponds to saying that the nonlinear term in eq. (6.27) is much larger than the diffraction term. Under this condition the main beam propagates in the selfmade uniform waveguide ; f is an oscillatory function of z between the values f,,, = 1 and ,fmin = 1. Consequentlythe growth rate ofperturbation is also an oscillatory function z. It is also interesting to see from eq. (7.36b) that rmax shows a maximum as a function of A , (at z = 0) for some optimum value of A , . A plot of rmax is given in Fig. 6.1. Corresponding to the optimum value of A,(z = 0)

- 5- - 3

Fig. 6.1. Variation of rmax with the (axial) intensity of the beam at z = 0 in a collisional plasma where the nonlinearity arises due to the heating and redistribution of electrons.

260

SELF F O C U S I N G OF LASER B E A M S

Fig. 6.2. Variation of fmar with z in a collisional plasma (when the main beam propagates in an oscillatory waveguide) f o r m = 10" sec- ', i o= 30 cm, m:/m' = 0.25, aAa(x = 0, z = 0) = 1.0.

a plot of rmax as a function of z is given in Fig. 6.2.The behaviour of growth rate in a collisionless plasma where the nonlinearity arises due to ponderand TRIPATHI [1974b]). motive force is similar (SODHA,SHARMA 6.3. GROWTH O F A GAUSSIAN PERTURBATION OVER A PLANE UNIFORM WAVEFRONT

To have a little more insight into the growth of a local perturbation it would be useful to visualize the growth of a Gaussian perturbation over a uniform wavefront. Let the perturbation at z = 0, have the amplitude of the form El = El, exp ( - y 2 / a 2 )

where

V,

o 71

26 I

EXPERIMENTAL INVESTIGATIONS

is the Fourier transform of E , .At some finite value of z , E , can be written as

1-

1 " EAY, z ) = (271)" $k;)

exp { - i(k;y+ k;,4>d 6

where k;l is a function of kl as given by eq. (6.15) (when k: = O).Thenumerical computation of the above expression is displayed in Fig. 6.3. It is obvious from the figure that the amplitude of the perturbation on the axis O., = 0) increases with increasing z (i.e. as the beam advances in the plasma). Further, the width of the perturbation also increases with increasing z, i.e., the power content of the perturbation increases as the wave propagates. I

1

I -.

-

-

wI

0.0

I

0.2

I

0.4

0.6

aa

Fig. 6.3. Amplitude profile (in the transverse plane) of a Gaussian perturbation over a uniform plane wave-front at various values of z ; /lo is the freespace wavelength. The otherparameters are aEi = 0 . 1 6 , w ~ / w 2= 0.5, ka =

0

7. Experimental Investigations on Self Focusing

In contrast to the extensive experimental work on self focusing of electromagnetic beams in dielectrics, the expedmental studies on self focusing in gaseous and semiconductor plasmas are very much limited. In the case where detailed experimental results have been reported, the inherent nonuniformity and nonstationary nature of the plasma make the interpretation of results extemely complicated. Nevertheless an experimental evidence for

262

cv

S E L F F O C U S I N G OF L A S E R B E A M S

the phenomenon of self focusing of electromagnetic beams in plasmas can be found from the novel experiment by EREMIN, LITVAK and POLUYAKHTOV [1972]. The experiment was carried out with a decaying discharge plasma (initial electronic concentration 1 0 ~ ~ - 1 0cm~ ’, electron temperature 5000 OK) having decay time of the order of a microsecond. The plasma was found to be uniform to an accuracy of 10% when the electron concentration had fallen to a value N , 10” ~ m - At ~ . one end of the plasma, a converging electromagnetic beam (of free space wavelength I , N 3 cm) is made incident from a lens. The initial radius of the beam is 10 cm and the distribution of intensity in the transverse plane is Gaussian. The receiver was mounted at a longitudinal distance of k 20 cm from the transmitting lens. In the absence of the plasma the beam is focused at a distance z N 20 cm and its radius is 3 cm. When the space is filled by the decaying discharge plasma, the transmission is obstructed in the beginning. However, when the electronic concentration decreases below the critical value, some power is received at the receiver. The characteristic parameters of the plasma at this instant may be taken as follows :

-

-

-

-

-

N,

=

5 x 10” ~ r n - ~ , NJneutral particles) = 4 x 1015 ~ r n - ~ ,

v,,,,~,~, = 7 x 10’ sec-’,

v , - ~=~9 ~x 10’ sec-

’

The critical power for self focusing due to heating and redistribution of carriers is P,,N 10 watt. The power used in the experiment was 5 100 watt, the corresponding focusing length could be computed as k 70 cm. This shows that the focusing length is not much affected by the presence of the plasma but the dimension of the focal spot should be influenced by the self focusing effect. This speculation was supported by observations. Further, the plasma on the axis of the beam was observed to decay more rapidly with time (- 5 x sec). sec) than the plasma away from the axis (- 9 x This shows that the plasma in the high field region is appreciably depleted on account of diffusion. A satisfactory quantitative interpretation of these transient results is however, quite involved and has not yet been attempted.

References AKHMANOV, S. A. and R. V. KHOKHLOV, 1972, Problems of Non Linear Optics (Gordon and Breach, New York). AKHMANOV, S. A., A. P. SUKHORUKOV and R. V. KHOKHLOV, 1966, Self-focusing and Selftrapping of Intense Light Beams in a Nonlinear Medium, Sov. Phys. JETP 23. 1025.

vl

REFERENCES

263

AKHMANOV, S. A. and A. P. SUKHORUKOV, 1967, Nonstationary Self-focusing of Laser Pulses in a Dissipative Medium, JETP Letters 5, 87. AKHMANOV, S. A,, A. P. SUKHORUKOV and R. V. KHOKHLOV, 1968, Self Focusing and Self Trapping of Intense Light Beams in a Non Linear Medium, Sov. Phys. Uspekhi 10,609. BALDWIN, G. C., 1969, An Introduction to Non Linear Optics (Plenum Press, New York). BLOEMBERGEN, N., 1965, Non Linear Optics (W. A. Benjamin Inc., New York). BLOEMBERGEN, N., 1973, Picosecond Non Linear Optics, in: Fundamental and Applied Laser Physics: Proc. Esfahan Symp., eds. M. S. Feld, A. Javan and N. Kurnit (John Wiley, N.Y.). BOOT,H. A. H., S. A. SELFand R. B. R. SHERSBY-HARVIE, 1958, Containment of a Fully Ionized Plasma by Radio Frequency Fields, J. Electr. and Control. 4,434. BORN,M. and E. WOLF,1970, Principles of Optics (4th ed., Pergamon Press, London). CHAPMAN, S. and T. G. COWLING,1952, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press). CONWELL, E. M., 1967, High Field Transport in Semiconductors, Solid State Phys. Suppl. 9 (Academic Press, New York). DEMCHENKO, V. V. and A. M. HUSSEIN,1973, On the Theory of TM-Wave Self-focusing in an Anisotropic Plasma, Physica 65, 396. DESLOGE, E. A. and S. W. MATTHYSSE, 1960, Collision Term in the Boltzmann Transport Equation, Am. J. Phys. 28, 1. 1972, New Proposition in the Mechanism of SelfDUBEY,P. K. and V. V. PARANJAPE, focusing of Laser Beams in Semiconductors, Phys. Rev. B6, 1321. 1973, Self Action of Laser Beams in Semiconductors, DUBEY,P. K. and V. V. PARANIAPE, Phys. Rev. BS, 1514. DYSTHE,K. B., 1968, Self-trapping and Self-focusing of Electromagnetic Waves in a Plasma, Phys. Lett. 27A, 59. EHRENREICH, N., 1957, J. Phys. Chem. Sol. 2, 131. EREMIN,B. G., A. G. LITVAK and B. K. POLUYAKHTOV, 1972, Investigation of Thermal EM Wave Self Focusing in Plasmas, Izv. VUZ. Radiofiz. 15, 1132. GHATAK,A. K., I. C. GOYAL and M. S. SODHA,1972, Series Solution for Steady State Selffocusing with Saturating Nonlinearity, Optica Acta 19, 693. GINZBURG, V. L., 1970, The Propagation of Electromagnetic Waves in Plasmas (2nd ed., Addison Wesley, Reading, Mass.). GUHA, S. and V. K. TRIPATHI,1972, Laser Focusing in GaAs: Intervalley Transfei Mechanism, Phys. Stat. Solidi 13, 981. HORA,H., D. PFIRSCH and A. SCHLUTER, 1967, Beschleuniging von inhomogenen Plasmen durch Laserlicht, Z. Naturforsch. 22a, 278. HORA,H., 1969, Self-focusing of Laser Beams in a Plasma by Pondermotive Forces, Z. Physik 226, 156. HORA,H., 1972, Nonlinear Forces in Laser Produced Plasmas, in: Laser Interactions and Related Plasma Phenomena, Vol. 2, eds. H. J. Schwarz and H. Hora (Plenum, New York) p. 341. JAIN,M., J. I. GERSTEN and N. TZOAR,1973, Magnetic Field Enhancement of Self Focusing of Laser Beams in Semiconductors, Phys. Rev. BS, 2710. KANE,E. O., 1957, Band Structure of Indium Antimonide, J. Phys. Chem. Sol. 1,249. KAw, P., G. SCHMIDT and T. WILCOX,1973, Filamentation andTrappingofElectromagnetic Radiation in Plasmas, Phys. Fluids 16, 1522. LEVINSTON, M. E. and M. S. SHUR,1972, The Gunn Effect Review, Sov. Phys. Semiconductors 5, 1561. LITVAK,A. G., 1968, On the Possibility of Electromagnetic Wave Self Focusing in the Ionosphere, Izv. VUZ Radiofiz. (USSR) 11, 1433. LITVAK,A. G., 1969, Finite Amplitude Wave Beams in a Magnetoactive Plasma, Zh. Eksper. Teor. Fiz. (USSR) 57,629. MORSE,P. M. and H. FESHBACH, 1953, Methods of Theoretical Physics (McGraw Hill, Inc.,

264

S E L F F O C U S I N G OF L A S E R B E A M S

[v

New York). PRASAD, S. and V. K. TRIPATHI,1973, Redistribution of Charged Particles and Self Distortion of High Amplitude Electromagnetic Waves in a Plasma, J. Appl. Phys. 44,4595. RASHAD,A. R. M., 1966, Self-focusing of a Laser Beam in Plasma, Proc. National Electronics Conf., Chicago, Vol. 22 (Chicago, Nat. Electron. Conf. Inc. 1966) p. 19. REINTJES,J., R. L. CARMAN and F. SHIMIZU,1973, Study of Self-focusing and Self-PhaseModulation in the Picosecond-time Region, Phys. Rev. AS, 1486. SCHLUTER, A,, 1969, Pondermotive Action of Light, Plasma Phys. 10, 471. SHEARER,J. W. and J. L. EDDLEMAN, 1973, Laser Light Forces and Self-focusing in Fully Ionized Plasmas, Phys. Fluids 16, 1753. SHKAROFSKY, I. P., T. W. JOHNSTON and M. P. BACHYNSKI, 1966, The Particle Kinetics of Plasmas (Addison Wesley, Reading, Mass.). SHOCKLEY, W., 1951, Hot Electrons in Germanium and Ohm's Law, Bell Sys. Tech. J. 30,990. SODHA,M. S., 1973, Theory of Nonlinear Refraction: Self Focusing of Laser Beams, J. Phys. Education (India) 1, No. 2, 13. and V. K. TRIPATHI,1974, Self Focusing of Laser Beams in SODHA,M. S., A. K. GHATAK Dielectrics, Plasmas and Semicdnductors (Tata McGraw-Hill Publ. Co., New Delhi). SODHA,M. S., S. C. KAUSHIK and V. K. TRIPATHI,1974, Self Focusing of EM Waves in Degenerate Electron-Hole Plasma, Appl. Phys. (W. Germany) 4, 141. SODHA,M. S., R. K. KHANNAand V. K. TRIPATHI,1973a, Self Focusing of EM Waves in InSb: Predominance of Nonlocal Effects, Appl. Phys. (W. Germany) 2, 39. and V. K. TRIPATHI, 1973b, Self Focusing of a Laser Beam in a SODHA,M. S., R. K. KHANNA Strongly Ionized Plasma, Opt0 Electronics 5, 533. @€IDA, M. S., R. K. KHANNA and V. K. TRIPATHI,1974, The Self Focusing of EM Beams in a Strongly Ionized Magnetoplasma, J. Phys. D: Appl. Phys. 7,2188. SODHA,M. S., A. KUMAR, V. K. TRIPATHI and P. KAw, 1973, Hot Carrier Diffusion Indiced Instability and Filament Formation in Plasmas and Semiconductors, Opt0 Electronics 5, 509. SODHA, M. S., R. S. MITTAL, S. K. VIRMANIand V. K. TRIPATHI, 1974, Self Focusing of Electromagnetic Waves in a Magnetoplasma, Opt0 Electronics 6, 167. and V. K. TRIPATHI,1974, Self Distortion of a Gaussian ElectroSODHA,M. S., S. PRASAD magnetic Pulse in a Plasma, Appl. Phys. (Germany) 3, 213. and V. K. TRIPATHI,1974a. Self Distortion of an Amplitude SODHA,M. S., R. P. SHARMA Modulated Electromagnetic Beam in a Plasma : Relaxation Effects, Appl. Phys. (Germany) 5, 153. SODHA, M. S R P. SHARMA and V. K. TRIPATHI. 197411. lnstahilitv 0 1 Inten$it> Distrihution, of a Laser Beam 111 A Plasma, J. Phys. D: Appl. Phys. 7,2471. SODHA,M. S., D. P. TEWARI, J. KAMAL, H. D. PANDEY, A. K. AGARWAL and V. K. TRIPATHI, 1973, Nonlinear Mechanisms for Self Focusing and Propagation of Microwave Pulses in Semiconductors, J. Appl. Phys. 44,1699. SODHA,M. S., D. P. TEWARI,A. KUMARand V. K. TRIPATHI, 1974, Saturating Nonlinear Dielectric Constant and Self Focusing of EM Waves in Plasmas: Kinetic Approach, J. Phys. D: Appl. Phys. 7, 345. SODHA,M. S., D. P. TEWARI, V. K. TRIPATHI and J. KAMAL,1972, Self Focusing of Microwaves in n-InSb, J. Appl. Phys. 43, 3736. SVELTO, O., 1974, in Progress in Optics, vol. X11, ed. E. Wolf (North-Holland, Amsterdam) pp. 1-51. TAW(T I . T. and H. WASHIMI, 1969. Self Focusing of a Plasma Wave along a Magnetic Field, Phys. Rev. Lett. 22,454. TEWARI, D. P. and A. KUMAR, 1974, Periodic Focusing of a Gaussian Electromagnetic Beam in a Magnetoplasma, Proc. Symp. Quantum and Opto-Electronics, B.A.R.C. Bombay (India). TEWARI, D. P., H. D. PANDEY, A. K. AGARWAL and V. K. TRIPATHI, 1973, Microwave Faraday Rotation and Self Focusing of Helicon Waves in n-InSb, J. Appl. Phys. 44,3153.

