PROGRESS I N OPTICS VOLUME VIII
E D l T O R I A L ADVISORY BOARD M. F R A N ~ O N ,
Paris, France
E. INGELSTAM,
Stockholm, Sweden
K. KINOSITA,
Tokyo, Japan
A. LOHMANN,
S a n Diego, U.S.A.
W. MARTIENSSEN
Frankfurt a m M a i n , Germany
G. SCHULZ,
Berlin, Germany
M. E. MOVSESYAN,
Erevan, U.S.S.R.
A. RUBINOWICZ,
Warsaw, Poland
W. H. STEEL,
Sydney, Australia
G. TORALDO DI FRANCIA, Florence, Italy
W. T. WELFORD,
London, England
PROGRESS I N O P T I C S VOLUME VIII
EDITED B Y
E. WOLF University of Rochester, N.Y.. U . S . A .
Contributors J . W. G O O D M A N , G. A. F R Y ,
H. 2. C U M M I N S , H. L . S W I N N E Y , A. M U S S E T , A. T H E L E N , H. R I S K E N , T. Y A M A M O T O , L. L E V I , C. L . M E H T A
1970 NORTH-HOLLAND P U B L I S H I N G COMPANY - AMSTERDAM
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C O N T E N T S O F V O L U M E I (1961) THE MODERN DEVELOPMENT OF HAMILTONIAN OPTICS,R. J . 1-29 PEGIS. . . . . . . . . . . . . . . . . . . . . . . . . WAVEOPTICSA N D GEOMETRICAL OPTICSIN OPTICAL DESIGN, 11. 31-66 K. MIYAMOTO. . . . . . . . . . . . . . . . . . . . . DISTRIBUTION A N D TOTALILLUMINATION OF 111. THE INTENSITY 67-108 ABERRATION-FREE DIFFRACTION IMAGES, R. BARAKAT . . . . LIGHTA N D INFORMATION, D. GABOR . . . . . . . . , . . 109-153 IV. V. ON BASICANALOGIES AND PRINCIPAL DIFFERENCES BETWEEN 155-210 OPTICAL A N D ELECTRONIC I N F O R M A T I O N , H. WOLTER . . . . COLOR,H. KUBOTA. . . . . . . . . . . . 211-251 VI . INTERFERENCE V I I . DYNAMIC CHARACTERISTICS OF VISUAL PROCESSES, A. FIOREN253-288 TIN1 . . . . . . . . . . . . . . . . . . . . . . . . . . V I I I . MODERN ALIGNMENT DEVICES,A . C. S . V A N HEEL . . . . . 289-329 I.
C O N T E N T S O F V O L U M E I1 (1963) I. 11. III. IV.
V. VI.
RULING,TESTINGA N D USE OF OPTICALGRATINGS FOR HIGH1-72 RESOLUTION SPECTROSCOPY, G. W. STROKE . . . . . . . . THEMETROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS, 73-108 J . M. B U R C H .. . . . . . . . . . . . . . . . . . . . . DIFFUSION THROUGH NON-UNIFORM MEDIA,R. G. GIOVANELLI109-129 CORRECTIONOF OPTICALIMAGES BY COMPENSATION OF ABERRATIONS A N D BY SPATIAL FREQUENCY FILTERING, J . TSUJI131-180 UCHI . . . . . . . . . . . . . . . . . , . . . . . . . FLUCTUATIONS OF LIGHTBEAMS, L. MANDEL . . . . . . . . 181-248 METHODSFOR DETERMINING OPTICALPARAMETERS OF THIN FILMS, F. ABELES. . . . . . . . . . . . . . . . . . . . 249-28 8
C O N T E N T S O F V O L U M E I r r (1964)
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THE ELEMENTS OF RADIATIVE TRANSFER, F. KOTTLER . . . I. APODISATION, P. JACQUINOT A N D B. ROIZEN-DOSSIER . . . II. TREATMENT OF PARTIAL COHERENCE, H. GAMO. . . 111. MATRIX
1-25
29-186 187-332
C O N T E N T S O F V O L U M E I V (1965) 1-38 HIGHERORDERABERRATIONTHEORY, J . FOCKE . . . . . . 37-83 APPLICATIONS O F SHEARING INTERFEROMETRY, 0. R R Y N G D A H L SURFACE DETERIORATION OF OPTICALGLASSES, K. KINOSITA 85-143 PPTICAL CONSTANTS OF THINFILMS, P. ROUARD A N D P. Bous145-1 97 QUET . . . . . . . . . . . . . . . . . . . . . . . . . THE MIYAMOTO-WOLF DIFFRACTION WAVE,A. RUBIXOWICZ199-240 V. THEORY OF GRATINGS AND GRATING MOUNTINGS, VI . ABERRATION W. T. WELFORD. . . . . . . . . . . . . . . . . . . . 241-280 AT 4 BLACKSCREEN,PARTI : KIRCHHOFF’S VII. DIFFRACTION THEORY, F. KOTTLER. . . . . . . . . . . . . . . . . . 281-314
I. 11. III. IV.
CONTENTS O F VOLUME V (1966) I. 11. 111. IV. V. VI. VII.
OPTICALPUMPING, C. COHEN-TANNOUDJI A N D A. KASTLER. . 1-81 NON-LINEAROPTICS,P. S. PERSHAN . . . . . . . . . . . 83-144 TWO-BEAM INTERFEROMETRY, W. H. STEEL . . . . . . . 145-197 INSTRUMENTS FOR THE MEASURINGOF OPTICAL TRANSFER FUNCTIONS, K. MURATA. . . . . . . . . . . . . . . . . 199-245 LIGHTREFLECTION FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE TNDEX, R. JACOBSSON . . . . . . . . . . . . 247-286 X-RAY CRYSTAL-STRUCTURE DETERMINATION AS A BRANCH OF PHYSICAL OPTICS,H . LIPSON A N D C. A. TAYLOR . . . . . 287-350 THEWAVEOF A MOVINGCLASSICAL ELECTRON, J. PICHT. . . 351-370
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CONTENTS O F VOLUME VI (1967) 1.
RECENT ADVANCES IN HOLOGRAPHY, E. N. LEITHA N D J . UPAT-
. . . . . . . . . . . . . . . . . . . . . . . . . 1-52 SCATTERING OF LIGHTBY ROUGH SURFACES,P. BECKMANN 53-69 MEASUREMENT OF THE SECOND ORDERDEGREE OF COHERENCE, M. FRANCON AND s. MALLICK. . . . . . . . . . . . . . 71-104 K. Y A M A J I . . . . . . . . . . . 105-170 IV . DESIGNO F ZOOM LENSES, SOME APPLICATIONS O F LASERSTO INTERFEROMETRY, I). R. V. HERRIOTT . , . . . . . . . . . . . . . . . . . . . . . 171-209 STUDIESOF INTENSITY FLUCTUATIONS IN V I . EXPERIMENTAL LASERS, J . A. ARMSTRONG AND A . w. SMITH . . . . . . . . 211-257 V I I . FOURIER SPECTROSCOPY, G. A. VANASSE,H. SAKAI. . . . . 259-330 V I I I . DIFFRACTION AT A BLACK SCREEN,PART 11: ELECTROMAGNETIC THEORY,F. KOTTLER. . . . . . . . . . . . . . 331-377 NIEKS
11. 111.
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C O N T E N T S O F VOLUME V I I (1969) MULTIPLE-BEAM INTERFERENCE AND NATURALMODES IN OPEN RESONATORS, G. KOPPELMAN. . . . . . . . . . . . 1-66 METHODSOF SYNTHESIS FOR DIELECTRIC MULTILAYER FILTERS, 11. E. DELANOA N D R. J . PEGIS. . . . . . . . . . . . . . . 67-137 111. ECHOES AT OPTICALFREQUENCIES, I . D. ABELLA. . . . . . 139-168 IMAGE FORMATION WITH PARTIALLY COHERENTLIGHT, B. J . IV. THOMPSON. . . . . . . . . . . . . . . . . . , . . . . 169-230 V. QUASI-CLASSICALTHEORYO F LASERRADIATION, A. L. MIKAELIANAND M. L. TER-MIKAELIAN. . . . . . . . . . 231-297 VI. THEPHOTOGRAPHIC IMAGE, S. OOUE. . . . . . . . . . 299-358 V I I . INTERACTION OF VERYINTENSE LIGHTWITH FREE ELECTRONS, J. H. EBERLY.. . . . . . . . . . . . . . . . . . . 359-415 I.
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PREFACE This volume, like its seven predecessors, presents review articles dealing with various aspects of optics. Some of the articles cover relatively new topics, such as synthetic aperture techniques, light beating spectroscopy, statistical properties of laser light and the theory of photoelectric counting. The other articles deal with multilayer antireflection coatings, the optical performance of the human eye, vision in communication and some aspects of interference microscopy. It is hoped that the comprehensive nature of all these review articles, written by experts in the respective fields, will ensure that they will become useful contributions to the optical literature.
Department of Physics and Astronomy, University of Rochester, N . Y., 14627 September, 1970
EMILWOLF
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CONTENTS I . SYNTHETIC-APERTURE OPTICS by J . W . GOODMAN (Stanford. Calif.) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . INTERFEROMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Image formation from interferometric data . . . . . . . . . . . . . 2.3 Practical difficulties in interferometric measurements . . . . . . . . 2.4 Fringe detection and sensitivity . . . . . . . . . . . . . . . . . . 2.5 Intensity interferometry . . . . . . . . . . . . . . . . . . . . . 3. FEEDBACK-CONTROLLED OPTICS. . . . . . . . . . . . . . . . . . . . 4 . IMAGING WITH PARTIALLY FILLED APERTURES . . . . . . . . . . . . . 4.1 General properties of partially filled apertures . . . . . . . . . . . . 4.2 Temporal synthesis of a filled aperture with a double-objective telescope 5. APERTURE SYNTHESIS WITH COHERENT ILLUMINATION . . . . . . . . . . 5.1 The active interferometer . . . . . . . . . . . . . . . . . . . . . 5.2 Doppler-spread imaging . . . . . . . . . . . . . . . . . . . . . 5.3 Holographic arrays . . . . . . . . . . . . . . . . . . . . . . . 5.4 Optical synthetic-aperture radars . . . . . . . . . . . . . . . . . 6 . OBJECTI~ESTORATION BEYOND THE DIFFRACTION LIMIT . . . . . . . . . 6.1 Noiseless object restoration . . . . . . . . . . . . . . . . . . . . 6.2 Restoration in the presence of noise . . . . . . . . . . . . . . . . 7 . APERTURE SYNTHESIS BY USE OF A PRIORI INFORMATION . . . . . . . . . 7.1 Objects restricted in polarization . . . . . . . . . . . . . . . . . 7.2 Objects restricted in spatial structure . . . . . . . . . . . . . . . 7.3 Temporally restricted objects . . . . . . . . . . . . . . . . . . . 7.4 Chromatically restricted objects . . . . . . . . . . . . . . . . . . 8 . CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I1. THE OPTICAL PERFORMANCE O F THE HUMAN EYE by GLENNA . FRY(Columbus. Ohio) 1 . INTRODUCTION. INVESTIGATIVE APPROACHES. . . . . . . . . . . . . . 53 2 . THEANATOMY A N D PHYSIOLOGY OF THE RETINA . . . . . . . . . . . . 55 3 BASICCONCEPTS: LINES.POINTS A N D BORDERS . . . . . . . . . . . . . 62 POWER A N D CONTRAST AND MODULATION SENSI4 . VISUALACUITY.RESOLVING TIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 66 4.1 Measurement and specification of visual acuity . . . . . . . . . . . 4.2 Resolving power . . . . . . . . . . . . . . . . . . . . . . . . . 68 70 4.3 The concepts of contrast and modulation . . . . . . . . . . . . . .
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5 SPREAD A N D TRANSFER FUNCTIONS . . . . . . . . . . . . . . . . . . 5.1 Spread functions . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Convolving a line with a point spread function . . . . . . . . . . . 5.3 Convolving a pattern of parallel strips with a line spread function . . . 5.4 Sine-wave techniques . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary of spread and transfer functions used for the eye . . . . . . 5.6 Index of blur . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . PSYCHOPHYSICAL TECHNIQUES . . . . . . . . . . . . . . . . . . . . 6.1 The role of coherence . . . . . . . . . . . . . . . . . . . . . . 6.2 Special devices for measuring resolving power. modulation thresholds and transfer factors . . . . . . . . . . . . . . . . . . . . . . . 7 . DETERMINATION OF THE MODULATION TRANSFER FUNCTION FOR THE RE-
71 71 71 73 74 75 80 84 84 86
89 TINA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.1 Derivation from the combined function . . . . . . . . . . . . . . . 98 7.2 Direct measurement with a double slit interference pattern . . . . . . 8. PERCEIVED MODULATION OF A SINE-WAVE GRATING AT SUPRATHRESHOLD 99 LEVELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 . DETERMINATION OF THE MODULATION TRANSFER FUNCTION OF THE OPTICAL 100 SYSTEM OF THE EYE . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Campbell-Green method . . . . . . . . . . . . . . . . . . . 100 104 9.2 The Arnulf-Dupuy method . . . . . . . . . . . . . . . . . . . . 10. KESOLVING P O W E R AT UNIT MODULATION . . . . . . . . . . . . . . . 106 10.1 Effect of pupil size on the resolving power . . . . . . . . . . . . . 106 10.2 The effect of luminance level on resolving power . . . . . . . . . . 108 10.3 Resolving power for double slit interference patterns . . . . . . . . 110 11. DEPENDENCE OF THE RESOLVING POWER ON WAVELENGTH COMPOSITION. . 110 12. VISIBILITYOF SQUARE-WAVE GRATINGS. . . . . . . . . . . . . . . . 116 120 13. THE VISIBILITY OF A BAR . . . . . . . . . . . . . . . . . . . . . . 123 14. THE VISIBILITYOF BORDERS . . . . . . . . . . . . . . . . . . . . . 124 15. RETINAL REFLECTOMETRY . . . . . . . . . . . . . . . . . . . . . . 126 16. ABERRATIONS O F T H E EYE . . . . . . . . . . . . . . . . . . . . . . 17. MOTORADJUSTMENTS O F THE EYE . . . . . . . . . . . . . . . . . . 127 128 18. GENERAL REVIEWS. . . . . . . . . . . . . . . . . . . . . . . . . 128 APPENDIXI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111. LIGHT BEATING SPECTROSCOPY by H . Z . CUMMINSand H . L . SWINNEY (Baltimore, Maryland) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . 1.1 Historical review . . . . . . . . . . . . . . . 1.2 Statistics and spectra . . . . . . . . . . . . . 2 . THE THEORY OF LIGHTBEATING SPECTROSCOPY . . 2.1 Classical coherence theory . . . . . . . . . . . 2.2 Homodyne or self-beat detection . . . . . . . 2.3 Heterodyne detection . . . . . . . . . . . . . 2.4 Forrester’s approach . . . . . . . . . . . . . . 2.5 Quantum theory . . . . . . . . . . . . . . . 3 . LIGHTSCATTERING THEORY. . . . . . . . . . . . 3.1 Scattering by a dilute solution of particles . . . 3.1.1 Spherical scatterers . . . . . . . . . . . . 3.1.2 Nonspherical scatterers . . . . . . . . . 3.1.3 Rigid rods . . . . . . . . . . . . . . . . 3.2 Scattering by pure fluids and liquid mixtures . .
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135 135 138 141 142 143 146 147 150 154 154 156 158 158 159
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4. COHERENCE A N D SIGNAL TO NOISECONSIDERATIONS . . . . . . . . . . . 4.1 Spatial coherence . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Source spectrum and statistics . . . . . . . . . . . . . . . . . . 4.2.1 Phase fluctuation . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Amplitude fluctuation . . . . . . . . . . . . . . . . . . . . 4.3 Light beating with a multimode laser source . . . . . . . . . . . . 4.4 Signal to noise . . . . . . . . . . . . . . . . . . . . . . . . . 5. APPARATUS A N D PROCEDURE . . . . . . . . . . . . . . . . . . . . . 5.1 Homodyne spectroscopy . . . . . . . . . . . . . . . . . . . . . 5.2 Heterodyne spectroscopy . . . . . . . . . . . . . . . . . . . . . LINEWIDTH EXPERIMENTS . . . . . . . . . . . . 6. REVIEWOF RAYLEIGH 6.1 Dilute solutions of macromolecules . . . . . . . . . . . . . . . . . 6.2 Simple fluids near the critical point . . . . . . . . . . . . . . . . 6.3 Binary critical mixtures . . . . . . . . . . . . . . . . . . . . . 6.4 Fixman's modification . . . . . . . . . . . . . . . . . . . . . . 7 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES
163 163 167 168 169 170 174 184 185 188 190 190 191 194 195 196 197
I V . MULTILAYER ANTIREFLECTION COATINGS by A. MUSSETand A . THELEN (Santa Rosa. Calif.)
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1. INTRODUCTION 203 2. SINGLELAYERANTIREFLECTION COATINGS . . . . . . . . . . . . . . . 205 206 3. TWO-LAYER ANTIREFLECTION COATINGS ON GLASS. . . . . . . . . . . . 4. THEDESIGNMETHODOF EFFECTIVE INTERFACES . . . . . . . . . . . . 210 5. TWO-LAYER ANTIREFLECTION COATINGS ON HIGH-INDEX SUBSTRATE . . . 212 6. THREE-LAYER ANTIREFLECTION COATINGS O N HIGH-INDEX SUBSTRATE . 214 217 7 . THREE-A N D FOUR-LAYER ANTIREFLECTION COATINGS O N GLASS. . . . . . 8. SYNTHESIZED LAYERS . . . . . . . . . . . . . . . . . . . . . . . . 222 9. THE COATING OF REFLECTION REDUCING MULTILAYERS . . . . . . . . 224 225 10. ENVIRONMENTAL STABILITY OF MULTILAYER ANTIREFLECTION COATINGS . 11. OPTICALPERFORMANCE OF COATINGSAND INCREASE IN TRANSMISSION THROUGH AN OPTICAL SYSTEM . . . . . . . . . . . . . . . . . . 225 230 12. PHOTOGRAPHIC APPLICATIONS . . . . . . . . . . . . . . . . . . . . . OF STRAY LIGHTIN AN OPTICALSYSTEM . . . . . . . . . . 231 13. SUPPRESSION 236 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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V . STATISTICAL PROPERTIES OF LASER LIGHT by H . RISKEN(Stuttgart)
. 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . 2. SEMICLASSICAL THEORY . . . . . . . . . . . . . . . . . . . 2.1 Laser equations without noise . . . . . . . . . . . . . . . 2.2 Laser equation near threshold with noise (Langevin method) . 2.3 Laser equation with noise (Fokker-Planck equation method) . 3 SOLUTION OF THE LASERFOKKER-PLANCK EQUATION . . . . . . 3.1 Stationary solution and its expectation values . . . . . . . . 3.2 Expansion in eigenmodes . . . . . . . . . . . . . . . . . 3.3 Correlation functions . . . . . . . . . . . . . . . . . . 3.4 Transient of the laser oscillation . . . . . . . . . . . . . .
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4 . PHOTOELECTRON COUNTINGDISTRIBUTION . . . . . . . . . . . . . . . 4.1 General relationships between the photoelectron distribution and intensity distribution . . . . . . . . . . . . . . . . . . . . . . . 4.2 Counting distribution for short intervals . . . . . . . . . . . . . . 4.3 Expectation values for arbitrary intervals . . . . . . . . . . . . . 4.4 Condensation effect of the counting distribution . . . . . . . . . . . 5. FULLY QUANTUM MECHANICAL THEORY. . . . . . . . . . . . . . . . 5.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . 5.2 Model and derivation of the laser master equation . . . . . . . . . . 5.3 Laser master equation near threshold . . . . . . . . . . . . . . . 5.4 c-number equation of the laser master equation . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
264 264 265 271 271 272 272 273 279 286 291
VI . COHERENCE THEORY O F SOURCE-SIZE COMPENSATION IN INTERFERENCE MICROSCOPY b.y T . YAMAMOTO (Tokyo)
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . GENERAL THEORY OF TWO-BEAM INTERFERENCE MICROSCOPES . . . . . . . 2.1 General theory of two-beam interferometers . . . . . . . . . . . . . 2.2 Localized fringes . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Interference microscope and source-size compensation . . . . . . . . 2.4 Delay compensation . . . . . . . . . . . . . . . . . . . . . . . 2.5 Use of laser as a source for interference microscopes . . . . . . . . . 3 . COHERENCE DIFFRACTION THEORY O F I M A G E FORMATION A N D TWO-BEAM I N TERFERENCE
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3.1 Steel’s unified theory . . . . . . . . . . . . . . . . . . . . . . 3.2 Extension to the polarization interferometer . . . . . . . . . . . . 3.3 Shearing elements and tilting elements . . . . . . . . . . . . . . . 4 . LOCALIZATION O F F R I N G E S WITH P A R T I A L L Y C O H E R E N T LIGHT. . . . . . 5 . SOURCE-SIZE COMPENSATION. . . . . . . . . . . . . . . . . . . . . 6 . P R A C T I C A L METHODSO F SOURCE-SIZE COMPENSATION I N SHEARING I N T E R FERENCE MICROSCOPE WITH POLARIZED LIGHT . . . . . . . . . . . . . 6.1 Methods for obtaining fringed field of view . . . . . . . . . . . . . 6.2 Methods for obtaining uniform field of view . . . . . . . . . . . . . 6.3 Objective compensation and pupillary compensation . . . . . . . . . 7 . I M A G E S O F SOURCE-SIZE C O M P E N S A T E D I N T E R F E R E N C E MICROSCOPES . . . . 7 . 1 Image intensity . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Harmonic analysis of image intensity . . . . . . . . . . . . . . . 7.3 Effects of thickness of the object under observation . . . . . . . . . 8. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
297 300 300 302 303 305 305 306 306 311 312 315 317 321 323 325 330 331 332 334 336 338 338
VII . VISION I N COMMUNICATION by L . LEVI (New York. N.Y.) 1. BASICCONCEPTS . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Problems of psychophysical investigations . . . . . . . . . . . . 1.2 The fundamental characteristics . . . . . . . . . . . . . . . . . 2 . BRIGHTNESS FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Psychophysical measurement technique (GREENand SWETS[1966]) . 2.2 Instantaneous brightness function . . . . . . . . . . . . . . . . 2.3 Steady-state brightness function . . . . . . . . . . . . . . . . .
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3 . SPATIALFREQUENCY RESPONSE . . . . . . . . . . . . 3.1 Subsystems of the visual system . . . . . . . . . 3.2 Optical subsystem . . . . . . . . . . . . . . . . 3.3 Retina-brain portion . . . . . . . . . . . . . . . 3.4 Total visual system . . . . . . . . . . . . . . . 3.5 Comparison of results . . . . . . . . . . . . . . 4 . NOISEIN THE VISUAL SYSTEM. . . . . . . . . . . . 4.1 Thrcshold measurements . . . . . . . . . . . . . 4.2 Noise sources and luminance dependence . . . . . 4.3 Spatial spectrum of noise . . . . . . . . . . . . . 4.4 Amplitude distribution of noise . . . . . . . . . 4.5 Noise on the object . . . . . . . . . . . . . . . 5. SHAPEOF M T F , LINEARITY A N D STATIONARITY . . . . 5.1 Linearity of brightness function . . . . . . . . . 5.2 Shape of the visual mtf . . . . . . . . . . . . . 5.3 Non-linearities of area effects . . . . . . . . . . 5.4 Stationarity. homogeneity, or isoplanatism . . . .
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . APPENDIX: METHODSO F MEASURINGTHE MTF OF ABOVE
THRESHOLD . . . . . . . . . . . . .
KEFERENCES . . . . . . . . . . . . . . . . .
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351 351 351 354 354 356 359 359 . 361 363 . 363 365 . . . 365 . . . 365 . . 365 . . . 366 . . . 368 . . . . . . 368 THE TOTAL VISUALSYSTEM . . . . . . . . . . . . 368 . . . . . . . . . . . . 370
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VIII . THEORY O F PHOTOELEC.TRON COUNTING by C . L . MEHTA (Rochestcr. N.Y.) 1. INTRODUCTION . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . DISTRIBUTION . . . . . . . . . . 4 . PHOTOCOUNTING 4.1 Polarized thermal light . . . . . . . . . . . . 4.2 Partially polarized thermal light . . . . . . . 4.3 Laser light . . . . . . . . . . . . . . . . .
. . . . 2 . PHOTOELECTRON COUNTINGFORMULA 3. INTENSITY . . . . . . . . . . FLUCTUATIONS 3.1 Polarized thermal light . . . . . . . . . . 3.2 Partially polarized thermal light . . . . . 3.3 Laser light . . . . . . . . . . . . . . . 3.4 Harmonic signal mixed with thermal field
4.4 Bunching effects
. . . . . . . . . . . . . . . .
5 . DEADTIMEEFFECTS . . . . . . . . . . . . . 5.1 General theory . . . . . . . . . . . . . . 5.2 Intensity stabilized laser light . . . . . . . 5.3 Thermal light . . . . . . . . . . . . . . . 6 . MULTIPLECORRELATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . INVERSION PROBLEM 8. Two PHOTON ABSORPTION. . . . . . . . . . .
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375
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371 384 384 391 395 397 399 399 404 406 407
. . . . . . . . . . . . APPENDIXA : SOMEPROPERTIES O F COMPLEXGAUSSIANDISTRIBUTIONS . . . . APPENDIXB: SOLUTION OF AN INTEGRAL EQUATION . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
411 411 415 417 418 423 429 431 434 437
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECTINDEX
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I SYNTHETIC-APERTURE OPTICS BY
J. W. G O O D M A N Stanford University, Stanford, California, U S A
CONTENTS INTRODUCTION. . . . . . . . INTERFEROMETRY . .
. .
. . . . . . . . .
3
. . . .
4
.
21
IMAGING WITH PARTIALLY FILLED APERTURES
26
APERTURE SYNTHESIS WITH COHERENT ILLUMINATION . . . . . . . . . . . . . . . . . .
32
OBJECT RESTORATION BEYOND THE DIFFRACTION LIMIT. . . . . . . . . . . . . . . . . . .
39
APERTURE SYNTHESIS BY USE O F A PRIOR1 INFORMATION . . . . . . . . . . . . . . . . .
44
CONCLUDING REMARKS . . . . . . . . . . . .
47
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . .
47
REFERENCES . . . . . . . . . . . . . . . . . . . . .
48
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PAGE
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FEEDBACK-CONTROLLED OPTICS. . . . . . .
Q 1. Introduction The most important practical limitation to the resolving power of any optical system is diffractioii of light by the finite size of the primary collector. Often this limitation is not reached, for imperfections of the optical components may introduce fixed wavefront aberrations, and atmospheric turbulence may introduce dynamic distortions, both of which restrict resolution to less than the “diffraction limit”. Nonetheless, with sufficient cost, time and care, it is possible to manufacture large-aperture corrected optical systems which are, for all practical purposes, free from fixed aberrations. Furthermore, in space applications (KOPAL[1968]), or in earth-based applications which utilize appropriate image processing (MORGAN [ 1967]), atmospheric effects can be negligible and diffraction-limited performance can be closely approached. Thus for a significant class of problems, the primary limit to resolution is aperture size. In such cases the most obvious means to obtain higher resolution is the construction of larger diffraction-limited optics. However, as aperture size is increased, a number of factors make further increases less and less attractive. First, the sheer weight and size of the optics make their use all the more awkward and difficult. Second, the cost and time associated with the fabrication of largeaperture systems soon become prohibitive for many applications. It is, therefore, important to consider alternative means for obtaining high resolution without the necessity of constructing ever larger optical components. Our purpose in this review article is to present a survey of a variety of techniques for increasing optical resolution by what may be termed synthetic-aperture optics. This term is defined here in a very broad sense to include a n y technique for achieving, with one or moYe small apertures, the resolution normally associated with a single large aperture. To date, the largest body of experience with aperture synthesis has been in the microwave region of the spectrum, where synthetic-aperture 3
4
SYNTHETIC-APERTURE OPTICS
[I.
s
2
radars (CUTRONA,VIVIAN, LEITHand HALL[1961]) and radio-astronomy telescope arrays (BRACEWELL [1962]) have been used for a number of years. It is significant that most previous experience has been at microwave frequencies, for indeed the primary technical difficulties associated with many optical aperture-synthesis techniques can be traced to the minute size of optical wavelengths, typically some five orders of magnitude shorter than microwave wavelengths. In the material to follow, the various approaches to optical aperture synthesis have been divided into six classes as follows: (1) interferometry; (2) feedback-controlled optics; (3) imaging with partially filled apertures; (4) aperture synthesis with coherent illumination; ( 5 ) object restoration beyond the diffraction limit; and (6) aperture-synthesis techniques based on a priori object information. In several cases a given technique could be placed logically in two or more of these categories, and in such cases we have chosen the particular category which seemed t o us to be most appropriate.
Q 2. Interferometry The oldest embodiments of the principles of optical aperture synthesis are found in the field of interferometry. For completeness, and for the purpose of establishing notation, we first briefly review the properties of interference phenomena (WOLF [1954, 19551, MANDEL and WOLF [1965]). 2.1. BASIC PRINCIPLES
Suppose that, as a result of some distant source, or system of sources, there exists across a certain plane a statistically stationary optical wave. Neglecting polarization effects (which can be included without changing the basic results to follow), we may represent the wave by a complex-valued analytic signal V ( x ,t ) , which is a function of position x in the plane and time t . The quantity of chief interest in interferometry experiments is the complex degree of coherence, y l z ( t ) . This quantity is defined in terms of the time-averaged product of the field amplitudes at positions x, and x, separated by vector spacing s = xz-xl,and at time instants t, and t, separated by delay z. Thus
1,
s
21
5
I N TE R F E R O M E T R Y
where the angle brackets signify an infinite time average. The complex degree of coherence has a direct physical interpretation in terms of the Young’s interference experiment illustrated in Fig. 2.1. Let S be an arbitrary source, and suppose that two pinholes are pierced at points x, and x2 in an otherwise opaque screen. The intensity of light observed at a point P behind the screen can be expressed as
Intensity
Fig. 2.1. Young’s interference experiment.
where I, and I, are the intensities that would be produced from the two pinholes individually, t is the relative delay introduced by the difference in propagation times over paths from x1 to P and x, to P, c is the velocity of light, and Re{ } signifies “real part of”. As indicated in Fig. 2.1, the pattern of interference consists of relatively fine fringes under a coarse spatial envelope. The envelope has fallen appreciably when the delay time z approaches a value equal to the reciprocal of the optical bandwidth of the interfering waves. The fine structure of the pattern changes by one full period when z changes by one period of the optical oscillation. When the optical wave is quasi-monochromatic (narrowband) and when t is much smaller than the reciprocal of the optical bandwidth, the fringes are of approximately constant depth, and the complex degree of coherence may be written
where G
=
clx is the mean optical frequency of the waves and
6
SYNTHETIC-APERTURE OPTICS
[I,
9 2
p l z = ylz(0) is known as the “complex coherence factor” (also known as the “complex visibility function” in radio astronomy). The modulus lplzl of the complex coherence factor may be identified as the “visibility” of the fringes in the usual sense; that is,
where I,,, and Imin signify the maximum and minimum intensities of the fringes for a given spacing s. The phase of the complex coherence factor is identical with the spatial phase of the sinusoidal fringe pattern, relative to the phase of the fringe pattern that would be produced by a monochromatic point source of frequency fi, located in the plane of S, equidistant from points x, and x,. Aside from the basic definitions above, the relation of most importance to us here is the van Cittert-Zernike theorem (BORNand WOLF [ 1964]), which relates the complex coherence factor to the intensity distribution across an incoherent source of illumination. Let I(,) represent the intensity distribution of that source, where I ( u )is normalized such that J I (u)du equals unity, and u is an angular position measured in radians, as seen from the origin of the x plane. The van Cittert-Zernike theorem specifies that I (u)and ,ul, are related by
1 , .
where, to take care of scaling factors that would otherwise arise, 1sJ is interpreted as the spacing of the two pinholes measured in wavelengths. The phase angle y,, depends on the distances D, and D, from the origin of the u plane to the points x1 and x,, respectively, in accord with the formula
This angle may be neglected when the source of interest is very distant from the x plane (D,-D, << J), or when all measurements of p1, are made with x1 and x2 symmetrically located about the origin of the x plane.
1,
s
21
7
INTERFEROMETRY
2.2. IMAGE FORMATION F R O M I N T E R F E R O M E T R I C DATA
The stellar interferometer discussed by MICHELSON [1890] and successfully used by MICHELSON[1920], MICHELSONand PEASE[1921] and PEASE[1931], is the first historical example of an optical aperturesynthesis instrument. The interferometer is essentially a reconfiguration of Young’s interference experiment, as indicated in Fig. 2.2. A crossbeam mounted at the upper end of a telescope carries a pair of mirrors to sample the incoming wavefront and to bend the light into
B
Fig. 2 .2 . Michelson stellar interferometer
two pencil beams traveling toward the telescope axis. Near the axis the pencil beams are redirected onto the telescope by two mirrors, and merged by the telescope optics in a region where the interference pattern may be observed. From the amplitude and spatial phase of the fringe pattern observed near the telescope axis, it is possible to estimate the complex coherence factor ,ulz for the particular spacing s of the two mirrors, a distance which can considerably exceed the physical dimensions of the telescope aperture. The Michelson stellar interferometer, or indeed any interferometric imaging device, may be used with varying degrees of complexity, depending on how much information one attempts to extract from the measurements. In order of increasing complexity, the possibilities are as follows: (1) determine the particular spacing at which the visibility
8
SY N T H E T I C - A P E R T U R E 0 P T I C S
[I,
9 2
lpi2\ first vanishes; ( 2 ) measure [p12(over an entire range of spacings; or (3) measure both the modulus and the phase of p12over an entire range of spacings. A common use of interferometers is for the measurement of stellar diameters, for which only knowledge of the spacing of the first zero of 1,uI2/is required. (For a review of methods for measuring stellar diameters, see BROWN [1968].) If the object of concern is assumed to be a uniformly bright circular disk of angular diameter 2tc, then from eq. ( 2 . 5 ) we obtain
If so represents the spacing at which lplzl first vanishes, then tc
= 0.61/~,
(2.8)
is the angular radius of the star. A more ambitious undertaking, with correspondingly richer rewards, is the measurement of lp121 over an entire range of spacings. However, even this more complete information is in general insufficient to determine the intensity distribution across the object, a task for which the phase distribution associated with p12 is indispensable. For certain symmetrical objects, p12is both real and positive, in which case knowledge of (p121alone does allow full image formation. In other cases, Fourier transformation of Ipl2I2to yield the autocorrelation function of I ( u ) may provide much useful information about the object. I n general, however, full image formation is not possible from values of IPlZl alone. It was first pointed out by WOLF [1962] that p12,as a function of a complex-valued argument, has certain analyticity properties which in some cases should allow the phase of piZ to be uniquely determined Irom measurements of its modulus alone. This question has been studied by several authors (e.g., ROMANand MARATHAY [1963], WALTHER[1963]), but to date no systematic method for determining the phase has been devised. However, if a suitable reference source with a known intensity distribution is present, the ambiguities can be removed and in principle the phase of p12 can be specified (MEHTA [ 19681) . The highest degree of experimental complexity is encountered when both the modulus lp121 and the phase Q12 of the complex coherence factor are to be measured directly. Full image formation is possible in
1.