vl

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265

V. K., M. S. SODHAand D. P. TEWARI, 1973, Nonlinear Interaction of a Gaussian TRIPATHI, Electromagnetic Beam with Germanium : Kinetic Treatment, Phys. Rev. B8, 1499. TRIPATHI,V. K., D. P. TEWARI,H. D. PANDEY and A. K. AGARWAL, 1973a, Damping Criterion fer the Focusing of Laser Beams in Semiconductors, J. Phys. D: Appl. Phys. 6, 363. TRIPATHI, V. K., D. P. TEWARI, H. D. PANDEY and A. K. AGARWAL, 1973b, Effect of Self Focusing on the Self Distortion of Amplitude Modulated Microwaves in Nonparabolic Semiconductors, Opt0 Electronics 4, 131. TZOAR,N. and J. I. GERSTEN,1971, Calculation of the Self Focusing of Electromagnetic Radiation in Semiconductors, Phys. Rev. B4,3540. TZOAR,N. and J. I. GERSTEN,1972, Calculation of the Self Focusing of Electromagnetic Radiation in Semiconductors, Phys. Rev. Lett. 26, 1634. H., 1933, Self Focusing of Transverse Waves in a Magnetoplasma, J. Phys. SOC. WASHIMI, (Japan) 34,1373.

This Page Intentionally Left Blank

E. WOLF, PROGRESS IN OPTICS XI11 0 NORTH-HOLLAND 1976

VI

APLANATISM AND ISOPLANATISM BY

W. T. WELFORD Physics Department, Imperial College, London SW7 2 8 2 , U.K.

CONTENTS PAGE

$ 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . 269 $ 2. THE ABBE SINE CONDITION . . . . . . . . . . . . . 271

5 3 . AXIAL ISOPLANATISM . . . . . . . . . . . . . . . . 273 5 4. ISOPLANATISM WITH NO AXIS OF SYMMETRY . . . 283 9 5 . ISOPLANATISM IN HOLOGRAPHY . . . . . . . . . . 289 REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

291

ADDENDUM . . . . . . . . . . . . . . . . . . . . . . .

292

9

1. Introduction

Apart from a short paper by HOFMANN [1973] to mark the centenary of the publication by Abbe of the sine condition, there seems to have been no review of the present topic. In what follows we shall be concerned with conditions for stationarity of aberrations in image-forming systems to which Fermat’s principle can be applied; this includes, in addition to ordinary lens and mirror systems, diffraction gratings and holograms, since in all these the point image is formed by a wavefront with a clearly defined phase distribution ; however, devices such as Fresnel lenses are excluded since in these no attempt is made to get a definite phase distribution and the system depends on ray optics only. The modern accepted meanings of the terms aplanatism and isoplanatism are as follows : a) Aplanatism. This term is thought of as applying to axisymmetric optical systems near the centre of the field; such a system is said to be aplanatic if it is free from spherical aberration, i.e. the axial image is aberrationless, and if there is no aberration varying linearly with field angle, i.e. no linear coma. b) Isoplanatism. This can apply to any small region of the field of an optical system with any or no symmetry ;the system is said to be isoplanatic in this region if the aberrations are stationary for smalldisplacements of the object point, i.e. if the aberration has no component depending linearly on the object position. Before elaborating on these definitions - in particular the aberrations and the kind of object displacement envisaged in the second definition must be specifiedmore precisely - we note that the word aplanatism was originally not used in quite the above sense; it derives from the Greek and means simply freedom from wandering or aberration. It seems to have been coined by BLAIR[1791], who used it to mean improved correction of secondary spectrum in achromatic doublets. Subsequently, according to KING[1955], it was used by J. Herschel in 1827 to mean freedom from 269

270

APLANATISM A N D ISOPLANAT ISM

[VI,

D

1

spherical aberration. HAMILTON [18441 referred to “direct aplanaticity”, meaning freedom from spherical aberration, and “oblique aplanaticity”, or freedom from linear coma as it would now be called. It was not until ABBE’S publication [I8731 that the term aplanatism was used in its present-day accepted sense. The term isoplanatism is more recent ; it was introduced by STAEBLE [1919], who coined the German noun Isoplanasie to mean the condition of a symmetrical optical system with spherical aberration but with zero linear coma. More recently “isoplanatism” has been applied to any situation in an optical system with no symmetry where the aberrations are stationary for small displacements of the object point; for example, FELLGETT and LINFOOT [19551 spoke of an isoplanatism patch, meaning an area in the field over which this condition holds to a suitable approximation. DUMONTET 119551 discussed this matter in detail and showed that even if an optical system had the same intensity point spread function over a certain field, there would in general be phase variations which could affect isoplanatism for partially coherent illumination ; however, in the present review we shall be concerned with isoplanatism as a geometrical optics concept only. The sine condition, as originally stated by ABBE[1873], referred to the constancy of transverse magnification by different zones of the aperture of .an optical system, but this is only one possible point of view. CLAUSIUS [18641 considered radiative equilibrium between a black body radiator and its optical image and he found a condition for the image formation to be perfect from which the Abbe sine condition can be easily deduced. The optical designer’s viewpoint, that the condition refers to zero aberration linear with field angle, was, of course, also apparent to Abbe but his original derivation referred to zonal magnification. Yet another viewpoint, which is concerned mainly with isoplanatism, is that the aberrations must be substantially constant over a certain $ze of field in order that images of extended objects may be calculated by integrating the effects of a suitable array of point images. Such images were first computed by STRUVE[1882] but it was not until the widespread use of Fourier transform methods of computing images that the need for the condition of isoplanatism to be fdfilled was stated (e.g. FELLGETT and LINFOOT [1955]). More recently still, the concept of isoplanatism has been extended [1973]). to certain cases of holographic image formation (WELFORD In the Abbe sine condition small displacements of the object point perpendicular to the axis are considered and the same applies to the various forms of axial isoplanatism condition (see section 3). HERSCHEL [1821] had, long before Abbe, given the corresponding condition for axial dis-

VL

§ 21

T H E ABBE S I N E C O N D I T I O N

27 1

placements of the object point. From the present viewpoint this is very similar to the sine condition, since both are concerned with stationarity of aberrations for small object displacement, but Herschel considered it as a condition for constancy of aberrations with change of magnification. Much stress has been laid on the fact that the Abbe sine condition and Herschel’s condition are mutually inconsistent except at certain special magnifications. However, it is now clear that in general isoplanatism at given conjugates can only hold for certain directions of object displacement; this applies to systems of any symmetry and to holographic systems, so that it is always necessary to state the direction of object displacement in discussing isoplanatism. The same applies, mutatis mutandis, to the position of the effective aperture stop, unless the imagery is aberration-free. The conditions to be discussed in this review take the form of tests to be applied to individual rays traced through an optical system, so that in the first instance they apply only to these rays and perhaps others associated with them by symmetry. In addition, there is sometimes implicit in the conditions the statement that they must hold for all rays through the chosen aperture. Also the rays concerned will be finite or exactly traced rays; thus we shall not discuss conditions for freedom from coma in the primary or Seidel approximation, which is based on paraxial raytracing. Some of the results to be described, e.g. the Abbe sine condition itself, are necessary and sufficient conditions for aplanatism or isoplanatism to hold but they do not give any information about the magnitude of the effects if the conditions do not hold. Others, such as the celebrated “offense against the sine condition” of CONRADY [1929], do give explicitly the change in aberration for a movement of.the object point. Since the practical optical designer is always concerned with tolerances for aberrations rather than with reducing aberrations precisely to zero, results of the second kind are [19711) that more useful than those of the first. It was suggested (WELFORD a distinction could be made by talking of isoplanatism conditions and nonisoplanatism formulae respectively, thus emphasising the more important role of the second class of result.

9

2. The Abbe Sine Condition

The history of what has come to be known as the optical sine condition [1905J and by or Abbe’s sine condition has been discussed by CONRADY CZAPSKI and EPPENSTEIN [1924]. It is often stated that CLAUSIUS [I18641 discovered the condition ; in the course of studies on radiative equilibrium he obtained by thermodynamical arguments the following result for perfect

212

APLANATISM A N D ISOPLANATISM

[VI, §

2

imagery of a surface element do onto an element do' by a narrow pencil of rays : n2 cos 8 do dS2 = nf2 cos 8' do' dS2' ;

(1)

in this equation dB is the solid angle of the rays, 8 is the angle which the imaging pencil makes with the normal to the surface element and y1 is the refractive index. The same result was obtained independently by STRAUBEL [1902]. If the element do is on the axis of a symmetrical optical system, it is possible to integrate over the solid angle and obtain

n2 sin2 8do

=

nr2 sin2 0 do'

(2)

and if this is paraxialized and the ratio taken, we obtain the square of the sine condition. It is because he gave eq. (1) that Clausius is generally regarded as the discoverer of the sine condition. ABBE[1873] stated the sine condition without proof as the condition for aplanatism, using the word in its modern sense. Immediately afterwards HELMHOLTZ [1874] showed that if do and do' are black body radiators in equilibrium and if they are aplanatic conjugates, then the sine condition must hold. The approach of Clausius and Helmholtz is reproduced in English by DRUDE[1917]. Abbe did not give a proof of the sine condition in his 1873paper and there are no references in the paper. However, six years later he published a proof (ABBE[1879]), which essentially derived the condition that all zones of the aperture should give the same magnification for a small object on the axis. The proof, which is reproduced by CZAPSKI and EPPENSTEIN [1924], involves the enumeration of different cases to deal with tangential and sagittal rays, since Abbe did not invoke the relationships between tangential and sagittal linear coma. It was from the publication of Abbe's 1879 paper that the sine condition became well known and understood in relation to image formation. HOCKIN [18843gave a proof using optical path differences along tangential rays; this proof is simple and often reproduced in elementary texts. BORN and WOLF [1959] gave a proof on similar lines but not restricted to tangential rays; the same applies to CONRADY'S [1905] proof, although it is not expressed so elegantly as that given by Born and Wolf. The generally accepted formulation of the sine condition is sin U' - sin U

~~

U'

U

(3)

for a stigmatic axial pencil, where U and u are finite and paraxial con-

VI, §

31

273

AXIAL ISOPLANATISM

vergence angles; occasionally it is stated as a magnification law for small objects and large apertures : n'v' sin U' = nq sin U ,

(4) where q is the object height. Equations (3) and (4) are obviously equivalent through the Lagrange invariant. The modification if object or image is in star space is obvious. There are three pairs of conjugates for refraction at a single spherical surface at which the image formation is aplanatic, (a) object and image coinciding at the surface, (b) object and image coinciding at the centre of curvatures and (c) conjugates (n + 1)r/n and (n' 1)r/n'. The first two are obvious once the concept of aplanatism is available and it does not seem possible to ascribe the third case clearly to a definite author. It is known that G. B. Amici used components apparently based on principle (c) in his microscope objectives as early as 1840-50 (CZAPSKIand EPPENSTEIN [1924]), i.e. a quarter of a century before the sine condition was enunciated ; possible the explanation is that Amici knew they had perfect spherical aberration correction and that he established the coma correction by finite raytracing. It is amusing to note that recently (KNUTSEN and STRAND [1964]) a proof was actually published of case (c) and the point was made that only a spherical refracting surface can have a pair of aplanatic conjugates; the authors can hardly claim priority.

+

0

3. Axial Isoplanatism

If a symmettical optical system has spherical aberration, it may still be useful to require that it shall have no linear coma. As will be seen in this review, several different meanings have been ascribed to this requirement, some concerned with ray aberrations and other with wavefront or optical path aberrations, and although most of the results are similar in form, they have somewhat different meanings. CONRADY [1905] made the first definite statement, that the requirement was equal magnification for all zones of the aperture, and he gave a diagram showing an off-axis pencil with all rays intersecting in pairs on the principal ray. Other authors discussed the matter in terms of aberration expansions, i.e. taking account only of Seidel [1909,1910] in a paper published in two differentjournals coma. CHALMERS but with identical wording, attempted to show the effect of noncompliance with the sine condition in the presence of spherical aberration, but his results were eventually expressed as a rather confused expansion in

214

APLANATISM A N D ISOPLANATISM

[VI, §

3

powers of the sine of the convergence angle and he did not produce any clear [19171carried out a similar calculation but again statement. Later on, SMITH did not get to the point of a clear statement, and it was not until STAEBLE [19191and LIHOTZKY [19191published almost simultaneously that a precise statement in terms of finite rays was given of the condition for freedom for coma in the presence of spherical aberration. 3.1. THE STAEBLE-LIHOTZKY CONDITION

The condition given by Lihotzky is

where 1' is the paraxial intersection length in the image space, t' is the exit pupil coordinate, L' is the intersection length of the finite ray considered, U and U' are finite convergence angles and m is the paraxial transverse magnification. Lihotzky's proof, which is rather involved, was reproduced by CZAPSKI and EPPENSTEIN [19243 and by BEREK[19303; only rays in the tangential section were considered and the condition ensures that a pair of finite rays intersect on the principal ray for small object heights: the derivation is such that rays of all convergence angles up to U' must satisfy the condition, since it is obtained by integrating a differential condition. In fact, if eq. (5) is suitably interpreted, this restriction is not necessary, as was found by later authors. Staeble gave a slightly more general formulation which included spherical aberration in the object space. In the important special case when the object is at infinity, eq. ( 5 ) takes the form

where f ' is the image side focal length and y is the object side incidence height. BEREK[1930] gave some variations on the basic Staeble-Lihotzky condition (eqs. (5) and (6)).If eq. (5) is written in the form J!-2' ~

( n sin Uln' sin U ) - m

-

r-7 m

(7)

it can be seen that the right-hand side contains only Gaussian quantities and if therefore the left-hand side is constant over the aperture, the system will be isoplanatic. Berek called this and the corresponding form of eq. (6) the proportionality condition since it required that the deviation from the

VI,

0 31

AXIAL ISOPLANATISM

215

Abbe sine condition (i.e. the denominator) shall be proportional to the longitudinal spherical aberration. If eq. (5) is written as -

l‘-l‘ z+-.--

n sin U - 7 = const., n‘sin U’

m

or correspondingly for eq. (6), this states that the graphs of longitudinal spherical aberration and of the ratio of sines (the second term on the lefthand side) must have the same shape; Berek called eq. (8) the coincidence criterion. Clearly, in order to apply this condition it is necessary to choose a position of the exit pupil, i.e. a value of 7. Finally, Berek noted that if the exit pupil coincides with the second principal plane, as is sometimes nearly the case for photographic objectives, and if the object is at infinity, the coincidence criterion reduced to the form

C-

nY n’ sin U‘

~

=

const.,

(9)

which he called the simplified coincidence criterion. Graphs of the two terms on the left-hand side of eq. (9) have frequently and VON ROHR[1932]) in summaries been used (e.g. by Mmd, RICHTER of designs of photographic objectives; however, it is usually not very clearly explained that the difference between the two terms is a true measure of coma only if the exit pupil actually is at the image-side principal plane.

3.2. CONRADY’S THEOREM

The importance of Conrady’s contributions to the subject of isoplanatism cannot be overestimated. Although others had recognized that variation of the ratio of sines indicated a variation of magnification by sagittal rays from. different zones of the aperture, it was Conrady who realized clearly the need for a measure of coma, i.e. a nonisoplanatism formula in the sense of section 1 above, rather than merely a null test. Also Conrady discussed sagittal ray coma as well as coma in the tangential section and showed how the two were related. As mentioned earlier, CONRADY had suggested [1905] how spherical aberration might be taken into account in a discussion of isoplanatism but it was not until the publication of his book (CONRADY [1929]) that he gave the complete treatment in print. Conrady’s nonisoplanatism formula was very similar in appearance to the StaebleLihotzky condition; it was an expression for “offense against the sine condition”, abbreviated to OSC by Conrady. The beautifully simple

216

A P L A N A T I S M A N D ISOPLANATISM

CVI,

03

derivation depended on the use of the sagittal magnification formula

nu, sin U

=

n'qi sin U',

(10)

which give the magnification of a small object on the axis by rays at large convergence angle in a sagittalplane with respect to the object and image. This result, true for any symmetrical optical system, is a simple consequence of the skew invariant (see, e.g. WELFORD [1968]); Conrady called it, rather misleadingly,the optical sine theorem and it has occasionally been confused in the literature with the Abbe sine condition. Referring to Fig. 1, let P and Q' be the axial image points for a paraxial and a finite ray and let P, be Gaussian

irnagy plane

n Fig. 1. Definition of Conrady's offense against the sine condition

the off-axis paraxial image point. Equation (10) above gives the position of Q: , the intersection of sagittal finite rays from P, , at a distance q', from Q . If there were no linear coma Q: ought to lie on the principal ray, at Q', ,but Conrady showed that if this were the case, the tangential rays need not be coma-free also. Conrady took the ratio Q',Q',/QQ', as a dimensionless measure of coma, i.e. the famous OSC'; it can then be shown by a simple geometrical argument that

OSC = 1-

n sin U .-/I-H mn' sin U' L: -7 '

Obviously if OSC is identically zero, then all of the various forms of the StaebleiLihotzky condition are fulfilled; if not then the value of OSC has a simple meaning, but this is not true of the Staeble-Lihotzky condition. We note also that Conrady's expression for offence against the sine condition.has direct meaning when applied to finite rays from a single zone in the pupil; it is also possible to apply eqs. (5) to (9) in this way but the significance is not so obvious. Conrady applied his results in a variety of ways, e.g. to finding the coma-freepupil position, and his formulaehave been used by optical designers as useful design tools. He demonstrated clearly

VI,

6 31

AXIAL ISOPLANATISM

277

that eq. (11) gives the sagittal coma due to the sum total of linear coma terms, i.e. all terms for which the optical path aberration has the form anvY(x2+Y2)". 3.3. LINEAR COMA AS AN OPTICAL PATH ABERRATION

A direct expression for the linear coma as an optical path or wavefront aberration when the axial pencil has spherical aberration can be obtained from Conrady's OSC formula by noting that the sagittal coma (QiQ; in Fig. 1) is very simply related to the total linear optical path coma; for if Yoma(x7

Y , v')

=

1anvrY(x2+y2Y

(12)

n

then

where p is the radius in the pupil of the finite rays in question. From this it follows that (WELFORD [19671)

the factor outside the bracket is, of course, the Lagrange invariant. This result shows clearly in terms of wavefront aberration the significance of nonisoplanatism. 3.4. LINEAR COMA AS RAY ABERRATION OR WAVEFRONT ABERRATION

Equation (14), which is an expression for linear coma as a wavefront aberration, can be obtained directly in other ways (see section 4.4) and it applies to all points in the pupil at the same radius; i.e. if the finite ray to which the test is applied leaves the exit pupil at a distance p from the axis, there will be no optical path coma at any azimuth even though, due to the interaction of the different order terms in eq. (12), there. is some residual coma at other pupil radii. On the other hand, if the OSC as given by eq. (1 1) vanishes for a particular finite ray at aperture p but not for others, it can be shown that the sagittal ray coma for aperture p is zero but the tangential ray coma and that for intermediate azimuths is non-zero ; this follows by differentiatingeq. (12) to find the ray aberration components in the usual way and Conrady himself pointed out that eq. (11) only applied to sagittal ray coma. There is no contradiction here, since it is well known

278

APLANATISM A N D ISOPLANAT ISM

cvs § 3

that optical path aberration may be zero and transverse ray aberration nonzero, or vice-versa, since the second is the derivative of the first; however, this point is usually made only in connection with spherical aberration of different orders. To put this explicitly, eq. (12) can be written Koma =

VY

Cn anp2"

(15)

and the two components of transverse ray aberration are proportional to

aw ~

ax

aw

=

__ =

aY

v C2~1a,xyp~~-', n

v

C an(x2+(2n+ l)y2)p2"n

If the right-hand side of eq. (14) is zero for a certain finite ray, i.e. for a certain value of p, this means, from eq. (15) that C a n p Z n= 0. n

Now the tangential component of ray aberration for rays in the sagittal section, i.e. the thing measured by OSC', is given by putting y = 0 in the second of eq. (16) and it is

so that equally if WComa= 0 then OSC' = 0. On the other hand, the tangential component of ray aberration for rays in the tangential section is given by putting x = 0 in the second of eq. (16 ) :

and this obviously does not necessarily vanish if W,,,, = 0 at a certain pupil zone, as pointed out by Conrady. The Staeble-Lihotzky condition was derived, as mentioned in section 3.1, by integrating a differential condition, i.e. by considering all zones of the aperture; if it holds strictly for all zones this must mean that all the coma terms are identically zero, i.e. all the a" of eq. (12) are zero. Clearly in that case both tangential and sagittal coma are zero and all rays unite strictly on the principal ray. If, however, eq. (6) or one of its variations is only true

VI,

6 31

AXIAL ISOPLANATISM

279

for one zone in the pupil, then only sagittal coma, i.e. the tangential component of transverse ray aberration for sagittal rays, is zero ;in other words, OSC’ = 0 at one zone. The above should clarify an apparent contradiction, that the StaebleLihotzky condition was derived by considering tangential rays and Conrady’s OSC formula by considering sagittal rays, yet both apparently have the same analytical form. Similar remarks apply to the strict Abbe sine condition, eq. (3): if it holds for all zones of the aperture then all orders of linear coma are identically zero but if it holds for an isolated convergence angle, the comatic wavefront aberration will vanish at this aperture and also the sagittal ray coma (eq. (18)), but not the tangential ray coma. We now recall the discussion by Abbe, Conrady and others of coma as a variation of magnification by different zones of the aperture; from what has been said in this section it is clear that if OSC vanishes for all zones of the aperture, then the magnification of all zones will be the same, but if there is isoplanatism at one zone only, the sagittal magnification for that zone will equal the paraxial magnification but this is not the case for the magnification by a pair of tangential rays, or indeed rays from any other azimuth in the pupil. Thus a discussion of coma in terms of magnification of different zones of the pupil can be misleading; this point was discussed at considerable length by SMITH[19171. 3.5. SOME DIFFERENT DEFINITIONS OF AXIAL ISOPLANATISM

It will be clear from what was said in section 3.4 that the interpretation of eq. (14) in terms af optical path aberrations or, alternatively, wavefront shapes, offers a simple concept of axial isoplanatism which has the merit of being directly applicable to diffraction calculations and tolerance systems based on diffraction. Nevertheless we should mention some ray-theoretic and other variations on the definition of axial isoplanatism. MARX[1959] noted that the definition of axial isoplanatism implied by the strict Staeble and Lihotzky condition, i.e. fulfillment at all apertures, meant that the off-axis pencil of rays would be strictly axisymmetrical about the principal ray, or, as it is sometimes expressed, the off-axis caustic would be the same shape as the axial caustic; however, Marx pointed out, the principal ray is inclined to the axis (unless the system is telecentric) and thus the pattern of ray intersections on the image plane is not exactly symmetrical. He suggested an alternative condition, based on tangential ray intersection only, which would ensure that the tangential ray inter-

280

A P L A N A T I S M A N D ISOPLANAT ISM

cv1,

43

sections remained symmetrical about the principal ray ;this condition is, in our present notation 1-

1 nsinu c-1’ 1 - -.r-7 cos U’ m n’sin U”

(20)

which differs from eq. ( 5 ) and eq. (11) by the factor on the left-hand side*. This condition could differ appreciably from the others at large relative apertures. However, FOCKE [19603 showed by using the classical aberration expansion theory that aberration correction of the type proposed by Marx, i.e. constancy of the ray intersection pattern, is impossible unless the system has no spherical aberration, so that Marx’s criterion is of little vahe. SINDEL [1960, 19621 in two lengthy papers considered the geometrical optics brightness distribution, i.e. the ray intersection density, in the image and formulated criteria for this to be stationary near the axis; he included also the effects of reflection and absorption losses. The expressions given involve repeated numerical integrations over the pupil radius with reflection and absorption losses included; we do not reproduce them here since it is unlikely that they would be used in practice. A more important objection to Sindel’s work and also to that of M a n is that they are concerned with very small differences between ray-optical criteria, whereas it is well known that diffraction effects cause the point spread function to be completely different from the geometrical optics prediction. It certainly does not follow that the diffraction point spread function will be symmetrical if the ray intersections on the image plane are symmetrical but the principal ray is oblique : equally, the diffraction point spread function for an oblique pencil which is geometrically truly of revolution symmetry about the principal ray need not be symmetrical if the principal ray is not normally incident on the image plane, but in this case the criteria of sections 3.1 and 3.2 are at least reasonably simple to use. TORALDO DI FRANCIA [I9521 made a curious attack on formulae of the type of those in sections 3.2 and 3.3, claiming that they were numerically inaccurate and offering an empirical formula which he claimed fitted better ; not many details are given but it seems as if he used an equivalent of eq. (1 1) to calculate tangential coma of uncorrected systems, whereas (section 3.4) in this case the formula is only valid for sagittal coma. HOPKINS [1946] gave a slightly different version of eq. (14) which in the * Actually Marx suggested also another equation which differs from eq. (20) only by quantities of the order of the square of the image height; to the approximation involved, i.e. linear coma, these must be the same.