§ 21
INTERFEROMETRY
9
such a case, with the resolution being limited solely by the maximum separation possible in each vector direction. Such measurements are commonplace in radio interferometry, but optical counterparts are to date nonexistent, a consequence of the very severe practical difficulties encountered at optical wavelengths. 2.3. PRACTICAL D I F F I C U L T I E S I N I N T E R F E R O M E T R I C MEASURE-
MENTS
2.3.1. Maintaining equal path lengths
One of the chief practical difficulties expected in the operation of an interferometer, whether in space or within the earth’s atmosphere, is the problem of maintaining equal path lengths in the two arms of the instrument. The accuracy with which path-length equality must be maintained depends on the amount of information to be extracted from the measurements, and in many cases on the brightness of the object itself. If the goal is to measure the first zero or the modulus of plZ, the path lengths in the two arms must be maintained sufficiently close to equality to assure maximum visibility of the fringes. If the bandwidth of the radiation is Av hertz, then differential delays of (Av)-l seconds or greater cannot be tolerated. For astronomical objects, which are typicaIly very weak, the widest possible bandwidth must be used to assure the highest possible measurement sensitivity. As an example, an instrument which utilizes all light in the bandwidth 5000A to 6000 A would require path-length equality to better than 3 microns. If, however, the object is relatively bright, as might be the case, for example, in satellite-to-satellite imagery, narrower bandwidths can be utilized and the path-length tolerances may be relaxed accordingly. It should also be mentioned that when active illumination of the object can be provided with a laser of very poor longitudinal and transverse coherence, the object may often be treated as incoherent, particularly if it has an extended depth and the resolution required is not too great. In such cases very narrow bandwidths are allowable and the pathlength tolerances become quite manageable, being typically on the order of a few millimeters. If the goal of the measurement is full image formation, then both the modulus and phase of plZmust be measured. In this case pathlength equality must be maintained to a fraction of a wavelength in order to measure the spatial phase of the fringes, regardless of the
10
SYNTHETIC-APERTURE OPTICS
[I.
5 2
bandwidth utilized. I n the case of the conventional Michelson interferometer of Fig. 2.2, the accuracies required are so great as to make measurement of phase totally impractical. Only when more sophisticated (and expensive) interferometers are envisioned can any hope be given to direct measurement of visibility phase. One possible approach (MILLER [1966]) is to couple the Michelson interferometer with a laser system which would monitor the path lengths within the interferometer to a fraction of a wavelength, and by means of a servomechanism, move the mirrors to maintain equality. While at first glance this appears to be an enormously complicated task, recent progress in active figure control of mirrors (see § 3) suggests that the task may not be impossible, particularly for interferometers of modest length. If we consider an interferometer of 20 foot length operating in a spectral region centered around 5000a, the path lengths must be maintained to one part in about los. A laser source with a frequency stability of one part in los is commercially available, and with proper precautions far greater stabilities may be achievable. The chief practical difficulty may lie not with the monitoring and servo system itself, but rather with the actuators, which must be capable of positioning the mirrors with a high degree of accuracy. In any case, laser monitoring methods do offer some hope for overcoming one of the most severe practical problems. 2.3.2. Limitations imposed by atmospheric seeing
I n the preceding discussions we have entirely neglected atmospheric turbulence and its effects on various types of interferometers. I n practice, for earth-based interferometers the atmosphere cannot be neglected, for at optical frequencies its inhomogeneities have very significant effects on the waves reaching the measurement instrument. We can classify the major effects of atmospheric turbulence as follows: 1. Phase translation - an overall advance or retardation of the wavefront reaching each aperture of the interferometer; 2. Wavefront tilt - a variation of the apparent angular location of the source; 3. Wavefront warping - a distortion of the intercepted wavefront more complicated than phase translation or tilt; and 4. Scintillation
variations of the intensity of the light waves reaching the interferometer. -
1,
s
21
INTERFEROMETRY
11
Classical measurements of atmospheric seeing yield considerable data regarding wavefront tilt, wavefront warping, and scintillation (MEYER-ARENDT and EMMANUEL [1965]). When a telescope with a variable aperture size is used to view a star, the visible effects of atmospheric seeing change as the aperture size is increased. Under typical seeing conditions, for aperture sizes smaller than 4 or 5 inches the primary effect of seeing is observed to be image dancing, caused by the changing tilt of the wavefront striking the aperture. The amount of image dancing typically amounts to 1 to 2 seconds of arc. For aperture sizes larger than 4 or 5 inches, wavefront warping becomes important, introducing a more or less continuous blurring of the image. Studies of the scintillation of stellar images are also quite extensive (eg., PROTHEROE [1955]). For a telescope with an aperture size smaller than 3 or 4 inches, the scintillation frequency spectrum contains considerable power in the 50 to 100 Hz range. Classical measurements do not yield data on the amount of phase translation encountered due to atmospheric seeing. Estimates based on theoretical models range from a few wavelengths to seven wavelengths, and calculations using a “frozen in” model of the turbulence suggest a frequency spectrum concentrated around 100 Hz for this phenomenon (CURRIE [1968a]). The effect of wavefront tilt on the performance of an interferometer is particularly easy to understand. If the separation of the two apertures is larger than about 15 cm, the tilts encountered will be nearly independent. In the focal plane of the instrument the two images of a point source will be displaced and will only partially overlap. If electronic fringe detection techniques are used, the measured visibility of the fringes will be reduced by this phenomenon. The solution to this problem is to reduce the size of the two apertures until diffraction spreading exceeds the amount of image dancing. Such will be the case for aperture sizes of 2 inches or smaller. Thus the light-gathering power of the instrument is severely constrained by atmospheric seeing considerations. The use of larger apertures requires application of a correction factor which depends on local seeing conditions, and which may be difficult to determine accurately. If the apertures are as small as 2 inches, wavefront warping should be quite negligible and will hereafter be ignored. Scintillation of the intensity of the light striking the two apertures will also hinder the measurement of visibility. However, in this case reasonably accurate correction factors can be obtained if the scintilla-
13
SYNTHETIC-APERTURE
OPTICS
[I, §
2
tion at each of the two apertures individually is monitored. The remaining atmospheric effect is phase translation. As the differential path length through the atmosphere to the two apertures changes, the interference fringes will translate, the translation amounting to a few fringe periods. These atmospherically induced path-length differences are of serious concern in interferometry, for if white light fringes are to be observed at a fixed detector, the equality of path lengths to the two detectors must be maintained to within a few wavelengths. Thus it is conceivable that the region of high fringe visibility, moving a t a reasonably rapid rate over a distance which might be larger than its own full extent, may never be detectable, at least by the human eye. Fortunately the experimental evidence provided by the work of Michelson and Pease demonstrates that the fringes are detectable with the human eye, at least at the interferometer spacings allowed by their 20 foot instrument. However, while fringes can be seen, this by no means implies that all of the information available from the interferometer in the absence of atmospheric effects can be easily extracted when the atmosphere is present. While fringes can be detected, and with sufficient care their amplitude can be estimated, their phase varies randomly with time, adding further difficulty to any attempt to determine the phase of pI2. If in the future laser monitoring methods can be used to maintain equality of path lengths within the interferometer, phase translation due to the atmosphere will then become a limiting factor in the determination of visibility phase. With electronic fringe detection this difficulty can in principle be overcome, for with an electronic detection system (see section 2.4) it would be possible to measure and record the phase of the fringes as a function of time. From a record of one or a few seconds duration it would be possible to estimate the mean phase, which presumably would not differ greatly from the phase that would be obtained in the absence of the atmosphere. However, such a technique applies only to rather bright objects. A second possible method for eliminating the effects of phase translation is by means of a three-element interferometer which makes redundant measurements. Because of the important properties of this interferometer, we now consider it in more detail. 2.3.3. Jennison’s triple interferometer
The imaging operation performed by a lens (or mirror) has built-in redundancies which are highest for low spatial frequencies and decrease
1,
P
21
13
INTERFEROMETRY
with increasing spatial frequency. The redundancy a t any one spatial frequency arises because the particular spacing s appropriate for that spatial frequency is embraced in a multitude of ways by the aperture. The redundant information is combined before detection in the usual method of image formation. However, if redundant interferometric measurements are made at a particular spacing s, the results of the measurements are available individually and can sometimes be utilized to advantage, as we shall see.
Combining Optics
Combining Optics
Detectors
Detectors
Combining Optics Detectors
Fig. 2.3. Jennison’s triple interferometer.
A method for utilizing redundant measurements of fringe visibility to overcome phase errors introduced by variations of cable length in radio telescopes was proposed by JENNISON [1958, 19671. The technique has not found widespread use in radio astronomy, for at radio frequencies it is more economical to temperature-stabilize the cables than to add a thirdinterferometer element. At optical frequencies the constraints are quite different; the possible application of this technique in optical interferometry was proposed by ROGSTAD [19681. The general configuration of the triple interferometer is shown
14
SYNTHETIC-APERTURE OPTICS
1'. 9
schematically in Fig. 2.3. It is assumed that within each box is contained a detection system capable of measuring the amplitude jp (s)1 and phase @ ( s ) of the complex coherence factor, up to phase errors d introduced by atmospheric phase translation (for simplicity of notation in this section, we replace pl, by p(s) and QI2 by @(s) and treat the separation s as a scalar). With reference to Fig. 2.3, we suppose initially that the two shorter spacings are of equal length sl, and therefore that the longer spacing s2 is simply 2s,. We use the phase delay of the left-hand light path as a phase reference, and represent the relative phase translations of the central and right-hand paths as d, and A , , respectively. The true phases of the complex coherence factor that would be measured in the absence of atmospheric seeing are represented by @ (8,) and @(s,). The phase of the fringe pattern formed by the left-hand interferometer is 0 , = @(%) +dl (2.9) while the phase measured by the right-hand interferometer is or =
@(s1)+(d,--d,).
(2.10)
Finally the phase measured by the central interferometer is
Bc
=
@(2s,) + A 2 .
(2.11)
If (2.9) and (2.10) are added and their sum is subtracted from (2.11), we obtain
e,-(e,+o,)
= @(2s1)-@(s,)
(2.12)
which determines @(2s1)up to an unknown additive constant @(sl).It is a simple matter to show that by changing the separations of the interferometer elements it is possible to determine @(Ns,), the phase for spacing Ns,, up to an additive undetermined constant N @(sl). The effect of the unknown additive phase shifts is to introduce a linear phase function of unknown slope across the visibility function. Since the visibility function is Fourier transformed to obtain the image intensity distribution, the unknown linear phase function leads to an uncertainty as to the absolute angular position of the source relative t o the interferometer. However, the intensity distribution across the source is obtained, free from the limitations usually imposed by phase translations. Implicit in the above arguments is the assumption that the atmospheric phase translations d, and A , are identical for all points on the
I.
s
21
I N T E K F E R O M E TR Y
15
object, or equivalently, that the object lies within a single “isoplanatic patch” of the atmosphere. This assumption will be valid only if the object has a sufficiently small angular diameter. The exact size of this angular region is unknown, but probably does not exceed a few seconds of arc under average seeing conditions. Application of the triple interferometer scheme to optical interferometry is not a simple task, for it requires that the three separate interferometers be simultaneously controlled by laser monitoring and servo systems. The geometry is particularly awkward in that changes of the mirror separations of the short interferometers require movement of the combining optics to maintain path-length equality. Nonetheless the method offers an interesting potential solution to the phase translation problem, should methods of maintaining path-length equality ever be successfully developed. In addition, it may find application with existing large-aperture telescopes in certain special cases. 2.4. FRINGE DETECTION AND SENSITIVITY
In the experimental work of Michelson and Pease, fringe amplitudes were estimated visually, with appropriate correction factors applied to account for atmospheric seeing conditions. Since the time of their experiments, detector technology has undergone enormous changes, making it now possible to consider very sensitive electronic detection of fringes. Several methods of electronic fringe detection will now be outlined. 2.4.1. Detection of fringe amplitude When only the amplitude of the fringes is to be determined, the detection methods take their simplest form. The most common arrangement is that shown in Fig. 2.4 (see BEAVERS[1963], ELLIOT and SCHERB [1967], and BEAVERS and SWIFT[1968]).The fringe pattern is allowed to fall on a “barred grid” (or “picket fence”) device, consisting of alternate transparent and opaque strips, separated at the expected fringe spacing. The light transmitted by the barred grid is collected and detected by a sensitive photomultiplier tube. In laboratory experiments the fringes may be swept across the barred grid, for example by the use of a rotating prism (BEAVERSand SWIFT [1968]), or the barred grid may be periodically translated across the fringes (ELLIOT and SCHERB[1967]). For experiments conducted in the presence of atmospheric seeing, the phase translation introduced by the atmo-
16
SYNTHETIC-APERTURE OPTICS
P
T Detector
Fig. 2.4. Detection of fringe amplitudc
sphere moves the fringes across the barred grid, additional motion serving only to incrrdse the frequency of the a.c. detector fluctuations. In all cases the visibility of the fringes can be estimated from the measured amplitude of the a.c. detector current and from the measured d.c. currents produced by the interferometer apertures individually. An alternate geometry for detecting the same information is shown in Fig. 2.5 (KULAGIN [1967]). In this modification of Michelson’s interferometer, the two beams are merged with zero angle, producing a
Fig. 2.5. Modified Michelson stellar interferometer.
1,
s 21
I N T E R F E R 0 M E T RY
17
uniformly bright or dark interference field, rather than interference fringes. In fact, two output beams are produced, one of which may be detected and the second used for visual monitoring. In a more sensitive version of the instrument, both outputs are detected (see section 2.4.3). I n any case, if a variable phase shift is introduced in one of the interferometer paths, for example by atmospheric phase translation, the intensity at either one of the outputs will alternate between light and dark. If ,i and ,i represent the resulting maximum and minimum detector currents, the fringe visibility is given by zmax -$,in
liU12l
= . $,ax
(2.13)
+irnin
2.4.2. Detection of fringe phase
The interferometer geometry shown in Fig. 2.6 will in principle allow determination of both the modulus and the phase of the complex coherence factor. By means of a beam splitter two identical fringe patterns are produced, each of which is allowed to fall on a separate
I i2 Fig. 2.6. Detection of fringe phase.
18
SYNTHETIC-APERTURE
[I,
OPTICS
9: 2
barred grid detector. The two barred grids are adjusted to lie in phase quadrature, such that their transmittances are
zl(x) = 9++sgn [sin 276~x1
z2(x)= where
ct
+ +a
(2.14)
sgn [cos 2 m x ]
is the fringe frequency, and sgn
[yl 4
i
+1 0 -1
y>o, y = 0, y
(2.15)
If I , and Ir represent the intensities produced by left and right interferometer apertures individually, the detector currents i, and i, obey the proportionalities 21
& [ I +, I r l
2 -
+;dI,&IF12 sin @12 (2.16)
2
i 2 0~ ~ [ I L
+~i-l+ ; d G 1 , 4 2 1 C
O ~~ 1
2
where Q12 = arg p12. From the currents produced by the individual arms of the interferometer separately and simultaneously, both the modulus and phase of p12can be estimated. While the method of fringe detection is straightforward in principle, the vcry sevcre practical difficulties discussed in section 2.3 have prevented its successful application to date. 2.4.3. Sensitivity of the Michelson stellar ifiterferometer
There is a remarkable lack of published data to date concerning the limiting sensitivity of the Michelson stellar interferometer. Studies of a vibrating barred-grid detection device have been reported (ELLIOT [1965], ELLIOT and SCHEKB [1967], SCHERB and ELLIOT [1967]), with predictions that electronic fringe-visibility measurements should be possible for sources as weak as +12th or +13th magnitude. More recently, CURRIE [1968b] and MILLER 119681 have analyzed the sensitivity of the detection method shown previously in Fig. 2.5. I n this case both outputs of the interferometer are detectedwithsensitive photomultiplier tubes, and fringe visibility estimates are based on a comparison of the photoelectron counts produced by the two detectors. The limiting sensitivities predicted are in the +loth to +14t11 magnitude range. No information is available with regard to the limiting sensitivity
I,
s
2:
INTERFEROMETRY
19
of the Michelson stellar interferometer when both the modulus and phase of plz are to be estimated, nor has the sensitivity of the threeelement interferometer of Fig. 2.3 been studied. In both cases, sensitivities considerably poorer than those mentioned above are expected. Finally, it should be mentioned that the possibility of utilizing heterodyne detection at each of the interferometer elements might be considered because of the resulting ease with which the two low-frequency signals could be combined. While this technique is often used for radio frequency interferometers, it is in general not applicable for optical interferometry of thermal sources [see SIEGMAN [1966] and KINGSTON [1968]). The difficulty arises from the fact that an optical heterodyne receiver responds only to radiation lying in a frequency passband of width comparable to the width of the post-detection (IF) bandwidth. At visible frequencies a radiometer using incoherent detection responds to all optical power in a bandwidth of about 1014 Hz, while an optical heterodyne system would seldom have an effective optical bandwidth exceeding lo9 Hz. Since the (r.m.s.) S / N ratio achieved in a radiometric measurement is proportional to the square root of the effective optical bandwidth, heterodyne detection can be expected t o be far less sensitive than incoherent detection in this application. 2.5. INTENSITY INTERFEROMETRY
We have previously assumed that the light incident on the two collecting apertures is first combined and then detected. As first proposed by BROWN and TWISS[1956], it is also possible to use the alternative procedure of first detecting the light incident at each of the apertures and then combining the two detector outputs. The I
I
Mirror
S-
Multiply
Time
Time
output
Fig. 2.7. Intensity interferometer.
20
SYNTHETIC-APERTURE
OPTICS
LI,
s
2
result is the so-called “intensity interferometer”. The basic geometry of an intensity interferometer is illustrated in Fig. 2.7. The two primary collectors focus light on the detectors D, and D,. The photocurrents are (after suitable amplification) passed on to an electronic correlator where they are multiplied and their product is time averaged. If I , and I, represent the intensities of the light striking detectors D, and D,, then the average output of the correlator obeys the proportionality R12(r)
(2.17)
where z is the differential delay time for the two paths. Furthermore, if the light in question is of thermal origin, the average normalized correlator output yl, may be expressed in terms of the complex degree of coherence as follows (see, for example, MANDELand WOLF[1965]):
Thus from the output of the correlator it is possible to estimate the modulus of the complex coherence factor. The phase, however, is not obtainable with the usual intensity interferometer, and therefore full image formation is not possible in most cases. There are very important practical advantages attendant with the use of intensity interferometry. First, the primary collectors serve only to gather light energy and need not be high optical quality. Second, moderate atmospheric turbulence does not significantly affect the results of the measurement. Third, the combination of the two electrical channels is far simpler and less critical than the combination of the two optical paths in the Michelson stellar interferometer, for the accuracy required of the delay is determined by the electrical bandwidth rather than the far wider optical bandwidth. While the practical advantages of intensity interferometry are considerable, they are achieved only at a certain price. The chief limitation is imposed by sensitivity, or equivalently, by the integration times required to achieve usable SIN ratios. The r.m.s. SIN ratio at the output of an intensity interferometer may be written in simplest form as (c.f. BROWN [1968]) (2.19)
where nT is the total number of photons received per second by each
1,
9 31
FEEDBACK-CONTROLLED OPTICS
21
of the identical detectors, Av is the optical bandwidth (assumed rectangular), ci is the quantum efficiency of the detectors (assumed independent of frequency), 4f is the electrical bandwidth (assumed rectangular), and T ois the integration time. Unfortunately for thermal sources the number of photons per unit bandwidth, n T / 4 v , may typically be of the order of lop3; assuming reasonable numbers of ci = 0.10 and Af = lo7 Hz, then for Ip121= 1 an integration time of To = 1000 seconds is required to achieve an SIN of 10. The intensity interferometer currently in use at Narrabri, Australia, has mirrors 6.5 meters in diameter and is limited to stars brighter than +2.0 magnitude (see BROWN[l968] and Twiss [1969]). While the sensitivity of the intensity interferometer is extremely poor for thermal sources, the situation is changed when the use of active illumination is considered. The amplitudes of the waves produced by a single-mode laser do not obey gaussian statistics, and as a consequence eq. (2.19) is no longer applicable. However, when a laser is oscillating in several independent modes simultaneously, the field amplitude is very nearly gaussianly distributed (HODARA[1965]). Even for only a few independent modes, eq. (2.19) is a reasonable approximation. Most important, for such light the value of n,/4v may typically be of the order of 103, and therefore the sensitivity of the technique is vastly improved. A number of other modifications of the intensity interferometer may prove useful for some applications in the future. Without discussing the techniques in detail, we mention the three-element intensity interferometer of GAMO[1963] which allows measurement of both the amplitude and the phase of p,,, and the “compound” interferometer of MACPHIE [1966] which combines many of the advantages of both the Michelson interferometer and the intensity interferometer.
Q 3. Feedback-controlled Optics A considerable partical difficuIty encountered in the use of largeaperture optical systems is the tendency of the primary collector to change its shape under conditions of gravitational and thermal stress. A possible approach to overcoming this problem is the use of a closedloop servo-control system for regulating the figure of the mirror. For several years, studies of the feasibility of active figure control have been made by the Perkin-Elmer Corporation (MARKLE [1968], CRANE [1969]). One of the specific goals of this work has been the servo
22
SYNTHETIC-APERTURE
OPTICS
[I,
9 3
control of a number of independent mirror segments t o form a single large mirror surface of high quality. Such a task lies within our very broad definition of aperture synthesis. Segmented
Primary Mirror
Fig. 3.1. Servo-controlled segmented mirror (after MARKLE[1968]).
The study of techniques for active figure control has been largely motivated by the anticipated requirements for the second and successive generations of space telescopes. These orbiting instruments are anticipated t o have apertures larger than 1 meter. Attainment of diffraction-limited performance from such telescopes is made extremely difficult by the unfavorable acceleration and vibration environment experienced during launch, as well as by the severe thermal enviroiiment encountered while orbiting in and out of the earth’s shadow. It appears possible that significant savings in weight may be achievable if some provision is made for controlling the figure of the mirror while it is in orbit. The essential components of a servo-controlled segmented mirror are shown in Fig. 3.1. The system consists of mirror elements, a figure analyzer, an electronic error computer and control logic, and actuators to make the necessary corrections to the alignment of the mirror elements. The figure analyzer forms the heart of the system, for it performs the essential function of determining the positions of points on the
FEEDBACK-CONTROLLED
23
OPTICS
Mirror Segments
Reference Mirror
Fig. 3.2. Twyman-Green interferometer for figure scnsing (after MARKLE[196S])
mirror segments relative to an ideal spherical surface. Spatial positions are determined with the help of two different types of interferometers. First is the modified Twyman-Green interferometer shown in Fig. 3.2. A single-mode He Ne laser is used for illumination. Lens 1 converts the plane wavefront in the interferometer into a spherical wavefront. This wave strikes the spherical mirror segments, which are to be positioned such that their centers of curvature coincide with the focus of lens 1. If the spherical segments are in their proper positions, the returning spherical wave is collimated by lens 1 and the interference occurring a t the beam splitter is uniform. If the center of curvature of a segment is laterally displaced from the interferometer focus (tilt misalignment), then the corresponding portion of the interference pattern contains straight lines. A longitudinal displacement between the focus and the center of curvature produces circular interference fringes. If the uniformity of brightness of the interference pattern is to be used as a measure of figure error, extreme care must be taken to assure that the illumination across the mirror segments and reference flat is uniform, and in addition that the sensitivity of the final detector is uniform. For accuracies of the order of A j l O O , the degree of uniformity
24
S Y N T H E T I C-A P E R T U RE 0 P T I C S
i1, I 3
required becomes impractical. This difficulty may be overcome by performing a phase comparison instead of an intensity comparison. A frequency shifter is inserted in the reference arm to introduce a small frequency offset. A scanning image dissector (nonintegrating) detects the beat frequency between the light returning from the mirror segments and the reference light. The phase of the current detected at each point in the field of the image dissector is indicative of the position of a corresponding point on the segmented mirror relative t o an ideal reference sphere. The uniformity of the phase of the detected currents is thus directly related to the accuracy of the total mirror surface. Errors of A/200 can be detected in this manner. Light Chopper Arc Source \
\
Koesters Prism
Surface
Mirror Segments
Fig. 3.3. Ambiguity sensors (after MARKLE[1968]).
Unfortunately the phase measurement described above is ambiguous in multiples of 232 radians. Since the initial positions of the segments of their ideal positions, some means of removing will not be within these ambiguities is required. This task can be accomplished with the help of a number of small equal-path white-light interferometers which are placed immediately above the gaps between the mirror segments, as shown in Fig. 3.3. With the help of these so-called “ambiguity sensors” it is possible to align adjacent segments to within rt 5 microinches, which is sufficiently close to allow unambiguous interpretation of the phase measurements from the laser interferometer. The figure error measured in this manner is finally converted by
I,
s 3:
FEEDBACK-CONTROLLED
26
OPTICS
the control electronics to a number of drive signals, and the corresponding actuators on the back of each segment are driven to null the error. Experimental work to date has utilized a 20-inch-diameter segmented mirror. For simplicity, a single 20-inch mirror was cut into three pie-shaped segments. An experimentally obtained error profile of the composite mirror with the servo loop closed is shown in Fig. 3.4.The average figure error is 2/40,which is not greatly different than that of the original nonsegmented mirror. Work is now underway to apply active control and sensing techniques to improve the figure of a thin, single-piece, deformahle mirror.
-
-- -
w 0
Fig. 3.4. Measured error profile with servo loop closed (after MARKLE[1968]).
The techniques employed in the sensing and active control of a large mirror are important in a very broad sense to the entire field of synthetic-aperture optics. In particular, the experimental results obtained provide the first practical demonstration that the spatial rigidity of a large optical system can be abandoned, provided it is replaced by the temporal stability of a high-quality laser source. This possibility is of considerable importance in interferometry, and will also be of great
26
S Y N T H E TI C - A P E R T U R E 0
PT I C S
[I,
§ 4
interest when the problems associated with constructing partially filled apertures are considered in the section to follow.
Q 4. Imaging with Partially Filled Apertures Many of the chief practical problems anticipated in the use of large optical systems, particularly in space, arise from the massive weight of the primary mirror. One conceptually simple method of alleviating this problem is to construct a system with a primary aperture that is not completely filled. Thus a number of small optical elements may be spread dilutely throughout the full area normally occupied by the primary mirror, yielding a parti..lly filled aperture. 4.1. GENERAL PROPERTIES O F PARTIALLY FILLED APERTURES
The interferometry techniques discussed in section 2 are in a sense techniques for imaging with partially filled apertures. However, there are important distinctions between interferometry and the techniques to be discussed now. Perhaps the most basic distinction arises from the Aperture
-1-
Point-Spread Function
Modulotion Transfer Function
(C)
Fig. 4.1. Point-sprcad functions and modulation transfer functions of partially filled apertures.
I,
9
41
IMAGING WITH PARTIALLY FILLED APERTURES
27
different physical quantities measured in the respective cases - the complex coherence factor on the one hand and an actual image on the other hand. To obtain an image directly (i.e., before detection) it is necessary to place the individual optical elements such that they fall on the surface of an ideal mirror figure. Thus all the elementary aperture segments must share a common focus in space, a requirement not encountered in interferometry. In the future, therefore, when we refer to “partially filled apertures”, we shall always mean a collection of sr-all optical e l c y x t s sharing (either simultaneously or sequentially) a common focus. When utilizing a partially filled aperture, the advantage of lower weight is obtained only at a certain price. The most obvious price paid is the lower energy collected: when a fraction of the aperture is filled, only a fraction of the incident light is collected by the instrument. In some cases, a more subtle price is paid in terms of resolution; i.e., the resolving power of a partially filled aperture may be lower than that of a completely filled aperture of the same size. However, whether such a price is paid depends very much on the nature of the object and on the particular definition chosen for “resolution”. This point is illustrated in Fig. 4.1. For simplicity, a one-dimensional aperture of total extent I is considered. Part (a) of the figure shows the filled aperture with its associated point-spread function and modulation transfer function. Part (b) illustrates the same quantities for a partially filled aperture composed of two elements, each of width +l, separated by an empty space of width +l. Finally, part (c) shows the limiting case of two tiny elements separated by the distance 1. Several important trends can be noticed in this figure. First, as the fraction of the aperture filled is decreased, the “sidelobes” of the pointspread function increase, while the width of the central lobe decreases somewhat. This trend is important when the ability of the instrument to resolve a multitude of point sources is considered. Suppose, for example, that a multitude of point sources is spread over a region which is comparable with the full extent of the point-spread function. In such a case, high sidelobes are very harmful, and we would say that the resolving power of the partially filled aperture is less than that of the filled aperture. On the other hand, suppose that we are attempting to resolve two point sources that are separated by a distance comparable with the width of the central lobe of the point-spread function, and that there are no other point sources in a distance corresponding t o the
28
S Y N T H E T I C -AP E R T U R E 0PT I C S
[I,
I 4
full width of the point-spread function. Then owing to the narrower width of the central lobe, the partially filled aperture must be said to provide better resolution than the filled aperture. We conclude that the resolving power of a partially filled aperture depends very much on the nature of the object. A similar conclusion is reached by examination of the modulation transfer functions of Fig. 4.1. The contrast of low-frequency sinusoidal components of an image is smaller for a partially filled aperture than for a filled aperture, but the contrast of high-frequency components is actually improved in the partially filled case. Thus, whether the partially filled aperture gives a “better” or a “worse” picture depends on the exact frequency content of the object. It should also be noted that recently coherent optical data processing methods for restoring any desired weighting to the modulation transfer function have been developed and show considerable promise (STROKE [1969]). There are certain advantages to be obtained when the elements are distributed randomly over a partially filled aperture (O’NEILL [1968]), for the symmetries which lead t o high sidelobe levels are thereby destroyed. If N elements are placed at random positions on the ideal figure of a full aperture, the ratio of the peak intensity of the central lobe of the point-spread function to the mean intensity of the sidelobes is N-I. For largeN a very significant suppression of sidelobes is achieved. Undoubtedly the most difficult practical problem anticipated in the realization of a partially filled aperture is associated with the requirement that the surfaces of the individual elements coincide with the appropriate portions of the ideal figure of a filled aperture. To provide the extra structural rigidity required to hold a multitude of separate elements precisely in their proper positions, much of the potential savings in weight afforded by a partially filled aperture may in fact be lost. The possible use of active figure sensing and control techniques with partially filled apertures is an important area for future consideration, for the combination of the two techniques may provide savings in weight which far exceed the savings afforded by either technique individually. 4.2. T E M P O R A L S Y N T H E S I S O F A F I L L E D A P E R T U R E WITH A
DOUBLE-OB J E C T I V E TELESCOPE
If the elements of a partially filled aperture are moved sequentially to occupy a variety of different relative positions, then by combination of the sequence of images so obtained it is possible to synthesize a
1,
4
41
IMAGING WITH PARTIALLY FILLED APERTURES
Phose ond Position
29
Ports of o Hyperboloid Phose ond Position
Aplonotic Double Eeom Telescope
Fig. 4.2. Configuration of the double-objective telescope (after WILCZYNSKI~ [1968]).
completely filled aperture. A particular realization of this sequential aperture-synthesis technique has been suggested by WILCZYNSKI [1967a, 1967b, 19681 for astronomical observations from space. This aperture-synthesis technique would utilize two optical elements sharing a common focus, as shown in Fig. 4.2. Separate images are recorded with the elements located at various relative positions. If all of the relative positions allowed by a full aperture were actually used for separate observations, there would be considerable redundancy in the collected data. Thus just as the full aperture would make redundant measurements of the low-spatial-frequency components of the image, so the synthesized aperture would contain redundant observations at all but the furthest spacings of the two elements. In order to minimize the total time required for a complete measurement, it is desirable to remove as much of this redundancy as possible. The “clock” telescope shown in Fig. 4.3provides a means for collect-
Fig. 4.3.The “clock” telescope (after WILCZYNSKI [1968]).
30
SYNTHETIC-APERTURE OPTICS
[I>
4 4
ing the data with low redundancy. The two mirrors at the circumference are off-axis parts of the main hyperboloidal mirror of an aplanatic telescope. All possible vector spacings of the two mirrors can be realized by simple rotations on the perimeter of the circle. The mirrors are masked by hexagonal apertures which are translated (rather than rotatecl) t o the desired positions. Translation of the hexagonal apertures to retain an invariant orientation with respect to a single direction in space assures that the regions explored in the spatial frequency do(TI or main are contiguous at all but the lowest frequencies, whcrc can be tolerated. Two alignment requirements are esselitial for the success of this technique. First, it is necessary that the focal points of the two mirrors coincide during any one measurement. Second, it is necessary that the same focal point be common to every one of the sequence of measuremcnts, or equivalently, that the entire synthesized aperture have a single focal point in space. To align the telescope, a convenient “reference” star is chosen, and the two images of this star are in all cases made to coincide, To adjust the focal points of the two elements, movable compensating wedges and gimbaled plane-parallel plates are envisioned, as shown in Fig. 4.2. An auxiliary interferometer provides superposition of the light received from the reference star via the two paths. The wedges are driven until maximum fringe visibility is observed, minimizing the path-length difference to better than 2/20. Tilt fringes are present if the images are angularly displaced, and are removed by motion of the gimbaled plates. In this way the foci of the two elements are brought into coincidence. To assure that the common image is at the same point for each of the sequence of observations, a very sensitive star tracker is estimated to provide sufficient accuracy for an aperture extension by a factor of 10. Thus the star tracker may be regarded as establishing a single reference point in space, with respect to which all alignments are made; this reference point is the focal point of the full synthesized aperture. The measurement technique described above collects spatial frequency information commensurate with the full range of spacings embraced by the telescope, and does so without unnecessary redundancy of the spacings of the two elements. Nonetheless, the lowest spatial frequencies, namely those corresponding to spacings embraced by the component elements individually, are present in every one of the sequence of meaurements and therefore are measured with high redundancy. With reference to Table 4.1, which shows thenumber of separate-
1,
9
-1:
IMAGING WITH PARTIALLY FILLED APERTURES
31
observations required to increase the linear dimension of the aperture by various factors, we see that 165 observations are required to increase the aperture size by a factor of 11. In such a case the lowfrequency components (i.e., those passed by a single element) are present with 330 times their proper relative amplitudes. To remove this undesired emphasis of the low-frequency components, it is necessary to perform certain data processing operations. There are several methods which will, in principle, allow the necessary operations to be performed. If each of the images obtained during the sequence of measurements is scanned and digitized, then a composite image may be obtained by simple addition of the component images. To remove the undesired low-frequency redundancy, the composite image may be Fourier analyzed digitally, and the low-frequency components properly attenuated. A second digital Fourier transformation then yields an image with a more proper balance of spatial frequency information. TABLE 4.1 Number of observations required to extend the aperture by various sizes [1968] (hexagonal elementary apertures assumed), after WILCZNSKI Aperture extension factor 3 5 7 9 11
Number of observations required 9 30 63 108 165
Alternative methods of data processing utilize coherent optical analog techniques. If photographic transparencies of each of the component images are produced, their Fourier spectra may be obtained by placing the transparencies sequentially in a coherent optical processing system. To retain phase information in the frequency plane, a coherent reference wave is added in a manner quite similar to that employed in holography. If a multiple-exposure recording is made in the frequency plane with each of the images present sequentially, a composite frequency-plane transparency containing the full spatial-frequency data is produced. An appropriate attenuating mask placed in contact with the film can be used to reduce the exposure produced by the low spatial frequency components. Fourier transformation of the developed frequency-plane transparency, again by means of a coherent optical
32
S Y N T H E T I C-APE R T U R E 0PT I CS
[I,
5 5
system, yields the final processed image. The chief technical difficulties with this approach arise from the requirement that the absolute phase of the reference wave not change during the sequence of exposures, and from the requirement that the original images be positioned in sequence in the optical processor with very high precision.