VI,

0 31

A X I A L ISOPLANATISM

28 1

present notation takes the form

the difference being the inclusion of the factor cos U' in the spherical aberration term; this was also given by M A R I ~ H A [1952]. L Equation (21) was obtained as the condition that the wavefront aberration at the point in the pupil corresponding to convergence angle U' shall remain constant for small off-axis movements; this implies a slight expansion of the wavefront on one side of the tangential section and a corresponding contraction on the other and thus the term cos U'arises, as in Fig. 2. The difference would only be significant at large apertures and for large amounts of spherical aberration. Exit pupil

Fig. Change in pupil width according to eq. (21). The pupil is PIP2;the off-axis wavefront ought to extend to A l , where P I A l is an arc with centre at the pupil centre, so that the extra portion A I B l of wavefront is included. Similarly on the other side a portion A2B2 ismissing.

BUCHDAHL [19701 came via a different route to the same conclusions as Focke, that the ray intersection pattern on the Gaussian image plane could not be truly symmetrical in off-axis images unless the spherical aberration were identically zero over the whole aperture; as he put it, an isoplanatic system is necessarily aplanatic, if that particular definition of isoplanatism is adopted. For the classical definition, which in Buchdahl's terms amounts to no non-zero linear coma coefficients in the aberration

282

APLANATISM A N D ISOPLANATISM

[VI, 4 3

expansion, he gave the Conrady OSC formula and he pointed out that, as noted above, OSC or equivalent quantities calculated at a certain aperture give information about the sagittal ray coma but not about the tangential coma at this aperture. It can be seen from the above discussion that there is some confusion about how axial isoplanatism should be defined and this applies even more strongly to off-axis isoplanatism (see section 4). Broadly, there are two possibilities: (a) to take isoplanatism to mean that the shape of the geometrical optics pencil, i.e. the rays and the wavefronts, shall stay the same for a small movement of the object point away from this axis, or (b) to discuss ray or wavefront aberrations in relation to points in the image plane and in the pupil or aperture stop. The numerical differences would be small in most practical situations but it is desirable to have the basic concepts clearly defined. Definitions based on (a) were used by all authors up to and including Conrady and this case seems to lead to unique formulae ; on the other hand, (b) can lead to different formulae, as we have seen; the complications get worse in dealing with non-symmetrical systems since it becomes increasingly difficult to say clearly where diaphragms and pupils are because of aberrations of pupil imagery and vignetting, whereas it is always possible to attach an unambiguous meaning to (a). Furthermore, isoplanatism conditions based on (a) are simpler in form, again particularly when the system has no symmetry. Finally, the differences between the ray intersection patterns for (a) and (b) may change again when the physical optics point spread function is considered and it is not possible to say which definition gives better results from a physical optics standpoint. It seems clear that for practical reasons of simplicity and the possibility of actually obtaining usable results that an (a) definition should be chosen and we make this choice in the present review. 3.6. ISOPLANATISM AT VARYING MAGNIFICATION

The fact that the Abbe sine condition and the Herschel condition for stationarity of spherical aberration with change of conjugates (eq. (29) in section 4) are not. compatible except at certain magnifications has been cited as a proof that a system cannot be aplanatic for more than one pair of [1936]); that this cannot be strictly true can be conjugates (see e.g. FLINT seen from the example of the three pairs of aplanatic conjugates of a single [1952]) that for any optical system refracting surface. It is known (WYNNE Seidel coma can vanish for more than one conjugate. SMITH[1927] gave a curious theorem for the necessary and sufficient

VI, 8 41

I S O P L A N A T I S M W I T H NO AXIS OF SYMMETRY

283

condition that any (nonsymmetrical) optical system should have a pair of aplanatic conjugate surfaces ; the condition is that it should be possible to express the eikonal or angle characteristic of the system as a homogeneous function of the first degree of three linear functions of the object and image space direction cosines. It is in fact very unlikely that any systems except those with spherical symmetry could be found to satisfy this condition. No other results on isoplanatism at different conjugates seem to be available.

0

4. Isoplanatism with no Axis of Symmetry

Axisymmetric optical systems in the axial region are clearly a special case because of the high degree of symmetry; it might appear worthwhile to consider symmetric systems in regions away from the axis and after that systems with less intrinsic symmetry, such as holograms, but in fact this usually offers no appreciable simplification and it turns out to be best to go immediately to systems with no symmetry. The intermediate systems then appear as special cases. The discussion in section 3.5 led to the conclusion that isoplanatism criteria should be based on the notion of constancy of wavefront shape either over the whole aperture or in a small region near a traced finite ray. This approach by-passes questions of limiting apertures to a certain extent but it is necessary to bear in mind that in most cases when a pencil passes obliquely through an aperture or a series of apertures, a shift of the object point will inevitably change the extent or size of the transmitted pencil. Thus we may not be able to have a situation in which the wavefront shape and its extent are unchanged for small object movements. This is illustrated in Fig. 3, which shows how a vignetted and aberrated wavefront in star space is changed in width by an object shift. Under these conditions it would be impossible for the point spread function to be unchanged

/\

Fig. 3. The effect of vignetting in changing the shape and width of a wavefront.

284

A P L A N A T I S M A N D ISOPLANATISM

rw § 4

since no change in aberrations can compensate for a change in size of the wavefront, i.e. a change in resolution limit. Nevertheless, the concept of a change in shape of the image-forming pencil is simple and, as we shall see, it can be expressed by simple and unambiguous conditions, so that we shall use this concept in what follows. 4.1. THE SMITH OPTICAL COSINE LAW

The first attempt at an isoplanatism theorem for systems of no symmetry was the optical cosine law of SMITH[1922,1923] :Fig. 4 shows the aim of this condition. Let C be an incident, possibly aberrated wavefront and let C’

Fig. 4. Smiths optical cosine law. The vectors 6s and 6s’ represent small translations of the pencils.

be a wavefront of the pencil emerging from any optical system. If C is bodily translated a small distance 6s then C‘ will in general be both translated and distorted; the optical cosine law gives the condition that C’ shall be translated without distortion through 6s’. It is n6s . r

=

n’6s’ . r’,

(22)

where r and r‘ are unit vectors along incident and refracted parts of a ray and where the condition is to hold for all rays of the part of the pencil to be tested. Smith’s proof, which is reproduced with minor variations by STEWARD [1928], did not make clear the significance of the theorem as an isoplanatic condition*, although he pointed out that it could be specialised to give, e.g., the Abbe sine condition. However, from our present viewpoint it appears as a very general result since there are no restrictions either on symmetry of the optical system or on the magnitude of the aberrations. * PEGIS [1961] described Smith as a “most prolific and difficult writer”, a delicious understatement.

vx, 8 41

I S O P L A N A T I S M W I T H N O A X I S OF SY-MMETRY

285

In spite of the apparent generality of Smith’s optical cosine law, it is found to be limited in scope in a curious way because of the restriction to translations as the displacements of the object and image pencils; for example, it will be found to be impossible to derive the Staeble-Lihotzky condition from it, unless it is assumed that there is no spherical aberration, when it reduces simply to the Abbe sine condition. The reason is that we want to consider small rotations of a pencil in order to have the most general situation possible*; for example, in the derivation of the StaebleLihotzky condition and of the Conrady OSC’ formula the aberrated imageside pencil is rotated about an axis through the centre of the exit pupil and perpendicular to the optical axis. In other cases, as will be seen below (section 4.3), we may need to consider an axis of rotation which is at some distance from the pencil. The publication by SMITH [1922] of his optical cosine law led to some curious controversy about its meaning and validity (HERZBERGER [1923] ; BOEGEHOLD [1924a1; SMITH[19241) but there was no permanent disagreement. BOEGEHOLD [1924b] claimed that it was possible to deduce the Staeble-Lihotzky condition from the optical cosine law, but his proof seems to be in error; in a note appended to Boegehold’s paper, Smith remarked guardedly that he had tried to extend the optical cosine law to include rotations of pencils but this seemed to be “on a quite different footing”. The theorem which enables this to be done is given in the next section. 4.2. THE MOST GENERAL ISOPLANATISM THEOREM

We consider incident and refracted wavefronts C and C‘ as in section 4.1, but now we are concerned with small rotations about axes given by the unit vectors p and p’, as in Fig. 5. The most general isoplanatism theorem (WELFORD [1971]) states that a small rigid rotation E of C about the axis p will produce a small rigid rotation E’ of the imaging pencil about p’ if for all rays m ( p , D,r ) =

n’&’(p‘, D‘,r’) +const.,

(23)

where, as before, r is a unit vector along a ray and D is a vector from any point on the axis p to any point on the ray specified by r. The scalar triple product {p, D,r ) is in fact equal to S sin 8, where S is the shortest distance

* The most general displacement of a rigid body is of course a screw movement but a rotation is adequate to represent the kind of displacement met with in light optics.

286

A P L A N A T I S M A N D ISOPLANAT ISM

4

Fig. 5. The most general isoplanatism theorem; p and p’ are unit vectors along rotation axes.

between p and r and 8 is the angle between these directions. The theorem furthermore states that if the condition of eq. (22) is not fulfilled then dW = n’&’{p’,D‘, r’} - ne{p, D , r }

(24)

is the change in wavefront aberration due to the rotation E , as a function of the particular ray r - r’ which is considered. In eqs. (23) and (24) terms of order are neglected, this being the equivalent of restricting axial isoplanatism conditions to linear coma. The most general isoplanatism theorem was so called because it included as special cases all previously known results; also it has led to some useful new special cases. We note immediately that if the axes p and p’ recede to an infinite distance the rotations become translations 6s and 6s‘and we recover Smith’s optical cosine law ; however, we now have also a nonisoplanatism theorem, rather than merely an isoplanatism condition, since the version of eq. (24) with the axes at infinity:

dW

=

n‘6s’ ‘ Y’ - n 6 s . r

(25)

gives the change of aberration if the condition is not fulfilled.

Fig. 6. Translations are equivalent to rotations for an aberration-free pencil: (a) a transverse translation is equivalent to a rotation about an axis in the pupil; (b) a longitudinal translation is equivalent to a rotation about an axis at infinity.

W § 41

1,SOPLANATISM WITH N O A X I S OF SYMMETRY

287

If a pencil is aberration-free a small rotation about an axis is equivalent to a certain small translation, as in Fig. 6, since all parts of the wavefront have the same curvature; we can neglect the difference between the lateral positions of the shifted wavefronts in the two cases, since there is no change in the (zero) aberration. Thus for aberration-free incident and refracted pencils we can use Smith's optical cosine law as equivalent to the most general isoplanatism theorem. 4.3. OFF-AXIS ISOPLANATISM IN A SYMMETRICAL OPTICAL SYSTEM

Let the centre of the exit pupil of a symmetrical optical system be P (Fig. 7) and let the (finite) principal ray of an off-axis pencil be PP; where PIis on the image plane. We may ask under what conditions the aberrations Pupil

n

\

l~mage plane

Fig. 7. Off-axis isoplanatism; the effect of a transverse shift of the image point Pi is obtained by a rotation about p'.

will remain unchanged for a small change dq in object height. In this case

PI will move vertically along the image plane and the principal ray will rotate about 0; ,the tangential image on it of the centre of the entrance pupil; thus the axis of rotation is at the intersection of the perpendicular to the principal ray through 0;and the horizontal line through P i . The [1971]): resulting change in wavefront aberration is found to be (WELFORD

in this equation (0, A', F ) are direction cosines of the principal ray, (L', M', N') are direction cosines of another ray of the pencil, for which isoplanatism is being tested, dq'ldq is the local magnification according to the principal ray and 61' is the tangential component of transverse ray aberration of the ray being tested with respect to the principal ray.

288

rv1, §

A P L A N A T I S M A N D ISOPLANAT ISM

4

If in eq. (26) we set ti?‘ = 0 and d W = 0 we obtain d?’ M‘- nM = const., n’ d?

(27)

which is a condition for aberration-free imaging by an off-axis pencil. This is one of a pair of equations known variously as the “cosine conditions”, the “cosine relations” and the “extended sine law”; the second one, which is a trivial consequence of the skew invariant, is d?‘ n‘-L-nL d?

= 0.

(28)

These two conditions for aberration-free imagery were probably first and EPPENSTEIN [19241 given in 1892 by Thiesen, according to CZAPSKI and they were given subsequently by many authors, including Smith, Steward and Buchdahl.

\

\ 1

Image plane \ \

I -

\

A x i s p‘

Fig. S. Off-axis isoplanatism with a focal shift; the vector d[ indicates the direction of displacement of the image point Pi and this is obtained by a rotation about the axis p’.

We can also consider longitudinal displacements of an off-axis object point, i.e. displacements parallel to the optical axis. The effective axis of rotation p’ for the imaging pencil is then at the intersection of the same perpendicular to the principal ray with the image plane, as in Fig. 8, and for a displacement d[ of the object point the change in aberration is (WELFORD [1971]): dW If we set

=

-d[[n‘dr(N-N.d[‘ -

m = R’ = 1 and

= 0, so

6 4 ’ ) - n(m - N)] 0; PI

__ M N

.

(29)

that the principal ray moves in to

VI,8

51

289

ISOPLANATISM IN HOLOGRAPHY

become the optical axis, eq. (29) reduces to

1

,

which yields the classical Herschel condition for zero change in aberration with a small change in magnification (HERSCHEL [1821]); we see that the form of the condition does not depend on the presence or otherwise of spherical aberration. It is easy to show also how the Conrady OSC condition, the StaebleLihotzky condition and eq. (14) above can be derived from the most general isoplanatismtheorem. The relationshipsbetween all these results are shown in Fig. 9.

Fig. 9. Relationships between isoplanatism conditions and theorems; the arrows indicate the relation of implication of one result by another.

0

5. Isoplanatism in Holography

The geometrical optics of holographic image formatipn has not been studied in great detail but it is clear from several papers that, just as for diffraction gratings, we can use geometrical optics concepts of diffracted rays and wavefronts and we can consider the aberrations which occur in image formation. If the reconstruction geometry is the same as in the formation of a hologram and if the wavelength of the light is unchanged the

290

APLANATISM A N D ISOPLANATISM

[VI,

§5

holographic image formation is aberration-free*. Aberrations occur through changes in the wavelength and geometry used in reconstruction. In the most general case of sideband Fresnel holography there is no symmetry at all in the system and this has led to considerable complication in the attempts at aberration expansions on the lines of classical Seidel theory. In the present review we are, as stated in the introduction, concerned only with finite conditions and here there is very little published work. WELFORD [1973al considered the effects of shifts of the reconstruction source on the aberrations and showed that for plane holograms there are certain configurations of object and reference point such that no aberrations appear in the reconstruction for small shifts of the reconstruction source ; thus such configurations can be called aplanatic. The results are obtained by considering holograms of single points ; the reconstructed wavefront from the image point can be found by the application of Fermat’s principle even though it is formed by diffraction rather than refraction and thus isoplanatism conditions are applicable, just as when considering image formation by diffraction gratings (WELFORD [1965)).

Fig. 10. Aplanatic configuration for a planar Fresnel hologram of a point object. The object and reference foci are mirror images in the plane of the hologram.

The results are that the imagery is aplanatic for rotations about any axis in the plane of the hologram (a) if the object and the reference point are mirror images in the plane of the hologram, as in Fig. 10, (b) if the object

’ There can be, of course, effects due to shrinkage and distortion of the holographic medium (photographic emulsion, photoresist, etc.) but these can be regarded as analogous to manufacturing aberrations in lens systems; also there may be variations in diffraction efficiency across the aperture in a hologram corresponding to varying transmission and reflection losses in lens systems; we do not consider either of these classes of effect here, as is customary in aberration theory.

v11

REFERENCES

29 1

point is in the hologram plane (image plane hologram), or (c) in Fourier transform holography, when both object and reference are at infinity. No other cases of strict aplanatism seem to exist for plane holograms although there are many geometries for which the lower degree, terms in the aberration expression vanish, i.e. Seidel type aplanatism. In the same reference it was shown that if a hologram is formed on a spherical surface with reference at infinity and object at the centre of the sphere (or vice versa) the image formation will be aplanatic. This led to a study of holograms formed on spherical surfaces and used as lenses (WELFORD [1973b]); it was shown that such elements would be aplanatic if they were made according to the condition 1

1

1

T+j=R’ where R is the radius of curvature of the surface and 1 and I‘ are the distances of the object and reference. Moreover, such holograms and also the plane hologram geometries mentioned above comply with the condition of DUMONTET [1955] mentioned in section 1, that there should be no overall phase shift in the point spread function. Imageforming elements of this kind could have uses in certain scanning microscopy applications. References ABBE,E., 1873, Schultze’s Archiv fur mikroscopische Anatomie IX,4 1 U 8 . ABBE,E., 1879, Sitzungs Berichte der Jenaer Gesellschaft fur Medizin und Naturwissenschaft, pp. 129-142. BEREK,M., 1930, Grundlagen der praktischen Optik (W. de Gruyter, Berlin). BLAIR,R., 1791, Trans. Roy. SOC.Edinburgh 3, S 7 6 . BOEGEHOLD, H., 1924a, Zentral-Zeitung fur Optik und Mechanik 45, 107, 295. BOEGEHOLD, H., 1924b, Trans. Opt. SOC.26, 287. BORN,M. and E. WOLF,1959, Principles of Optics (1st ed., Pergamon, London, New York). BUCHDAHL, H., 1970, Hamiltonian Optics (Cambridge University Press). CHALMERS,S. D., 1909, Proc. Phys. SOC.London 22, 1-10. S. D., 1910, Phil. Mag. 19, 356. CHALMERS, R., 1864, Poggendorf s Annalen 121, 1 4 . CLAUSIUS, CONRADY, A. E., 1905, Monthly Notices Roy. Astron. SOC.65, 50149. CONRADY, A. E., 1929, Applied Optics and Optical Design, Part I (Oxford University Press). CZAPSKI,S. and 0. EPPENSTEIN, 1924, Grundziige der Theorie der optischen Instrumente (3rd ed., J. A. Barth, Leipzig) (Ch. VII, by H. Boegehold). DRUDE,P., 1917, Theory of Optics, English transl. by C. R. Mann and R. A. Millikan (Longmans Green, London). DUMONTET, P., 1955, Optica Acta 2, 5343. 1955, Phil. Trans. Roy. SOC.Series A 247, 369407. FELLGETT, P. B. and E. H. LINFOOT, FLINT,H. T., 1936, Geometrical Optics (Methuen, London). FOCKE,J., 1960, Optik 17, 51tL-521.

292

APLANATISM AND ISOPLANATISM

[VI

HAMILTON, Sir W. R., 1844, Manuscript of a paper read to the Royal Irish Academy, June 24, 1844, but never published. The MS is reproduced in: The Mathematical Papers of Sir William Rowan Hamilton, Vol. I, Geometrical Optics, eds. A. W. Conway and J. L. Synge (Cambridge, 1931). HELMHOLTZ, H., 1874, Poggendorf s Annalen, Jubelband, pp. 557-584. HERSCHEL, J. F. W., 1821, Phil. Trans. Roy. SOC.111, 222-267. HERZBERGER, M., 1923, Zentral-Zeitung Wr Optik und Mechanik 44,21 1. HOCKIN,C., 1844, J. Roy. Microscop. Soc. (2) 4, 337-346. Ch.,1973, Jena Review 18, 164170. HOFMANN, HOPKINS,H. H., 1946, Proc. Phys. SOC.LVIII, 92-99. KING,H. C., 1955, The History of the Telescope (Charles Griftin & Co. Ltd ., London). KNUTZEN, J. and A. STRAND,1964, Optik 21, 128-129. LIHOTZKY, E., 1919, Wiener Sitzungs-Berichte 128, 85-90. MARECHAL, A,, 1952, Imagerie geometrique (Revue d’optique, Paris). MARX,H., 1959, Optik 16, 610616. MERTB,W., R. RICHTER and M. VON ROHR,1932, Das photographische Objektiv (Springer, Wien). PEGIS,R. J., 1961, in Progress in Optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam), p. 3. P., 1960, Optik 17, 289-314. SINDEL, SINDEL, P., 1962, Optik 19, 369-384; 3 9 7 4 8 . SMITH,T., 1917, Proc. Phys. SOC.London 29,293-309. SMITH,T., 1922, Trans. Opt. SOC.London 24, 3140. SMITH,T., 1923, Optical Calculations, in: Dictionary of Applied physics, Vol. IV, ed. Sir. W. Glazebrook (MacMillan, London). SMITH,T., 1924, Trans. Opt. SOC.London 26, 281-284. F., 1919, Miinchener Sitzungs-Berichte 163-196. STREBLE, STEWARD, G. C., 1928, The Symmetrical Optical System (Cambridge University Press). STRAUBEL, R., 1902, Phys. Zeitschr. 4, 11&117. H., 1882, Wiedemann’s Annalen 25,407; 27, 1008. STRUVE, TORALWDI FRANCIA, G., 1952, Revue d’Optique 31, 381-392. WELFORD, W. T., 1965, in: Progress in Optics, Vol. IV, ed. E. Wolf (North-Holland, Amsterdam) pp. 241-280. WELFORD, W. T., 1967, Handbuch der Physik 29 (Springer, Heidelberg) pp. 1 4 2 . WELFORD, W. T., 1968, Optica Acta 15, 621423. WELFORD, W. T., 1971, Optics Communications 3, 1-6. WELFORD, W. T., 1973a, Optics Communications 8, 239-243. W. T., 1973b, Optics Communications 9, 268-269. WELFORD, WYNNE,C. G.. 1952, Proc. Phys. SOC.London 65B, 429437.

Addendum at proof stage It was noted by H. Rull and H. Kiemle (1973, Optics Communications 7, 158-162) that if a hologram is made as in Fig. 10 the imaging properties are invariant for displacements of the hologram in its own plane. This may be regarded as another way of stating an aplanatism property. A description of a method of design of a practical aplanatic hologram objective based on the result of eq. (31) is given by W. T. Welford (1975, Optics Communications 15, 46-49). This shows how the aberrations introduced by a meniscus support and by using different wavelengths for formation and reconstruction may be eliminated.

AUTHOR INDEX A ABBE,E., 270, 272, 291 ACKERHALT, J. R., 51, 53, 66 ADAM,A., 34, 39,61, 66 AGARWAL, A. K.,217,227,228,235,264,265 AGARWAL., G. S., 51, 52, 66 AHARANOV, Y., 54, 66 AKHMANOV,S.A., 171,174,213,214,262,263 ALLEN, L., 51, 66 h, A,, 79, 90 ANDERSON, D. K., 48, 68 ANDRADE,J., 62, 66 ARECCHI, F.T., 32, 61, 66 ARMITAGE,J. D., 142, 162 ASHBY, N., 3, 18, 19, 21, 23, 24 ATABEK, O., 53, 68

B BACHYNSKI, M. P., 172, 181, 199, 200, 204, 236, 264 BALDWIN, G. C., 171, 263 R. R., 105, 162 BALDWIN, BALLIK,E. A., 62, 67 BALTES,H. P., 5, 22, 24 H., 119, 162 BARRELL, BATES, W. J., 141, 162 BAUER,A,, 3, 9, 12, 15, 20, 23, 24 BEAMS,J. W., 36, 67 BELL,J. S., 56, 66 G., 113, 162 BENEDETTI-MICHELANGELI, BENNETT,J. M., 158, 163 V. P., 105, 156, 166 BENNETT, BEREK,M., 274, 291 BERNHARD, E., 101, 162 BERNY,F., 86, 90 BIDDLES,B. J., 101, 161, 162 BIER,N., 73, 90 BIRCH,K. G., 106, 116, 142, 154, 155, 162, 167

BLAIR,R., 269,291 BLANKE,R., 111, 162 BLEULER, E., 55, 66 N., 59, 66, 171, 172, 263 BLOEMBERGEN, B~EGEHOLD, H., 285, 291 Born, D., 54,66 Born, N., 54, 66 Born, W. L., 62, 67 BOOT, H. A. H., 178, 263 BORN,M.,31, 66, 95, 96, 105, 108, 109, 128, 141, 16CL162, 175, 263, 272, 291 Bomm, M., 115, 162 BOm,T. H., 30, 66 BRADT,H. L., 55, 66 E., 32, 61, 66 BRANNEN, BREIT,G., 53, 66 BRIERS,J. D., 144, 162 BROSSEL, J., 160, 162 BROWN,B. R., 150, 155, 162 R. J., 141, 165 BRUENING, BRUNINC,J. H., 118, 162, 166 BRUNNER, W., 51, 62, 67, 68 O., 114, 141, 162 BRYNGDAHL, BUCCINI,C. J., 105, 107, 164 BUCHDAHL, H., 281,291 B m o v , G. N., 153, 155, 162 R. K., 50, 66 BULLOUGH, BUNNAGEL,R., 100, 119, 163 BURCH,J. M., 144, 163 D. G., 4, 8, 17, 20, 23, 24 BURKHARD, BURNHAM, D. C., 59, 60, 66 BUROW,R., 167 C