5 5.
Aperture Synthesis with Coherent Illumination
In the preceding discussions we have been primarily concerned with the formation of images of self-luminous objects and of objects illuminated by natural incoherent light. With the present state of laser technology, it is also possible to provide intense active illumination of objects which are not too distant. While many present laser sources provide illumination which, due to multimoding and temporal instabilities, is at best only partially coherent, nonetheless sources do exist that can provide illumination with nearly perfect coherence (e.g., the 10.6 micron CO, laser). Attention is now turned to aperture-synthesis techniques that rest critically on the use of active coherent illumination. 5.1. THE ACTIVE INTERFEROMETER
When active illumination is employed, it is possible to gain spatial resolution by the use of sophisticated transmitting optics. An example of such a technique is the active interferometer proposed by MURRAY [l968], and illustrated in Fig. 5.1. A laser source illuminates the two elements of what might normally be considered a Michelson stellar interferometer. Interference of the two mutually coherent transmitted waves may be viewed as establishing a transmitted power pattern which varies sinusoidally with angle at a spatial frequency determined by the separation of the two transmitting elements. Imagine a distant object passing through the transmitted power pattern with constant angular velocity. If the size of the object is small compared with the width of a single lobe of the power pattern, then the power reflected from the target will undergo sinusoidal variations. If the power collected by a large “light-bucket’’ receiver is detected, the resulting photocurrent will likewise oscillate sinusoidally. Suppose, however, that the object is somewhat larger in angular diameter than is a single lobe of the power pattern. The oscillations of the reflected power will then be partially smoothed by the finite extent of the object, and the detected photocurrent will have a correspondingly smaller a.c. component. Thus by measuring the degree of modulation
SYNTHESIS WITH COHERENT ILLUMINATION
"
33
Light-Bucket" Receiver
Fig. 5.1.The active interferometer concept (after MURRAY[1968]).
(or the "modulation index") of the photodetector output, information concerning the size of the object passing through the multilobed beam is obtained. The advantages of an active interferometer are that a collector of very low optical quality may be used to gather the returned energy, and that for vertical paths, atmospheric turbulence may not have the same resolution-degrading effects encountered in conventional imagery. While the atmospherically induced phase translations will swing the transmitted power pattern randomly over some angular extent, the use of transmitted beams which are small in cross section is expected to prevent serious distortion of the pattern itself. Thus while the detected sinusoidal photocurrent may suffer a random phase modulation, the a.c. power should remain relatively unchanged, allowing spatial information to be extracted. 5 . 2 . DOPPLER-SPREAD IMAGING
An active technique which has strong analogies with certain Doppler mapping radar astronomy methods (eg. see EVANSand HAGFORS
34
S Y N T H E T I C -A P E
RT U R E 0 PTIC S
[I,
§ 5
round Receiving Optical
(Displayed as x Position)
Fig. 5 . 2 . Dopplcr-spread imaging (after HUFNAGEL [1967]).
[1968]) has been proposed by HUFNAGEL [1967]. With reference to Fig. 5 . 2 , a CW laser illuminates a distant object which is assumed to be rotating at a certain angular velocity about some axis of rotation. An optical system casts an image of the object onto a line array of photodetectors, where a local oscillator signal passed directly from the laser is also introduced. The outputs of these heterodyne detectors are passed to spectrum analyzers, which present displays of power vs. frequency for each detector output. For a rotating object there is a one-to-one relationship between the Doppler shift of the reflected radiation and projected distance from the rotation axis. Thus displays of received power vs. frequency are equivalent to displays of reflected power vs. distance from the rotation axis. The optical system provides resolution in the dimension parallel to the rotation axis, while the Doppler mapping provides resolution in the dimension normal to the rotation axis. Under many conditions the resolution achievable by Doppler mapping can exceed the diffraction limit (or the atmospheric limit) to the resolution achievable with a conventionalimaging system of the same aperture size. The prime candidate for the laser source for this technique is the 10.6 micron CO, laser. 5.3. HOLOGRAPHIC ARRAYS
When an object is illuminated by coherent light, it is possible to form images holographically, as well as with conventional optics. One
I.
S
51
SYNTHESIS WITH COHERENT ILLUMINATION
35
Retroref lector
[Computer J
Fig. 5.3. Holographic array.
possible holographic approach involves an array of small telescopes operating side by side to form demagnified holograms of the wavefronts existing across their respective collecting pupils, as shown in Fig. 5.3. The reference wave could be supplied externally (eg. by a retroreflector), or, for long coherence length sources, it could be supplied locally. For image fields with only a modest number of resolvable elements (e.g., 256 x 256), scanning electronic detectors could be used and the images could be formed with the help of a digital computer (see GOODMAN and LAWRENCE [1967]). The resolution achieved in the final image would be appropriate to the aperture covered by the full array, rather than that dictated by one of the component telescopes. To achieve the full resolution of the array, the lateral positions of the various detectors must be known to a fraction of the period of the finest interference fringe. For small fields of view this requirement can be met. I n addition, the relative phases of the reference waves a t the various apertures must be incorporated in the digital computations.
36
SYNTHETIC -APERTURE
0PT IC S
LL § 5
If the reference wave is supplied externally, longitudinal displacements of the detectors affect the reference and object waves identically, and the relative phase of the two waves is unchanged by such displacements. A second practical difficulty is associated with radial object motion, which causes the interference fringes on the detectors to translate. The fringes will be “frozen” in time if a short pulse of illumination is used. To further minimize this problem, the object could be illuminated at those instants when its radial velocity is minimum. Holographic arrays have the interesting and important property that the collected data are combined after detection rather than before detection. Thus while feedback-controlled optics and partially filled apertures must be properly aligned before the detection process, the holographic system can effectively align the elements a posteriori during the final image-forming computation. 5.4. OPTICAL SYNTHETIC-APERTURE RADARS
The highly successful use of synthetic-aperture radar principles in the microwave region of the spectrum (CUTRONA,VIVIAN,LEITHand HALL[1961]; SHERWIN, RUINAand RAWCLIFFE [1962]; CUTRONAand and VIVIAN[1966]) leads HALL [1962]; CUTRONA,LEITH, PORCELLO us to consider the possible application of analogous principles at optical frequencies. As indicated in Fig. 5.4, a moving vehicle (e.g., an aircraft
Fig. 5.4. Synthetic-aperture radar geometry.
or a satellite) carries an active source of coherent illumination over a straight-line path. The illumination is directed toward a distant object of interest, which is assumed to lie to one side of the flight path. The moving vehicle also carries a small-aperture coherent receiver, which
1,
s
51
S Y N T H E S I S WITH C O H E R E N T ILLUMINATION
37
detects and records both the amplitude and phase of the waves returned from the object along the flight path. The record of the amplitude and phase of the radar returns is ultimately processed to yield an end product which is a two-dimensional image of the reflectivity distribution of the illuminated object. The two dimensions of the map are in the direction normal to the flight path and in the direction parallel to the flight path. Resolution in the “range” dimension (normal to the flight path) can be obtained by pulse-echo timing, the usual radar range measurement technique. Similar information can be obtained with a CW system if the illuminating beam is made narrow in the range dimension and is rapidly scanned in range. Azimuthal resolution may be obtained by processing the entire azimuthal record received from each range resolution element. It is in this dimension that aperture synthesis takes place. A given azimuthal record may be regarded as a recording of the amplitude and phase distribution of the radiation along the flight path due to a single range cell. The resolution achieved in this dimension is determined by the distance the vehicle travels along the flight path while illuminating and receiving radiation from a given point on the object. If we assume a common angular beamwidth Q for the transmitting and receiving apertures, then for an object point at range R, the vehicle travels an approximate distance QR while receiving returns from that point. The maximum azimuthal resolution obtainable is that of a diffraction-limited aperture of length 2 9 R , where the factor of 2 arises because both the sensor and the source move along the path. The minimum resolvable dimension at range R may now be written
When the spread Q of the transmitting and receiving beamwidths is due solely to diffraction by the finite extent D of the apertures, then Q M AID and ps = 4D. (5.2) Thus the minimum resolvable dimension is equal to one-half the physical diameter of the receiving and transmitting apertures, and is quite independent of range. Such resolution may far exceed the diffraction limit of the receiving and transmitting apertures themselves. Having discussed the basic principles, we now turn to the problems encountered in attempting to design an optical version of this wellknown microwave technique. The most promising optical source is
38
S Y N T HE T I C - A P E RTU R E 0 P T I CS
[I,
s
5
again the 10.6 micron CO, laser, which can have both adequate coherence length and adequate output power for the task, assuming modest ranges (e.g., a few kilometers). The most important practical problem encountered is associated with the accuracy with which the positions of the source and sensor must be predictable, controllable, or measurable along the flight path. If position errors in excess of a fraction of a wavelength occur in a distance OR along the path, the recorded information will contain aberrations that ultimately reduce the resolution in the final imaging process. I n the microwave radar case, the above problem can be overcome by the use of accurate position sensors on board the vehicle, which allow appropriate compensation of the detected data. However, the sensing of the position of a moving platform to a fraction of an optical wavelength is not currently feasible, and therefore the problem is an extremely serious one for an optical system. A partial saving grace in the optical case is the fact that the path lengths required to synthesize an aperture of a given resolution are far shorter than those required for comparable microwave resolution. Thus, since the ratio of wavelengths in the two cases is about lo4 (assuming a 10.6 micron optical source), the requirement for lo4 times greater accuracy of source and sensor position in the optical case is partially compensated by the fact that the synthetic apertures may typically be lo4 times shorter. For example, a 1-meter synthetic aperture would be a reasonable goal at optical frequencies, but is of little interest at microwave frequencies. As a consequence of the rather short synthetic apertures allowable at optical frequencies, it soon becomes evident that a major source of difficulty may often be vibration caused by moving parts of the vehicle. Vibration amplitudes as small as 2 or 3 microns can potentially destroy the resolution of a 10-micron-wavelength system. In some cases a vibration mount providing isolation in the direction normal to the flight path may be a suitable solution. However, there is extra weight and cost associated with the use of such a mount, and in many cases that weight and cost might be more profitably invested in larger optical elements for conventional imaging with the required resolution, A second solution possible in some cases is to provide a phase reference from a cooperative retroreflector which maintains a fixed position with respect to the object (e.g., it may be physically attached to the object). In this case variations of the path of the moving vehicle affect the phase of the object and the phase reference identically, and
1,
§ 61
OBJECT RESTORATION
39
consequently do not introduce aberrations. In summary, the use of the synthetic-aperture radar rechnique in the optical region of the spectrum is made very difficult by the minute size of the wavelengths involved. While solutions can be envisioned for special cases, no widespread or general use of this technique can be envisioned at this time.
Q 6. Object Restoration beyond the Diffraction Limit For many years it was believed that the Rayleigh limit to resolution (i.e., an angular resolution of 1.22 A/D, where D is the diameter of the optics) represented a fundamental and immutable limitation to the resolving power of any optical system. More recent studies (TORALDO DI FRANCIA [1952], WOLTER[1961], HARRIS[1964], BARNES[1966] and FRIEDEN [1967]) have shown that, for a bounded object and a noiseless imaging system, analytic continuation in the frequency domain will in principle allow restoration of unlimited detail, even beyond the usual limits imposed by evanescent waves. It was simultaneously realized that .noise provides the ultimate obstacle to postand GUSTINCIC detection object restoration. More recent studies (BUCK [1967], RUSHFORTH and HARRIS[1968], TORALDO DI FRANCIA [1969]) have examined the effects of noise in some detail, and have made clearer the particular circumstances under which object restoration can be usefully employed. In this section we briefly outline the restoration problem, both with and without noise. 6.1. NOISELESS OBJECT RESTORATION
While several methods of object restoration have been proposed in the past, the most elegant formulation utilizes an integral equation approach. Assuming incoherent illumination and an object with intensity distribution O ( x ) which is nonzero only within the interval (-R, z), then under most conditions the image intensity distribution I (x)may be expressed by the convolution equation
I ( x ) = J:a O(6) S(x-6) d5
(6.1)
where S( x ) represents the point-spread function of the imaging system. If the eigenfunctions +$(x)of the integral equation
40
S Y N T H E T I C-APERTU R E 0 P T I C S
[I,
5
6
are complete in the class of objects of interest, then any such object can be expanded in the eigenfunction series
where
and
Thus the image intensity 1 (x)may be written
I(%)=
/
a
-a
m
2 O ibi(t)S ( x - t ) i=O
dt.
(6.6)
In practice it is only possible to work with a finite number M of eigenvalues, which is chosen sufficiently large to yield an adequate approximation to the actual 0 ( x ) .Using this fact, and interchanging orders of integration and summation, we obtain
ra
M
I (x)=
1 oi J - a
i=o
$bi
(8 S(x-t)dE
M
=
1 O,Ai
+i
(x).
i=O
Thus the effect of the imaging system is to multiply each eigenfunction component of 0 (x)by a corresponding eigenvalue Ai. Restoration may be accomplished by dividing the itheigenfunction component of I (x) by li,yielding l
O i= &
a
f-,
ci
(El
dt
(6.8)
and a “restored” object intensity distribution
‘s^
d(x) = i2= ~ ili
-a
I(E) $d ( E ) dt.
(6.9)
I n the unprocessed image, resolution is limited by the fact that the eigenvalues Ai drop to very small values for i greater than some
1,
§ 61
OBJECT RESTORATION
41
particular threshold. Division by these small eigenvalues, as called for by eq. (6.9), amplifies the previously attenuated eigenfunction components of the object, restoring image detail, at least to the limit commensurate with the maximum number M of eigenfunction components chosen a priori as a satisfactory “restored” approximation to the true object. 6.2. RESTORATION IN THE P R E S E N C E O F N O I S E
I n practice, the assumption of a noise-free imaging is unrealistic. The point-spread function S ( x ) may be known imperfectly, and film grain noise or electronic detection noise are invariably present. A more fundamental source of noise is the statistical fluctuation of photon arrivals, which ultimately contributes uncertainty to any measurement. Analyses of the effects of noise on the restoration process have adopted the assumption of an additive gaussian noise N ( x ) , yielding the imaging equation
I ( % )=
O(5) S(x--5) dt+N(x).
(6.10)
While this particular model is unrealistic in the many applications for which the noise is multiplicative and/or nongaussian, nonetheless it is the only model found tractable to date. The results are believed to be indicative of those that would be obtained from more general models. Using the same restoration procedure implied by eq. (6.9), the restored object distribution may be written (6.11) where (6.12)
Thus a noise term (6.13) results from the restoration process. Division of the noise components
N iby what eventually become (for large i) very small eigenvalues ili provides amplification of noise, and poses the ultimate limitation to
42
S Y N T H E TI C - A P E R T U R E 0 P T I C S
[I,
5 6
the restoration process. The mean-square error resulting from this noise may be defined by <e2> =
where
(N&,)
=
fjk
M
(6.14)
i=O
Ia I' --a
dx = 2 (1/4)z(N3
+i(x) +j (u)R N (x,u)du dx
(6.15)
--a
with R N ( x ,u)= (N(u) N(x)) being the autocorrelation function of the noise process. RUSHFORTH and HARRIS [1968] have applied the above analysis to a n imaging system which is a perfect lowpass filter, i.e., with pointspread function
S ( x ) = 2vC sinc 2vcx
(6.16)
for which the eigenfunctions are the prolate spheroidal wave functions studied by SLEPIAN and POLLACK [1961]. While no incoherent imaging system with such a point-spread function can be constructed, nonetheless a system with a more general point-spread function can be reduced to one with the point-spread function (6.16) merely by dividing by the actual optical transfer function in the spatial-frequency interval ( - y e , v,) and giving proper attention to the system noise (see FRIEDEN [ 19671). Four types of system noise have been considered in the studies reported. I n all cases the power spectral density of the noise process (at its origin) has been assumed constant over frequency. Results for various types of noise are as follows: (a) Background noise on (-a, CX).The mean square error is found in this case to be equal to M +l. Thus, as more eigenfunction components of the object are included to improve the resolution, the meansquare error in the image increases. However, if the average distance between the zeros cf + M ( x ) is taken as a measure of the size of a resolution element in the restored image, the number of resolvable elements after processing is found to be M + l . Thus for this type of noise the mean-square error per resolvable element does not change as more eigenfunction components of the object are added. (b) Background noise on (- co, co). The mean-square image error for such noise can be shown to be lint. But the A, are found to vanish very rapidly for i > 4v,a. Using the one-dimensional version of
1,
§ 61
43
OBJECT RESTORATION
the Rayleigh criterion, the number of resolvable elements in the unprocessed image is 4v,a. Thus when resolution beyond the diffraction limit is attempted, very large contributions to the mean-square image error are introduced. It is therefore very important that any background noise be confined to the same spatial region occupied by the object. (c) White measurement noise (i.e., noise with a constant power spectral density over all frequencies). The mean-square image error is in this case zEo(1/AJ2. Thus the errors increase even more rapidly than in the preceding case. (d) Bandlimited measurement noise. If the spectral density of the measurement noise is uniform over the passband of the system and zero outside that passband, the mean-square image noise is again CEOI/&. Comparison of cases (c) and (d) shows that it is extremely important to artificially bandlimit any noise produced during the detection process. However, even when such precautions are taken, the mean-square image error increases dramatically when resolutions beyond the diffraction limit are attempted. Figure 6.1 shows a plot of the r.m.s. image error (e2)* vs. the parameter d,lR, where d, is the size of a resolution element in the
5 d,/R
Fig. 6.1. r.m.s. image error (after RUSHFORTH and HARRIS[1968])
S Y N 'I' H E T I C -A P E R T U R E 0 P T I C S
44
[I,
5
7
processed image (as determined by the average spacing of the zeros of and R is the usual Rayleigh limit to resolution ( R = i v c in this one-dimensional analysis), The parameter c = 4vc tc represents the space-bandwidth product of the unprocessed image. For simplicity, a noise spectral density of unity is assumed in this figure. Lower SIN ratios in the unprocessed image imply correspondingly higher values of (e2)* than indicated here. Each different value of c yields a different curve, and each curve is characterized by two regions, one of low slope and one of high slope. As the complexity of the unprocessed image increases (i.e., c increases), the curves shift to the right, indicating that to improve the resolution in the processed image to some fixed fraction of the Rayleigh limit requires greater and greater signal-to-noise ratio as the space-bandwidth product of the unprocessed image grows. These results indicate that, if the object is very poorly resolved by the optical system a t the start, a significant improvement in resolution (i.e., the addition of a few resolvable elements) can be accomplished if reasonably high signal-to-noise ratios are available before processing. If, however, the object is extremely complex at the start (e.g., an aerial reconnaissance photograph), improvement of resolution by even a small fraction of the Rayleigh limit requires signal-to-noise ratios that are unrealistically high. There are, of course, applications in which the object is very poorly resolved at the start and the signal-to-noise ratio is high. For such applications, object restoration beyond the diffraction limit could play an important role. In addition, in microscope imagery of time-invariant specimens, rather enormous signal-to-noise ratios can be achieved by scanning the image slowly with a very sensitive detector, and therefore object restoration may find some application in this area. Finally, it should be appreciated that when information less complete than full object restoration (e.g., discrimination, counting, determination of size and general shape) is desired, significant improvements appear achievable by post-detection processing with far less stringent requirements on signal-to-noise ratio.
+M(x))
tj 7. Aperture Synthesis by Use of a priori Information A wide variety of techniques for utilizing a priori object information to obtain resolution beyond the usual diffraction limit are described in the literature. Indeed several techniques described in previous sections
1,
s
71
S Y N T H E S I S BY USE O F A P R I O R 1 I N F O R M A T I O N
45
could be placed in this category. An object is seldom a completely general function of position, polarization and time; when one or more characteristics of the object are specified a priori, such restrictions can often be used to obtain improved resolution (see LOHMANN [1968], LUKOSZ[1966]). In this section we give some examples of the important gains that can be achieved by proper use of a priori information. 7 . 1 . OBJECTS RESTRICTED I N POLARIZATION
When a nonbirefringent object is illuminated by polarized light, proper encoding of the object information into two separate polarization “channels” can double the resolution of the optical system. For example (see LOHMANN [1956], GARTNERand LOHMANN [1963], LOHMANN and PARIS[1964]), as shown in Fig. 7.1, a Wollastonprism can be used to encode spatial frequencies vz > 0 into one polarization and spatial frequencies v, < 0 into the orthogonal polarization. The full aperture of the optical system is used for each polarization component. A second Wollaston prism recombines the two channels to produce an image with doubled resolution in the x-direction. Wol laston Prism
Wollaston Prism
Imaging Elements
(Source
(Polarizer
‘Object
Analyzer1 (Image
Fig. 7.1. Aperture synthesis with Wollaston prisms (LOHMANN and PARIS[1964]).
7.2. OBJECTS RESTRICTED I X SPATIAL STRUCTURE
When the spatial structure of the object is highly constrained, significant improvements of resolution can be attained by taking advantage of these constraints. An example is the use of the moirC effect by Lord Rayleigh to test gratings. When a grating of known periodicity and unknown defects is placed in contact with a perfect grating of the same period, deformations of the coarse moirC fringes indicate the very fine grating defects. A second example is provided by the so-called “super-gain aperture” (TORALDO Di FRANCIA [1952]). If the amplitude and phase transmis-
46
S Y N T H E TI C - A P E R T U R E 0 P T I C S
[I,
s:
'i
sion across the pupil of an imaging system are varied in the proper prescribed manner, the central maximum of the point-spread function is made narrower, a t the price of higher secondary maxima, which may be several resolution units from the central maximum. Such a technique can be used to resolve two point sources separated by less than the Rayleigh limit, at the price of inefficient use of energy and a very limited allowable object field. Other examples are the double star methods of LACOMME [1954] and LAU 119371, which may be used to resolve two closely spaced point sources, and the grating technique utilized by BACHLand LUKOSZ [1967] which increases resolution a t the price of a limited object field. The reader may consult the references cited for further details. 7.3. TEMPORALLY RESTRICTED OBJECTS
If the object is time independent, or changes only slowly with time, a scanning system may be employed to transfer spatial information to the time domain. The temporally modulated wave may be sent to the image plane by an optical system with essentially no resolving power, and a second, synchronous scanning system may then be used to reconstruct the image. A second example is the moving-grating technique of LUKOSZ and MARCHAND [1963] which is illustrated in Fig. 7.2 (see also LUKOSZ [1967]). In this case a moving grating is used to deflect the object light through a multiplicity of angles, each angle of deflection being simultaneously coded with a different temporal frequency. The various temporal frequency components each carry through the system a different segment of the spatial-frequency spectrum of the object. At the output of the system a second grating, located in a plane conjugate to that of the first grating, moves in the opposite direction with the same speed. The result is a decoding of the temporal frequency information. The image produced by the first lens is then relayed t o a detector
t"
vO-2Av
Fig. 7.2. Aperture synthesis with a moving grating (LUKOSZ [1967]).
ACKNOWLEDGMENTS
47
which has associated with it a bandpass temporal filter, responding only to frequency v,,. The final image resolution can far exceed that produced by the unaided optical system. 7.4. CHROMATICALLY RESTRICTED OBJECTS
If the object is chromatically restricted, i.e., contains only various shades of gray, then spatial information can be transferred to chromatic information. For example, if white light is sent into a prism spectroscope and the displayed spectrum is used for illuminating the black and white object, a specific wavelength is assigned to each spatial x-coordinate (see, for example, KARTASHEV[19601). The chromatically coded light may be sent through an optical system with spatial resolution in the y-direction only. A second prism spectroscope again displays the various wavelengths in the x-direction, thus recovering spatial resolution in that dimension. Q 8. Concluding Remarks In the previous sections we have attempted to outline the present status of the field of synthetic-aperture optics. Some of the techniques mentioned are old, while some are relatively new. Some of the techniques discussed have been put to practical use on a limited basis (e.g., the Michelson stellar interferometer and the intensity interferometer) ; however, the majority have not yet reached a state of development that allows their use 011any significant scale. I t is premature to attempt a guess as to which of these various methods will be important in the future and which will remain a theoretical curiosity, for the results will undoubtedly rest on future developments in optical technology and on the ingenuity of optical engineers.
Acknowledgments The material presented in this article has been drawn largely from a report entitled “Synthetic-Aperture Optics”, which was the product of a Summer Study held in 1967 by the National Academy of Sciences, under the sponsorship of the U.S. Air Force Systems Command, to whom we express our gratitude. While it is impossible t o mention the names of all 56 scientists who participated in the Study, special mention should be made of the following: David D. Cudaback, Douglas G. Currie, Richard H. Miller, and David H. Rogstad, who made major
48
S Y N T H E T I C - A P E R T U R E 0 PTI C S
I1
contributions to section 2 of this article; David A. Markle, for his contributions to section 3; Janusz S. Wilczynski, for his contributions to section 4;Craig K. Rushforth, who is responsible for much of section 6; and Adolf W. Lohmann, whose contributions are represented in very abbreviated form by section 7. The good qualities of this article are reflections on the competence and enthusiasm of the participants in the Study; the author accepts responsibility for any shortcomings.
References BACHL,A. and W. LUKOSZ, 1967, J . Opt. soc. Am. 5 7 , 163. BARNES, C. W., 1966, J . Opt. SOC.Am. 5 6 , 575. BEAVERS, W. I., 1963, Astron. J . 6 8 , 273. BEAVERS, W. I. and W. D. SWIFT,1968, Appl. Opt. 7 , 1975. BORN,M. and E. WOLF, 1964, Principles of Optics (second ed., Pergamon Press, London, N.Y.). R. N., 1962, in: Handbuch der Physik, Vol. LIV, ed. S.Fliigge BKACEWELL, (Springer-Verlag, Berlin) pp. 42-129. BROWN, R. Hanbury and R. Q. TWISS,1956, Nature 1772 27. BROWK, R. Hanbury, 1968, in: Annual Review of Astronomy and Astrophysics 6 , ed. L. Goldberg (Annual Reviews, Inc., Palo Alto, Calif.) pp. 13-38. BUCK, G. J . and J . J . GUSTINCIC, 1967, I E E E Trans. Ant. Prop. AP-15, 376. CRANE,R . , 1969, Appl. Opt. 8, 538. CURRIE,D. G., 1968a, in: Synthetic-Aperture Optics, Vol. I1 (National Academy of Sciences, Washington, D.C.) pp. 79-83. CURRIE,D. G., 196813, i b i d , pp. 35-41. CUTRONA,L. J . , W. E. VIVIAN,E. N. LEITHand G. 0. HALL, 1961, I R E Trans. Mil. Electr. MIL-5, 127. CUTKONA, L. J. and G. 0. HALL,1962, I R E Trans. Mil. Electr. MIL-6, 119. CUTRONA,L. J., E. N. LEITH,L. J. PORCELLO and W. E. VIVIAN,1966, Proc. I E E E 5 4 , 1026. ELLIOT, J . L., 1965, M.S. Thesis, Dept. of Physics, MIT. ELLIOT, J. L. and F. SCHERB, 1967, Astron. J. 72, 794. EVANS,J . V. and T. HAGFORS, 1968, Radar Astronomy (McGraw-Hill Book Co., New York). FRIEDEN, B. R., 1967, J . Opt. SOC.Am. 5 7 , 1013. GAMO,H., 1963, J. Appl. Phys. 3 4 , 875. GARTNER, W. and A. LOHMANN, 1963, Z. Physik 1 7 4 , 18. GOODMAN, J. W. and R. W. LAWRENCE, 1967, Appl. Phys. Letters 11, 77. H A R R I S , J . L., 1964, J. Opt. SOC.Am. 54, 931. HODARA, H., 1965, Proc. IEEE 53, 696. HUFNAGLE, R. E., 1967, in: Restoration of Atmospherically Degraded Images, Vol. I1 (National Academy of Sciences, Washington, D. C.) pp. 239-246. JENNISON, R. C., 1958, Monthly Notices Roy. Astron. SOC.1 1 8 , 276. JENNISON, R. C., 1967, Introduction to Radio Astronomy (Philosophical Library, Inc., New York). A. I., 1960, Opt. Spectry. 9, 204. KARTASHEV,
I1
REFERENCES
49
KINGSTON,R. H., 1968, in: Synthetic-Aperture Optics, Vol. I1 (National Academy of Sciences, Washington, D.C.) pp. 73-78. KOPAL,Z., 1968, Telescopes in Space (Faber & Faber Ltd., London). KULAGIN, E. S., 1967, Opt. Spectry. 2 3 , 459. LACOMME, P., 1954, Opt. Acta 1, 33. LAU,E., 1937, Phys. 2 s . 3 8 , 446. LOHMANN, A. W., 1956, Opt. Acta 3, 97. LOHMANN, A. W. and D. P. PARIS, 1964, Appl. Opt. 3, 1037. LOHMANN, A . W., 1968, in: Synthetic-Aperture Optics, Vol. I1 (National Academy of Sciences, Washington, D.C.). LUKOSZ, W. and M. MARCHAND, 1963, Opt. Acta 10, 241. LUKOSZ, W., 1966, J . Opt. SOC.Am. 56, 1463. LUKOSZ, W., 1967, J . Opt. SOC.Am. 5 7 , 932. MACPHIE,R. H., 1966, IEEE Trans. Ant. Prop. AP-14, 369. MANDEL, L. and E. WOLF, 1965, Rev. Mod. Phys. 37, 231. MARKLE, D. A,, 1968, in: Synthetic-Aperture Optics, Vol. I1 (National Academy of Sciences, Washington, D.C.) pp. 133-167. MEHTA,C. L., 1968, J . Opt. SOC.Am. 5 8 , 1233. MEYER-ARENDT, J . R. and C. B. EMMANUEL, 1965, Optical Scintillation: A Survey of the Literature, N.B.S. Tech. Note No. 225 (National Bureau of Standards, Washington, D.C.). MICHELSON, A,, 1890, Phil. Mag. 3 0 , 1 . MICHELSON, A., 1920, Astrophys. J . 5 1 , 257. MICHELSON, A. and F. PEASE, 1921, Astrophys. J . 5 3 , 249. MILLER,R. H., 1966, Science 153, 581. MILLER,R. H., 1968, in: Synthetic-Aperture Optics, Vol. I1 (National Academy of Sciences, Washington, D.C.) pp. 95-105. S.P., 1967, Nature 2 1 3 , 465. MORGAN, MURRAY, B. C., 1968, in: Synthetic-Aperture Optics, Vol. I1 (National Academy of Sciences, Washington, D.C.) pp. 239-262. O’NEILL,G. K., 1968, Science 160, 843. PEASE, F. G., 1931, Ergeb. Exakt. Naturw. 10, 84. PROTHEROE, N. M., 1955, Contrib. Perkins Obs. 2, 127. ROGSTAD, D. H., 1968, Appl. Opt. 7, 585. ROMAN,P. and A. S. MARATHAY, 1963, Nuovo Cimento 30, 1452. RUSHFORTH, C. K. and R. W. HARRIS,1968, J . Opt. SOC.Am. 5 8 , 539. SCHERB, F. and J . L. ELLIOT, 1967, Astron. J . 7 2 , 826. SHERWIN, C. W., J . P. RUINAand R. D. RAWCLIFFE, 1962, I R E Trans. Military Electron. MIL-6, 11 1 . SIEGMAN, A. E., 1966, Proc. IEEE 5 4 , 1350. SLEPIAN, D. and H. 0 . POLLAK, 1961, Bell Syst. Tech. J. 40, 65. STROKE, G. W., 1969, Opt. Acta 16, 401. TORALDO Dr FRANCIA, G., 1952, Nuovo Cimento Suppl. 9, 426. TORALDO DI FRANCIA, G., 1969, J. Opt. SOC.Am. 59, 799. TWISS,R. Q,, 1969, Opt. Acta 16, 423. WALTHER, A,, 1963, Opt. Acta 10, 41. WILCZYNSKI, J. S., 1967a, J . Opt. SOC.Am. 5 7 , 579. WILCZYNSKI, J. S., 1967b, J. Opt. SOC.Am. 5 7 , 1415.
50
S Y N T H E T I C-A P E R T U R E O P T I C S
[I
WILCZYNSKI, J . S., 1968, in: Synthetic-Aperture Optics, Vol. I1 (National Academy of Sciences, Washington, D.C.) pp. 211-233. WOLF,E., 1954, Proc. Roy. Soc. (London) A 2 2 5 , 96. WOLF,E., 1955, Proc. Roy. Soc. (London) 230, 246. WOLF,E., 1962, Proc. Phys. Soc. (London) 8 0 , 1269. WOLTER,H., 1961, in: Progress in Optics, Vol. I, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) pp. 155-210.
I1
THE OPTICAL PERFORMANCE OFTHEHUMANEYE BY
G L E N N A. F R Y College of Optometry, T h e Ohio State University Columbus, Ohio, U S A
CONTENTS PAGE
9 3
9
s 9:
INTRODUCTION. INVESTIGATIVE APPROACHES . 2 . THE ANATOMY AND PHYSIOLOGY O F THE RETINA . . . . . . . . . . . . . . . . . . . . . 3 . BASIC CONCEPTS: LINES. POINTS AND BORDERS 4 . VISUAL ACUITY. RESOLVING POWER AND CONT R A S T A N D MODULATION SENSITIVITY . . . .