CAMPBELL, F. W., 89, 90 CARMAN, R. L., 172, 264 CARMICHAEL, H. J., 51, 66 am, A. M., 30, 66 CHALMERS,S. D., 273, 291 CHANG,C. S., 51, 66 293

294

AUTHOR INDEX

CHAPMAN, S.. 204.263 CHURCHILL, G. G., 53, 67 CLAUSER, J. F., 34, 39, 55, 56, 58, 59, 61, 66 CLAUSIUS, R., 270, 271, 291 C~MMINS, E. D., 55, 67 c o r n y , A. E., 271-273, 275, 291 CONWELL, E. M., 172, 183, 184, 263 C~RNEJO, A., 164, 167 COSYNS, M. G. E., 100, 163 COWLING, T. G., 204, 263 C~WNIE, A. R., 111, 163 Cox, 167 CRANE, R., 117, 163 CRISP, M. D., 29, 43, 47, 48, 51, 66 CUMMINGS, F. W., 29, 43, 47, 51, 67 CURRIE, D. G., 40,41, 67 CUTRONA, L. J., 138, 163 CZAPSKI, S., 271-274, 288, 291

D DAVIDSON, F., 34, 39, 61, 66 DAVIS, S. P., 32, 61, 68 DAVIS, W., 31, 38, 66 DE BROGLIE, L., 62, 66 DE LA €'ERA-AWRBACH, L., 30, 66 DEMCHENKO, V. V., 235,263 DE METZ,J., 107, 164 DFSLOGE, E. A., 204, 263 DEW,G. D.,99, 100, 116,119, 121, 157,158, 163, 167

DIETZ,R. W., 158, 163 DWC, P. A. M., 29, 43, 63, 66 DRUDE, P., 272, 291 DUBEY, P. K., 228, 229, 263 DUCHOPEL, I. I., 101, 163 DUCHOPEL, J. J., 101, 164 DUCHOPEL, 167 DUMONTET, P., 270, 291 DYSON, J., 116, 144, 146, 163, 166 DYSTHE, K. B., 178, 263 E

EBERLY, J. H., 46, 51, 53, 66, 67 EBERSOLE, J. F., 167 EDDLEMAN, J. L., 217, 264 EHRENREICH, N., 182, 263 EINSPORN, E., 119, 163 EINSTEIN, A., 6, 11, 15, 23, 24, 29, 30, 54, 66 EMERSON, W. B., 157, 163 EPPENSTEIN, O., 271-274, 288, 291

EREMIN, B. G., 262, 263 EZIN,U. Sch., 115, 159, 160, 164

F FARRANDS, J. L., 111, 163 FAULDE, M., 167 FEDINA, 167 FELLGETT, P. B., 270, 291 FERCHER, A. F., 146, 154, 163 FERGUSON, H. I. S., 32, 61, 66 FERMI, E., 43, 66 FESHBACH, H., 202, 263 FISHER, R. F., 76, 90 FLINT,H. T., 282, 291 FOCKE, J., 280, 291 FORRESTER, A. T., 62, 66 F o ~ R BJ., C., 167 Fox, D. G., 103, 165 FRANCON, M., 141, 163 FRED,M., 116, 166 FREEDMAN, S. J., 55, 58, 59, 66 FUCHS, R., 22, 24 FURRY, W. H., 54, 66 G GABOR, D., 31, 66 GALLAGHER, J. E., 118, 162, 163 GATES,J. W., 101, 115, 143, 162, 163 GATTI,E., 32, 61, 66 GAVIOLA, E., 48, 66 GEIST,J., 5, 24 GERLACH, W., 36, 67, 68 GERSTEN, J. I., 228, 235, 263, 265 GHATAK,A. K., 171, 172, 178, 185, 188, 191, 214, 217, 218, 263, 264

GHIELMETTI, F., 62, 67 GIBBS,H. M., 48, 49, 53, 66, 67 GING,J. L., 62, 67 GINZBURG,~. L., 172,177,179,196,204,263 GIRARD, A., 109 GLAUBER, R. J., 32, 33, 40,41, 63. 67 GUODMAN, J. W., 167 GORDON, J. P., 50, 67 GOYAL, I. C., 318, 263 GREEN, F. J., 106, 142, 154, 155, 162 GRIMM, T. C., 20, 23, 24 GROPPER, L., 32, 68 GROTCH, H., 53, 67 GUBEL, N. N., 101, 163 GUBISH, R. W., 89, 90 GUDMUNDSEN, R. A., 62, 66

AUTHOR INDEX

GUHA,S., 174, 263 GULLSTRAND, A., 73,91

H HAAS,C., 22, 24 HABERMANN, E., 101, 162 E. C., 142, 164 HAGEROTT, HAHN,E. L., 48, 49, 68 HAIG,N. D., 62, 67 HAKEN, H., 62, 67 HAMILTON, W. R., 270, 292 HANBURY BROWN,R., 32,33,61,67,68 HANFS, G . R., 163 HANSCH,T., 51, 67, 68 HARIHARAN, P., 167 HARRIS,J. S., 101, 104, 127, 128, 162, 163 HEINTZE, L. R.,102, 163 HELMHOLTZ, H., 73, 91, 272, 292 M., 49, 51, 52, 68 HERCHER, HERMAN, R. M., 53, 67 HERRIOTT,D. R., 102,107-109,118,127,162, 163 HERSCHEL, J. F. W., 269,270,289,292 HERTZ,J. H., 51, 67 HERZBERG, G., 48, 67 HERZBERGER, M., 285, 292 HESSE,G., 111, 163 HILBERT, D., 6, 24 R. S., 107, 163 HILBERT, HILLER,C., 125, 157, 165 HISANO, K., 22, 24 HOCKIN,C., 272, 292 HOFFMAN, K., 51, 67 Ch., 269, 292 HOFMANN, HOLLERAN, R.T., 107, 163 HOLT,R. A,, 56, 66 HOPKINS, H. H., 102, 104, 127, 128, 161-163, 167, 280, 292 HORA,H., 178, 190, 263 HORNE,D. F., 158, 164 HORNE,M. A., 56, 66 HOUSTON, J. B., 105, 107, 164 HUNT,G. R., 5, 24 HUSSEIN, A. M., 235, 263 I ICHIOKA, Y., 156, 164 IGNATOWSKY, W. v., 161, 164 INGELSTAM, E., 160, 164 IVANOFF, A,, 81, 91

295

J JACQUINOT, P., 109 JAIN,M., 235, 263 JAMES, W. E., 107, 164 JANOSSY, L., 34, 39, 61, 66 JASPERSE, J. R., 22, 24 JAVAN, A,, 62, 67 JAYNES, E. T., 29,43,44,47,48,51,6&68 JOHNSON, P. O., 62, 66 T. W.,172, 181,199,200,204,236, JOHNSTON, 264 JONES,R. A., 100, 106, 117, 164 T. F., 62, 67 JORDAN, K KADAKIA, P. L., 100, 106, 117, 164 A., 22, 24 KAHAN, W I N , R., 22, 24 KAMAL, J., 184, 192, 228, 264 KANE,E. O., 182, 263 KARTASHEV, A. J., 115, 159, 160, 164 KAUSHIK, S. C., 264 KAw, P., 190, 250, 263, 264 KEOL,R., 51, 67 KETELAAR, J. J. A,, 22, 24 KETTERER, G., 156, 164 R. K., 178, 181, 190, 193, 199,225, KHANNA, 235, 264 KHOKHLOV, R . V . , 171,174,213,214,262,263 KICKER, B., 125, 157, 165 KIEMLE, H., 292 H. J., 51, 52, 67 KIMBLE, KIMMEL, H., 111, 164 KING, C. M.,103, 165 KING,H. C., 269, 292 KZRCKHOJT,G., 4, 5, 23,24 KLAUDER, J. R., 40,41,67 KLEIMAN, H., 32, 61, 68 KLEIN,M. V., 107, 164 KLIEWER, R. L., 22, 24 A. A., 62, 67 KLOBA, KNEULI~JHL, F. K., 5, 24 KNOLL,H., 73, 91 KNUTSEN, J., 273, 292 KOCH,H., 111, 164 KOCHER, C. A,, 55, 67 KOKKEDEE, J. J. J., 48, 68 KOLOMITZOV, J. W., 101, 164 KONTIEWSKY, Ju. P., 115, 164 KOOMEN, M., 83, 91

296

AUTHOR INDEX

KOPPELMANN, G., 113, 117, 158, 160, 164, 167 R., 53, 67 KORNBLITH, KOTSCHKOVA, 0. A,, 115, 164 J., 89, 91 KRAUSKOPF, KREBS, K., 113, 117, 158, 164 KRIESE, M., 154, 163 KRUG, w., 95, 113, 164 K-, J., 154, I63 KUMAR, A,, 203,209,214,223,233,250,264

L LAMA,W. L., 30, 46,67 LAMBJr., W.E., 30, 31, 67 LANG,J. G., 160, 164 LANGENBECK,~., 111, 112,139,144,164 LANYI,G., 29,43,47,51,67 LARIONOV, N. P., 153,155, 162, 164 LAU,E., 113, 164 LAWRENCE, E. O., 36, 67 LEE,P. H., 111, 165 LE, W. H., 155, 164 LEGRAND, Y., 77,91 LEITH, E. N., 138, 146, 149, 163 LELES, B. K., 13,22, 24 LEVINSTON, M. E., 172, 263 LICHTENECKER, K., 36, 67 LIEBOWITZ,B., 62, 67 E., 274, 292 LMOTZKY, LMFOOT,E. H., 270, 291 LIPSETT,M. S., 62, 67 LITI-LE,A. G., 32, 61, 68 LITVAK,A. G., 178, 199, 262, 263 LOCHHEAD,J. V. S., 4, 8, 17, 20, 23, 24 LOHMANN,A. W., 111, 114, 142, 150, 155, 156, 162, 164 F., 32, 67 LONDON, LUKIN,A. W., 153, 155, 162, 164, 167 M MACGOVERN, A. J., 15S155, 164 MAGYAR, G., 62, 67 MAHE,C., 166 MALACARA, D., 164, 167 MALIE,C., 107, 164 MALLICK, S., 141, 163 MANDEL, L., 3C34, 37-39,46,51, 52,6143, 6668, 159, 164 MANDELL, R. B., 74, 75, 91 MAW~CHAL, A., 281, 292 MARIOCE,J. P., 107, 164, 166

R., 119, 162 MARRINER, MARSHALL,T. W., 30, 61 MARX,E., 36, 67 MARX,H., 279, 292 MATSUMOTO, K., 114, 156, 164 S. W., 204, 263 MATTHYSSE, MAYER,E., 36, 67, 68 MCCAFFREY, J. W., 3, 18, 19, 21, 23, 24 MCCALL,S. L., 48, 49, 68 MCKENNA,J., 40,41, 67 MERT!~, W., 275,292 J. R., 77 MEYER-ARENDT, MILLET,F., 107, 164 MILONNI,P. W., 51, 68 MINKWITZ, G., 100, 114, 121, 125, 161, 165, 166 MIRWAIZ, S. O., 153, 162 MITRA,S. S., 22, 24 MITTAL,R. S., 197, 199, 264 MJASNIKOV, Ju. A,, 101, 163 MOLLOW,B. R., 51,61, 68 MORGAN, B. L., 32, 61, 68 MORSE,P. M., 202, 263 MURTY,M., 111,142, 145, 164, 166, 167 MUSTAFIN,K.S.,114, 153, 155,162,164,167

N NADAI,A,, 157, 165 NASH, F. R., 50, 67 NEW, V. D., 13,22,24 C. L., 118, 162 NENNINGER, R. K., 30, 68 NESBET, NEWSTEIN, M., 51, 68 NOMARSKY, G., 144, 165 0

OFFNER,A., 107, 165 G., 53, 68 OLIVER, O’NEILL,P. K., 105, 107, 164, 167 P PALERMO, C. J., 138, 163 PANDEY, H. D., 217, 227, 228, 235, 264, 265 PARANJAPE, V. V., 228, 229, 263 PARIS,D., 150, 164 J., 111, 146, 150, 165 PASTOR, PAUL,H., 51, 62, 67, 68 PEGIS,R. J., 284 PEN CHINA,^. M.,4,8, 17,20,23,24 A. Ja., 115, 164 F’ERFSCHOGIN,

AUTHOR I N D E X

PPIRSCH, D., 178, 263 PFLEEGOR, R. L., 62,63,68 PHILLIPS,D. T., 32, 61, 68 PIC&, J. L., 61, 68 PILSTON, R. G., 111, 165 PLANCK, M., 6, 25 PLJNDL,L. N., 22, 24 B., 54, 66 PODOLSKY, POLSTER, A. D., 102, 163 POLSTER,H. D., 103, 165 POLWAKHTOV, B. K., 262,263 PORCHELLO, L. J., 138, 163 POUND,R.V., 32, 61, 68 PRASAD,S., 178, 244, 264 h l b l A K , w., 116, 165 ~ I N G S H E I M E., , 6, 25 PROCTOR, C . A,, 79,90

R RADLOFF, W., 62, 68 m K O V , R. A,, 153, 156, 162, 167 RANCOURT, J. D., 117, 165 RASHAD,A. R. M., 178,264 RASMUSSEN, W., 52, 68 RAYLEIGH,Lord, 119,120,165 REBKA,G. A., 32, 61, 68 REINTJES,J., 172, 264 RICHTER,G., 51, 62, 67, 68 RICHTER, R., 275, 292 b E , G., 101, 165 RE=, R., 100, 113, 114, 121, 125, 158, 165, 166 WNITZ, J., 95, 164 RMMER,M. P., 103, 107, 163, 165, 167 ROBERTS,103 ROBISCOE, R. T., 48, 68 ROFSLER,F. L, 113, 165 RONCHI,V., 141, 144,'165 ROSEN,N., 54, 66 ROSENBERG, G. V., 160, 165 Ross, W., 111, 163 ROYCHOUDHURI, C., 167 R€JLL,H., 292 S

SAKURAI,T., 108, 113, 165 SALAMO, G. J., 53, 67 J. B., 104, 108, 113, 141-143, 165 SAUNDERS, SCARL,D. B., 32, 61, 68 SCHABERT, A., 51, 67 SCHIEDER, R., 52, 61, 68

297

SCHLUTER,A,, 178, 263, 264 SCHMELZER, Ch., 51, 67 SCHMIDT,P. G., 190, 250, 263 SCHOBER, H., 115, 165 SCHbIROCK,O., 100, 112, 120, 121, 157, 165 E., 43, 68 SCHR~DINGER, SCHUDAM, F., 49, 51, 52, 68 SCHULZ,G., 95, 100, 105, 114, 121, 122, 125-129, 135, 157, 161, 164-166 SCHULZE, R., 115, 165 SCHWIDER,J., 100, 105, 106, 109, 114, 121, 125, 126, 135, 137, 143, 147, 151, 156, 157, 159, 165-167 SCOLUIK,R., 91 SCOTT,G. D., 160,164 SCOTT,R. M., 145, 166 SCULLY, M. O., 30, 31, 67 W. A., 114, 164 SELFSNEV, SELF,S. A., 178, 263 I. R., 39, 51, 68 SENITZKY, SERIES, G. W., 30, 68 SHAHIN,I. S., 51, 68 SHAKNOW, I., 55, 68 SHANNON, R. R., 117, 165 SHARMA,R. P., 247, 260, 264 SHEARER, J. W., 217, 264 SHERSBY-HARVIE, R. B. R., 178,263 SHIMIZU,F., 172, 264 SHIMONY, A., 56, 66 SHISHIDO,K., 108, 113, 165 SHKAROFSKY, I. P., 172, 181, 199, 200, 204, 236,264 S H O C ~ NK., , 3, 18, 19, 21, 23, 24 SHOCKLEY, W., 183, 264 SHOEMAKER, A. H., 145, I66 SHUR,M. S., 172, 263 SILLITO, R. M., 62, 67 SILVA,E., 62, 66 SIMONS, C. A. J., 105, 166 SINDEL,P., 280, 292 S., 87, 90 SLANSKY, S u n , R. N., 167 S m , T., 274,279,282,284,285,292 SNOW, K., 146, 166 SODHA,M.S.,171,178,181,184,185,190-193, 197, 199, 203, 209, 214,217, 218, 223, 225, 228, 235, 244, 247, 250, 260, 263, 264 SONA,A,, 32, 61, 66 STAEBLE, F., 270, 274, 292 STEEL,W. H., 141, 145, 166 STEHLE,P., 51, 66 STEINBERG, G. N., 111, 165 P., 22, 24 STETIZER,

298

AUTHOR INDEX

STEUDEL, H., 51, 67 STEWARD, G. C., 284, 292 STONER Jr., J. O., 54, 68 STRAND, A,, 273, 292 STRAUBEL, R., 272, 292 STROUD Jr., C. R., 29,43,44, 46,47,49, 51, 52, 68 STRWE,H., 270,292 SUDARSHAN, E. C. G., 30, 31, 39-41, 67, 68 SUKHORUKOV, A. P.,.171, 174, 213,214,262, 263 SUZUKI,T., 156, 164 SVELTO, o., 171 T TAKASHIMA,M., 114, 164 TANIun, T., 235, 264 TAYLOR, G. N. A., 99, 166 TEWARI,D. P., 184, 192, 203, 209, 214, 217, 223, 227, 228, 233, 235, 264, 265 THOMAS,L. K., 15, 18, 20, 23, 25 TIMOSHENKO, S., 157, 166 TOLANSKY, S., 95, 160, 166 F. S., 116, 166 TOMKINS, TORALDO DI FRANCIA, G., 280, 292 TORGE,R.,146, 163 TOSCHEK, P., 51, 67 TOUSEY,R., 91 TRIP AT HI,^. K., 171, 172,174, 178,181,184, 185, 19@193, 197, 199,203,209,214,217, 223, 225, 227, 228, 235, 247, 250, 260, 26S265 Twrss, R. Q., 32, 33,61,67,68 TYLER,C. E., 36, 68 TZOAR,N., 228, 235, 263, 265

U

VAN HEEL,A. C. S., 105, 166 VARGA,P., 34, 39, 61, 66 VIALLE, J. L., 61, 68 VINCENT, R. K., 5, 24 S. K., 197, 199, 264 VIRMANI, VOLKMANN, A., 79, 91 VON BAHR,G., 79,91 VONROHR,M., 275, 292 J., 102, 163 VRABEL,

W WAETZMANN, E., 141, 166 WALKUP,J. F., 167 WALLS,D. F., 51, 66, 72 H., 52, 61, 68 WALTHER, WASHIMI,H., 235, 264, 265 WATERS,J. P., 160, 166 WATERWORTH, M. D., 107, 164 WEHLAN, W., 32, 61,66 WEINBERG, D. L., 59, 60, 66 WEINSTEIN, M. A,, 3, '9, 1 1, 15, 18, 20,23,25 WEINSTEIN, W., 105, 166 WELEORD, W. T., 270, 211, 276, 277, 285, 287, 288, 29G292 WENTZEL, G., 30, 68 WESSNER, J. M., 48, 68 WILCOX,T., 190, 250, 263 WOHLER,J. F., 107, 166 WOINOWSKY-KRIEGER, S., 157, 166 WOLF,E., 3%33,66,67,95,96, 105, 108, 109, 128, 141, 159-162, 164, 175, 263,272, 291 WOLTER,H., 160, 166 WOSTE,L., 61, 68 Wu, C. S., 55, 68 WYANT,J. C., 105, 142, 144, 153-156, 164, 166, 167 WYNNE, C. G., 282,292

Uhlenbeck, G. E., 32, 68 ULBRICH, C. W., 62, 67 UPATNIEKS, J., 146, 149, 164 URNIS,I. E., 101, 163

T., 95, 166 YAMAMOTO, YOUNG,T., 77, 91

V

Z

VAN DEN BRINK,G., 85, 91 VANDEN DOEL,R., 48, 68 VANDEWARKER, R., 146, 166

ZIEGLER, G. F., 101, 165 ZOLLNER,F., 111, 163, 166 ZQRLL,U., 113, 166

Y

SUBJECT INDEX A Abbe condition, see sine condition aberration, diopter of spherical, 82 -, expansions, 273, 280 - free imaging, 288 -, laterial spherical, 77 - measurements of the human eye, 69 et seq., 79, 86 et seq. -, negative spherical, 79 -, optical path, 277 -, positive spherical, 77, 79 -, ray, 277 -, spherical, 83, 273, 274, 275, 278, 282 -, stationarity of, 269 absorption coefficient, 8, 10 absorptivity, 5, 7, 9, 23 -, thermodynamic definition of, 21 et seq. achromatic doublets, 101, 269 acoustical phonon, 183, 227 - - scattering, 226 Airy distribution, 108, 113 ambipolar diffusion, 173, 174, 183, 241 aplanatic menisci, 101 aplanaticity, direct, 270 -, oblique, 270 aplanatism, 267 et seq. -, Seidel type, 290 aqueous humor, 76 aspheric surfaces, testing of, 131 astigmatism, 76 atomic cascade, polarization correlations in, 54 et seq. autocorrelation function, 33

B Bessel function, 184, 202 blackbody, 5 Bloch equations, 51 - space, 44

vector representations, 44,45, 48 Boltzmann distribution, 11 - equation, 174, 203, 209 Bose-Einstein statistics, 33 boson bunching, 33 Bouguer’s law, 9, 11, 13, 23 -

C

calcite crystal, 116 carriers, diffusion of, 185, 188 -, heating of, 178, 181 et seq. -, redistribution of, 185, 240 - transport, 172 cascade decays of Ca atoms, 54 Cer-Vit, 158 coherence time, 33 coincidence criterion, 275 - method, 115 - -, simplified, 275 collision loss mechanism, 190, 201 collisional heating of a plasma, 171 collisionless magnetoplasma, 197, 232 - plasma, 190, 217, 220 common-path interferometer, 141 Common test, 103 comparing a surface with itself, 140 et seq. compensated semiconductor, 208, 235 concave surfaces, testing of, 101 concept of local temperature, 179 conduction loss, 201, 223 conductivity tensor, 194, 195, 212 Conrady’s theorem, 275, 289 continuity equation, 176, 241 controlled fusion, 171 convex surfaces, testing of, 101 copper phtalocyanine, 13, 22 cornea, 72 et seq. corneal axis of symmetry, 74 - lens, 71 - surface, curvature of corneal surface, 73

299

300

SUBJECT INDEX

Coulomb attraction, 188, 239, 241 critical power for self focusing, 222,225,226, 227,236, 262 crystaline lens, 71, 76 - -, refractive index of, 76 - -, thickness of, 76 current density, 195, 238, 239

D damping, 51 Debye length, 179, 182, 185, 187, 188 - temperature, 184 decay of an excited atom, 45,48 - of a multi-level atom, 53 decaying discharge plasma, 262 defocusing, 218 density operator, 56 - -, diagonal representation, 40 detection efficiency, 42 dielectric constant, 193 --, effective, 175,189,191,192,201,207,212, 250 --, field dependent, 172, 174, 185, 196, 198, 201, 217, 251 - -, non linear part of, 191, 197, et seq. - - of a magnetoplasma, 194, 229 et seq. - tensor, 194 diffraction effects, 173, 220 - gratings, 141 dioptric power, 85 Dove prism, 142 drift velocity of electrons, 177, 179, 185, 194, 204,206, 238

-, thermodynamic definition of, 21 et seq. energy accumulation time, 36 energy balance equation, 35, 199, 200, 201, 239 - condition, 37 - conservation, 39 - current, 37 - dependence of camer mass, 173, 174, 182, 188,235 - relaxation time, 240 equation of continuity, 176,241 - of transfer, 9, 11

F Fabry-Perot interferometer, 51, 112, 158,159 filter, 109 - _ _ , eigen-wavelength of, 109 Fermat’s principle, 269, 290 Fizeau interferometer, 99, 101, 106,109, 110, 122, 127 - fringes, 161 flats, testing of, 119 fluorescence in multi-level atoms, 52 et seq. Fock space, 40 - states, 41, 65 Foucault’s test, 86 Fourier analysis, 144 - plane, 138 - transform methods, 241, 270, 290 fovea, 76 free carrier nonlinearity, 171 Fresnel lenses, 269 - zone plates, 145 fully ionized plasma, 233 et seq. --

E G effective dielectric constant, nonlinearity in, 173 eikonal, 283 Einstein coefficients, 7, 11,20,46 - photoelectric condition, 31 electric current density, 42 transition dipole moment, 44 electromagnetic field, quantization of, 29 - - vectors, 40 electron cyclotron frequency, 194, 210 - positron annihilation radiation, 54 electro-optic shutter, 37 emission coefficient, 8, 10, 11 -, definitions of, 8, 20, 21 emissive power, 5 emissivity, 6, 7, 16, 19 ~

Gaussian intensity distribution, 174, 185,250 statistics, 33, 198, 203 germanium semiconductors, 191 gross reflexion, 8 Green’s function technique, 202 group velocity, 243

~

H Haidinger rings, 98, 112 Hanbury Brown-Twiss effect, 32 harmonic oscillator, 13, 14 Heisenberg uncertainty’relations, 160 Hermitian operators, 53 Herschel’s condition, 271, 282, 287

SUBJECT I N D E X

Hilbert space, 29, 40,56, 63 hologram, carrier frequency, 154, 156 -, comparing a surface with a, 146 -, Fresnel zone type, 155 -, Gabor type, 154 - interferometer, 146 -, synthetic, 105, 131 holographic image formation, 270 - two frequency grating, 144 hot carriers, ,172 - electrons, 178 human eye, 72 - -, achromatic axis of, 82 - -, axis of revolution of, 76 - -, dioptric power of, 89 - -, refractive power of, 83 hyperfinestructure ofsodium, 49,51 hyperopic eyes, see myopic eyes

indium antimode semiconductors, both type of carriers, 193 - - -, n-type, 192 induced absorption 7, 20 - - for a freely radiating metal, 17 et seq. - emission, 7,9, 11,20,21 instability, growth of, 249, 256 interference experiments, 61 - patterns, measurements of, 115 interferometric testing of surfaces, 93 et seq. - -, relative, 96 intervalley transfer of carrier mass, 174 ionized impurity scattering, 184 ionosphere, D region of, 180 isoindicial surfaces, 76 isoplanatism,267 et seq., 279 - at varying magnification, 282 -, axial, 273 et seq., 279 et seq. - conditions, 282 - in holography, 289 et seq. -, most general theorem of, 285 et seq., 289

---

for a weakly absorbing body, 10 et seq. for opaque bodies, 6

L Lagrange invariant, 273 Lamb shift, 47, 48 - -, dynamic, 47 Langmuir probes, 61 laser, amplitude stabilized, 37 -, dye, 51 -, retinal coagulation, 90 -, single mode, 34 - unequal path interferometry, 105, 107 lattice, 183, 184 - vibrations, 183 LiF single crystals, 22 light filamentation, 250 linear coma, 277 liquid mirror, 119, 138 - -, reflectivity of, 120 local concentrations of electrons, 21 1 Lorentzian spectral distribution, 34,47 M

joint detection probability, 57

Mach-Zehnder interferometer, 150, 152 magnetoplama, 229 -, strongly ionized, 235 -, weakly ionized, 233 master wave, 150 Maxwell equations, 35,37,42,44,175 Maxwellian distribution of carriers, 184 Michelson interferometer, 97, 105, 107 microscopic interference methods, 95 Milne’s principle of detailed balance, 12 Moire, 112, 114, 138, 148, 151, 154 - fringes, 143 momentum-spin space, 29 - transfer experiments, 59 et seq. moving focus, 249 multi-level atom, 53 - pass interferometry, 111 multiple-beam interferometry, 108, 160 - wavelength interferometry, 109 myopic eyes, 79

K

N

keratometer, 74 keratometry, small-mire, 75 Kirchhoffs law, 8, 14, 20, 21 - -, deviations from, 3 et seq., 18 et seq., 22

negative absorption, 9 et seq. neoclassical theory, 29, 43 net absorption coefficient, 11,12,23 - - for freely radiating metals, 15 et seq.