53
5.
71
1.
SPREADANDTRANSFERFUNCTIONS . . . . . . 3 6 . PSYCHOPHYSICAL TECHNIQUES . . . . . . . . 9 7 . DETERMINATION O F THE MODULATION TRANSFER FUNCTION FOR THE RETINA . . . . . . . . 9 8. PERCEIVED MODULATION O F A SINGLE-WAVE GRATING A T SUPRATHRESHOLD LEVELS . . . 3 9 . DETERMINATION O F THE MODULATION TRANSFER FUNCTION O F THE OPTICAL SYSTEM O F THE EYE . . . . . . . . . . . . . . . . . . . . . . . 9 10. RESOLVING POWER A T U N I T MODULATION . . 9: 11 . DEPENDENCE O F THE RESOLVING P O W E R ON WAVELENGTH COMPOSITION . . . . . . . . . . 9 12. VISIBILITY O F SQUARE-WAVE GRATINGS . . . . 5 13. THE VISIBILITY O F A BAR . . . . . . . . . . . 5 14. THE VISIBILITY O F BORDERS . . . . . . . . . 3 15. RETINAL REFLECTOMETRY . . . . . . . . . . . 9 16. ABERRATIONS O F THE E Y E . . . . . . . . . . . 9 17 . MOTOR ADJUSTMENTS O F THE E Y E . . . . . . . 9 18. GENERAL REVIEWS . . . . . . . . . . . . . . . APPENDIX I . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . .
55 62 66 84
89 99
100 106 110 116 120 123 124 126 127 128 128 129
Q I . Introduction, Investigative Approaches The human eye (Fig. 1) is an optical device which forms an optical image on its retina. The retina (Fig. 2) has many layers, but the first layer to become involved in the act of seeing is the layer of photoreceptors which are called rods and cones. The rods respond at low levels of irradiance so that if two neighboring rods absorb one quantum each within the same tenth of a second, the two rods acting together can generate a signal which can be transmitted through the retina to the brain and can be detected. Two quanta absorbed by a single rod can also generate such a signal. The cones are much less sensitive than the rods, but at the center of the fovea only cones are found, and the sensitivity of this region of the retina has to depend on the cones. A single cone has to absorb many
OPT1
CILIARY BODYT]-
ORA S E R R A T A l
Fig. 1. Horizontal section of the human eye. (From FRY[1965a].)
53
54
THE OPTICAL PERFORMANCE OF THE HUMAN EYE
111,
9
1
CHOROID PIGMENT E P l T H t L l U M
RODS AND CONES /OUTER LIMITING MEMBRANE C E L L BODIES OF RODS
(0)AND
CONEStb)
~
SYNAPSES BIPOLAR (d,e.f,h) AND HORIZONTAL ( C ) C E L L S SYNAPSES AMACRINE C E L L S (i) GANGLION CELLS (rn,n,o,p,s) GANGLION C E L L AXONS VITREOUS
t I
9 1 INNER LIMIT I NG ,'MEMBRANE
Fig. 2. The retina. A schematic reconstruction showing pathways through the retina and their lateral interconnections. (Drawing from POLYAK [I9571 and relabeled by FRY[1965a] .)
more than two quanta during a short flash of light in order to generate a detectable signal. The same number of quanta can also produce a detectable signal if they are distributed over a small cluster of cones. In dealing with theoretical problems of optical image formation on the retina, one can substitute for the layer of rods and cones a hypothetical retinal layer which is smooth and uniform, infinitesimal in thickness, and capable of absorbing all the light that falls on it. Although it is true that many problems can be solved by this approach, one can also use other approaches. One can enucleate the eye and cut a window in the back of the eye as DEMOTT[1959] and others have done with eyes of lower animals, and investigate directly the quality of the image formed in the vitreous humor. So far the results obtained in this way have not correlated well with data obtained by other kinds of measurements. Problems have been encountered in preventing changes in the eye between enucleation and measurement. Such eyes are deprived of their blood supply and mechanisms for maintaining pressure, posture and transparency. Another approach is to study the image formed by light reflected from the retina. This is useful in retinoscopy and objective refractometry where measurements are needed for prescribing glasses to correct refractive errors and to compensate for loss of accommodation. The light is diffusely reflected, and the pigment epithelium appears to be
11.
§ 21
T H E ANATOMY AND PHYSIOLOGY O F T H E RETINA
55
the layer most involved. Fortunately, this layer lies next to the photoreceptors so that no allowance needs to be made for discrepancies between subjective and objective measurements of the refraction of the eye (FRY[ 19491). The refraction of the eye is the reciprocal of the distance measured along the primary line of sight from the spectacle point (14mm in front of the cornea) to the point conjugate to the retina. It is not possible as of now to evaluate the role played by the retinal elements in degrading the image formed on and reflected back from the pigment epithelium, and it has been difficult to relate the quality of this image to the image picked up by the photoreceptors as the light passes through the retina on the way to the pigment epithelium. Progress, however, is being made in this direction. Vos [1963] in particular has studied various ramifications of this problem. Many investigators have measured the distribution of flux in the images formed by light reflected from the retina and have used this information to assess the quality of the optical images formed on the retina. These measurements will be considered in more detail later. In the psychophysical approach the eye is left intact and the subject describes what he sees when certain stimuli are applied to the retina. Among other things, he can make the judgment that he did or did not see something during a specified interval of time, or that he is now seeing this or that, or that two patches are equal in brightness or can be discriminated from each other. If we utilize such judgments in the study of the optical performance of the eye, it is necessary to understand the retinal mechanisms for transforming the optical image to a physiological impression that can be transmitted along isolated optic nerve fibers to the brain. It is very proper, therefore, to include in this discussion the eye-brain mechanisms for processing and transmitting retinal impressions. § 2. The Anatomy and Physiology of the Retina
If we are to use the psychophysical approach, it is essential to understand the structure and function of the retina. It is not a matter simply of reviewing well known facts. There has been continuous progress toward a fuller understanding of these mechanisms. These recent developments are the things that I propose to emphasize. Figure 1 shows a cross section of the human eye. Light entering the eye passes through the cornea on its way to the pupil, which constitu-
56
T H E OPTICAL PERFORMANCE O F T H E HUMAN EYE
[II,
P
2
tes the aperture-stop, and the light is then refracted by the crystalline lens of the eye, which lies behind the pupil, and proceeds to the retina. As shown in POLYAK'S [1957] diagram (Fig. 2 ) , the photoreceptors lie on the back side of the retina so that light passing through the pupil must also pass through the entire retina before entering the photoreceptors. The photoreceptors funnel the light into their outer segments. This funnelling process amplifies the problem of self-filtering of photopigment that absorbs light and generates signals. In the case of the cones, the funnelling accounts for the directional sensitivity in which cones pointed toward the center of the pupil absorb much more light than those pointing toward the edge of the pupil. This is known as the STILES-CRAWFORD[1933] effect and is illustrated in Fig. 3. I.0-
0.80.6 0.4
-
0.24 2 0 2 4 r r I DISTANCE FROM THE CENTER OF THE PUPIL
Fig. 3. The Stilcs-Crawford effect. Data for the horizontal meridian of the right eye of Subject B.H.C. (From FRY[1955].)
Since the cones have a center-to-center distance of about 3p, the diameter of a cone is small with respect t o the wavelength of light, and it is necessary to treat the photoreceptors as wave guides; thus a special branch of optics called retinal optics (ENOCH [1963]) has grown up to cover this phase of the problem. The important question to be asked in this connection is whether the photoreceptors that fall in the different parts of the area covered by the image of a point respond in proportion to the flux per unit area falling in these parts (FRY[1955]; ENOCH and FRY[1958]). Retinal densitometry is an objective method of studying the bleaching of the photopigment by light. It has been demonstrated (RUSHTON [1956 and 19611; FRY[1969a]) that the rate of bleaching in rods for white light (2750' K) is
11.
s
51
THE ANATOMY A N D PHYSIOLOGY O F THE RETINA
ds/dt
=
- 1 . 6 2 ~ 10-7 Es,
57
(1)
where s is the fraction of the maximum amount of photopigment in a given unit area of the retina, and E is the illuminance falling on the retina, expressed in photopic trolands. When a given molecule of photopigment absorbs a quantum, it decomposes and generates a signal; but the molecule eventually regenerates so that in darkness the total amount of photopigment per rod always returns to its maximal value. As a matter of fact, at ordinary levels of luminance only a small fraction of the pigment is bleached, so that the rate of bleaching is proportional to the retinal illuminance. This fact is of considerable importance in analyzing the effect of optical blur on the response of the eye. Note that when s = 1, eq. (1) becomes a statement of the basic photochemical (Roscoe-Bunsen) principle, that the response is proportional t o the intensity times the time, i.e., ds
N
Edt.
(2)
The situation is somewhat more complicated with cones. Microspectrophotometry has made it possible to demonstrate that there are three kinds of cones - red-sensitive, green-sensitive and bluesensitive. Furthermore, retinal densitometry has made it possible to measure the rates of bleaching of the red-sensitive and green-sensitive pigment in the fovea of the eye where only cones are found. For white
Fig. 4. Relative luminosity curves: (a) photopic; (b) scotopic. (From Science of Color.)
5s
T H E OPTICAL PERFORMANCE OF T H E HUMAN E Y E
[II,
§ 2
light (2750' K) the rate of bleaching (RUSHTON [1958]) is the same for red-sensitive and green-sensitive pigments:
dsldt
=
- 2 10-7 ~
ES.
(3)
The amount of blue-sensitive pigment is so small that up to now its rate of bleaching has not been measured by retinal densitometry. In studies of image processing by the eye, it is customary to use an artificial aperture-stop which restricts the size of the beam entering the eye to about 2mm. This eliminates any complications from the
w 0.4
>
w
a 400
500
450
0.008
-
0906
-
OLOO)
-
600
550
650
roo
0.001 -
0!002
400
450
WAVELENGTH
550
500
(mu1
Fig. 5. The red (R), green (G) and blue (B) components of the ICI photopic luminosity curve @). The graph a t the right for the blue end of the spectrum is magnified vertically 100 times. (From FRY[1965b].)
11,
8 21
T H E ANATOMY A N D PHYSIOLOGY O F THE RETINA
ANGULAR
DISPLACEMENT
FROM T H E CENTER OF T H E
59
FOVEA
Fig. 6. Concentration of rods and cones a t different distances from the center of the fovea (based on the data of GSTERBERG [1936]).
Stiles-Crawford effect and keeps the rate of bleaching proportional to the amount of light entering the eye. The concept of rate of bleaching gives new meaning to the photopic (cone) and scotopic (rod) luminosity curves. These luminosity curves (Committee on Colorimetry of the Opt. SOC.Am. [1953]) are shown in Fig. 4.The retinal illuminance can be re-expressed in terms of retinal irradiance; luminosity for the rods and for the different kinds of cones can be re-expressed in terms of rate of bleaching per unit of retinal irradiance for the different dil bands of the spectrum. The laws of color mixture indicate that the signals generated by the different cones are proportional to the rates of bleaching and are additive; hence, it is assumed that brightness information from red-, green-, and blue-sensitive cones is combined and summated and transmitted over the same channels. Therefore, it may be said that the brightness response of the cones in a given region of the retina is proportional to the sums of the rates of bleaching in the three kinds of cones. I n the case of the peripheral retina, the response of the rods is integrated with that of the cones.
60
THE OPTICAL P E R F O R M A N C E OF T H E HUMAN E Y E
tII,
9
2
The splitting of the photopic luminosity curve into its red, green, and blue components must conform to the absorption curves found by retinal densitometry and microspectrophotometry of individual photoreceptors, but one can analyze color mixture data by locating the three fundamental colors on a color mixture diagram according to specific criteria and can use this approach to make a precise evaluation of the red, green, and blue components of the luminosity curve. My own analysis (FRY[1965]) of the luminosity curve into its three components is shown in Fig. 5 . It is important to keep in mind the relative number of rods and cones at different parts of the retina. ~ S T E R B E R G ’ S [1935] data are shown in Fig. 6. At the center of the fovea only cones are found. The number of ganglion cells per unit area is higher here than elsewhere, and on this account the fovea is better designed for the perception of fine detail than any other part of the retina. Thus, to see most effectively one must point his eye so that the critical details which need to be seen project their images on the fovea. This pointing skill is a feature of the eye’s performance as an optical instrument which will be considered later. The important thing is that when the eye is used in conjunction with a microscope or telescope or for viewing a photograph made with a microscope or telescope or for viewing a photograph made with a camera, it is the foveal region of the retina which is all important. It is proposed, therefore, to cover only foveal vision in this review. Another feature of the retina is that each photoreceptor is not
I .O
-
0.8 -
\ I
R I Roe-kt
I=Ioe-k2(i/cTf
I1
I
p0.6 z
kr0.8 0-=8
; pR I 1 I I
20.4 -
I Q W
0.2 -
2 GO.0 -
------=’
_I
w c0.2
I
1
I
I
I,
‘./
I
I
I
I
I
Fig. 7. Line spread functions for inhibition ( I ) and excitation ( R )in the retina. The area under the I curve is 0.8 times the area under the R curve.
11,
5
21
T H E ANATOMY AND PHYSIOLOGY O F T H E RETINA
61
connected by an independent channel to its private optic nerve fiber. Although the number of optic nerve fibers supplying the fovea is about as large as the number of cones in the fovea, the interconnections are very diffuse. As shown in Fig. 2, the rods and cones have to transmit their signals to the ganglion cells. l k ganglion cells give rise to an intermittent all-or-none “spike” response, whereas the photoreceptors give rise to steady potentials. By means of lateral branches or by means of the horizontal and amacrine cells the rods and cones can transmit their excitatiorl to a large cluster of ganglion cells. On the other hand, the rods and cones can also generate inhibitory potentials which are even more widespread. I n the following analysis, which applies to the human retina, it has been assumed that the line spread functions for excitation and inhibition are bilaterally symmetrical and conform to the spread functions shown in Fig. 7 . The spread function for excitation has a sharp peak which prevents the spread of excitation from restricting acuity. The wide bottom provides widespread lateral summation which makes it possible for large objects to be seen a t low levels of illumination. As will be seen later, the diffuse spread function for inhibition enhances contrast. It is assumed that the excitatory and inhibitory effects generated by cones are proportional t to the stimulus applied to the cones and that these effects are strictly additive when they converge at a common pathway. These assumptions make it possible to predict response patterns by convoluting the stimulus pattern with the physiological spread functions. The assumptions that have been made about excitation and inhibition cannot be claimed to have been fully substantiated, although they do provide an explanation of what the eye sees. The amount of excitation and inhibition generated in the retina need not be proportional to the stimulus, and the inhibition need not be of the simple forward type in which one part of the retina inhibits activity in another region without inhibiting the capability of that region for inhibiting other regions. The processes of excitation and inhibition could be much more complicated than have been assumed, and in the end these assumptions may have to be modified or revised.
t This relation could be non-linear without affecting too much the calculation of the distributions of excitation and inhibition, but the evidence indicates that the relation is actually linear.
62
THE OPTICAL P E R F O R M A N C E O F T H E HUMAN EYE
III.
s
3
Electrophysiological techniques have been used by ENROTH-CUGELL and ROBSON[1966] to study the mechanisms of spread of inhibition and excitation. The activity of a single ganglion cell was recorded and this technique was used to map out the region of the retina which contributes signals to it. This region is known as the “receptive field”. The effects of stimulating isolated areas can be studied or these various areas can be made to interact with each other. Mechanisms for the spread of excitation and inhibition were demonstrated. Both spread functions, which included blur of the optical image, appeared to be Gaussian with the spread being greater for inhibition than for excitation. Not all of the ganglion cells behaved in the same way. It is to be hoped that further studies along these lines will eventually provide us with a complete picture of the mechanisms of the excitation and inhibition in the retina. The paper by Enroth-Cugell and Robson includes references to many previous investigators who have contributed to our knowledge of retinal spread functions. The ganglion and bipolar cells that supply the center of the fovea have been pushed sidewise from the center of the fovea so that the cones at the center interact with other cones in only three directions, sidewise and away from the center, but never across the center of the fovea. No specific neural interaction effects have yet been found to be associated with this arrangement. The close packing of the cones is achieved with a hexagonal array, but such a pattern is not systematically maintained over any large region (WSTERBERG [1935]). No physiological effect has been definitely shown to be associated with this arrangement, although the possibility for involvement in a kind of retinal astigmatism must be recognized.
Q 3. Basic Concepts: Lines, Points and Borders The visual field of the eye can be described as a cone which contains all the lines of sight that can pass through the center of the pupil and penetrate the portions of the retina that respond to light. One can relate the distribution of the luminance in the field of view to the distribution of flux on the retina which is characterized by the presence of lines, points, sharp borders, gradients and uniform patches. What the observer sees can also be described as a subjective field of view broken up by points, lines, sharp borders and gradients. The terms point and line are figurative words that have no real meaning in terms
11,
S 31
BASIC CONCEPTS:
LINES, POINTS AND BORDERS
63
of the optical image formed on the retina; a point is merely a small patch bounded by a border, and a line is a strip bounded on opposite sides by borders. Furthermore, since each border is a special kind of gradient, the distribution of flux on the retina can be described as an array of uniform patches and gradients. The point concept, the line concept and the border concept all have meaning in terms of object space. So far as the perceived distribution of color in the subjective field of view is concerned, there is some justification for differentiating between a sharp border and a gradient, because in the process of degrading a border between two adjacent areas, one can find a threshold level at which the border is perceived as either sharp or blurred.
Fig. 8. Pattern for demonstrating photometric matching of average brightncss when the grating is fine and the matching of individual bars when the grating is coarse.
In the case of uniform arrays of points or parallel lines, the eye has the remarkable capacity of assessing directly the average brightness. In Fig. 8 one can vary the luminance of the uniform patch to match the “average” brightness of the grating. The eye can also perform an additive mixing of different juxtaposed hues and saturations while at the same time being able to detect the hues, saturations and brightnesses of the elementary patches of color. Advantage is taken of this remarkable capability of the eye by such artists as Seurat and van Gogh, and it is also put to use in connection with line drawings, etchings, stipling, half-tone reproductions, photographic grain and T V rasters. At a grosser level of detail different contrast borders in the field of view take on different meanings. We regularly speak of striped zebras and spotted cows. These arrays of stripes and spots are considered to be an aspect of the color covering the surfaces of these animals. The borders of the spots on a spotted cow (Fig. 9) have a different meaning than the border of the cow separating the cow from the background.
64
T H E OPTICAL P E R F O R M A N C E O F T H E H U M A N EYE
111,
§ 3
Fig. 9. Camouflage involving confusion between the border of a n object (a cow) and borders bctwcen patches of color covering t h e object and the background. Courtesy of S. Kenshaw.
A part of the border of the cow may be invisible and still filled in with an imaginary border by the observer. All of this merely illustrates the point that human performance based on information extracted from displays formed on the retina can be much more complicated than the problem of coding the information transmitted along optic nerve fibers. Thus in this discussion of the optical performance of the eye, we can and we have restricted consideration to the visibility of lines, points, sharp borders, gradients and uniform patches without reference to the meaning they convey. The problem of transmitting the impression of a gradient or a uniform patch along the optic nerve must not be confused with the more general problem of transmitting information. It is helpful to point out at the outset the difficulty which may be
11, §
31
BASIC CONCEPTS: LINES, POINTS AND BORDERS
A
65
B
Fig. 10. Perception of border contrast and the difference in brightness between two isolated patches.
encountered in using an expression such as the “perception of a brightness difference” or the “brightness difference threshold”. One can compare the brightnesses of two independent patches of color (Fig. 10A) and subjectively assess the brightness difference, or one can have two contiguous patches (Fig. 10B) separated by a contrast border, as in the case of the border line between the two compared patches of color in a photometer. I n this case one can interpret the perception of a border as the perception of a brightness difference, but it ought to be recognized that the physiological mechanisms involved differ in this case as compared with two separate patches of color. In order to avoid confusing the two problems, phraseology such as border contrast sensitivity, border perception, and border contrast thresholds must be used in dealing with the visibility of a contrast border. It must also be recognized that many of the physiological ramifications involved in the perception of borders and gradients can be set aside if we are interested only in the role played by the quality of the optical image. Consequently, in the following discussion, I shall not attempt to deal with the problems that arise from changes in curvature and length of a border, or the effects of borders crossing each other or forming sharp angles with each other. The expression “perception of fine detail” can include all kinds of combinations of curved and straight borders, but it has become customary in the study of the optical performance of the eye to deal primarily with straight borders which are long enough to make a further increase in length of no importance and with combinations of straight borders to form lines, bars and gratings. Occasional reference will be made to targets of other configuration, such as points, disks and annuli, when information obtained with such targets can throw light on the visibility of borders, bars and gratings.
66
T H E OPTICAL PERFORMANCE O F T H E H U M A N EYE
111.
§ 4
Q 4. Visual Acuity, Resolving Power and Contrast and Modulation Sensitivity 4.1. MEASUREMENT AND SPECIFICATION O F VISUAL ACUITY
In various branches of human endeavor, it becomes necessary to rate the performance of an observer in terms of a single measure of his capacity for perceiving fine detail. The ophthalmologist and optometrist are confronted with the necessity of rating observers who have to be compensated for eye injury or who have to be classified for the purpose of payment of income tax. This is done by specifying the central (foveal) acuity of each eye. Visual acuity is usually measured with Snellen letters (Fig. 11) or with similar targets, such as the Landolt C (or broken ring). Visual acuity can be rated by specifying
SNELLEN
FOUCAULT
LANDOLT
KOENIG
IVES
Fig. 11. The critical dimension of commonly used visual acuity test patterns.
the size of some critical detail in minutes of arc. In the case of the broken ring, it is the width of the break in the ring, although in this case the width of the break is equal to width of the stroke forming the ring and is also equal to one fifth of the outside diameter. In the case of Snellen letters, all of which are capitals, the critical detail is the width of the “strokes” that make up the letters; this is also equal to one fifth of the overall height of the letters. Another popular target is the Koenig bar target (Fig. l l ) , which consists of two bars separated by an interval equal to their width and having a length equal to three times their width. I n this case, the width of the interval between the bars is taken to be the critical detail. In using various kinds of targets for the measurement of visual acuity, it is sometimes necessary to be arbitrary about what constitutes the critical detail. For example, in the case of Bodoni lower-case type, the critical dimension is considered t o be one fifth of the height of
11,
s
41
67
V I S U A L ACUITY
lower-case letters like e and o as opposed to 1 and j . However, one can specify also the size of the type as 8-point, 10-point, etc. This avoids the problem of specifying some critical detail. One must always specify what it is about the target that the observer must detect, identify, or see in a given way in order to pass the test. One must also specify how the results on several trials are to be averaged or assessed in terms of a single measure. The ability to see fine details involving a critical dimension of one minute of arc is considered to be a normal performance. One of the problems is that the measurement of visual acuity with different types of target does not always yield the same answer. The remarkable thing is that a complex, civilized society can get by without insisting on a higher degree of standardization than exists at the present time.
AVERAGE LUMINANCE (A) SINE WAVE
( 6 ) SAW TOOTH
( C ) SQUARE WAVE
Fig. 12. Specification of period, resolving power, visual acuity, modulation and contrast for various types of gratings.
Gratings can also be used for measuring visual acuity. When a square-wave grating (Fig. 12), which is also called a Foucault target (Fig. l l ) ,is used, the bright and dark bars are usually of equal width. As in the case of the Koenig bar pattern, the width of the individual bars is taken to be the critical detail for specifying visual acuity. This is illustrated by the visual acuity measurements reported by SHLAER [1937-381. When the dark-light ratio is other than one to one, the critical detail is taken to be one half of the cycle or period or center-tocenter distance. This is important because the temptation here is to use the width of the cycle as the index of performance, and this is what is done in the recent studies of visibility of gratings. But this should not be described as a measure of visual acuity. COBB [1911] used an Ives pattern (Fig. 11) produced with two transparent gratings to measure visual acuity. This pattern is equivalent to a sawtooth distribution (Fig. 12), and one half of the width of
68
THE OPTICAL PERFORMANCE O F THE HUMAN E Y E
[II,
ss
4
a tooth is used as the measure of visual acuity. Similarly, in the case of a sinusoidal grating (Fig. 1 2 ) , one half of the cycle is the proper dimension to be used in specifying visual acuity. Although one can specify the linear measurement of the critical dimension, provided that the distance of the test object from the eye is also specified, it is more customary to use the angular measurement expressed in minutes of visual angle measured at the entrancepupil of the eye. The reciprocal of the critical dimension expressed in minutes of arc is the measure of visual acuity. This is also called decimal acuity. Another specification of visual acuity is the Snellen notation. In this case the size of the test object is specified by giving the distance at which the critical detail subtends one minute of arc. The visual acuity is specified by the use of a fraction, the denominator of which indicates the distance at which the critical detail subtends one minute of arc and the numerator of which indicates the actual distance at which the critical detail is at the threshold of visibility. The distances can be specified in either feet or meters. The test is usually made at 20 feet and the size of the test characters is varied to measure acuity. The resulting Snellen fractions become 20/20, 20/40, etc. It is also important to be aware of other specifications of visual acuity. Whenever it is necessary, the Snellen fraction can be immediately reduced to the decimal notation of visual acuity. The decimal notation can easily be converted to a percentage notation known as percentage visual acuity. It is easy to confuse visual acuity with what is known as the visual efficiency (Am. Med. Assoc. Council on Industrial Health [ 19551). Visual efficiency is related to decimal acuity ( A ) by the following formula: Visual efficiency = 0.836(11A-1).
(4)
The idea behind the use of this scale is that a person whose percentage visual efficiency has dropped from 100 percent (20/20) to 50 percent is able to earn only one half as much and should, therefore, in the case of accidental impairment of vision, be compensated in proportion. 4.2. RESOLVING POWER
Physicists are accustomed to the use of the concept of “resolving power” for specifying the performance of an optical instrument. This concept is used in connection with a grating target and represents the
11,
P
41
69
RESOLVING POWER
number of cycles per millimeter in the image that can be resolved. The number of cycles per millimeter is called the spatial frequency of the image of the grating. The same concept can be applied to the eye, but instead of specifying the number of cycles per millimeter, it is more usual to specify the resolving power in terms of the number of cycles per degree or per minute of visual angle. In converting from visual angle to distance across the retina, it may be assumed for most purposes that the distance from the second nodal point of the eye to the retina is 17 mm. The concept of resohifig power is also used in connection with a two-point target. The concept is based on the capability of the human eye for resolving the image of a double star. I t is assumed that the eye can resolve the images of two stars when the angular separation is such that the distance between the centers of the two images on the retina is equal to the radius of the first dark ring in the Fraunhofer diffraction pattern of a single star (Fig. 13). This is known as Rayleigh’s criterion. Hence, the theoretical capability of two stars of a given wavelength (A) being resolved by an eye can be computed from the radius g of the entrance-pupil: Resolution threshold in radians = 0.61 A/g.
(5 1
There is a certain amount of interest in the ability of the eye to resolve two points, and OGLE [1951] had some hope of using the twopoint approach in fundamental studies of the optical performance of the human eye. However, gratings and bars appear to be the preferred targets for the study of the performance of the eye.
I.o
0
.I.o
2.0
DISTANCE FROM CENTER (MINUTES)
Fig. 13. Rayleigh criterion for resolving the images of two monochromatic points.
70
T H E OPTICAL P E R F O R M A N C E O F T H E HUMAN E Y E
111, I 4
4.3. THE CONCEPTS O F CONTRAST AND MODULATION
Contrast (C) at a border is defined in several ways, but the most widely used definition is in terms of an object (0)and its background (B): Contrast
=
(LB-LO)/LB,
(6)
where L, and Lo represent the luminances of the background and object respectively. In measuring the threshold it is usually found that the value of (L,-L,( is the same for objects brighter and darker than the background, and hence the contrast threshold may be specified as Contrast threshold
=
1 (LB-LO)/L,l = jAL/L1.
(7
1
In a disk-annulus pattern such as is used in photometry one can determine when the disk is darker or brighter than the annulus, but not the point at which the two are equal; thus, the range between the two levels equals 2(AL(,and L has to be computed from the relation
In the case of a bipartite pattern or square-wave grating, there is no basis for deciding what parts constitute object and background, and hence contrast has t o be defined in terms of the bright and dark parts of the pattern as follows: Contrast
=
(Lmax-Lmin)/Lmi,.
(9)
Contrast can also be expressed in terms of modulation ( M ) : Modulation
=
(Lmax--Lmin)/(Lmax +Lmin)
= Amplitude/Average luminance.
(10)
I n the case of sine waves, as well as sawtooth waves and square waves (see Fig. 12), 4(Lmax-Lmin) equals the amplitude (AC component), and +(Lmax +L,,,) equals the average luminance (DC component). Since ccntrast thresholds are apt to be of the order of 0.01, it follows that the modulation threshold is numerically about one half as large as the contrast threshold, and it is necessary for visual scientists working in this area to be ever mindful of this difference. The concepts of contrast and modulation can also be applied to
11,
9 51
SPREAD A N D TRANSFER FUNCTIONS
71
distributions of retinal illuminance on the retina as well as to distributions of luminance in the field of view. These concepts can also be applied to the demodulated image transmitted through the retina to the brain. § 5. Spread and Transfer Functions 5.1. SPREAD FUNCTIONS
We have already considered the spread functions of excitation and inhibition in the retina. We can apply the same concept to a radially symmetrical optical image of a point source formed on the retina. Once the distribution of flux for a point is given, the spread function for a line or for a contrast border can be computed. This procedure has come t o be known as convolving (or convoluting) a line or contrast border with a point spread function. In the case of a border one can go from a point spread function to a line spread function and then convolve the border with the line spread function. When the image of any object is formed on the retina, the object is said to be convolved with the point spread function for the optical image. When such an image is in turn transmitted through the retina, it is further convolved with the spread function for excitation and inhibition. The spread of excitation in the retina is superimposed upon the spread involved in the formation of the optical image, and in this kind of situation where two spreading mechanisms occur in tandem, one can convolve the point spread function of the first with that of the second and obtain a single point spread function that represents both. The procedure for doing this has been described by FRY[1955]. If the point spread function is radially asymmetrical, the line spread function can still be bilaterally symmetrical in special cases, but it can also be bilaterally asymmetrical. In this review there has been no need to consider asymmetrical spread functions because we have been primarily concerned with axial chromatic aberration, spherical aberration, and the effect of changing pupil size and throwing the eye out of focus. In the study of these effects consideration can be limited to radially symmetrical beams of light. 5.2. CONVOLVING A L I N E W I T H A POINT SPREAD FUNCTION
The concentration of flux at a given part of the retinal image of a point can be described in terms of the lumens per square minute measur-
72
THE OPTICAL PERFORMANCE OF T H E HUMAN E Y E
[II.
5
5
ed at the second nodal point. Flux per square minute can also be expressed in trolands: lumens per square minute. One troland = 8.46 x In the case of a radially symmetrical retinal image of a point the ] a function of the distance (Y) from flux per square minute [ E ( Y ) is the center of the image (Fig. 14), where Y is measured in minutes of angular length t subtended at the second nodal point.
FRAUNHOFER
0
DISTANCE FROM CENTER (MINUTES) Fig. 14. Gaussian and Fraunhofer point spread functions which approximate each other; g = 1 mm, L = 492 nm, u = 0.4 min (from FRY[1965c]).
Once the distribution of flux for a point is given, the distribution for a straight line of constant candlepower per unit of length can be computed [FRY(1955)J.The first step is to normalize E ( r ) by dividing it by F , the total amount of flux in the image. The normalized value of E ( Y )is designated G ( Y )and is defined as where
F
= 2nIOmrE(r)dr.
The distribution of flux across the image of a line can be described in terms of flux per unit area E (t) as a function of the distance ( t ) from the center of the image. The formula for this distribution is
E ( t ) = 2.1,;
[G(r)r/Z/(r2-t2)]dr,
(13)
7 Distance across the retina can be expressed in minutes or in microns (p). If the secondary nodal point lies 17 millimeters in front of the retina, one minute = 4.92 microns.
11,
I
51
SPREAD AND TRANSFER FUNCTIONS
73
where D is the flux per minute of length of the image. The next step is to normalize E ( t ) by dividing it by the flux per minute of length of the image (0). The normalized value of E ( t ) is designated H ( t ) and is defined as
H(t) = E(t)/D,
(14)
where
The quantity D can also be described as the flux entering the eye from a segment of the object line one minute long measured at the first nodal point. Actually, the normalized spread function for a line can be determined from the point spread function without giving any consideration to either D or E (t):
H(t) = 2
[G(Y)Y/~(Y'-~')]~Y.
J;,
Procedures for working backward to determine the point spread function from a line spread function have been described by MARCHAND [1964, 19651. 5.3. CONVOLVING A PATTERN O F PARALLEL STRIPS WITH A L I N E SPREAD FUNCTION
We can now use the image of a line to convolute a pattern, such as a bar or a straight contrast border or a grating, which can be divided into long parallel strips, each having constant luminance from end to end. The distribution of luminance across such a pattern can be de scribed as a function L(s) of s, where s represents the distance in minutes from some arbitrary starting point. The distribution of flux in the convoluted image on the retina is given by the following general formula :
I-, 00
E(s)= A
H ( t ) L(s-t) dt,
(17)
where A is the area of the pupil expressed in square millimeters and where E is expressed in trolands and L in nits. In the case of an abrupt contrast border (step function) between a
74
T H E OPTICAL PERFORMANCE OF T H E HUMAN E Y E
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9 5
dark border on the left and a bright border on the right, eq. (17) reduces to
E ( s )= A L
J-:
H(t)dt.
(18)
One can differentiate eq. (18) to obtain the line spread function from the border spread function:
H ( t ) = [dE(s)/ds]/(AL).
(19)
5.4. SINE-WAVE TECHNIQUES
A sine-wave grating continues to be a sine wave after it has been convolved with any point spread or line spread function, and the only effect is a loss in amplitude. This remarkable property of a sine wave is the basis for the development of a large set of sine wave techniques used in the study of the performance of optical systems and other spreading mechanisms. The uses of these techniques in the study of image formation and transmission of images through the retina constitutes the major wave of progress in the optics and the related physiology of the eye during the past decade. This era was initiated by SCHADE[1956]. The general procedure for convolving a sine wave with a line spread function is illustrated below for the case of an optical image formed on the retina. The formula for the distribution of luminance ( L )across a sinusoidal grating is
L (s)
=
4(Lmax +I-min) +B
where 4(L,,, +I-,,,) 2(Lmax-Lmin) 1
= =
(Lmax--Lmin)
c o ~(zns/:) 9
(20)
average luminance, amplitude,
d = wavelength (or peak t o peak distance) in minutes,
l / S = spatial frequency in cycles per minute, and s = distance across the grating from a given peak used as an
arbitrary reference point. As explained by PERRIN [1960], the general formula [eq. (17)] for convolving a strip pattern with a symmetrical line spread function H ( t ) reduces in the case of a sinusoidal grating to
where
SPREAD AND TRANSFER FUNCTIONS
15
J’rW
T = modulation transfer function = H ( t ) cos (antis)dt, t = distance across the retina from the center of the line spread function in minutes, A = area of the pupil in square millimeters and is the factor for converting from luminance ( L ) in nits to retinal illuminance ( E ) in trolands, and H ( t ) = normalized line spread function. As long as s and t are expressed in minutes measured at the primary and secondary nodal points, we do not have to be concerned about differences in s in object space and on the retina. It should be noted that T(l/S) is the Fourier transform of the normalized line spread function H ( t ) . When a sine wave is convolved with a line spread function, the ratio of the after and before modulations represents the modulation transfer factor ( T ) : Modulation after convolution T = Modulation before convolution’ (22) If these factors for different spatial frequencies are plotted on a graph, the resulting curve is called the modulation transfer function. This function can be computed directly from the point or line spread function by the use of the Fourier transforms (GIVENS[1966]). When the line spread function is bilaterally symmetrical, the inverse Fourier transform can also be used to work backward from the modulation transfer function to compute the line spread function. When the line spread function is bilaterally asymmetrical, special techniques (PERRIN [1960]; LAMBERTS [1958]) can be used to go back and forth between it and the modulation transfer function. Various psychophysical methods have been developed for determining experimentally the modulation transfer functions and from these deriving the line spread functions for optical blur and the spread of excitation and inhibition. 5.5. SUMMARY O F SPREAD AND TRANSFER FUNCTIONS USED FOR
THE EYE
In this section of the review I have summarized some point spread functions involving radial symmetry and the corresponding modulation transfer functions which have been found useful in dealing with probems related to the eye. When an eye is free from spherical and other monochromatic
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T H E OPTICAL PERFORMANCE O F T H E H U M A N E Y E
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aberrations, it is called a diffraction-limited eye. When a point source of coherent light is conjugate to the retina of a diffraction-limited eye, the image formed on the retina is a Fraunhofer diffraction pattern. The point spread function for the Fraunhofer diffraction pattern (Fig. 14) is given by the following formula (FRY[1955]):
where
-
E [(2/7)J1(7>I2 is the first order Bessel function of (7) and where
J,(r)
7 = (1/3438)( 2 @ / 3 , ) ~ .
(23)
(24)
I n eq. (24) Y is the distance from the center of the image measured in minutes of arc a t the second nodal point, 3, the wavelength, g the radius of the entrance-pupil, and 3438 the factor for converting from radians to minutes. The same units are used for g and A. The formulas for the corresponding line and contrast border spread functions can be found elsewhere (FRY[1955]). The corresponding modulation transfer function (Fig. 15) is
T
= (Z/n)(e-sin
8 cos O ) ,
(25 )
where cos
e = 3438 2/(2gq,
(26)
and where S is the center-to-center distance between the bright lines in the retinal image of the grating measured in minutes of arc at the second nodal point. The derivation of eq. (25) has been explained by GIVENS[1966]. In a diffraction limited eye out of focus, the point spread function 1.0
0.8
0.6
T
0.4
0.2 0
0
20
40
60
CYCLES PER DEGREE Fig. 15. Gaussian and Fraunhofer modulation transfer functions corresponding to the two line spread functions in Fig. 14.
11,
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77
SPREAD AND TRANSFER FUNCTIONS
0
20
SPATIAL FREQUENCY
40
60
00
(c/deg)
Fig. 16. Modulation transfer functions for a diffraction-limited eye. The pupil diameter is 2.5 mm and the wavelength is 578 nm. The different curves are for the eye in focus (0) and for the eye out of focus to various degrees. The numbers indicate diopters (from CAMPBELLand GUBISCH[1967]).
for monochromatic light is called a Fresnel image. The formula for a Fresnel point spread function can be found elsewhere (FRY[1955]). Linfoot has worked out modulation transfer functions for Fresnel images slightly out of focus. A graph showing certain of these functions (Fig. 16) has been presented by CAMPBELLand GUBISCH[1967]. It would be helpful if tables for these functions were available. As shown in Fig. 14, the Fraunhofer diffraction pattern can be approximated by a Gaussian spread function:
E
=
E , exp[-&(~/o)~]],
(27 )
where r is the distance from the center of the image and CJ is the standard deviation of the distribution. The Gaussian spread function can also be used to express the spread of inhibition (Fig. 7 ) , although the value for sigma is higher. The line spread function has the same form as the point spread function, and the border spread function corresponds t o the integral of the probability curve, values for which are widely available in table form. The corresponding modulation transfer function (GIVENS[1966]) is
T = exp [-2(n0/S)~].
( 2 8)
Figure 15 shows the modulation transfer for both the Fraunhofer and the Gaussian point spread functions in Fig. 14, which approximate each other.
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T H E OPTICAL PERFORMANCE O F THE HUMAN E Y E
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The line spread function shown in Fig. 7 , which is used to represent the spread of excitation, conforms to the following equation:
R
=
R, exp(-kt),
(29)
where k is a constant which varies with the luminance level and where t is the distance from the center of the image. The corresponding modulation transfer function is as follows:
T
=
[1+(2~/kB)~]-’.
(30)
The curve for this function is shown in Fig. 17. The “pillbox” distribution representing the out-of-focus blur circle is an approximation or substitute for Fresnel distributions. The image of a point source represents a disk-shaped area over which the flux is uniformly distributed.
0
20 40 60 CYCLES PER DEGREE
80
Fig. 17. Modulation transfer functions for physiological spread of excitation and inhibition corresponding t o the line spread functions in Fig. 7.
The formulas for the corresponding line and border spread tions are:
*
func-
H ( t ) = [2/(76P)]2 / ( 1 2 - t 2 ) , (31) where P is the radius of the blur circle and t is the distance from the center of the image; for a border E ( w ) = ( A L / n )[&c+(w/P) d ( P 2 - w z )
+ sin-’
(w/P)],
(32)
where L is the luminance on the bright side of the border, A is the area of the pupil, and w is the distance from the border. * Equation (32) for the border spread function includes corrections for the errors which appear in eq. (7.10) in Fry [1955].
79
SPREAD AND TRANSFER F U N C T I O N S
0.2
-0.2
0.4
0.6
0.8
1.0
27r i ( I & 1
Fig. 18. Modulation transfer function for a n out-of-focus blur circle. The symbol i represents the radius of the blur circle.
The formula for the modulation transfer function (GIVENS[ 19661) is
T
= 2 [J1(2nf/iB)]/(2nP/d),
(33)
where f is the radius of the blur circle and J1(27tf/S) is the first order Bessel function of ( 2 n f / s )This . function [eq. (33)] is shown in Fig. 18. The phenomenon of spurious resolution which occurs with an eye (FRY[1953, 19611) out of focus finds its counterpart in the oscillation of the modulation transfer function above and below the zero level. For over a century the concept of the blur circle has been widely used in providing approximate answers to out-of-focus imagery in the
I PLANE OF THE EXIT PUPIL Fig. 19. Geometry involved in computing the size of the blur circle for anout-of-focus eye (from FRY[1955]).
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THE OPTICAL PERFORMANCE O F THE HUMAN E Y E
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5
eye, and it is still used for quick checks in the most sophisticated studies. It is assumed that when the eye goes out of focus, the distance ( v ) from the retina to the point of best focus (M’) in millimeters is three tenths of the number of diopters out of focus and the radius ( F ) of the blur circle is related to the radius (8‘) of the exit-pupil by the following equation: ~
F = [~’/(O’M’)][DI,
__
(34)
where O’M’ is the distance from the exit-pupil to the point of best focus. The relationships are shown in Fig. 19. See also Appendix I. In astigmatism the blur ellipse is the counterpart of the blur circle. Problems in astigmatism can be solved by using approaches (FRY [1955]) similar to those used for simply throwing the eye out of focus. 5.6. INDEX O F BLUR
The FRY-COBB[1935] index of blur is useful in dealing with various types of problems. It can be defined in terms of point, line, or border spread functions (FRY[1955]), but the most meaningful definition is in terms of the spread function for a border (Fig. 20). It represents the ratio of the slope (dE/ds) at the midpoint to the difference in retinal illuminance on the two sides of the border. In terms of the normalized point spread function, I/# = 2IOwG dr.
(35)
I n terms of the normalized line spread function,
POINT
LINE
BORDER
Fig. 20. Relation of the index of blur to the point, line and border spread functions of the eye (from FRY[1959b]).
11.
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SPREAD AND TRANSFER FUNCTIONS
81
It turns out also that the index of blur can be related to the modulation transfer function. T ( I / S ) d(l/S).
I/+ = 2
(37)
/om
This means that 46 is the reciprocal of the area under the curve representing the modulation transfer function. For example, the two transfer functions shown in Fig. 15 have the same +-value because the areas under the two curves are equal. I n the case of the Fraunhofer distribution for a point, it turns out that in radians = 0.591/g. (38)
+
If we compare this equation with eq. (5),it is apparent that by chance the value of I$ is almost numerically equal to the radius of the first dark ring. Hence, it may be said, in accordance with the Rayleigh criterion, that two points can be resolved when they are separated by an amount equal to 4. One of the advantages of the Fry-Cobb index of blur is that it can also be easily computed for a monochromatic Fresnel image, and thus it can be used to assess the performance of the eye for any wavelength, pupil size, or lack of focus. The formula for for a monochromatic Fresnel image (FRY [1955]) is
+
46 in minutes = 3438 Ac/(f’g’V),
0
20
10
(39)
30
lGl
Fig. 21. The V(o)function used in computing the index of blur (from FRY[1955]).
82
T H E OPTICAL PERFORMANCE O F T H E HUMAN E Y E
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where c is the distance from the exit-pupil to the retina, g’ the radius of the exit-pupil, f’ the secondary focal length of the eye, ilthe wavelength of the light, and V a function of & defined as follows:
2 2.5! 3.5!
(G)
+
2
4
4 ~
4.5!5.5!
..
.],
(40)
where
0=
[2JG(g’)2](1z’/A) (l/a-l/c),
(41)
and where a is the distance from the exit-pupil to the plane of the Fraunhofer image; n‘ is the index of the vitreous; and 0 is a measure of the extent to which the eye is out of focus. For an emmetropic eye in focus for an infinitely distant object,
g’ = ( c i f ’ k
(42)
where g is the radius of the entrance-pupil. Using this equation, eq. (39) reduces to
4 in minutes = 3438 A/(gV).
(43)
Fig. 22. The effect of pupil size and improper focus on the amount of blur as measured b y 4 (from FRY[1955]). .$expressed in microns: 6‘= radius of exit-pupil. See Appendix I.
TABLEI Values of V for different values of
la 0 2 4 6
W
v
I4
V
14
V
I4
V
1.6977 1.5973 1.3314 0.9881
8 10 12 14
0.6677 0.4397 0.3208 0.2825
16 18 20 22
0.2741 0.2617 0.2322 0.2017
24 26 28 30
0.1663 0.1522 0.1494 0.1472
11,
5
5j
SPREAD A N D TRANSFER FUNCTIONS
When the Fraunhofer image falls on the retina, and eq. (43) reduces to
$ in minutes
=
2028 Alg.
83
CO reduces to zero (44)
This is equivalent t o eq. (38). The constant V has been evaluated for various values of G, and the results are tabulated in Table 1 and plotted in Fig. 21. Figure 22 shows for monochromatic yellow light how a change in pupil size affects the depth of focus and how it affects the amount of blur when the eye is in or out of focus. According to eq. (44),I/+ varies in proportion to the radius of the entrance-pupil when the eye is in focus.
-1.00
0
+I00
DIOPTERS OUTOF FOCUS Fig. 23. Comparison of the 4-values for the physical and the geometrical images of a monochromatic point for an eye thrown out of focus to various degrees (from FRY [1955]). expressed in microns. 6‘ = radius of exit-pupil. See Appendix I.
The index of blur can also be used to demonstrate how well the “pillbox” distribution representing a blur circle approximates a Fresnel image. For the pillbox distribution,
$
= &nV = g[ng’/O’M’]la-~(,
(45 )
__
where f is the radius of the blur circle and O’M’ is the distance from the exit-pupil to the optical image. Figure 23 shows the $-values for the pillbox distributions compared with the +-values for Fresnel images. For small values of (a-c), Fresnel images are needed to assess the amount of blur; but for large values of (a-c), the pillbox distributions serve as useful approximations. In the case of the Gaussian spread function,
4 = 2.5066 U.
(46)
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THE OPTICAL PERFORMANCE
O F THE HUMAN EYE
[II, $ 6
$j6. Psychophysical Techniques 6 . 1 . THE ROLE O F COHERENCE
Since in most situations the eye has t o cope with images formed by coherent light from point sources, it is desirable to evaluate the performance of the eye in terms of such images. At any rate, the interpretation of data dealing with the performance of the eye must take into consideration the coherence of the light. Some of the problems are illustrated by SHLAER’S [1937-381 study. The pupil of the eye was made conjugate to a source of light (the incandescent ball of a tungsten arc). Two plates of ground glass were placed between the source and the condensers to diffuse the image of the source. A transmitting grating which served as the test object was placed in this beam at a point conjugate to the retina. In this arrangement coherence and diffraction are difficult to assess, and the principles involved in interpreting the performance of the eye differ considerably from the situation in which the grating used as a test object is made up of points of coherent light. VAN NES and BOUMAN[1967] used a dispersing prism type uf monochromator to obtain monochromatic light. They focused the source on the entrance-slit of the monochromator and used a pair of lenses to form an image of the exit-slit in the plane of an artificial pupil placed just in front of the eye. A transparent sinusoidal grating was placed close to the primary focus of the lens system used to focus the exit-slit of the monochromator on the artificial pupil. This made the source conjugate to the plane of the pupil and the grating conjugate to the retina. This arrangement could have been modified so that the data could be interpreted in terms of ordinary seeing if a ribbon filament had been used as the source and if lenses had been mounted at the entrance and exit slits to make the source conjugate to the plane of the grating. This would have made the grating consist of points of coherent light. This method of illuminating a monochromator has been described by SAWYER [l966]. LE GRAND[1936] called attention to a method of viewing objects which he called directed vision. He used a lens to focus a point source at the center of the pupil of the eye and then placed a small opaque object between the lens and the eye. This produced a sharper image than when a diffusing surface constituted the background for the object. According to Le Grand, this avoids the possibility of having the image degraded by spherical aberration, He showed that the re-
11,
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61
P S Y C €1 0 P H Y S I C A L T E C H N I Q U E S
85
solving power for two parallel threads was much better in directed vision than in normal vision. In Fig. 24 the light from the source at A is focused on the slit at C by the lens B. The lens D focuses an image of the slit at the entrancepupil of the eye. This arrangement conforms to Le Grand’s condition of directed vision. BERGER-LHEUREUX-ROBARDEY [1965] considers the beam entering the eye in this case to be coherent. According to Berger-Lheureux-Robardey, one can transform such a beam to an incoherent beam simply by increasing the width of the slit so that the image of the source completely covers the pupil. This raises the question of what constitutes coherence in a beam of light.
Fig. 24. Arrangement for “directed vision”. The source A is conjugate to the pupil, and the target a t E is conjugate to the retina.
Berger-Lheureux-Robardey claims that the arrangement in Fig. 24 with a wide slit is equivalent to ordinary seeing. This does not appear to be correct. She could have modified the arrangement as shown in Fig. 25 to make it correspond to ordinary seeing. This arrangement makes the source (A) conjugate to the target (E), so that each point in the target which transmits light becomes a source of coherent light. The images of these points formed on the retina are Fraunhofer images. If they are slightly out of focus, we call them Fresnel images of coherent point sources; but when the source itself is conjugate to the pupil of A
B C
D E
Fig. 25. Modification of the arrangement in Fig. 24 which makes the source (A) conjugate to the target E and to the retina. The diaphragm a t C is conjugate to the pupil. This corresponds to ordinary seeing in that each point in the target plane is h. point source of coherent light.
86
THE OPTICAL PERFORMANCE O F THE HUMAN E Y E
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6
the eye, objects placed in the plane at E conjugate to the retina can only produce shadow images on the retina. Let us look again at Fig. 24. If instead of putting fine threads in the plane conjugate to the retina as Le Grand did, we place a grating in this plane, this restructures the beam entering the eye, and one can no longer assume that the beam entering the eye is a simple image of the slit (C). If the arrangement shown in Fig. 24 is used for stimulating the retina, then introducing into the beam a Babinet compensator between two polarizers, a slit, a transmitting grating, an opaque point or bar, or a dispersing prism will affect the distribution of flux in the plane of the pupil, as well as in the plane of the retina. Furthermore, the distribution of flux in the plane of the retina can be manipulated by spatial filtering in the plane of the pupil. No attempt has been made to cover all of the ramifications of this problem in the present review. However, this problem is involved in some of the various arrangements for measuring the optical performance of the eye, and we need to be aware of when and how the measurements are being affected. Needless to say, if we place in the plane conjugate to theretina a self-luminous surface as the face of a TV tube or a print illuminated from in front, or a diffusely reflecting or transmitting surface upon which falls an optical image of a grating, any of these arrangements simulates or constitutes ordinary seeing. All of these arrangements have been used for measuring the resolving power and modulation thresholds of the human eye. DEPALMAand LOWRY[1962] used transparent gratings placed at some distance in front of a diffusely reflecting surface which formed a luminous background. The interpretation of the results in this case presents somewhat of a problem. 6.2. SPECIAL DEVICES
FOR MEASURING KESOLVING POWER, MODULATION THRESHOLDS AND TRANSFER FACTORS
LE GRAND[1935] was apparently the first investigator to use interference fringes formed on the retina to measure the image degradation produced by the retina and brain independently of the image-forming mechanism of the eye. In the discussion that follows, the degradation produced by the combination of the retina and brain will be referred to as the retinal degradation to simplify phraseology, if for no other reason. There is also reason to believe that most of this
11,
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PSYCHOPHYSICAL TECHNIQUES
87
degradation does occur at the retina, but we do not have to consider this issue settled. Le Grand looked at a point source through two small apertures of variable separation placed in front of one eye and was able to produce fringes of various spacings on the retina. He was able to measure the resolving power using fringes of high contrast and variable spatial frequency, and he was also able to test the effect of reducing the contrast by attenuating one of the beams. He compared the results obtained in this way with the results obtained with a grating formed by a Babinet compensator placed between two polarizers and illuminated by light from a source which was conjugate to the pupil and which produced an image large enough to cover the pupil. With the latter arrangement the contrast could be varied by manipulating one of the polarizers. This grating pattern is not equivalent to a grating made up of points of coherent light
Fig. 26. Westheimer’s arrangement for producing interference fringes on the retina [1960]). (from WESTHEIMER
WESTHEIMER [19601 has developed an ingenious arrangement (Fig. 26) for producing sinusoidal interference fringes on the retina. The lens L, produces an image of the source V on a slit S, and this is re-imaged at M by the lenses at L, and L,. The interference filter at IF makes the light monochromatic (555 nm), and the grating (6 lines per mm) at RR forms a grating diffraction image at M. A diaphragm is used to screen out all but the zero and the two first order images. These are covered with polaroids with the axis for the zero order image perpendicular to the axes for the first order images. The lens L, and the eye can be moved in a fore and aft direction to change the spacing, and the Polaroid analyzer (A) can be rotated to control the modulation. This device permits the measurement of the resolving power and the modulation thresholds at various spatial frequencies. By means of a field splitter ARNULFand DUPUY[1960] presented
88
T H E OPTICAL P E R F O R M A N C E O F THE H U M A N E Y E
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t o the eye beams of light which produced sinusoidal grating patterns with the lines of both grating patterns perpendicular to the dividing line of the field splitter. In this way the modulations of the two gratings could be compared and could be made to be perceived as equal by manipulating the modulation difference. One of the grating images formed on the retina was an interference pattern produced by two rectangular beams entering the eye at two different parts of the pupil. The separation of the two beams was varied to manipulate the spatial frequency, and the long dimension of one of the beams was varied to degrade the modulation. The second fringe pattern was a multiple interference pattern produced by a wedge of two plane surfaces. The spacing of the fringes in this case could be varied by changing the angle of the wedge. For various spacings, Arnulf and Dupuy were able to measure the contrast loss produced by the optical system of the eye for the multiple reflection pattern by reducing the contrast of the double slit pattern to the same level. Using the double slit pattern by itself, they were able to measure the threshold of visibility for various spacings by manipulating the modulation. They also measured the resolving power using the multiple reflection fringe pattern. Unfortunately, the grating pattern produced by multiple reflection does not conform to ordinary seeing because the source is conjugate t o the plane of the pupil.
m TL Fig. 27. Arrangement used by Campbell and Green to produce interference fringcs on the retina (from CAMPBELLand GREEN[1966]).
CAMPBELLand GREEN[1965] used the arrangement shown in Fig. 27 to produce a set of sinusoidal interference fringes on the retina, the spacing and modulation of which can be varied. The lens L, in front of the laser source produces a divergent beam of coherent light from a point source a t the secondary focal point. The pellicle beam splitter RP and the first surface mirror M, produce a second source, and the
II,
3 ‘71
MODULATION TRANSFER FUNCTION
89
two sources are imaged by the objective lens L, in the plane of the pupil. The spacing of the fringes can be varied by changing the lateral displacement of the two sources, and the modulation can be manipulated by superimposing a patch of veiling luminance. With this arrangement they were able to measure the modulation thresholds for the retina and brain. They also used a TV display for generating a sinusoidal grating pattern consisting of points of coherent light. This is the same technique which had been introduced by SCHADE[1956]. The spacing of the grating could be varied and the modulation could be manipulated to measure the threshold. The threshold in this case is determined by the retina and the brain combined with the imageforming mechanism of the eye. By comparing the thresholds obtained by the two methods and by assuming that the modulations arriving at the brain are the same in the two cases for a given frequency, Campbell and Green were able to assess the contrast transfer function for the optical system of the human eye. Q 7. Determination of the Modulation Transfer Function for the
Retina 7 . 1 . DERIVATION F R O M THE COMBINED FUNCTION
One can start with a modulation threshold curve for the retina and brain combined with the image-forming mechanism of the eye and then divide the ordinates of this curve by the modulation transfer factors for the optical system of the eye. The resulting curve should represent the modulation threshold obtained when the optical system is by-passed. I n making this analysis, I have used data obtained in my own laboratory (FRY[1969b]). The grating pattern was generated by starting with a square-wave grating and convoluting it with a Gaussian spread function to produce a sine wave having a modulation of 0.73. ENGEL [l968] and FRY[1968a] have demonstrated that a sine-wave grating of known modulation can be produced in this way. The contrast was reduced to the threshold level by a patch of veiling luminance. The grating pattern was 87‘ high and 174’ wide and had a constant average luminance of 21.7 fL. The surround had the same luminance, and the whole pattern was viewed through a 2-mm artificial pupil at a distance of 1 meter. This gives an average retinal illuminance of 233 trolands. The results for subject DK are shown in Fig. 28. The modulation thresholds have been plotted as a function of
90
THE OPlICAL PERFORMANCE OF THE HUMAN EYE
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9 7
Fig. 28. Modulation threshold data of DK. The dashed curve shows the modulation of the retinal image. These are the values for threshold which would be expected if the optics of the eye were by-passed (from FRY[1969b]).
center-to-center separations ( S ) of the bright bars in minutes as measured a t the second nodal point of the eye. The reason for using B rather than the spatial frequency (1/B) is that threshold data for bars and disks, etc. are all plotted as a function of size with size increasing from left to right, and hence this facilitates comparison.
-DIFFRACTION w
0 0 0
0
-
THEORY
GAUSSIAN APPROXIMATION
2
2z -
30.54
-
-I
a
-
z -
I - W
a
-
MINUTES
0
0
I DISTANCE FROM CENTER OF IMAGE
2 (r)
Fig. 29. Image formed on the retina by a point source of white light with a 2-mm exit-pupil and the Gaussian spread function which approximates i t (a = 0.4 min). (From FRY[1969b].)
11,
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MODULATION TRANSFER FUNCTION
91
White light was used in the experiment. The point spread function for white light for a diffraction-limited eye with a 2-mm exit-pupil has been computed by Fry and is shown in Fig. 29. This differs somewhat from the spread function for a 2-mm artificial pupil placed at the anterior focal point of the eye, but this difference was ignored. The theoretical point spread function can be approximated by a Gaussian spread function as shown in Fig. 29. Equation (27) represents this function. The corresponding modulation transfer function [eq. (2S)l is included in Fig. 28 and is used in deriving the modulation threshold curve for the retina. The advantage in using logarithmic scales for plotting this type of data is that a shift sidewise of a modulation transfer curve without a change in form corresponds to a simple change in the value of 0. A shift of the threshold curve up or down indicates a general change in the threshold level at all frequencies. Furthermore, the threshold curve can be inverted to show its relation to the modulation transfer curve. The same ordinate scale can be used for modulation thresholds and transfer factors. The next step is to make allowance for the effect of the slope of the gradients between the troughs and peaks at low frequencies. The upturn of the curve in Fig. 28 at low spatial frequencies reflects the fact that the threshold depends not only upon the luminance difference between the peaks and troughs of the sine wave, but also upon the slope of the gradients between the peaks and troughs. One can demonstrate this by showing that the visibility of a single blurred border is dependent on the slope of the gradient at the border as well as the luminance difference on the opposite sides of the border. The same apparatus that was used to produce a sine-wave grating
Fig. 30. Basis for comparing the modulation threshold for a single blurred border to the modulation threshold for a sine-wave grating (from FRY[1969b]).
92
T H E OPTICAL PERFORMANCE O F THE HUMAN E Y E
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was also used to produce a bipartite pattern with a dividing border that could be blurred to various degrees. The contrast threshold for such a blurred border can be compared with the thresholds for a grating. One can express the contrast for a blurred border in terms of modulation. Figure 30 shows the basis for comparing a blurred border to a sine wave grating. In the case of the single border (Lmax-Lmin). dL/ds = (0.40/0)
(47)
In the case of a grating dL/ds
= ( z l g ) (Lmax--Lrnin)*
(48)
Thus, in order for the (dL/ds)/(L,,,--Lmin) value for a single border to be equal to that for a sine-wave grating, the o value for the border must be such that o = 0.127 S. (49) In Fig. 31 the modulation thresholds for a single border have been plotted (curve A) as a function of o and those for a sine-wave grating (curve B) as a function of S. The cr and 3 scales have been adjusted so that the (dL/ds)/(Lma,--Lmin)values are equivalent. It is obvious that SIGMA VALUES FOR A SINGLE BLURRED BORDER
0.001
1
10
I MINUTES 100
CENTER TO CENTER SEPARATION
Fig. 31. Modulation thresholds for subject DK for a single blurred border (A) and a sine-wave grating (B). Curve C is the modulation threshold for a sine-wave grating corrected for the border gradient effect (from FRY[1969b]).
11,
g 71
MODULATION TRANSFER FUNCTION
93
the modulation threshold for the single border becomes independent of the amount of blur for small amounts of blur, and hence allowance
for blur can be made in the threshold values for a sine wave by multiplying the threshold values (curve B) by a factor defined as follows:
'
Threshold value for a sharp border = Threshold value for a blurred border'
(50)
The resultant curve (curve C) is shown in Fig. 31. Allowance for blur does not eliminate entirely the upturn in the modulation threshold curve. As we shall see later, part of the upturn can be attributed to the inhibitory network in the retina. Let us now examine the role played by physiological irradiation and inhibition. If it can be assumed that the excitation ( R ) transmitted from the photoreceptors to the ganglion cells in the retina is linearly related to the intensity of the stimulus ( E )and if part of the excitation spreads or irradiates laterally, one can treat irradiation in exactly the same way as optical blur. Let us assume that the line spread function for excitation conforms to eq. (29). If we convolute the distribution of retinal illuminance in the retinal image of a sine-wave grating with the line spread function defined in eq. (29), we obtain the distribution of excitation (I?) transmitted to the ganglion cells, which is
R
=pE
+pTR A E cos ( 2 n ~ / B ) ,
(51)
where E is the average retinal illuminance and A E is the amplitude of the modulation and s represents the distance across the grating from an arbitrary starting point a t the center of one of the bright bars. The symbol p represents a constant. The symbol T , represents the modulation transfer function for the spread of excitation which is given by eq. (30). The response of the ganglion cells is also controlled by the inhibition ( I )generated by the photoreceptors. RATLIFF[1965, p. 1571 has assumed that the inhibition ( I ) is proportional to the stimulus applied to the retina and has a Gaussian line spread function conforming to the following equation:
I
= I, exp [ - + ( t / ~ ) ~ ] ,
(52)
where t is the distance from the center of the line image. If the distribution of retinal illuminance in the retinal image of a
94
T H E OPTICAL PERFORMANCE O F T H E HUMAN EYE
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§ 7
sine-wave grating is convoluted with the spread function defined by eq. ( 5 2 ) , we obtain the distribution of inhibition, which is
I
= Y E +yTI
AE cos (Zns/B).
(53)
The symbol y represents a constant. The symbol TI represents the modulation transfer function for the spread of inhibition and is
T I = exp [ - 2 ( n ~ 7 / 8 ) ~ ] .
(54)
It may now be assumed that the net excitation transmitted to the ganglion cells is proportional to ( R - I ) : (R-I)
1
(,L-Y)E +AE(,LLTR-YTI)cos ( 2 ~ ~ / 8 ) .
(55)
It follows, therefore, that the modulation transfer function [T,,-,,] for physiological irradiation of excitation and inhibition is T(R-I)= ( ~ - Y / P ) - ' [TR- ( Y / P ) T I I -
(56)
The total modulation transfer function ( T ) for the human eye including optical blur as well as physiological irradiation of excitation and inhibition is the product of T , and T,.-,,,
T = TETU-I,,
(57)
where T , is the modulation transfer function for the image-forming mechanism of the eye. Figure 32 provides a useful way of visualizing the components which
CENTER TO CENTER SEPARATION IMINUTESJ
Fig. 32. Interrelations between the components contributing t o the modulation transfer function for the human eye (from FRY[1969b]).
11,
I
71
95
MODULATION TRANSFER FUNCTION
v)
W 3
' 0
20
40
60
80
100
120
DISTANCE ACROSS THE RETINA
140
160
180 200
220
(MINUTES)
Fig. 33. Distributions across the retina of excitation (I?) and inhibition ( I )produced by a sine-wave grating of constant modulation and variable frequency. Each half cycle is 1.25 times wider than the half cycle which precedes it. The numbers above and below the ( R - I )curve indicate the lengths of the half cycles. The transfer curves are shown in Fig. 32 (from FRY[1969b]).
demodulate the image of a sine-wave grating. In the same graph are shown plots of T,, T,, T I , ( y / p ) T , and [TR- ( y / p ) T I ] .The constants aE, k and a, shift the T,, T , and TI curves sidewise without changing their forms. The ratio ( y / p ) controls the ratio of inhibition to excitation and raises or lowers the values of T by changing the DC component of the excitation transmitted to the ganglion cells. Figure 33 shows the interrelation of excitation and inhibition at the various frequencies in the case of a sine-wave grating. A sine wave of variable frequency has been convoluted with the E and R spread functions to show the distribution of excitation, and it has been convoluted with E and I spread functions to show the distribution of inhibition. In the lower curve the two distributions have been combined by subtracting the inhibition from the excitation. It should be noted that at low frequencies the AC component of inhibition partially neutralizes the AC component of excitation. This effect accounts for
96
THE OPTICAL PERFORMANCE
OF THE H U M A N E Y E
[II,
4
7
part of the upturn in the modulation threshold curve at low frequencies. However, at high frequencies the AC component of inhibition is reduced t o zero, and it no longer has the effect of lowering the AC component of excitation. At all frequencies inhibition enhances modulation by lowering the DC component. It is possible that the visibility of each border in the grating is affected by lying close to other parallel borders.
Fig. 34. The relation of the modulation threshold curve (curve C in Fig. 31) to the modulation transfer curve of the eye ( T )defined by eq. (56). (From FRY[1969b].)
If the variation in the modulation threshold for the eye at different spatial frequencies represents compensation for the demodulation imposed by optical and physiological blur, the reciprocals of the modulation transfer values ( T ) at the various frequencies should be proportional to the modulation threshold values corrected for blur (curve C in Fig. 3 1 ) . Figure 34 shows the reciprocals of the modulation transfer values multiplied by a constant ( K ) plotted on the same graph with the corrected threshold values. The values of the constants K , aE, k , 0, and ( y i p ) have been chosen to obtain a reasonable agreement between the experimental and theoretical results. The failure to obtain perfect agreement at the high frequencies could indicate that the assumed spread function for physiological irradiation of excitationneeds to be modified. It could also be attributed to spherical aberration of the eye which would increase the value of a E . Another factor to be considered is the blur produced by the high frequency micronystagmoid movements of the eye.
11,
5
71
97
MODULATION TRANSFER FUNCTION
Figure 31 brings out the fact that the threshold for a single sharp border is much higher than the minimum threshold for a sine-wave grating which occurs at a frequency of about five cycles per degree (5 = 12 min). Figure 35 illustrates a possible basis for this difference. This figure shows a contrast border and a sine wave a t a frequency of five cycles per degree which have been convoluted with the total spread function for the human eye. The difference in excitation on the two sides of the contrast border is less than the difference between the peaks and troughs of the grating. CONTRAST BORDER
SINE WAVE =12min)
6
r---------
,,
10 min
I
___-_----A
J
\-I
\ f
.
Fig. 35. The effect of demodulation by the eye on the image of a sharp contrast border and the image of a sine-wave grating with a center-to-center spacing of 12 min. The transfer curves are shown in Fig. 32. (From FRY[1969b].)
Other investigators have attempted to analyze the transfer function for the optical image forming mechanism of the eye and retina into its components. RATLIFF[1965, pp. 146-1591 used only two components, optical blur and inhibition. What Ratliff calls “optical blur” includes physiological blur and optical blur. PATEL[1966] used data obtained by measuring the light reflected from the retina to deduce the role played by the retina in determining the spread function of the eye as a whole. SCHADE[1956] and NACHMIAS [1968] have used a Gaussian function to represent optical blur and excitation and a second Gaussian function to represent inhibition. Since the optical transfer function is not affected by changes in the luminance level and stimulus duration, the changes in the form of the modulation threshold curve which have been reported by VAN NES [1968], SCHOBER and HILZ[1965], NACHMIAS [1967], and others have to be explained by changes in (YIP), k and 0, and also by changes in the form of the excitation and inhibition spread functions.