J

301

302

S U B J E C T INDEX

reflection, 8 nonlinear dieelectric, 59 - optics, 171 - propagation, 242 nonlinearity, 222 -, heating, 255 -, pondermotive, 223, 255 saturation of, 174 nonparabolic semiconductors, 172, 227, 235 nonparabolicity of conduction band, 188 null lens, 98, 106, 150 -test, 103, 106 -

-.

0 ophthalmometer, 73 optical equivalence theorem, 39 - path coma, 277 - polarising methods, 1 12, 117 - sine theorem, 276 optometer, 77 oscillatory waveguide, 219,225,232,259

P parabolic equation approach, 172, 214 - semiconductors, 188, 225 parallax method, 81 parametric down conversion, 59 - frequency conversion, 60 paraxial magnification, 279 - ray approximation, 200,202,215, 23 1, 256 - ray trace techniques, 71 perturbation approximation, 184, 189, 193, 200, 203, 209, 220, 239 - Hamiltonian, 18 phase space probability density, 41 - velocity, 243 phoropter, 83 photoelectric effect, 30 photoemission, 39 photoemission probability, 31 photokeratoscopy, 74 photometric balacing, 1 15 photopigments, 72 Planck function, 6, 7, 11 point reference methods, 144 polar optical mode, 184 - - phonon, 184 pondermotive force, 173, 174, 177 et seq., 178, 179, 185, 189, 198,203,217, 222, 239 - mechanism, 190, 197

Poynting vector, 35, 36, 53 proportionality condition, 274

Q quantum electrodynamics, 29 et seq. quartz, 158 quasithermal equilibrium, 179, 206

R Rabi frequency, 52 ray intersection density, 280, 282 reaction field, 44 reflectance coefficients, 42 refractive index, 249 resonance fluorescence, 48. 50 et seq. resonance fluorescence, experimental information on, 51 - -, Fourier spectrum of, 51 reticule, 81 retina, 72 reversing interferometers, 143 rods, 12 Ronchi grating, 149

S Sabattier equidensities, 113 sagittal linear coma, 272,275,282 - magnification formula, 276 scatter plate, 144, 146 Schrodinger equation, 35, 44 Seidel approximation, 271, 290 self absorption, 15 - focusing, experimental investigations on, 261 - - in a nonlinear isotropic medium, 213 et seq. - length, 229, 232, 254 -, nonsteady, 238 et seq., 245 - of laser beams, 169 et seq. -, transient aspects of, 172 - induced transparency, 49 semiclassical methods, 29 - radiation theory, 27 et seq. shearing methods, 140, 141 et seq. sideband Fresnel holography, 290 sine condition, 269,270,271 et seq., 276,282, 285 -, offence against the, 275, 276, 279, 285 slightly ionized plasma, 190 Smith optical cosine law, 284,287 -

-

-

SUBJECT I N D E X

source-field theories, 30 space charge field, 185, 188, 241 spherical surfaces, absolute testing of, 127 - -, examination of, 100 spontaneous emission, 7, 11, 23, 43 Staeble-Lihotzky condition, 274, 276, 278, 279, 285, 289 steady statesolutions, 50,177,205,210 stimulated absorption, see induced absorption - emission, see induced emission strongly ionized plasma, 190 surface measuring interferometer, 105 T

transmission interference, 108 transmittance coefficients, 42 transverse thermal conduction, 178 two beam interference, 96 Twyman-Green interferometer, 105, 112, 118, 147, 149, 150, 151

V valley semiconductors, 172 velocity of diffusion, 241

W wave guide, 173 uniform selfmade, 258 Wien’s approximation, 12, 19 WKB approximation, 213, 231

- -,

TE wave, 253 thermal conduction, 200,203,205 - - mechanism, 190 - current density, 197 thermodynamic equilibrium, 3, 10 theta shearing, 142, 143 three-beam interference, 115 total emission for a freely radiating metal, 17 et seq.

303

Z zero-point field theories, 30 zonal magnification, 270

CUMULATIVE INDEX - VOLUMES I-XI11 11, 249 A B E L ~ F., , Methods for Determining Optical Parameters of Thin Film VII, 139 &ELLA,I. D., Echoes at Optical Frequencies XI, 1 AGARWAL, G. S., Master Equation Methods in Quantum Optics IX, 235 AGRANOVICH, V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion IX, 179 ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers IX, 123 AMMANN, E. O., Synthesis of Optical Birefringent Networks ARMSTRONG, J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers VI, 211 XI, 245 ARNAUD, J. A., Hamiltonian Theory of Beam Mode Propagation BALTES, H. P., On the Validity of Kirchhoff s Law of Heat Radiation for a Body in a Nonequilibrium Environment XIII, 1 R., The Intensity Distribution and Total Illumination of AberrationBARAKAT, Free Diffraction Images I, 67 XII, 287 BASHKIN,S., Beam-Foil Spectroscopy BECKMANN, P., Scattering of Light by Rough Surfaces VI, 53 IX, 1 BLOOM, A. L., Gas Lasers and their Application to Precise Length Measurements IV, 145 BOUSQUET, P., see P. Rouard BRYNGDAHL, O., Applications of Shearing Interferometry IV, 37 O., Evanescent Waves in Optical Imaging XI, 167 BRYNGDAHL, 11, 73 BURCH,J. M., The Metrological Applications of Diffraction Gratings CHRISTENSEN, J. L., see W. M. Rosenblum XIII, 69 v, 1 -HEN-TANNOUDJI, C., A. KASTLW, Optical Pumping &WE, A. V., Production of Electron Probes Using a Field Emission Source XI, 221 VIII, 133 H. Z., H. L. SWNNEY,Light Beating Spectroscopy CUMMINS, DECKERJr., J. A., see M. Harwit XII, 101 DELANO, E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 DEMAEUA, A. J., Picosecond Laser Pulses IX, 31 X, 165 DEXTER, D. L., see D. Y. Smith DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XII, 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 FRAN~N M., , S. MALLICK, Measurement of the Second Order Degree of Coherence VI, 71 FRIEDEN, B. R., Evolution, Design and Extrapolation Methods for Optical Signals, IX, 311 Based on Use of the Prolate Functions VIII, 51 FRY,G. A., The Optical Performance of the Human Eye 1, 109 GABOR,D., Light and Information 111, 187 GAMO,H., Matrix Treatment of Partial Coherence XIII, 169 GHATAK, A. K., see M. S. Sodha IX, 235 GINZBURG, V. L., see V. M. Agranovich 11, 109 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media 304

CUMULATIVE INDEX

305

GNIADEK, K., J. PEnmwcz, Applications of Optical Methods in the Diffraction IX, 281 Theory of Elastic Waves GOODMAN, J. W., Synthetic-ApertureOptics VIII, 1 GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission XII, 233 HARWIT,M., J. A. DECKERJr., Modulation Techniques in Spectrometry XII, 101 X, 289 HELSTROM,C. W., Quantum Detection Theory -on, 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, 247 111, 29 JACQUTNOT, P., B. ROIZEN-DOSSIER, Apodisation IX, 179 JONES,D. G. C., see L. Allen KASnw, 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: Kirchhoffi Theory IV, 281 VI, 331 K o ~ RF.,, Diffraction at a Black Screen, Part 11: Electromagnetic Theory KUBOTA,H., Interference Color I, 211 XI, 123 LEAN,E. G., Interaction of Light and Acoustic Surface Waves LEITH,E. N., J. UPATNIEKS, Recent Advances in Holography VI, 1 LEVI,L., Vision in Communication VIII, 343 LIPSON,H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch of Physical Optics V, 287 MALLICK, S., see M. Franqon VI, 71 MANDEL, L., Fluctuations of Light Beams 11, 181 MANDEL, L., The Case for and against Semiclassical Radiation Theory XIII, 27 MARCHAND, E. W., Gradient Index Lenses XI, 303 VIII, 373 MEHTA, C. L., Theory of Photoelectron Counting MIKAELIAN, A. L., M. L. TER-MIKAELIAN, Quasi-ClassicalTheory of Laser Radiation VII, 231 M I Y A M OK., ~ , Wave Optics and Geometrical Optics in Optical Design I, 31 K., Instruments for the Measuring of Optical Transfer Functions V, 199 MURATA, Multilayer Antireflection Coatings VIII, 201 MUSSET,A., A. THELEN, O~UE S.,, The Photographic Image VII, 299 PEGIS, R. J., The Modem Development of Hamiltonian Optics 1, 1 PEGIS, R. J., see E. Delano VII, 67 k m N , P. S., Non-Linear optics V, 83 ~ETYKIEWICZ, J., see K. Gniadek IX, 281 PICHT,J., The Wave of a Moving Classical Electron V, 351 RISKEN,H., Statistical Properties of Laser Light VIII, 239 111, 29 RoIzEN-DOSSIER,B., see P. Jacquinot ROSENEZLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye XIII, 69 ROUARD,P., P. BOUSQUET, Optical Constants of Thin Films IV, 145 RUBINOWCZ, A,, The Miyamoto-Wolf Diffraction Wave IV, 199 SAKAI,H., see G. A. Vanasse VI, 259 XIII, 93 SCHULZ, G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces Sc-ER, J., see G. Schulz XIII, 93 Tools of Theoretical Quantum Optics X, 89 SCULLY, M. O., K. G. WHITIWY, S I T ~ CE. , K., Elastooptic Light Modulation and Deflection X, 229 SLUSHER, R. E., Self Induced Transparency XII, 53 SMITH,A. W., see J. A. Armstrong VI, 211

306

CUMULATIVE INDEX

X, 165 SMITH,D. Y., D. L. DEXTER, Optical Absorption Strength of Defects in Insulators SMITH,R. W., The Use of Image Tubes as Shutters x, 45 SODHA, M. S., A. K. GHATAK, V. K. TRIPATHI, Self Focusing of Laser Beams in Plasmas and Semiconductors XIII, 169 V, 145 STEEL,W. H., Two-Beam Interferometry IX, 73 STROHBMN, J. W., Optical Propagation Through the Turbulent Atmosphere STROKE,G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution 11, 1 Spectroscopy SVELTO, O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams XII, 1 VIII, 133 H. H., see H. Z. CUMMINS SWINNEY, 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 VII, 169 THOMPSON,B. J., Image Formation with Partially Coherent Light XIII, 169 TRIPATHI,V. K., see M. S. Sodha TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and 11, 131 by Spatial Frequency Filtering VI, 1 UPATNIEKS,J., see E. N. k i t h VI, 259 VANASSE, G. A,, H. SAKAI,Fourier Spectroscopy I, 289 VANHEEL,A. C. S., Modem Alignment Devices IV, 241 WELFORD,W. T., Aberration Theory of Gratings and Grating Mountings XIII, 267 WELFORD,W. T., Aplanatism and Isoplanatism X, 89 WHITNEY,K. G., see M. 0. Scully WOLTER,H., On Basic Analogies and P h i p a l Differences between Optical and Electronic Information I, 155 X, 137 WYNNE,C. G.; Field Correctors for Astronomical Telescopes YOSHINAGA, H., Recent Developments in Far Infrared XI, 77 VI, 105 Y W I , K., Design of Zoom Lenses YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy VIII, 295 XI, 77 YOSHINAGA, H., Recent Developments in Far Infrared