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T H E OPTICAL PERFORMANCE O F THE HUMAN EYE
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§ 7
7.2.DIRECT MEASUREMENT WITH A DOUBLE SLIT INTERFERENCE PATTERN
One way to demonstrate that the spread of excitation in the retina constitutes a blur component independent of optical blur is to by-pass the optics of the eye altogether and measure directly the transfer function for the retina and brain. This is exactly what LEGRAND [1935], WESTHEIMER [1960], ARNULF and DUPUY[1960] and CAMPBELL and GREEN[1965] have done by measuring the modulation thresholds for Young interference fringes formed on the retina. The data for Westheimer, Arnulf and Dupuy, and Campbell and Green have been replotted in Fig. 36. The points representing the data for Campbell and Green are points taken from the smooth curves which they used to represent their data. The points for Arnulf and Dupuy represent average values for three subjects. Also, for each subject the experiment was repeated several times using artificial pupils of different size. Since the results are based on two small beams passing through the center of the artificial pupil, changing the size of the pupil should have no effect on the results. As a matter of fact, the size of the pupil was found to have no effect, and the data points for artificial pupils of 4, 2.5, 1.6 and 1 mm in diameter have been averaged. I
0.I
5- 0.01
3 3
n 0
=0.001
I CENTER
10
TO
100
CENTER SEPARATION
Fig. 36. Modulation threshold data for Young interference fringes formed on the retina so that the optical system of the eye is by-passed. Different levels of average retinal illuminance were used: Westheimer, 2200 trolands; Campbell and Green, 500 trolands; Arnulf and Dupuy, 50 trolands.
1,
I 81
MODULATION O F A SINE-WAVE GRATING
99
The dashed curve used by Westheimer to fit his data (Fig. 36) conforms to the following equation: Mthreshold
(58)
where K is a constant and T is defined by eq. ( 2 5 ) and represents the modulation transfer function for the optical system of a diffractionlimited eye with a 2-mm pupil exposed to 555 nm. Westheimer stated that his data could be fitted in this way, but since the measurements by-pass the optics of the eye, it is obvious that the spreading mechanisms have to be described in terms of physiological mechanisms in the retina. Westheimer’s data have also been fitted with a second curve B, which conforms to eq. (58), but this time T represents the modulation transfer function for the combination of physiological excitation and inhibition as defined by eq. (56). Westheimer concluded from his data that the modulation transfer function for the eye and brain is monotonic for the range of frequencies investigated, and that his data show no upturn in the modulation threshold at low spatial frequencies, which might indicate inhibition. The data of Arnulf and Dupuy and those of Green do show an upturn. The curve C used to fit the data of Arnulf and Dupuy is the same as curve B, and has merely been displaced down and to the right. Since logarithmic scales are used for i and M , sliding a curve down corresponds t o a change in K , and moving it sidewise involves a change in k and a change in cE and a,. The data for Campbell and Green do not conform to curve B. The difference in general threshold levels for the investigators is puzzling. They worked with different field sizes and luminance levels, but there appears to be no simple explanation for the differences in thresholds. $j8. Perceived Modulation of a Sine-Wave Grating at
Suprathreshold Levels I n deriving the modulation transfer function for the human eye, the writer and others, eg., MENZEL 119591 and PATEL[1966], have assumed that the modulation of the output from the retina at the threshold of visibility is constant from one spatial frequency to another, and that the variation in input required for the threshold reflects the different amounts of demodulation produced by the system at the different frequencies.
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T H E OPTICAL P E R F O R M A N C E O F T H E H U M A N EYE
[II,
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As shown above, this relation may be complicated at low frequencies by mechanisms other than those that can be described as spreading mechanisms, but in the range from 6 to 60 cycles per degree the basic assumption probably holds. One can do such obvious things as showing that an increase in optical blur increases the modulation threshold at the higher frequencies. Another approach to this problem is to check to see if the amplitude of the perceived modulations at suprathreshold levels correlate with the thresholds at different frequencies. BRYNGDAHL [1966] has measured the perceived brightness of bright and dark bars of a grating at suprathreshold levels for different frequencies. Heshowed that the perceived modulation varies with frequency in accordance with what might be expected from the changes in threshold. He also showed that the perceived modulation is greater than the actual modulation of the grating. This may be related to the lowering of the DC component of the eye’s response by inhibition. Bryngdahl used a stimulus pattern similar to that shown in Fig. 8 and photometrically matched the upper half first to the bright lines and then to the dark lines. He used center-to-center separations of 7 minutes and greater. With spacing finer than this, it becomes difficult for the eye to assess anything other than the average brightness (FRY[1959]). ARNULFand DUPUY[1960] had to match two suprathreshold sinusoidal gratings on the basis of equal perceived modulation. They reported measurements for center-to-center spacing as low as 1.74 minutes. § 9. Determination of the Modulation Transfer Function of the Optical System of the Eye 9.1. THE CAMPBELL-GREEN METHOD
CAMPBELL and GREEN [1965] simply measured the modulation threshold for Young interference fringes for different spatial frequencies and then separately measured the modulation thresholds for a sinusoidal grating generated with a TV tube and viewed through artificial pupils of different size. The same spatial frequencies were used in the two experiments. Since the only effect that the optical system of the eye can have on a sinusoidal grating stimulus exposed to the eye is to reduce its modulation, it may be assumed that if the
11,
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101
T H E MODULATION TRANSFER FUNCTION
image of the TV display on the retina and the Young fringes have the same spatial frequency, they must have the same modulation threshold. Thus, the ratio of the modulation reductions necessary to bring the two patterns to their thresholds must represent the optical transfer function for the image-forming mechanism of the eye. Unfortunately, the monochromatic light emitted by the laser (632.8 nm) differed from the wavelength composition of the TV display, which peaked at 530 nm and had a half-width of *30 nm. It can likely be assumed that a 632.8 nm set of interference fringes has the same threshold as a 530 nm grating. The average retinal illuminance for both types of gratings was kept constant at 500 trolands. The data for subject Green for a pupil 2 mm in diameter are given 1.0
0.5
; L
u w 4L -8
0.2
c
0.1
I
0
10
I
t
20 30 40 Spatial frcquency (cicleg)
I
50
60
Fig. 37. Threshold data for subject DG for Young fringes (solid line) and for a sinusoidal grating (open circles) viewed through a 2-mm artificial pupil with monochromatic light of 530 nm. The filled circles represent modulation transfer factors deduced from the data for the image-forming mechanism of the eye (from CAMPBELLand GREEN [1965]).
102
9 9
[IL
T H E OPTICAL PERFORMANCE O F THE HUMAN E Y E
1.0
0.9 08 -Y
.g 0.7 Q
0.6 bf.
.-E 0.5 d
..a
+
2
0.4
-Y
2
0.3
0.2 0.1
, I
0
0.1
I
I
0.3 0.4 0.5 Noniializrd spatial frcqmucy
0.2
I
I
0.6
0.7
Fig. 38.The data of subject DG for the modulation transfer factors for the image-forming mechanism of the eyc for a 2-mm pupil ( 0 )(same as in Fig. 37) and for three other pupil sizes: V , 2.8 mm; m, 3.8 mm; A,5.8 mm. The abscissavaluesrepresent normalized spatial frequency. The Fraunhofer transfer function is also shown. The wavelcngth (530 nm) was the same for all curves (from CAMPBELLand GREEN[1965]).
in Fig. 37. The accommodation of Green’s eye was paralyzed, and the retina was made conjugate to the TV display with a +l.SO lens mounted before the eye. The solid curve without dots represents the data obtained with double slit interference fringes. This figure shows how the modulation transfer factors for the optical mechanism are determined. The ratio for the two thresholds at a given frequency gives the transfer factor for that frequency. The curve representing these ratios a t different frequencies is the modulation transfer function. In Fig. 38 a normalized frequency scale is used: Normalized frequency =
[A/(2[)] (1/S),
(59)
where il and g are expressed in the same units and d is expressed in radians. This scale has the merit that all diffraction-limited eyes fall on the one theoretical line regardless of wavelength or pupil size. The curves for actual eyes of different pupil size are not to be compared
11,
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103
THE MODULATION TRANSFER FUNCTION
1.0
0.8
0.6
:0.2 M
z
..-d E
o
+2
*
3
0.8
c
c
6
0.6 0.4
0.2
0.1
0.2
0.3 0.4 0.5 0.6 Normalized frequency
0.7
0.8
0.9
.,
Fig. 39. The data of subject DG for the modulation transfer factors of the image-forming mechanism of the eye for the eye in focus (0) and for the eye out of focus to various degrees measured in diopters: 0.50; A, 1.00; 0, 2.00. The pupil diameter = 2 mm and 1 = 530 nm. The broken curves represent the Fraunhofer and the Fresnel transfer functions (from CAMPBELLand GREEN [ 1 9 6 5 ] ) .
with each other, but each is compared to the theoretical curve to determine whether it measures up to its own standard of performance. The theoretical curve in Fig. 38 is the same as the curve in Fig. 15 for the Fraunhofer image. The only difference is that in Fig. 38 a normalized frequency scale is used. The discrepancies which are found must be traced to spherical aberration and other factors. Campbell and Green also presented data for an eye thrown out of focus to various degrees by changing the lens mounted in front of the eye. The modulation transfer functions are shown in Fig. 39 along with the theoretical curves for a diffraction-limited eye. These curves were computed by Linfoot according to the theoretical method described by HOPKINS [1962].
104
T H E OPTICAL PERFORMANCE OF T H E HUMAN EYE
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9
9
The Campbell-Green method should be tested at different luminance levels. If it is a valid method for determining the transfer function for the image-forming mechanism, the results should not be affected by changing the luminance level. By working at a relatively low luminance one could avoid the problems encountered with the coarseness of the retinal mosaic at high frequencies. 9.2. THE ARNULF-DUPUY METHOD
ARNULF and DUPUY [1960] measured the modulation transfer function for the image-forming mechanism of the eye by comparing the fringes formed by interrreflection at a glass wedge with a set of Young fringes of the same frequency as described above. The beam forming the wedge fringe pattern fills the artificial pupil mounted in
0
2
4
x
6
8
10
m Fig. 40. Data of ARNULF and DUPUY[1960] for the modulation transfer function of the optical system of the eye for differentpupil sizes; 1 = 546. The Fraunhofer modulation tranfer curve is included for comparison.
front of the eye, and the modulation of this pattern is affected by changing the size of the artificial pupil. The modulation of the Young fringes can be varied from unity t o zero and can be used to measure directly the modulation of the wedge fringes. The data have been presented by Arnulf and Dupuy as shown in Fig. 40. The abscissa scale is a normalized scale in which the spatial frequency 1/S in lines per radian is multiplied by the normalizing factor A ( 2 c ) . One can represent on such a graph by a single curve [eq. (19)] the theoretical modulation transfer function for an optical system which is in focus and limited only by diffraction. This one curve applies t o pupils of all sizes and to all wavelengths.
T H E MODULATION TRANSFER FUNCTION
1 2 CENTER
5
10
105
50
20
TO CENTER SEPARATION ( 5 )
Fig. 41. Modulation transfer data of ARNULFand DUPUY[I9601 for pupils 2.5 mm in diameter and smaller plotted as functions of g.
I-0
0.5 T
0.2 minutes I
,
I
Fig. 42. Modulation transfer data of ARNULF and DUPUY[I9601 for 2.5-mm and 4-mm pupils plotted as functions of f .
The advantage in plotting the data in this way is that if the optical system is diffraction-limited, the curves for all pupil sizes must coincide with the theoretical curve. In fact, the curves for pupils under 0.87 mm do coincide, and for pupils of this size and smaller the eye functions as a perfect optical system. The discrepancies become progressively worse for larger pupils indicating that the eye is subject to spherical aberration and other sources of optical blur. In Figs. 41 and 42 the data for the different pupil sizes have been replotted with the modulation transfer factors as a function of the center-to-center separation. The curves used to represent pupil sizes of 0.76 mm and smaller are the theoretical Fraunhofer curves. The theoretical curves all have the same form and are merely displaced sidewise from each other. Displacement to the left indicates improved performance, and hence the amount of improvement produced by a change in pupil size is indicated. The optical system of the eye achieves
106
THE OPTICAL PERFORMANCE OF T H E HUMAN E Y E
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$ 10
its maximum performance when the pupil is 2.5 mm in diameter. The data in Fig. 41 can be compared directly with the data of Campbell and Green in Fig. 38.
Q 10. Resolving Power at Unit Modulation 10.1. EFFECT O F PUPIL SIZE ON THE RESOLVING POWER
COBB [1915] made a careful study of the effect of pupil size on visual acuity measured with an Ives patten illuminated with white light. As explained above, this pattern is equivalent to a sawtooth distribution. He used artificial pupils of various sizes and kept the average retinal illuminance constant at 148 trolands. The results are shown in Fig. 43. The results have been expressed in terms of resolving power ( l / S ) instead of in terms of visual acuity. Cobb pointed out that if the eye were diffraction-limited, it should perform much better than it does. The resolving power (1/B) should vary directly as the diameter of the pupil, as indicated by the dotted line. I n adding this dotted line to the figure, it has been assumed that the two curves should come together for small pupils. Cobb concluded that the performance of the eye with large pupils must be seriously impaired by spherical and chromatic aberration and other defects.
0.7 0.6 -
;;; 0.5 -
\ -
oI
0.4 -
w
n
0.3
Trolands
WHITE LIGHT
-
1 2 DIAMETER
3
4
5
6
OF ENTRANCE-WPIL
Fig. 43. Resolving power of the eye for a sawtooth grating for pupils of different size. The average retinal illuminance was kept constant a t 148 trolands. (Data from COBB [1915].)
11,
§ 10)
RESOLVING P O W E R AT U N I T MODULATION
107
BERGER-LHEUREUX-ROBARDEY [ 19651 has measured the resolving power of the eye at unit modulation with the Wollaston prism technique described below. The beam of light used in this measurement was obtained by focusing an enlarged image of a mercury source on the artificial pupil mounted just in front of the eye. The arrangement is similar to that shown in Fig. 24, except that in this figure the artificial pupil is not shown. A filter rendered the light monochromatic (546.3 nm). Artificial pupils of different size were used as indicated in Fig. 44, and the resolving power at unit modulation was measured by placing a Wollaston prism in the beam at the point conjugate to the retina and rotating the Wollaston prism around an axis parallel to the fringes. The grating pattern was sinusoidal.
0
1
2
3
4
5
6
DIAMETER OF ENTRANCE-PUPIL
Fig. 44. Resolving power of the eye for a sinusoidal grating for pupilsof different size and different levels of retinal illuminance. (Data from BERGER-LHEUREUX-ROBARDEY [1965].)
The results obtained at three luminance levels are shown in Fig. 44. Although the role of diffraction is different when the source is conjugate to the aperture-stop instead of to the retina as in Cobb’s case, the data do show that with monochromatic light the performance of the eye is still impaired by spherical aberration and other defects when the pupil is large; they also show that the performance is affected by
108
T H E OPTICAL PERFORMANCE OF T H E HUMAN EYE
-
[II,
10
10000
0 v)
1000
2
a 2
100
K
t
10
i -_1
I
t; K
0.1
> a
0.01
,
I
I
, 1 , , , , 1
2
5
10
20
RESOLUTION THRESHOLD (MINUTES)
Fig. 45. Effect of the level of retinal illuminance on the resolution threshold for the fovea of the eye for a square-wave grating. (White light with a 2-mm artificial pupil.) (Data from SHLAER [1937-381.)
.-,*
lo
,to I to 31 to 7
1- ' '
' "%\I
'I' ''\I
'I / ;/ I I f
0.1 -
OPAQUE
[TO CLEAR RATIOS
MODULATION THRESHOLD FOR THE RETINA
5
0.01 -
F Q _J
3
n 0
E 0.001
111 I
minutes I
I
11.
§ 101
109
RESOLVING POWER A T U N I T MODULATION
the source in the plane of the pupil. The grating was conjugate to the retina. The data for white light are shown in Fig. 45. The resolving power reaches an upper limit where further increase in retinal illuminance has no effect. Shlaer believed that this upper limit was dependent on the coarseness of the retinal mosiac. In order to check this possibility further, he investigated the effect of changing the modulation because if the resolving power were limited by the coarseness of the retinal mosaic, it should not be affected by changing the modulation. In order to manipulate the modulation, he changed the relative widths of the light and dark bars of his squarewave grating and found for the three black-white ratios which he used only a small difference, which he considered negligible. In a separate paper (FRY[1968c]) I have described a more detailed analysis of Shlaer’s modulation data, in which I have assumed that the image-forming mechanism of the eye has a Gaussian spread function with a sigma value of 0.212; and I have computed the effect of optical blur on the retinal images of his three gratings. I have plotted Shlaer’s data in Fig. 46, which shows the thresholds in terms of the modulation of the image formed on the retina. The curve through the data has a slope similar to that found for the retinal threshold curve in Figs. 28 and 36. Hence, it is assumed that the limit encountered by Shlaer is not due to the coarseness of the retinal mosaic. Furthermore, he was able to improve the acuity beyond the level indicated in Fig. 46 by increasing the size of his pupil, It must be noted that the TABLE2 Resolution thresholds measured with a double slit diffraction pattern ~~
~
Investigator
~~
Resolution threshold (minutes)
Retinal illuminance (trolands)
Le Grand
1.25 1.45
13 600 2.72
546
Campbell
0.98
500
633
Green
0.95
500
633
Westheimer
1.25
2 200
555
Byram
0.50
I
not specified
Wavelength (nm)
not specified
110
T H E OPTICAL P E R F O R M A N C E O F THE HUMAN E Y E
[II,
§ 11
lowest resolution threshold found by Shlaer, namely 0.94 min, is better than the best measurements (Table 2 ) reported by several investigators who used double slit interference patterns. 10.3,RESOLVING PATTERNS
POWER FOR DOUBLE SLIT INTERFERENCE
The resolving power as measured with a double slit interference pattern must be better than that obtained with normal seeing in order to use the modulation thresholds to measure the optical transfer function. This relation holds in Fig. 37, where the modulation threshold curves cross the unit modulation line at different points. We must face the possibility that the upper limit of resolution is determined by the spread of excitation in the retina when a double slit pattern is used. It may be noted that the threshold curves (Fig. 36) for the double slit pattern cross the unit modulation line at a slope, indicating that the limit of resolving power has not yet been reached. The values for resolving power at unit modulation as found by different investigators using the double slit interference pattern are shown in Table 2. BYRAM[1944] reported that whereas the transition from resolution to fusion was smooth in ordinary seeing, it was not smooth with a double slit pattern; as the lines approached the limit of resolution, they broke up into segments and appeared to shimmer or flutter. CAMPBELL and GREEN[1965] pointed out that just before fusion the central patch scintillates and appears brighter and more desaturated than the surround. These effects may be taken as evidence that the structure of the retinal mosaic is involved. HELMHOLTZ [1924] reported seeing similar effects when a fine grating is viewed with the natural pupil. WESTHEIMER[1960] reported that his subjects found the transition t o be smooth. LE GRAND[1935] reported that at 13 600 trolands the difference between the Young fringes and the Babinet fringes was negligible, but the Babinct fringes suffered more impairment than the Young fringes as the luminance level or contrast was reduced.
Q 11. Dependence of the Resolving Power on Wavelength Composition The existence of three types of cones in the fovea poses a problem for resolving power because they do not respond equally to all mono-
rr, S 113
D E P E N D E N C E ON WAVELENGTH COMPOSITION
111
chromatic stimuli, and thus the resolving power of the eye should depend on the extent to which the three types of cones are involved. Are the cones of a kind arranged in clusters or do the different kinds of cones interdigitate so that cones of each of the three types are uniformly distributed? Are the three types equally numerous? How do the responses of the three kinds of cones interact at the level of the bipolars and ganglion cells? O’BRIEN and MILLER [1952] investigated this problem by using double slit monochromatic interference fringes, but they were unable by this method to throw any light on the arrangement or relative numbers of the three receptors. HARTRIDGE [1947] used a single point and reported a scintillation of changing colors as the image moved across the retina. At ordinary luminance levels visual acuity with high contrast gratings is independent of the wavelength for a given pupil size when monochromatic stimuli are used, provided that allowance is made for the effect of wavelength on diffraction. Yellow light, which stimulates red- and green-sensitive cones, yields the same result as red light, which stimulates only red-sensitive cones. One way of explaining this independence is to say that the three kinds of cones channel their brightness responses through the same ganglion cells. As long as the luminance levels are above the thresholds of the three kinds of cones, it is reasonable to expect a simple summation of excitation and inhibition at the level of interaction between the retino-cortical pathways. The spread of excitation and inhibition is the same as if each cone had an ICI luminosity curve. In a previous publication (FRY[1955]), I have assumed that the retinal image of a polychromatic point source represents a superposition of the monochromatic images, each of which is dependent upon the total flux for that part of the spectrum and the extent to which the eye is out of focus for that wavelength. The ICI photopic luminosity curve was used in computing the amount of flux for the different parts of the spectrum. In accordance with this straightforward analysis of the problem, I have computed the distribution of retinal illuminance in an image for white light having a uniform energy spectrum. It was assumed that the eye was focused for 555 nm, and the data of WALD and GRIFFIN[1947] for the chromatic aberration of the eye were used in computing the extent to which the eye was out of focus for other wavelengths. The distribution of flux in such an image is shown in Fig. 47. Included in the same figure is the distribution of flux for the
112
T H E O P T I C A L P E R F O R M A N C E O F THE HUMAN EYE
DISTANCE ACROSS THE RETINA
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11
(MICRONS)
Fig. 47. The theoretical optical images formed on the retina for a white point and for a monochromatic point (555 nm) with a 2-mm exit-pupil and with chromatic aberration and diffraction taken into account. (From FRY[1955].)
image of a monochromatic point of 555 nm. The total flux in these two images is not the same; the amounts of flux have been adjusted so that the retinal illuminances at the two peaks are equal. It is obvious that the white image shows more spread than the yellow one. I proposed a simplified method of evaluating the effects of chromatic aberration for a diffraction-limited eye through the use of the Fry-Cobb index of blur, +. It turns out that for a polychromatic point image in a diffraction limited
-
0
0 +ID DIOPTERS CUT OF FOCUS FOR YELLOW LIGHT
-100
Fig. 48. The theoretical effect of throwing the eye out of focus on the value of (expressed in microns) for white light and yellow light (555 nm) with a 2-mm cxitpupil. (From FRY[1955].) Eye thrown o u t of focus by moving the retina.
11,
§
111
D E P E N D E N C E ON W A V E L E N G T H C O M P O S I T I O N
113
RADIUS OF EXIT PUPIL
Fig. 49. The theoretical effect of varying the pupil size on the value of 4 (expressed in microns) for white light and yellow light (555 nm). Eye focused for 555 nm. (From FRY[1955].)
where F A represents the total flux in the image per unit wavelength, and F represents the total flux in the image. Table 1, which gives values of V for various values of a, can facilitate the computation of 4. Figure 48 shows a plot of 4 expressed in micron units of distance across the retina as a function of diopters out of focus for yellow light and for white light. When the eye is in focus, the image for monochromatic yellow light is sharper than the image for white light; but for values of 4 greater than 12p, the depth of focus is better for white light than for monochromatic light. The index of blur approach can also be used to show the effect of pupil size on blur. The results are shown in Fig. 49. For monochromatic light, the reciprocal of 4 increases in proportion t o the radius of the exit-pupil, whereas for white light, the image becomes sharper as the radius increases up to about 1.5 mm and then becomes poorer as the effect of chromatic aberration offsets the effect of pupil size on diffraction. CAMPBELL and GUBISCH[1967] have approached the problem by treating the modulation transfer function for heterochromatic light as the weighted sum of the several transfer functions for the different parts of the spectrum. These component parts are weighted according to the luminosity curve, and allowance is made in the transfer functions for the extent to which the eye is out of focus, They have used the average measurements of IVANOFF [1947], HARTRIDGE [19471 and CAMPBELL [1957] for the chromatic aberration of the eye. Using this approach, Campbell and Gubisch showed that the
114
THE OPTICAL P E R F O R M A N C E O F T H E H U M A N E Y E
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modulation transfer functions for white and yellow light converge at high and low frequencies but diverge from each other in the middle of the frequency range with the maximum difference falling around 30 t o 40 cycles per degree, depending on the size of the pupil. Hence, one cannot expect to find differences in the performance of the eye for white and yellow light at unit modulation. One must look for the difference in the frequency range of about 30 cycles per degree. Campbell and Gubisch measured the modulation thresholds for white and yellow light at a frequency of 30 cycles per degree with the eye thrown out of focus various amounts. The pupil diameter was 2.5 mm. The results are shown in Fig. 50 and can be compared to the curves in Fig. 48. 10 r
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Fig. 50. Data for RWG for contrast sensitivity (1/S)for a sine-wave grating for various diopters out of focus for white light (filled circles) and yellow (555 nm) light (open circles) with a 2.5 mm artificial pupil and a spatial frequency of 30 cjdegree. (From CAMPBELLand GUBISCH[1967].)
They also showed that for a small pupil (1.5 mm) the difference between monochromatic yellow and white light is negligible, but is measurable with a 2.5 mm pupil and again disappears with a 4.0mm pupil. This agrees with the predictions in Fig. 49, except that at a pupil diameter of 4.0, it must be assumed that spherical aberration is so bad that the difference between white and yellow light is negligible. Although the use of the CIE photopic luminosity curve for weighting the different parts of the spectrum in computing the modulation transfer function for heterochromatic light is satisfactory, if one is looking for theoretical values to compare with the experimental data obtained by reflection from the retina, Campbell and Gubisch have questioned whether the CIE photopic luminosity curve can be used
11,
9
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D E P E N D E N C E ON W A V E L E N G T H C O M P O S I T I O N
115
in computing theoretical values to compare with the experimentally obtained values for modulation thresholds. By comparing the results obtained with white light with the results obtained with monochromatic light (546 and 578 nm), they concluded that it is better to use the green component of the luminosity curve for weighting the different parts of the spectrum when the eye is in focus for 546 nm and to use the red component of the luminosity curve when the eye is in focus for 578 nm. The theory involved is that the red-sensitive cones dominate the situation in one case and the green ones dominate in the other. This conclusion is based in part on VON BAHR’S [1946] finding that if grating acuity is measured with a mixture of two colors, the performance is determined by the color for which the eye is focused. One can raise the question as to what would happen if one could restrict the input to the retina to one kind of cone. I n the different types of dichroniatic forms of color blindness, it is often, but not universally, assumed that one of the three kinds of cones is missing. But the chances arc that these are replaced by cones of the other two types, and hence the resolving power need not be affected because the population of photoreceptors is more sparse. I n blue cone monochromatism (BLACKWELL and BLACKWELL [1961]), visual acuity is impaired; however, information about the distribution and number of photoreceptors in such retinas and about the neural organization is lacking. GREEN[1968] has approached this problem by using normal eyes and the two-color and bleaching techniques developed by STILES [1949] and BRINDLEY[1954] for isolating the responses of the red, blue, and green photoreceptors. In this technique, the modulation thresholds for red and green light were measured on red and green backgrounds. I t was found that the modulation threshold for green light was dependent upon the extent to which green light stimulates green receptors. Since red light stimulates only red receptors, the modulation thresholds measured with red light are dependent upon red-sensitive photoreceptors acting alone. The modulation curves for red and green light are very similar, and this helps to explain the eye’s response to yellow light, which involves a combination of the red and green photoreceptors. I n studying the isolated response of the blue receptors, Green first bleached the red and green receptors by exposing the retina to yellow light and then measured the modulation thresholds for blue light with a blue sinusoidal grating superimposed on a yellow background. The resolving power of the eye for high levels of modulation is no better
116
THE OPTICAL PERFORMANCE O F THE HUMAN EYE
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than 10 cycles per degree. One could conclude from this that the blue receptors are less numerous and more sparsely distributed than the red and green receptors, and the high levels of acuity achieved with blue light under ordinary conditions must be explained by the involvement of the red and green photoreceptors. The bleaching and two-color techniques used by Green involve high levels of luminance. At ordinary luminance levels, below 1000 trolands, the red and green photoreceptors involve little or no bleaching; thus, changes in adaptation dependent upon mechanisms at the level of the bipolars and higher could affect equally the responses of the red, green and blue photoreceptors regardless of the wavelength composition involved in producing the state of adaptation. An important point which may be deduced from Green’s data is that below its threshold a cone yields no response which can be summated with the responses from other photoreceptors. Consequently, it may be assumed that there is no such thing as a sub-threshold response of the red photoreceptors which can affect the supra-threshold response of the green photoreceptors. § 12. Visibility of Square-Wave Gratings
Although sine-wave gratings appear to be ideal from the point of view that no matter how much one demodulates such a pattern, it still retains the form of a sine wave, there are, however, things t o be VARYING MAG. oVARYlNG CONTRAST
MINUTES I
I
Fig. 51. iModulation threshold curve for a square-wave grating. Averagc data for two subjects. Whitc light and a 2-mm entrance-pupil. The dashed curve represents the modulation threshold for a single border. (From LEVYand FRY[1968].)
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VISIBILITY O F SQUARE-WAVE GRATINGS
learned from the study of the eye's response to square waves. At high frequencies, it may be assumed that the optical image of a square-wave grating formed on the retina has become transformed to a sine wave. Since it is approximately true that for an eye in focus the point spread function can be approximated by a Gaussian spread function, it follows that the modulation of the retinal image for a square wave of high frequency is precisely 41n times that of a sine wave. Using a formula derived by ENGEL [l968] for the modulation of a line grating convoluted with a Gaussian spread function, FRY[1968a] has shown that the modulation of a high frequency square wave convoluted with a Gaussian spread function is 41" times that for a sine wave. CAMPBELLand ROBSON [1964 and 19681 have verified by experiment that this is true. They had arrived at the conclusion that this relationship must exist by assuming that the eye filters out all but the first harmonic of the square wave. Figure 51 shows the modulation threshold curve (average for two subjects, DL and DK) obtained in a study by LEVYand FRY[1968]
MINUTES 0
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50
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Fig. 52. Distributions across the retina of excitation (R) and inhibition ( I )produced by a square-wave grating of constant modulation and variable frequency. Each half cylcc is 1.25 times wider than the half cycle which precedes it. The numbers below the ( R - I ) curve indicate the lengths of the half cycles. The transfer curves are shown in Fig. 32. (From LEVYand FRY[1968].)
118
T H E OPTICAL PERFORMANCE OF THE HUMAN EYE
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for a luminance level of 21.7 fL and an artificial pupil of 2 mm. Shown also by the dotted line is the level of the modulation threshold for a single border between a bright and a dark area. In order to make the comparison between the grating and a single border meaningful, the grating was always presented with a border between a dark bar on the left and a bright bar on the right a t the center of the pattern; the subject was asked to fixate a point on this border. This corresponds to ficating a point on a single border. It is interesting to show the effect of convoluting a square-wave grating with a Gaussian spread function, which simulates the optical spread function of the eye, and an exponential spread function, which represents physiological irradiation of excitation, and a negative Gaussian spread function, which represents the spread of inhibition. This is shown in Fig. 52 and represents the same kind of analysis as that shown in Fig. 33 for a sine wave. The thing that makes the square wave differ from the sine wave is that the effects a t the borders between the bright and dark bars eventually become the same from border to border and are equivalent to those for a single straight border between extended bright and dark areas; whereas in the case of the sine wave, the gradient between the peaks and the troughs gradually flattens, and eventually the blurred bars become washed out to a uniform patch of brightness. It is obvious from Fig. 5 2 that the improved visibility a t a frequency of about 3 cycles per degree is dependent on the fact that the modulation of the inhibitory component has been reduced to a low level, whereas at lower frequencies, the modulation of the inhibitory component reduces the modulation of the excitation transmitted through the retina. The upturn in the modulation curve shown in Fig. 51 can, therefore, be partially attributed to the inhibitory mechanism, but it actually rises above the level for a single border. If the spacing of the grating were gradually increased, the threshold for the grating would eventually have to return to the level for a single border. The question needs to be raised, therefore, as to whether the mechanism subserving border contrast makes it possible for the visibility of a given border in the grating t o be affected by the presence of adjacent borders. If we consider the case of a single border, we can convolute it with the spread functions involved in optical blur and in the spread of excitation and inhibition, the same as in the case of a square-wave or sine-wave grating, as shown in Fig. 35. The excitation transmitted through the retina is considerably enhanced on the bright side of the border and
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VISIBILITY O F SQUARE-WAVE
A
GRATINGS
119
R
Fig. 53. Cornsweet’s pattern (RATLIFF[1965]) (A), which when rotated gives the illusion illustrated in (B) of a uniform disk surrounded by a uniform annulus of lowcr brightness.
depressed on the dark side. Although some observers claim to see bright and dark bands on the two sides of a contrast border, these are probably caused by poor focus, which results in gradients which produce definite bright and dark Mach bands. If the eye is in focus, the observer sees a contrast border as an abrupt transition from bright to dark, and the color on the bright side of the border is perceived as uniform or else as gradually decreasing from the border outward. The counterpart of this occurs on the dark side of the border. The extreme case of this phenomenon can be demonstrated by rotating the pattern shown in Fig. 53A, which when rotated creates the illusion of a contrast border between uniform patches of color of different brightness, as shown in Fig. 53B. If we can think of frequency of impulses in the optic nerve fibers as the correlate of brightness, then it is necessary to postulate some kind of frequency-equalizing mechanism in the retina which tends to wash out gradients but which can maintain an abrupt frequency difference at the midpoint of an S-shaped gradient. It must be supposed that this frequency-equalizing mechanism operates at a higher level than the mechanism involved in the spread of excitation and inhibition. It is conceivable that this mechanism would permit the presence of one border to affect the visibility of a neighboring border. If this kind of mechanism operates in the case of a grating, the remarkable thing is that it must operate when the inhibiting border lies at its own threshold of visibility. This problem has been studied by FIORENTINI [1968] by measuring the effect of a single inhibiting border on a nearby, parallel, narrow bar used as a test stimulus. WILD 119591 has also studied this problem.
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THE OPTICAL PERFORMANCE OF THE HUMAN E Y E
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§ 13. The Visibility of a Bar
It Is interesting to compare the results obtained with a square-wave grating with those obtained with a single bar (LEVYand FRY[1968]). The open circles in Fig. 54 represent measurements made with single bright bars having various widths. Reducing the width of a single bright bar has no effect on its visibility down to a width of about 8 minutes. As width is further reduced, the threshold curve eventually conforms to a straight line with a slope of minus one. 0.20 oBRIGHT BAR 'DARK BAR
a
WIDTH OF BAR (MINUTES)
Fig. 54. Contrast thresholds for bright and dark bars of various widths. Bar height = 78 min. Background luminance = 21.7 fL. Artificial pupildiameter = 2 mm. Averaged data for two subjects. (From LEVYand FRY[1968].)
If a dark bar is used instead of a bright bar, the results show, as indicated by the filled circles in Fig. 54, that the dark bar has a minimum threshold at a bar width of about 10 minutes. The difference between bright and dark bars at the wide end of the scale indicates that two borders with their dark sides facing each other interfere more with each other than two borders with their bright sides facing each other. When the bars are narrow, the opposite relation exists. These findings are not consistent with the assumption that the contrast threshold is the same for dark objects on a given background as it is for bright objects. In this experiment the subject fixated the center of the bar both when the bar was dark and when it was bright. It has been proposed by FRYand COBB [1935] that the index of blur defined above can be assessed from a set of threshold data for bright bars on a darker background. The data in Fig. 54 for bright bars have been replotted in Fig. 55B t o illustrate the procedure. According to the theory, a straight line with a slope of minus one tangent
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T H E VISIBILITY O F A B A R
I
10
100
10
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100
WIDTH OF BAR (MINUTES)
Fig. 55. Data in Fig. 34 analyzed to determine the index of blur.
to the curve where the bars are narrow and a horizontal straight line tangent to the curve where the bars are wide must intersect at the point which has an abscissa value equal to 4. As a bar increases in width, the illuminance at the center of the image increases. For narrow bars this increase is proportional to the width, but as the bar gets wider, the illuminance increases less rapidly and eventually levels off to a constant value. Figure 56 shows the effect of increasing the width of a bar on the distribution of flux across the image. In particular, the change in concentration of flux at the center of the image can be noted. In this figure the line spread function is Gaussian, and the distance across the retina is expressed in microns.
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Fig. 56. Bars of various widths convoluted with a Gaussian spread function having a o-value of 5 microns. The abscissas (m)indicate distance from the center of the image measured in microns. (From FRY[1955].)
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Fig. 57. Distributions of excitation ( X ) and inhibition ( I )across images of bright bars on a bright background; A L / L = 0.286. The R - I curves represent the net excitation transmitted through the retina. The transfer curves are shown in Fig. 32. (From LEVY and FRY[1968].)
Fry and Cobb made the simple assumption that the threshold for a given background level depends on having a fixed amount of retinal illuminance a t the center of the image. For the case in which the bar is very wide, this can be easily computed. It was later found (FRY[1946, 1965dl) that the index of blur as measured by the Fry-Cobb method varies with the luminance level; it had to be concluded that physiological irradiation is involved. Figure 57 shows bright bars of various widths on a background of a given luminance convolved with spread functions which give curves for R, I and R-I. This figure indicates that for a very wide bar the threshold no longer depends upon the response at the center of the bar, but rather upon the peak and trough at each of the two borders. The implication of Fig. 57 is that as the bar increases in width, the threshold should go through a minimum before it becomes independent of width. With a bright bar (Fig. 55B) the penomenon of a minimum does not occur, but it does occur with a dark bar as shown in Fig. 55A. I n this case the horizontal line used to assess 4 is drawn tangent to the threshold curve a t its minimum.
THE VISIBILITY O F BORDERS
123
It should bz noted in Fig. 57 that at a width of 10 minutes the peak in the response is still a single peak and has not broken into a double peak. It is as if a response with a single pnak rises out of the center of a crater of inhibition; the floor of the crater is the background, and it is the extent of t h i rise from the floor of the crater to the peak of the excitation which determines the threshold. The floor-to-peak rise would be nearly the same with or without inhibition, and hence inhibition can be ignored in analyzing threshold data for bars narrower than 10 minutes. It should be noted in passing that 4 as derived from the threshold data for bright and dark bars is of the order of four minutes of arc; but in the analysis of the modulation data for a sine-wave grating it was assumed that the point spread function for the image-forming mechanism is Gaussian and has a standard deviation of 0.4minute, which gives it a +value of one minute. The implication is that the major source of the spread is physiological irradiation. One can use threshold data for a single bar not only to determine the value of but also to determine the entire line spread function (FRY[1955]) for the combination of optical blur and spread of excitation. The first step in deriving the line spread function is to plot the reciprocal of the threshold values of contrast as a function of the width G of the bar and then differentiate to obtain values of [d(l/C)/dG]for various values of G. Since in this case 3 = 2t, one can now compute the normalized line spread function [ H ( t ) ]for the combination of optical blur and spread of excitation:
+,
H ( t ) = Cmin [d(l/C)/dG].
(62)
I t is important to note that this whole procedure can be based on threshold measurements for bars that range in width from 0 to 10 minutes; and as indicated in Fig. 57, one can assume that in this range the threshold is dependent on a constant peak to floor difference in R. For a bright white bar on a background of 2293 trolands with a pupil 2.33 mm in diameter, FRYand COBB [1935] found that the line spread function approximates a Gaussian distribution having a standard deviation of 0.73 minute ($ = 2.92 minutes).
5 14.
The Visibility of Borders
The visibility of borders is an intriguing problem in itself, with many
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T H E O P T I C A L P E R F O R M A N C E O F T H E H U M A N EYE
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facets. It merits consideration here because LOWRYand DEPALMA [1961] have worked backward from measurements of the Mach band effectto determine the modulation transfer function of the eye, including the optical as well as the physiological components. Lowry and DePalma used a uniform gradient between two uniform areas and obtained bright and dark Mach bands a t the two edges of the gradient. RATLIFF[ 19651 has also used the Mach band phenomenon in connection with his study of the mechanisms of lateral inhibition in the limu111s eye (RATLIFFet al. [1963]). MARIMONT [1963] used the asymmetry of the bright and dark Mach bands studied by Lowry and DePalma as evidence that the system is non-linear. We could explain this effect by assuming that the part of the system beyond the retina is non-linear, but we must face the possibility that our concepts a t the retinal level may have to be revised to take non-linearities into account.
8
15. Retinal Reflectometry
As pointed out above, one of the methods of studying the quality of the image formed on the retina is to measure and analyze the flux reflected from the retina when the image of a line or border or grating is formed on the retina. Various investigators (WESTHEIMER and CAMPBELL119621; KRAUSKOPF 1119621; FLAMANT [1955]; ROHLER[1962]) have contributed studies of this kind, but one of the more recent and more sucessful of these studies is the one made by CAMPBELL and GUBISCH[1966]. White light was used, and the image of a narrow bar was formed on the retina with artificial pupils of different size. The distributions of flux deduced from light reflected from the retina are shown in Fig. 58. Shown also in each of the diagrams for the different pupil sizes is the theoretical perfect image of a white slit formed by a diffraction-limited eye corrected for chromatic aberration. As pointed out by Campbell and Gubisch, the measured image conforms to the theoretical image when the pupil size is reduced to 1. 5 mm, except a t the bottom of the distribution, where it flares out. This flaring out has been attributed to stray light. Campbell and Gubisch explained that the stray light component is not involved in the psychophysical approach described above because stray light is involved with the Young fringes formed on the retina as well as with the image of a sinusoidal grating formed in the ordinary way. Consequently, when the one modulation transfer
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RETINAL REFLECTOMETRY
function is subtracted from the other, the stray light is eliminated. One can make an analysis of the role played by chromatic aberration by computing the distribution of flux in the image on the basis of
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Angiilw distance (minutes of arc) Fig. 58. Optical line spread functions of the human eye for various pupil diameters. The continuous curves are derived from measurements of light reflected from the retina. The dotted curves are theoretical curves for white light for an eye corrected for chromatic and spherical aberration and limited only by diffraction. (From CAMPBELLand GUBISCH[1966].)
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THE OPTICAL P E R F O R M A N C E O F THE HUMAN E Y E
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diffraction theory on the assumption that the eye is diffraction-limited. The CIE luminosity curve can be used in this analysis, and one does not have to be concerned with the splitting of this curve into red, green and blue components as in the psychophysical method of measuring optical blur. In this way, one can account for about one half of the discrepancy between the theoretical and measured distributions. The remainder of the discrepancy must be attributed to spherical aberration, irregular astigmatism, stray light (FRY[1965c]), and the factors which could make the distribution of flux to which the photoreceptors respond differ from the distribution of flux in the reflected image. From the data in Fig. 58, GUBISCH[1967] has computed Strehl ratios for the eye for different pupil sizes and has found these to range from 0.025 for a 6.6-mm pupil to 0.65 for a 1.5-mm pupil. These ratios permit the eye to be compared to other optical instruments.
Q 16. Aberrations of the Eye Only a limited number of aberrations has to be studied in order to investigate image quality of the eye. As explained above, the eye turns toward whatever has to be seen critically, and we do not have t o pay attention to the quality of the images falling anywhere on the retina other than at the center of the fovea. Consequently, we do not have to be concerned with the variations in radial astigmatism and coma corresponding to the secondary lines of sight. The beam centered on the line of sight is not always radially symmetrical, but once the distribution of flux formed on the retina by this beam has been determined, the procedure does not have to be repeated for other beams. Since the primary line of sight deviates from the optic axis by about 5", it is oblique to each of the refracting surfaces; and the images formed by different wavelengths of light are laterally displaced from each other. Since only one beam is involved, it is more appropriate t o describe this type of aberration in terms of dispersion rather than in terms of chromatic differences in magnification. This aspect of image quality can be manipulated by changing the centration of the pupil and by placing dispersing prisms in front of the eye. The problems of chromatic dispersion have been ignored in this review, but they have been treated elsewhere (FRY[1955]). The problems of chromatic aberration have been handled in this review by collecting data relative to the axial chromatic aberration
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of the eye and using this data to compute the effect of combining images of different wavelength. The major problem which has not yet been covered in this review is that of determining the monochromatic aberrations of the beam corresponding to the primary line of sight. One obvious approach t o this problem is to trace beams of light through different parts of the pupil, which converge at the center of the fovea and then to use this data in reverse to determine the course of the rays transmitted through the exit-pupil from a point in front of the eye. One can then reconstruct the wave front emerging from the exit-pupil and from this compute the distribution of flux on the retina. In order to demonstrate the feasibility of the procedure, the writer [I9551 used IVANOFF’S data [1947] and computed the distribution of fluxin images of monochromatic light out of focus to various degrees. The results obtained have been useful in demonstrating many aspects of the role played by spherical aberration, but as pointed out by WESTHEIMER[1955], Ivanoff’s analysis of his data involves an artifact, and the whole procedure should be repeated using more valid data for spherical aberration. This kind of study should be extended to demonstrate the effects obtained with pupils of different size and with heterochromatic point sources, although experimentally this complication can be avoided by using monochromatic light. The changes in the spherical aberration of the eye with changes in accommodation should also be investigated. Attention needs to be called to the recent work of ARNULFet al. [1956] and of BERNY[1968] on the use of the knife-edge technique for the study and quantitative assessment of the spherical aberration of the eye. This technique supplements the ray tracing techniques used by the earlier investigators (AMESand PROCTOR [1923]; FRY [1949]; IVANOFF [1953]; KOOMEN et al. [1949]). Spectacle lenses have t o be maintained rigidly with respect to the head while the eyes move, and hence the problems of design are somewhat similar to those involved in the design of eyepieces for a roving eye. A contact lens mounted on the cornea could be designed to correct for the spherical aberration of a roving eye. I t also permits the use of spatial filtering. § 17. Motor Adjustments of the Eye
In physiology the words motor and sensory differentiate two major
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functions of a nervous system. In this sense, motor adjustments of the eye under nervous control include the pointing of the eye in a given direction, the focusing of the eye, and the regulation of the size of the pupil. These problems have been reviewed recently by FIORENTINI [1961], WESTHEIMER[1963], ALPERN[1962] and LOWENSTEIN and LOWENFIELD [1962]. All that needs to be said here is that micronystagmoid movements can either blur the retinal image or improve the visibility when they are slow enough. Researches in this field form a large and separate branch of visual science. The subject of the nervous control of accommodation and the art and science of pupillography have similarly been developed into well identified, separate branches of visual science.
Q 18. General Reviews Attention is also called to several reviews of both the sensory and motor mechanisms involved in the perception of fine detail (WESTHEIMER [1965]; LIT [1968]; BOYNTON [1962]; ONLEY 119641; RIGGS [1965]).
Appendix I Although the distance from the second nodal point to the retina is now generally assumed to be about 17 mm, it is 15 mm in the Helmholtz schematic eye. The secondary focal length f‘ is 20 mm and the index of the vitreous (%’) is f (FRY[1959b]). In Figs. 22 and 23, it has been assumed that R‘ coincides with F’, that f’ = 20 mm and that the distance from the second nodal point to the retina is 15 mm. The effect of throwing the eye out of focus can be explained in terms of Fig. 19. In Figs. 22 and 23 the eye without a lens is in focus for an object at R which lies at an infinite distance and is then thrown out of focus by a thin lens at the primary focal point (F). The number of diopters out of focus is the power of the lens, and it follows from the Newtonian formula for conjugate foci that Diopters out of focus = 11/(0.001ff’)= 3.33 II where f , f‘ and II are expressed in mm. __ In eq. (41) a = O’M’, c = O’R’ and (a-c) = 2).
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REFERENCES
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References ALPERN,M., 1962, Movements of the Eyes, in: The Eye, Vol. 3, ed. H. Davson (Academic Press, New York) Chs. 1-8. American Medical Association Council on Industrial Health, 1955, A.M.A. Archives of Industrial Health 12, 439-449. AMESJr., A. and C. A. PROCTOR, 1923, Am. J. Phys. Optics 4, 3-37. ARNULF,A. and 0. DUPUY,1960, C. R. Acad. Sc. 250, 2757-2759. ARNULF,A,, 0. DUPUYand F. FLAMANT, 1956, M6thode objective pour 1’6tude des de’fauts du systeme optique de I’oeil, in: Problems in Contemporary Optics (Instituto Nazionale di Ottica, Arcetri-Firenze) pp. 330-335. BERGER-LHEUREUX-ROBARDEY, S.,1965, Rev. Opt. 44, 294. 1 BERNY,F., 1968, Formation des Images RBtiniens: ddtermination de l’aberration spherique du systeme optique de l’oeil, Doctoral Thesis, University of Paris. BLACKWELL, H. R. and 0. M. BLACKWELL, 1961, Vision Research 1, 62-107. BOYNTON, R. M., 1962, Ann. Rev. Psych. 13, 171-200. BRINDLEY, G. S., 1954, J. Physiol. 124, 400-408. BRYNGDAHL, O., 1966, J. Opt. SOC.Am. 56, 811. BYRAM, G. M., 1944, J. Opt. SOC.Am. 34, 718-724. CAMPBELL, F. W., 1957, Optica Acta 4, 157-164. CAMPBELL,F. W. and D. G. GREEN,1965, J. Physiol. 181, 576-593. CAMPBELL, F. W. and R. W. CUBISCH,1966, J. Physiol. 186, 558-578. CAMPBELL,F. W. and R. W. CUBISCH, 1967, J. Physiol. 192, 345-358. CAMPBELL,F. W. and J . G. ROBSON,1964, J. Opt. SOC.Am. 54, 581. CAMPBELL, F. W. and J, G. ROBSON,1968, J . Physiol. (London) 197, 551. COBB,P. W., 1911, Am. J . Physiol. 29, 76. COBB,P. W., 1915, Am. J. Physiol. 36, 335. Committee on Colorimetry of the Opt. SOC.Am., 1953, The Science of Color (Thomas E. Crowell, New York) pp. 223-228. DEMOTT,D. W., 1959, J . Opt. SOC.Am. 49, 571. DEPALMA, J. J. and E. M. LOWRY, 1962, J . Opt. SOC.Am. 52, 328-335. ENGEL, G. R., 1968, J. Opt. Soc. Am. 58, 1416. ENOCH, J . M., 1963, J . Opt. SOC.Am. 53, 71. ENOCH, J. M. and G. A. FRY,1958, J. Opt. SOC.Am. 48, 899. ENROTH-CUGELL, C. and J. G. ROBSON,1966, J . Physiol. 187, 517. F I O R E N T I N I , A , , 1961, Dynamic Characteristics of Visual Processes, in: Progress in Optics, Vol. 1, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) Ch. 7. FIORENTINI, A,, 1968, Excitatory and Inhibitory Interactions in the Human Eye (to be published in the Trans. of the Intern, Conf. on Visual Science. Indiana University, April 2-4, 1968). FLAMANT, F., 1955, Rev. Opt. 34, 433. FRY,G. A,, 1946, Optometric Weekly 37, 1521. FRY,G. A,, 1949, The 0-Eye-0 15, 8. FRY,G. A , , 1953, Am. J. Optom. 30, 22-37. FRY,G. A,, 1955, Blur of the Retinal Image (The Ohio State University Press. Columbus). FRY,G. A., 1959a, The Relation of Blur and Grain to the Upper Limit of Useful
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Magnification, Report (RADC-TN-59-267) from the Ohio State University ltesearch Foundation to the Rome Air Development Center under Contract No. AF30(602)-1580. FRY,G. A., l959b, The Image-Forming Mechanism of the Eye, in: Handbook of Physiology, Vol. 1 (American Physiological Society, Washington, D.C.) pp. 647-670. FRY,G. A , , 1961, J . Opt. Sac. Am. 51, 560-563. FRY, G.A , , 1965a, The Eye and Vision, in: Applied Optics and Optical Engineering, Vol. 2, ed. I<.Kingslake (Academic Press, New York) pp. 1-76. FRY,G. A , , 1965b, Am. J . Optom. 42 , 271. FRY,G. A , , 1965c, J . Opt. Soc. Am. 55, 333-335. FRY, G. A , 1965d, J . Opt. Soc. Am. 5 5 , 108. FRY,G. A . , 1968a, J . Opt. Soc. Am, 58, 1415. FRY, G.A , , 1969a, Am. J . Optom. 46, 319-338. FRY,G. A., 1969b, J. Op. Soc. Am. 59, 610-617. FRY,G. A. and P. W. Conu, 1935, Trans. Am. Acad. Ophth. and Otolaryn., pp. 423-428. GIVENS,M. P., 1966, Notes on the Use of the Sine Wave Response of Optical Systems, in: Notes on Tmage Evaluation, ed. R. E. Hopkins (Institute of Optics, University of Rochester, New York) pp. 27-42. GREEN,D. G., 1968, J. Physiol. 196, 415-429. I<. W., 1967, J . Opt. Soc. Am. 57, 407-415. GUBISCH, HARTRIDGE, H., 1947, Phil. Trans. Roy. Soc. London B 2 3 2 , 519-671. HELMHOLTZ, H. VON, 1924, Helmholtz’s Treatise on Physiological Optics, Vol. 2, cd. J. P. C. Southall (Am. ed., Optical Society of America) pp. 34-35. HOPKINS, H. H., 1962, Proc. Phys. Soc. London 79, 889-919. IVANOFF, A , , 1947, Rev. Opt. Theor. Instrum. 26, 145-171. IVANOFF, A , , 1953, Les aberrations de l’oeil: leur rGle dans l’accommodation (editions de la Rev. Opt. Theor. Instrum., Paris). KOOMEN, M. T., R. T o u s ~ uand R. SCHOLNIK, 1949, J. Opt. SOC.Am. 39, 370. J . , 1962, J . Opt. Soc. Am. 5 2 , 1046-1050. KRAUSKOPF, LAMBERTS, R. L., 1958, J. Opt. SOC.Am. 48, 490. LEGRAND,Y., 1935, C. I<. Acad. Sc. 2 0 0 , 490. LIZGRAND,Y., 1936, Rdunions de 1’Institute d’Optique, 7 t h Series, pp. 6-11, LEVY,J . D. and G. A. FRY,1968, Interaction between Borders in a Square-Wave Grating (to be published). LIT,A., 1968, Ann. Rev. Psych. 19, 27-54. LOWENSTEIN, 0.and 1. E. LOEWENFELD, 1962, The Pupil, in: The Eye, Vol. 3 , ecl. H. Davson (Academic Press, New York) Ch. 9. LOWRY, E . M. and J. 1. DEPALMA, 1961, J . Opt. Soc. Am. 5 1 , 740-746. E. Mi., 1964, J . Opt. Soc. Am. 54, 915. MARCHAND, MARCHAND, E. W., 1965, J. Opt. Soc. Am. 55, 352-354. MARIMONT, R.B., 1963, J . Opt. SOC.Am. 53, 400-401. MENZEL,E., 1959, Die Naturwissenschaften 4 6 , 316-317. NACHMIAS, J., 1967, J. Opt. Soc. Am. 57, 421. NXHMIAS,J., 1968, J . Opt. Soc. Am. 58, 9. O’RIIIEN,B. ?nd N. D. MILLER,1952, J. Opt. SOC.Am. 4 2 , 289. OGLE,li. h-.,1951, J . Opt. Soc. Am. 41, 517.
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I11 LIGHT BEATING SPECTROSCOPY BY
H. 2. CUMMINS and H. L. S W I N N E Y Departmertt of Physics, T h e J o h m Hopkilzs Urtiversity, Baltimore, Marylalzd 21218, U S A
CONTENTS PAGE
S
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INTRODUCTION . . . . . . . . . . . . . . . . .
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THE T H E O R Y O F LIGHT BEATING SPECTROSCOPY . . . . . . . . . . . . . . . . . . . . . .
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4 9
3 . LIGHT SCATTERING T H E O R Y .
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4 . COHERENCE AND SIGNAL TO NOISE CONSIDER-
ATIONS . . . . . . . . . . . . . . . . . . . . .
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3
5.
APPARATUS AND PROCEDURE . . . . . . . . .
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$
6.
R E V I E W O F RAYLEIGH L I N E W I D T H EXPERIMENTS . . . . . . . . . . . . . . . . . . . . .
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3
7 . CONCLUSIONS .
. . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . .
196 197
§ 1. Introduction 1 . 1 , HISTORICAL REVIEW
During the past few years a new kind of optical spectroscopy has come into existence which permits the measurement of optical lineshapes and frequency shifts with resolution on the order of several Hz. The new “light beating” spectroscopy has been successfully applied to the study of a variety of physical phenomena for which the best resolution available with traditional dispersive spectroscopic apparatus (w 10 MHz) was completely inadequate. The phenomenon of light beating (or photoelectric mixing), which is a direct analog of the familiar mixing of a.c. electrical signals in nonlinear circuit elements, was first demonstrated in the classic experiment of Forrester, Gudmundsen and Johnson in 1955 (FORRESTER et al. [1955]). I n their experiment, a l98Hg lamp was placed in a mag3300 Gauss, and the Zeeman-split 5461 A light illuminetic field of nated the photocathode of a microwave phototube. The two strong CT components of the light, for which the Zeeman splitting was 1O1O Hz, beat with each other in the square-law photoelectric emission process, and the photocurrent was found to have an a.c. component at the Zeeman difference frequency. Although this experiment clearly demonstrated the light beating phenomenon, the measurement was extremely difficult primarily because of the low intensity andlarge linewidth of the light source, and the technique did not appear to hold great promise as a spectroscopic tool. A dramatic change occurred with the advent of lasers. When Javan, Bennett and Herriott first obtained 1 p oscillation in a He-Ne laser, they directed the output onto the cathode of a photomultiplier and examined the output current with a standard r.f. spectrum analyzer which readily exhibited the effect of intermode beats (JAVAN et al. [196l]). Similarly when the output of two independent lasers simultaneously illuminated a single photomultiplier, a beat note which was detected at the difference frequency was used to determine the stability
-
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of the lasers (JAVAN et al. [1962]). Whereas Forrester’s observation of a beat signal with the mercury source had constituted an experimental tour de force, the high intensity and narrow linewidth of laser sources quickly converted the light beating phenomenon into a standard laboratory technique. This aspect of light beating spectroscopy, the mixing of two narrowband optical components which produces an a.c. component in the photocurrent at the optical difference frequency, has been extensively employed in the analysis of the mode structure of lasers, a field which we will not further discuss. I t has also been employed in several experiments in which laser light is scattered in some process which produces a well-defined shift in the optical frsquency, and scattered light is then combined with unshifted lasx light at a photocathode. We consider three illustrative examples. (1) When light is diffracted by ultrasonic waves in a fluid (the D&ye-Sear; effect), the diffraction maxima contain components whose frequency is shifted from that of the incident light by integral multiples of the ultrasonic frequency. The shifts were measured by mixing the diffracted light with the incident laser light (CUMMINSet al. [1963]). (2) Light whichis scattered from a fluid has spectral components (the Brillouin components) which are shifted in frequency from the incident light by inelastic photon-phonon scattering. LASTOVKA and BENEDEK[1966a] mixed the Brillouin components in laser light which had been scattered from toluene, with unshifted laser light on a high-frequency photodiode, and observed a beat signal. (3) Light which is scattered by small impurities in a moving fluid will be Doppler shifted by an amount proportional to the local fluid velocity. Mixing of the scattered light from different regions of a fluid (in laminar flaw) with unshifted laser light permitted YEH and CUMMINS[1964] to map out the local fluid flow pattern. I n all of these experiments the quantity measured was the difference between the center frequencies of two optical lines. However, light beating spectroscopy is also capable of analyzing distributed optical spectra, and it is this application with which we will be primarily concerned. It was first pointed out by ALKEMADE [1959] that a single optical spectral line falling on a phototube would produce a photocurrent whose low-frequency noise spectrum (which could be measured with an r.f. spectrum analyzer) would provide information on the shape, and particularly the width, of the optical [1961] in a line. The same conclusion was reached by FORRESTER paper entitled “Photoelectric Mixing as a Spectroscopic Tool’’. For-
111.
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INTRODUCTION
137
rester considered a distributed spectral line as a series of “slices”, and treated the beating between all possible pairs of slices as a simple extension of the two-component case. With this approach (which we will discuss in detail in $2), Forrester deduced the photocurrent spectra which would be produced by several different types of optical spectra. This application of light beating spectroscopy was also utilized initially in the study of the output of lasers, an application which had been foreseen by FORRESTER [1961]. Unfortunately, much of the early experimental work was performed without proper consideration of the unusual statistical properties of laser radiation, and the interpretation of the data was frequently incorrect. (This point will be discussed further in $1.2.) High resolution light beating spectroscopy has proved to be particularly well suited to the study of quasielastic light scattering phenomena in which laser light is scattered from various nonpropagating thermodynamic fluctuations. Two closely related experimental approaches have developed which are now generally designated as heterodyne andhomodyne detection. (Both heterodyne and homodynedetection were discussed by FORRESTER 119611, who used the terms “superheterodyne” and “low-level” detection, respectively.) In the heterodynz approach, the scattered light whose spectrum is to be measured is mixed with some of the unscattered light (which acts as a local oscillator) on the photodetector. This technique, first suggested by TOWNES [1961], was utilized by CUMMINSet al. [1964] to study the broadening due to particle diffusion of laser light scattered from a dilute solution of polystyrene latex spheres, by ALPERT et al. [1965] to study the concentration fluctuations which cause critical opalescence in an aniline-cyclohexane mixture near the critical mixing temperature, by ALPERT et al. [1966] t o study critical point fluctuations in carbon dioxide, and by LASTOVKA and BEKEDEK[1966b] to study the spectrum of light scattered from a normal liquid (toluene). I n the homodyne detection scheme no local oscillator is used; the scattered light alone illuminates the photodetector. This technique was first utilized by FORD and BENEDEK[1965] in their investigation of light scattering in SF, near the liquid-vapor critical point. The homodyne scheme has since been widely employed in the analysis of critical scattering in pure fluids and binary mixtures. Both homodyne and heterodyne detection have also been utilized in an extension of diffusion broadening measurements to a variety of biologically in-
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teresting macromolecules (cf. DUBINet al. [1967], BERGEet al. [1967], CUMMINSet al. [1968]). The application of light beating spectroscopy to precision measurements of the spectrum of light quasielastically scattered from dilute solutions of macromolecules, or from pure fluids and binary liquid mixtures undergoing critical point fluctuations, will be a primary concern of this paper, and we will return to a review of the experimental literature in this area in $6. Several other important applications of light beating spectroscopy (which we will not further discuss) have been considered in a recent review article by BENEDEK[1968b] to which the interested reader should refer. Additional useful background information can be found in Professor Benedek’s 1966 Brandeis Summer Institute Lectures (BENEDEK [1968a]), and in the 1967 Varenna Lectures of CUMMINS[ 19681. 1.2. STATISTICS AND SPECTRA
Although we will be primarily concerned with the determination of the spectra of optical signals, we will frequently find that the relation between the optical spectrum and the photoelectric current spectrum depends crucially on the statistical properties of the optical signal. Before the advent of lasers, all available optical sources were chaotic and the intensity I ( t ) of a single mode of the radiation field when excited by a narrow band stationary light source could be described by the statistical probability distribution appropriate to a random Gaussian complex process:
$ ( I ) d1 = ( I ) + exp ( - I / ( I ) ) dI.
(1.1)
When a quasi-monochromatic optical signal with this intensity distribution interacts with a photoelectric surface, the probability of detecting n photoelectrons in a time T (where T << llAv, the reciprocal bandwidth of the optical spectral line) can be shown to be (cf. MANDEL [1963], KLAUDER and SUDARSHAN [1968])
which is the Bose-Einstein distribution. For a (hypothetical) light source which produces an optical field of perfectly constant intensity,
the associated photoelectron distribution would be
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INTRODUCTION
$(a,
T )=
(n)" exp (-(a)),
139
(1.4)
which is a Poisson distribution. (The general problem of deducing
p (I)from p (n, T ) has been discussed by WOLFand MEHTA [1964] and BBDARD[1967b].) With the constant intensity source, the variance or mean square fluctuation in the photoelectron emission rate is
which produces the shot noise in the photoelectron current spectrum ( ( n ) is the mean photoelectron emission rate). With the Gaussian source described in eq. ( l . l ) ,the variance in the photoelectron emission rate is (('n)')Gaussian
= (n> +(a>',
(1.6)
which contains both the shot noise term ( n ) , and a second "excess noise" term ( n ) z . It is through the electronic spectral analysis of this excess noiso in the photocurrent spectrum that the optical spectrum of the incident light can be determined, as ALKEMADE[1959] first pointed out. Fortunately, there is a simple relation between the optical spectrum and the photoelectron current spectrum if the optical field is a Gaussian random process, which is the case for the scattering experiments with which we will be primarily concerned. When one considers radiation from a laser oscillating in one or a few modes, however, one is dealing with an optical field which is not a Gaussian process. GOLAY [196l] first suggested that the probability distribution for the intensity of a laser would differ drastically from that of a chaotic light source. Thus the fluctuations in the photoelectric current produced by laser radiation must also differ drastically from those in the current produced by a thermal source with the same average intensity (MANDEL and WOLF [1963]). In fact, for a single mode laser operating far above threshold, the intensity distribution approaches the distribution of eq. (1.3), and the photoelectron distribution becomes essentially Poisson, as MANDEL[1964] has emphasized. I n this limit, the "excess noise" disappears altogether and one is left with just the shot noise spectrum characteristic of eqs. (1.4), (1.5) (cf. MARTIENSSENand SPILLER[1966]). The intensity distribution for a real single mode laser operating well above threshold can be approximated by adding a small Gaussian noise component onto a large coherent component. The spectrum of
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the excess noise then determines the spectrum of the small noise part of the laser radiation, and is unrelated to the main coherent component (FREED and HAUS[1966], HAUS[1968]). Because the non Gaussian statistical properties of laser radiation were not recognized in some of the early experiments, the spectrum of the excess photoelectron noise was incorrectly assumed to be related to the complete spectrum of the laser radiation rather than just to the small noise modulation component, which led to a gross underestimate of the monochromaticity of lasers. The problem arose from the apparently self-evident (but nevertheless erroneous) assertion that since photons obey Bose-Einstein statistics, they will always be distributed according to the Bose-Einstein distribution of eq. (1.2) (cf. BOLWIJN et al. [1963]). The Bose-Einstein distribution, however, only describes bosons in thermal equilibrium, and the laser is a distinctly nonthermal source. The statistics of the laser radiation exhibit the statistical properties of the generating process and not those of a thermal ensemble. Finally, we point out that the determination of the statistics of an optical field [in the sense of $ ( I ) or $(n, T)] does not, in itself, provide any knowledge about the spectrum. There are, however, several interesting techniques by which the statistical properties of the photoelectrons may be used t o determine the optical spectrum, and these techniques provide a coniplementary method t o the photoelectron current spectrum measurements which we will consider. We mention here three methods by which the optical spectrum may be deduced from analysis of the photoelectron statistics. (For more extensive reviews see WOLF [1966], ARECCHI[1968].) (1) The photoelectron distribution function p ( n , T ) for a Gaussian field is given by eq. (1.2) only if the sampling time T is << 1jAv. If the distribution is measured for different values of T, then the dependence of $(n, T ) on T can be used to determine Av (cf. JAKEMAN and PIKE[1968], JAKEMAN et al. [1968]). (2) In addition to the one time probability distribution $ (n, T), one can measure higher order joint probability distributions from which the spectrum can be determined (cf. ARECCHI et al. [1966], BEDARD[1967a]). (3) If the lineshape is known, the width can be determined from delayed photoelectron coincidence measurements (cf. MORGAN and MANDEL [1966], PHILLIPS et al. [ 19671).
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T H E THEORY OF LIGHT BEATING SPECTROSCOPY
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Q 2. The Theory of Light Beating Spectroscopy In this section we will discuss the theoretical connection between the optical spectrum of light incident on a photoelectric detector and the spectrum of the photoelectric current, from several different points of view. In order to simplify the discussion we will assume that the optical field is spatially coherent over the surface of the photodetector. The effects of imperfect spatial coherence will be discussed in $4 where we will consider the signal to noise problem in detail. Before beginning our discussion of the classical theory, we comment on the relation between the classical and quantum mechanical theories of optical coherence, a topic which has been the subject of considerable debate in recent years. When an optical field interacts with the electrons in a photodetector, the interaction term in the Hamiltonian which gives rise to first order transitions is
-z(eimc)A - p i . i
(2.1)
The probability of detecting photoelectron emission will be given in the usual first order perturbation theory approximation by a term proportional to the square of the matrix elements of (2.1) between initial and final states of the system, suitably summed over final states. The vector potential of the optical field A ( r , t ) can be discussed from two different points of view. On the one hand, A (r , t ) can be considered as a classical function and the perturbation theory performed in the semiclassical radiation theory limit. Alternatively, A (r, t ) can be considered as an operator, and the radiation field treated as a quantum mechanical system. The first choice (semiclassical) leads to a description of optical fields in terms of the classical coherence functions of Wolf (cf. MANDELand WOLF [1965]). The second choice (quantum mechanical) leads to a description of optical fields in terms of the quantum mechanical correlation functions of GLAUBER[1963a]. The quantum mechanical description provides a more fundamental approach to the problem since ultimately the radiation field must be considered as a quantum mechanical system. In general the two descriptions may lead to different conclusions. However, for a large and important class of optical fields the two descriptions lead to identical conclusions. The fields which we will consider are, in fact, of this class so that the classical description leads to correct predictions. We emphasize, however, that the correctness of the classical formulation which we present in this
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[III,
3: 2
section cannot be asserted a priori, but must be eventually demonstrated for the fields under consideration by comparison with the quantum theory. Arguments based on the correspondence principle are not relevant since we will usually consider fields for which the mean number of photons is small. 2.1. CLASSICAL C O H E R E N C E THEORY
We consider the (classical) optical field at the photocathode to be specified by a real vector function (MANDELand WOLF [1965])
1
+m
E(') (Y,t ) =
V(Y, w )
exp ( - i d ) dw,
(2.2a)
u --w
where E(') (Y,t ) is proportional to the vector potential (or the electric field), and V ( Y , w ) = v*(Y, - 0 ) since E(F)is real. We then define the complex analytic signal (2.2b) The probability per unit time of photoelectron emission from a photocathode illuminated by the field ( 2 . 2 ) is then found to be (cf. MANDELet al. [1964])
W'" ( t ) = 0 E* (1) E ( t ) ,
(2.3)
where E* (t) E ( t ) is defined as the instantaneous intensity I ( t ) and 0 is a suitably defined quantum efficiency. (We have suppressed the Y dependence in view of the assumed spatial coherence.) The photoelectric current i ( t ) = eW(l)( t ) = eoE*(t) E ( t ) , while the joint probability that one photoelectron will be emitted at time t (per unit time) and anotlzer at time t +t (per unit time) is
W ( ' ) ( t , t + ~=) +E*(t) E ( t )E*(t+z) E ( t + t ) .
(2.4)
We also will need the averages of i ( t ) and W ( 2(t, ) t + z ) , which for stationary fields are:
( i ( t ) ) = e(W(l) (t)> = ea(E* ( t ) E ( t ) ) = ea(I>, (W ( 2'( t,t + t ) ) = 02(E*(t) E ( t ) E*(t+z) E ( t + z ) ) = 0 2 ( 1 ) 2 g ( 2 )
(2.5) (t),
where (2.6)
111,
§ 21
143
THE THEORY O F LIGHT BEATING SPECTROSCOPY
The power spectrum P,(w) of the photocurrent is given by the Wiener-Khintchine theorem: 1
I-,
PZ(0)= 2n
-
eiwrci (t)dt,
(2.7)
where the current autocorrelation function is
C i(T) = ( i ( t ) i ( t + t ) )
= ez(W(l)(t) W ( l )( t + z ) ) ,
(2.8)
and C i(t)= Cl (-t). For distributed spectra (spectra with nonzero bandwidth) centered at cu = 0, P ( w ) in eq. (2.7) is multiplied by 2 for w > 0, and Pi(cu) is set equal to zero for cu < 0. Now the photocurrent i ( t ) actually consists of a series of discrete pulses which we will assume to be infinitely narrow. Therefore C i(t) has two distinct contributions: If the electrons at t and t +z are distinct,
(W(1)(t) W ( l ' ( t + t ) )= ( W y t , t + t ) )
= .2(1)2g(2)
(z),
while if the same electron occurs at t and t+z,
(W(1) (t) W ( l ) ( t + z ) )= (W (1)(t ) ) 6(z) = . ( 1 ) S ( t ) . Therefore,
C i(t)= e2o ( I ) S(T) + e 2 d
g @ )(t)
(2.9) =
e ( i ) S(T) +(ij2g(2)(z).
We consider below two different detection schemes which were analyzed from a different point of view by FORRESTER [1961]. 2.2. HOMODYNE OR SELF-BEAT DETECTION
Suppose that the optical field is a narrow band Gaussian random process. (For monochromatic light scattered by a dilute solution of scatterers, for example, the Gaussian statistics of the scattered field follows from the central limit theorem.) The field is characterized by an autocorrelation function C,(Z)
=
( E * ( t )E ( t + t ) )
=
(I)@')(t).
For random Gaussian fields, the second-order correlation function g(2)(T) is related t o g(l)(t)by (MANDEL [1963], REED [1962]): g(2)(t)= 1 +p (412.
(2.10)
144
LIGHT BEATIN G SPECTROSCOPY
[III,
s
2
Whence
C i(7)= e(i) d(z) +(i)z(1+Jg(l)( t ) J 2 ) .
(2.11)
In the experiments which we will be discussing, the normalized correlation function g(l)(T) will usually have the form gC1) (7)= e-iOoT e-7171.
(2.12)
The optical spectrum of a field described by eq. (2.12) is
which is a Lorentzian of half-width at half-maximum Aw, (optical) = y , centered at w = coo, with total intensity (I). The photocurrent spectrum associated with this field is found from eqs. ( 2 . 7 ) , (2.11) and (2.12): 1 0 3
Pi(@)= gs_03 eioT{e(i) d(z) + (i)2 + ( i ) 2 e-2YIT1)dT (2.14a) 1 - 4;)+(;>2d 2n
2Yb ( w ) +(02 w 2 +(2y)2'
Since P,(w) is symmetric about w = 0, we combine the positive and negative frequency parts to obtain a spectrum for positive frequencies only : (2.14b)
P;(w)w2 8 ' ( w ) , is the d.c. component. The third term is a Lorentzian of half-width
111,
5 21
T H E THEORY O F LIGHT BEATING SPECTROSCOPY
145
Aw+ (photocurrent) = 2y and total power ( i ) 2 centered at w = 0. The third component is the light-beating spectrum. The original optical spectrum of half-width Am+ (optical) = y has produced a Lorentzian photocurrent spectrum of twice the optical width centered a t w = 0. Thus, measurement of the low-frequency photocurrent spectrum (from w = 0 out to w 20 y ) permits an accurate determination of the width of the original optical line. The homodyne or self-beat technique was first employed by FORD and BENEDEK[1965] and is today the most frequently utilized form of light beating spectroscopy. It should be noted that eq. (2.10) which allowed the current autocorrelation function to be expressed in terms of g(l)(t)is only valid for fields with Gaussian statistics. I n general, for non Gaussian fields, eq. (2.10) is not correct and there is no simple connection between the optical spectrum I ( w ) and the photoelectric spectrum Pi ( w ). As an example, consider an amplitude stabilized field with random phase modulation, which is a reasonable model for a single mode laser operating well above threshold (cf., for example, ARECCHI[1965]) :
E ( t ) = E , exp { 4 ~ , t + + ( t ) l } > where $(t) is the stochastic phase modulation. For this field
The optical spectrum which is the Fourier transform of g ( l ) ( t )will be centered at w,, but will be broadened due to the decay of the phase autocorrelation (exp {id(t)}exp { -i+ (t +z)}) with increasing z.
(2.16)
so that from eq. (2.9), the photocurrent autocorrelation function is C i(t)= e(i) d(z) +(Q2, and the photocurrent spectrum is
e(i> P t ( w ) = __ n
+(i)2
d’(w).
(2.17)
Comparison of eqs. (2.17) and (2.14) shows that the source we are discussing here produces the same d.c. and shot noise terms as the Gaussian random field, but that the “light beating” term is completely missing ! Thus the photocurrent spectrum of an amplitude stabilized
146
LIGHT BEATING
SPECTROSCOPY
[III,
5
%
single mode laser exhibits no light beats even though the optical spectrum has a finite linewidth due to the random phase modulation. This result, which can be attributed to the lack of phase sensitivity in the photoemission process, is equivalent to the assertion in 5 1.2 that there is no “excess noise” in the photoelectric emission current produced by a constant intensity source. Thus the total power (neglecting shot noise) in the spectrum produced by a Gaussian field is 2 ( i ) 2 , while for the “coherent” source, the power is (Q2. This result also follows from the requirement that P ( o ) d o = C, = (i2>, and ( i 2 ) = (Q2 for a constant intensity source, while for a Gaussian source ( i 2 ) = 2(Q2.
Jr
2.3. HETERODYNE DETECTION
In 2.2, we considered the spectrum of the photocurrent with the detector illuminated only by the field under study. As an alternative procedure, the detector can be illuminated simultaneously by this field and by a coherent local oscillator signal. This heterodyne detection scheme was utilized in the original polystyrene diffusion broadening experiment of Cummins, Knable and Yeh (CUMMINSet al. [1964]) and was extended to the study of critical opalescence by ALPERTet al. [1965]. (Methods for producing local oscillator signals will be discussed in 5 5 . ) Let the field under study be E , and the local oscillator field be ELO( t)= E!, exp {-ioLOt}. The photocurrent produced by either Es or EL, in the absence of the other is i, or i,,, where (is(t)) (iL0
=
eo(E:
(4 E S ( t ) ) >
( 4 ) = eo(E$ (4 E L ,
( t ) > = e4EI,ol2,
since is constant. The current autocorrelation function C , ( T ) = e2od(z) ( E * ( t ) E ( t ) ) + e 2 0 2 ( E * ( t ) E ( t ) E*(t+t) E ( t + z ) ) simplifies considerably if the local oscillator field is much stronger than the scattered field so that I,,>> (Is). When ( E * (t) E ( t ) E* (t+r) x E ( t + t ) ) is expanded using E ( t ) = E , ( t ) +E$ exp (-icuLOt}, the result contains 16 terms of which ten are zero, and three are time independent terms whose sum is I f o +21L, (I,). The remaining three terms give ILo{rexP (ioLO.)l
(E:
(t) Es
(t+.)) +[exp (-ioLo.)l ( E s (t-t.))} +(E: (W,( t ) E: (t +.) Es (t +.)>.
111,
I 21
147
THE T H E O R Y O F L I G H T BEATING SPECTROSCOPY
If I,, >> ( I s ) ,we can neglect the last term in the preceding line (which is just the homodyne spectrum discussed earlier) and also keep only I,, of the d.c. terms. Thus:
C i( T )
= e2aI,,6(t)
+e2a2&
+e2a21,,
+
( I s ) {[exp (iwLoz)]gil) (t) [exp (-i%o
=
e i,,d(t)
+& +iLo(is)
T)lgP*(.I>
+
{[exp (iwL0z)]gF)( T ) [exp (-iwLot)]gA1)* ( T ) } .
(2.18)
The photocurrent spectrum produced by the combined signal and local oscillator field is again found from eq. (2.18) via the WienerKhintchine theorem, eq. ( 2 . 7 ):
+[exp ( - i ~ ~ ) ~ , t ) ] g ~ ~ ) * ((2.19) ~))d~. Again the photocurrent spectrum consists of three terms. The first is the shot noise, the second is the d.c., and the third term gives the heterodyne light beating spectrum. Notice that in this case the light beating spectrum depends on gJ1) rather than gb2), and is therefore determined uniquely by the spectrum of the signal field independent of
the statistics.
Characteristically gil) (T) = [exp (-iw0z)] (2.12)), whence
[exp
(-yltl)]
(as in eq.
(If wLo = w o , the last term should be doubled for w 2 0, and set = 0 for w < 0.) Note that the heterodyne light beating spectrum is a Lorentzian identical in shape to the optical spectrum but centered a t w = lwo-wLoI.
This characteristic is one major advantage of the heterodyne detection scheme. The light beating spectrum is an exact replica of the optical spectrum shifted t o a center frequency jwo-wLoI, and it does not depend on the statistics of the field. Thus, for example, two single mode lasers can produce a heterodyne beat signal even though, as we have seen, neither will produce a homodyne signal. 2.4. FORRESTER’S APPROACH
A useful heuristic discussion of the light beating effect was given by
148
LIGHT BEATING SPECTROSCOPY
[III,
§ 2
FORRESTER [1961]. Following Forrester, we consider an optical source of spectral density I @ ) which illuminates a photodetector. We will assume that the optical field is spatially coherent over the illuminated portion of the photocathode, and that the detector response is independent of frequency. Divide the frequency range into intervals of width Ag, as shown in Fig. 1. Then intensity in the mth frequency interval is given by I(flm)ag,and the corresponding electric field can be written
4
4 l
A N G U L A R FREQUENCY
p
Fig. 1. Distributed optical spectrum I@).
where the different phase factors +m are assumed to be random. (This assumption is equivalent t o asserting that the field is a Gaussian random process.) The photocurrent is then i = alE12, where a is the quantum efficiency, and E = cE==lE,,n. Hence the photocurrent is given by
' ( t ) = GAP 2 ['(Pm)'@n)i+
exp {i(~m-B,)t+i(+nL-~n)}.
(2.22)
m, n
We can write the sum over m and n as three separate sums:
c c + L: + 2.
m,n
=
m i n m=n
(2.23)
m i n
The m = n sum is the d.c. photocurrent, n M
(2.24) which we now drop since we are interested in the spectrum of the photocurrent. If we reverse the roles of m and x in the third sum in eq. (2.23) and then combine the first and third sums, we have
111,
s
21
THE THEORY O F LIGHT BEATING SPECTROSCOPY
;(4
=
243
2 [ W m ) W n ) I +cos [(Bm-Bn)t+(+m-+n)l.
149
(2.25)
m>n
Now let b = m--.n, w = /3,-B,, and y n = c # ~ + ~ - +Then, ~ . dropping the sum over 6, we have the component of the current at frequency w (in bandwidth AB) : OD
; ( w >t ) =
2cAg
C [I(Bn+W) n=l
I(Bn)I+
cos (wt+yn).
(2.26)
Since the phase factors y n are assumed to be random, the time average of the current squared is the sum of the squares of the terms in eq. ( 2 . 2 6 ) . Therefore, since (cosz (wt+y,)) = 4,the power P ( m , A/?) of the photocurrent in bandwidth Ag at frequency w is 03
P ( w , AB)
=
2 W V
2 I ( P n + w ) Wni.
(2.27)
.n=l
The experimentally determined quantity is the power spectral density of the photocurrent, which is obtained by dividing eq. (2.27) by the bandwidth Ag :
Pi (w> = 2a2 2 I ( P , + w ) I(@,)AB,
(2.28)
n
where P i ( @ )is the photocurrent power spectrum neglecting d.c. and shot noise components. In the limit A/? i 0, eq. (2.28) becomes (2.29) For a Lorentzian line of half-width y centered a t w o , we have (see eq. (2.13)) (2.30)
Substituting (2.30) in (2.29) and evaluating the integral, we obtain
which is just the light beating term in eq. (2.14). The convolution equation derived by Forrester (eq. 2.29) was also utilized by ALKEMADE[ 19591. Through the convolution theorem for Fourier transforms Forrester's result can be shown to be equivalent to the result of the preceding section,
150
LIGH1 BEATING SPECTROSCOPY
since I @ ) and ( E * ( t ) E ( ~ + T )are ) a Fourier transform pair (MANDEL [1963] p. 200). The convolution equation (eq. (2.29)) is often quite convenient for finding the light-beating component of the photocurrent when the optical spectrum is known. Three examples were considered by Forrester: Optical spectrum 1 ( B )
Light-beating spectrum P: (0)
A: Rectangular
R: Gaussian
C: Lorentzian
The rectangular spectrum (A), for example, is much easier to evaluate through the convolution equation than through the WienerKhintchine theorem, although for the Lorentzian (C), the converse is true. Note that in each case the total power in the light beating spectrum is equal to the d.c. power (i)2, a result which must always hold for fields with Gaussian statistics regardless of the actual spectrum. 2.5. QUANTUM THEORY
The quantum theory of optical coherence has been developed during the past five years by GLAUBER[1963a, 1963b, 1964, 1965, 1966, 19681. The starting point of the quantum theoretical treatment is the photoelectric interaction in the detection process (eq. (2.1)) but with the field quantities treated as operators rather than as classical variables.
111,
5
21
T H E THEORY O F LIGHT BEATING SPECTROSCOPY
151
GLAUBER[1963] separated the field operator E(r, t ) into positive and negative frequency parts which are photon annihilation and creation operators, and showed that the probability per unit time that a photon is absorbed at a point r a n d time t by an "ideal" photodetector is proportional to the intensity
I ( r , t ) = (ylE(-) ,.( t ) E'+' (r, t)ly),
(2.31)
where Iy) is the initial state of the radiation field. Similarly, the rate per unit (time). of detecting n-fold delayed counting coincidences is proportional to
W("'(Yltl. . . r n t n )= = ( y ] E ( - )(r1tl) .. . E ( - ) ( r n t n )E'+)(rntn). . . E(+)(rltl)1 ~ ) .
(2.32)
For the more general case where the field may not be in a pure state ly), the density operator p is used to define a series of quantum-me-
chanical correlation functions:
G1 (rt, r't') = Tr{pE(-) (rt) E(+)(r't')} (2.33)
G(")(r1tl . . . r 2 n t 2 n= ) = Tr{pE(-)(rltl) . . . E(--)(rntn) E(+)(Y~+ . .~. E~ (~++ )~()Y ~ ~ ~ ~ ~ ~ ) GLAUBER [19661 has specifically considered the problem of evaluating the photocurrent autocorrelation function for an N-atom model photodetector. If N is sufficiently large,
( i ( t )i ( t + z ) )
= e20d(z)G'l) (0) +e202G(2) (z),
(2.34)
where o is again the quantum efficiency of the detector and is assumed to be constant over the bandwidth of the signal. For the case of fields with Gaussian statistics, Gt2) (z) = G(l)(0)(1+Ig(l) (r)12)(GLAUBER [1965] p. 150), while oG(l) ( 0 ) is the average photoemission rate, whence
( i ( t )i ( t + z ) )
=
e(i)d(z)+(i)2(1+Igc1'
(z)12),
(2.35)
where g'l) ( r )is the normalized form of G(l)(z): gel) ( T ) = G(1)(z) /G(') (0).
(2.36)
The final step in evaluating (2.35) for a specific Gaussian field requires a relation between g(l)(z) and the optical spectrum. GLAUBER ([I9651
152
LIGHT BEATING SPECTROSCOPY
[III,
$ 2
(p. 168)showed thatgcl) (T)is just theFourier transformof thenormalized (T) = optical spectrum. For a Lorentzian spectrum of half-width y , l+exp ( - 2 y l t l ) , which together with (2.35) recovers our classical result (eq. (2.14)).Thus, for narrow band Gaussian random fields, the quantum theory and the semiclassical theory lead t o identical photocurrent spectra. We now consider the relation between the quantum theory and the semiclassical theory starting with eq. (2.34), which gives the current autocorrelation function (and thus the photocurrent spectrum) in terms of the two quantum mechanical correlation functions G(I)(0) and G(,) (T). For each mode of the radiation field, a convenient set of basis states are the coherent states \ak)which are related to the number states In) by
>.I
=
exp
(-a142)
1+)!.(
a”
n
. ) .1
(2.37)
The la) states are readily shown to be eigenstates of the annihilation operator:
atlat) = a,l%>>
(2.38)
and also,
(%la; = 4
(%I.
GLAUBER[ 1963bl discusses various representations of the density operator p using the coherent states la) as basis states. (See also KLAUDER and SUDARSHAN [1965].) A particularly important class of fields exists for which the density operator can be written in the “P-representation” which for a single mode has the form: (2.39) where P (R) is a “quasi-probability” function. Note, however, that P ( a ) is not restricted t o positive values. (For narrow band classical fields E ( r , t ) == A exp {-i(wot-ko r ) } , the (complex) amplitude which function A is described by a probability function W , is always positive.) For Gaussian fields, the QM quasiprobability function is (2.40a) while the classical probability distribution is
111,
5
51
THE THEORY O F LIGHT BEATING SPECTROSCOPY
163
(2.40b) Thus for this case P ( K )and W , ( A ) are formally identical and lead to the same photocurrent spectra, as we have seen. Now consider the coherent state. Classically, E (r,t ) = A , exp {-i(q,t-k, * r ) ) ,so
W , ( A ) = 6(2)(A-Ao).
(2.4la)
Since the intensity E * ( r , t ) E ( r , t ) of this field is constant, the photocurrent autocorrelation function is given by
( i ( t ) i(t+z)> = ei6(z)+i2. Quantum mechanically, the coherent state is described by
P(,)
(2.41b)
= 6'2'(c(--cco).
Thus
s
G ( ~ ) ( T=) P(cc)lx(0)]2d2ct=
Ic1012
=
(I),
so that, from eq. (2.34)
( i ( t )i ( t + z ) )
(I)+e2cr2 (I), = e(i>8(z)+(i)2, = e2a6(z)
(2.42)
which is just the result of our classical calculation. Similarly, the model of a laser which we have utilized in 0 2.2 (an amplitude stabilized field with random phase) is described classically by (2.43a)
while the QM equivalent is (GLAUBER[1965] p. 164) (2.43b) Since the classical distribution function (2.43~~) again represents a constant intensity source, the photocurrent spectrum is identical t o that for a coherent source of the same intensity. Equivalently, P ( K ) of (2.43b) and (2.41b) lead to identical photocurrent spectra.
154
LIGHT B E A T I N G SPECTROSCOPY
[m
§ 3
Thus, for all homodyne cases which we considered in 92.2, the semiclassical and quantum theories lead to identical predictions. Extension to the heterodyne problem for the case in which a large (coherent) local oscillator field is at the center frequency of the superimposed Gaussian random field has been discussed by CUMMINS [1968] and again leads to predictions which agree with the semiclassical theory. In conclusion, we point out once again that the quantum theory prediction for the current autocorrelation function, eq. (2.35), and the semiclassical prediction, eq. (2.9), cannot be automatically equated despite their similarity in form, because the quantum mechanical and the classical correlation functions need not be identical. However, for the cases of interest in this review the P-representation always exists, and the quasi-probability P ( a ) is found to be identical to the classical amplitude probability function W , ( A ) . For these cases, the quantum mechanical and the classical values of G(l)(0) and G(2)(T) are the same, and thus both theories lead to the same predictions for the photocurrent spectrum regardless of the intensity of the optical field. This result can, in fact, be shown to hold for any case in which the P representation exists if P ( a ) is nonnegative for all a (SUDARSHAN [1963]).
Q 3. Light Scattering Theory In the preceding sections we have discussed the photoelectric current spectra produced by various optical fields without specifying the source of the fields. We now specialize t o light scattering experiments and present a brief summary of light scattering theory which will serve as a basis for the discussion of recent experiments in 5 6. We will assume that the scattering medium under study is illuminated by a monochromatic exciting source E ( t ) = E, exp ( - h o t ) and that the detector is small enough so that the scattered field is spatially coherent over the entire detector. The effects of relaxing these conditions will be considered in § 4. 3.1. SCATTERING BY A DILUTE SOLUTION O F PARTICLES
Consider a large volume (filled with a solvent) which contains N identical scatterers. The volume is illuminated with a monochromatic plane wave of frequency GO, polarized perpendicular to the scattering plane, and light scattered at an angle 0 is observed at a distant point R, (see Fig. 2 ) .
111,
§ 31
LIGHT SCATTERING THEORY
155
b Fig. 2. Geometry of a light scattering experiment.
The field observed at R, due to the
it”scatterer will be
Ej = A j ( t ) ei@je-ioot
(3.1)
where the amplitude A may depend on the orientation of the scatterer. If we let the position of the jthscatterer be rj and we choose the phase 4 = 0 for a scatterer at the origin, then +j
M
(K,-Ks) . rj = q r j ,
(3.2)
where KOand Ks are the wave vectors of the incident and scattered light respectively. If the scatterer moves slowly (v << c), then (Kol M /Ksl so that ( q / M 2(K,(sin 4 0 = (476?2o/~,,,)
sin 40,
(3.3)
where no is the refractive index of the solvent. For the total scattered field at R, we have N
N
ES =
C 3=1
Ej =
C
j=1
A $ ( t ) [exp {iq * rj(t)}][exp {-iw,t>].
(3.4)
The average total scattered intensity is given by 1 s = (lEs12>,
where the angular brackets denote a time average. Since the scatterers are not correlated, all cross terms average to zero, whence
156
LIGHT BEATING
SPECTROSCOPY
We now invoke the statistical independence of the scatterers to eliminate cross terms ( j # m ) , the statistical independence of position and orientation to factor amplitudes and phases, and finally the fact that the N scatterers are identical so that each must have the same autocorrelation function. Thus
(A* (t) A (t+z)) x
C(z) = N exp (-iw,z)
([exp {-ia . r(411rexp {iq . r(t+t)}l). (3.7) Since we are now dealing with single particle correlation functions, the subscripts have been eliminated. The optical spectrum is then given by the Wiener-Khintchine theorem as: =
(N/2n)
r+m
J --oo
[exp
{i(co-wO)z}l
[cA(z)l[cq4
(z)l dz,
(3*8)
where
(z>l= > [C, (z)l = ([exp {-ia r(411 [exp {iq r(t+411>. ECA
*
*
3.1.1. Spherical scatterers
If the scatterers are spherical, then the scattering amplitude A ( t ) is a constant and the amplitude autocorrelation function [C,(Z)] = IA 12. From eq. (3.8), the optical spectrum is then given by 1
= NI4",/-_
+0°
[exp {i(w-Wo)z)l [C+I).(
dt-
(3.9)
The phase autocorrelation function [C, (z)] is easily analyzed for [1964], ARECCHI[1967], CUMMINS three important cases (cf. PECORA et al. [1968]): a. Static scatterers (fixed random positions),
[C,
(41= 1.
(3.10a)
b. Scatterers moving with constant velocity v,
[C, (z)] = exp (iq * vz).
(3.1Ob)
111,
9 31
157
LIGHT SCATTERING THEORY
c. Scatterers undergoing translational diffusion characterized by the diffusion constant D,, (3.10~) For each case, the phase autocorrelation function can then be put into eq. (3.9) to find the optical spectrum: a. Static scatterers,
I ( w ) = N J A I 2S(w--w,j,
(3.l l a )
so the total scattered intensity NIAj2 appears at the frequency of the incident light coo, and the scattering is perfectly elastic. b. Constant velocity v,
I ( w ) = NIA12G(w-wo+q
*
v).
(3.1l b )
Again the total scattered intensity appears at a single frequency, but now is Doppler shifted to w = w o - q . v. This case can be easily extended to the dilute gas where the velocities are distributed according to Maxwell-Boltzmann statistics, and leads to the usual Dopplerbroadened (Gaussian) spectrum. c. Translational diffusion, [exp {i(w-wo)z}] [exp {-DTq2jzl}]dt (3.1lc)
The quantity in brackets in eq. (3.11~)is a normalized Lorentzian centered at w = w o , with half-width at half-maximum of Aw;
= DTq2.
(3.12)
Equation (3.1lc) for the quasielastic scattering from scatterers undergoing translation diffusion has been utilized in the analysis of the experiments which will be discussed in 3 6. For the case of spheres in water at 20" C, assuming D, = kT/6zqr (Stokes' law; q = viscosity) and Avac = 6328 A, it was shown that the linewidth (eq. (3.12)) becomes
Am,
=
49n sin2 46 Y
,
(3.13)
where Y is the radius of the sphere in microns (CUMMINSet al. [1964]).
158
LIGHT BEATING SPECTROSCOPY
[111,
3
3.1.2. Nonspherical scatterers Many important macromolecules are physically anisotropic but optically isotropic. For optically isotropic particles which are very small compared to the wavelength, the scattering amplitude is independent of orientation, and the spectrum will be identical to that of spheres regardless of the shape (Rayleigh limit). If the scatterer is too large t o meet the Rayleigh criterion (L << A), but its refractive index (n) is not too different from that of the solvent (no), then it may meet the weaker Rayleigh-Gans criterion (VANDE HULST[1957]): 2K,L(n-n,) << 1. The Rayleigh-Gans procedure is simply to imagine that each element of an individual scatterer is subjected to the incident field E , x exp{i(K,.r-coot)), whichproducesalocaldipolemoment, and thensum up the scattered field by integrating over the scatterer. The amplitude of the scattered field at the distant point R, is then:
($J:
=Iv
A , exp (iq r ) d V = A,V exp (iq r ) d ~ ) , A where A , is the scattered amplitude per unit volume of scatterer in the Rayleigh limit L << A. lA,12 is evaluated in VAN DE HULST[1957]: 4n21,
ni (a-no) 2.
(3.14)
-
The factor {(l/V) JY exp (iq r)dV} = f , is a normalized “form factor”. In the limit g --f 0 ( 0 + 0 ) , f, = 1. In general, f , 5 1. For spherical scatterers, /,(sphere) = (sin qb)/qb, where b is the radius. 3.1.3. Rigid rods For rigid rod-shaped scatterers, the form factor f, can be readily evaluated. Let y be the angle between the rod axis and q. If a, the cross-sectional area of the rod, is << 22, then
= [sin
($qL cos y ) ] / ( @ cos y ) ,
where L is the length of the rod.
111,
s
31
159
LIGHT SCATTERING THEORY
As the orientation of the rod changes, f will change. If the rod changes its orientation randomly in time, f will be given by f = fo+fl ( t ) where f o = ( f ) is the time average of f , and (fl ( t ) ) = 0. The amplitude autocorrelation function [C, (z)] = (A* ( t )A ( t + ~ ) ) will thus be given by:
[C,
(TI1 = =
IAOV
(V,*+fT (t)l [ f O + f l
(t+z)l)
(3.15a)
I A O ~ ?{lfol2+(fT (t)f l @+.)>I.
Combining eqs. (3.8) and (3.15), we have 1
I ( w ) = N / A o V J2
+m
Ifo12
1dT +
ei(w-wo)T [C, ).(
(3.15b)
2n.L 1
+ NlAov122n
jimei(w-wo)T
+T))
-m
[C,
(z)l dT.
Thus the spectrum consists of two parts: The first (spherical) depends only on [C6 (z)] and is therefore equivalent to the spectrum of spherical scatterers considered earlier. The second (modified) depends on both [C, ( T ) ] and (fT ( t )f l (t + T ) ) andis therefore determined by both the translational and orientational dynamics of the scatterer. It has been shown by PECORA [1964, 19681 that for macro-molecules in solution the “modified” part of the spectrum depends on both the translational and rotational diffusion constants of the molecule. Thus the spectrum of light scattered by nonspherical molecules in solution is not a simple Lorentzian (as in the case of spheres - eq. (3.11c)), but is rather a complicated superposition of Lorentzians which can, under some conditions, be effectively resolved (cf. CUMMINSet al. [I 19691). The calculation of the optical spectrum in this section somewhat obscures the statistical properties of the scattered field, since translational diffusion appears to give rise to phase (rather than amplitude) modulation which would not be observed in a homodyne experiment. However, the random phase modulation in the individual scattered fields E , produces an amplitude modulation in the total field E, = C E , in the usual two-dimensional random walk fashion. Thus the optical fields whose spectra are given by (3.11~) and (3.15b) are also Gaussian random processes. 3.2. SCATTERING BY P U R E FLUIDS AND LIQUID MIXTURES
The theory of 0 3.1 breaks down in pure fluids and liquid mixtures because the assumption of particle independence is not valid. I t can
160
LIGHT BEATING SPECTROSCOPY
[IIL §
3
be shown (PECORA [1964], KOMAROV and FISHER[1962]) that the spectrum of light scattered from a fluid is formally determined by the space and time Fourier transform of the Van Hove two-particle space-time correlation function (VAN HOVE[1954]). However, the space-time correlation function cannot yet be derived from first principles, and so the usual procedure followed consists of describing the fluid by the classical hydrodynamic equations (cf.
MOUNTAIN[1966]). Since the interparticle separation in fluids is very small compared to the wavelength of light, it is possible to treat a fluid optically as a continuous medium in which the local dielectric constant is perturbed by the thermally excited collective excitations which are solutions of the hydrodynamic equations of motions. In this approximation, one finds that a scattering experiment of the sort indicated in Fig. 2 (but with a continuous scattering medium) “selects” normal modes (or collective excitations) of the system for which q = K,--K, (EINSTEIN [1910]). If this mode produces a fluctuation in the dielectric constant c , , then the light scattering cross section is given by (cf. CUMMINS [1968]):
(3.16)
(V is the scattering volume, c is the speed of light, Y is the angle between E, and K g . ) In pure fluids and in liquid mixtures there are density or concentration fluctuations which do not propagate. These fluctuations relax by diffusive processes and have time dependence exp ( - y t ) . Thus these fluctuations give rise to components in the scattered light spectrum centered at w = wo ; these components constitute the Rayleigh or quasielastic spectrum. First we consider a mixture of two liquids of different refractive indices. If the concentration of one component fluctuates away from its equilibrium value by an amount AC, the return to equilibrium will be governed by the diffusicn law: dC - = D O 2C,
at
where D is the binary diffusion coefficient
111,
9 31
LIGHT SCATTERING THEORY
If AC(t = 0)
=
I61
-
AC, exp (iq r ) ,then
ac
-
at
-q2D(AC),
so that the autocorrelation function for concentration fluctuations of wavevector q in fluid mixtures is
ccb:( t ) c, ( f + t ) = ) (IC,l2)
exp
(-%"~l).
(3.17)
The differential cross section for the scattering of coherent light by concentration fluctuations characterized by the autocorreiation function eq. (3.17) is then found from eq. (3.16) to be:
Similarly, in simple fluids, fluctuations in the density p can occur at constant pressure (isobaric entropy fluctuations), and the return t o equilibrium is governed by the thermal diffusion law:
where x is the thermal diffusivity = A/p c, ( A = thermal conductivity, c, = specific heat at constant pressure). The nature of the nonpropagating diffusive mode in simple fluids can also be deduced directly from the hydrodynamic equations (cf. MOUNTAIN[ 19661). The autocorrelation function for isobaric density fluctuations of wave vector q in simple fluids is therefore
( t b , ( t f z ) ) = (JP,I2) exp (-xqZl1.l)>
(3.19)
and the differential scattering cross section is
The two cases are therefore equivalent. If we let p represent the density for the simple fluid case and the concentration for the mixture case, then we have for the autocorrelation function
( ~ : , ( t ) ~ , ( t + z= ) ) (IP, (t)12)e-Y171J
(3.21)
162
LIGHT B E A T I N G SPECTROSCOPY
[III.
§ 3
where for simple fluids, y = xq2, and for liquid mixtures, y = Dq2. For the differential scattering cross section we then have
The spectrum of the scattered light is a Lorentzian, centered a t w = wo with half width at half maximum Aw, = y. For the following discussion, it will be convenient to express eq. (3.22) as an intensity distribution I ( w ) rather than as a cross section. Since d2G./dQddw= IRiII,, we have
I ( w ) = 21,
Yh
(m-w0)2+T
'
(3.23)
where
Equation (3.23) shows that the Rayleigh linewidth Aw, is proportional to the thermal diffusivity x for simple fluids, or to the binary diffusion coefficient D for fluid mixtures. In the vicinity of the critical point (critical mixing point for fluid mixtures), this coefficient approaches zero so that the Rayleigh linewidth must also approach zero. (Modifications t o these equations which are required very close to the critical point will be discussed in 5 6.4.) The Rayleigh linewidth in normal fluids is characteristically less than 100 MHz, and quickly falls to 100 kHz or less as the critical point is approached. Therefore the usual dispersive spectroscopic techniques are of insufficient resolution in the critical region, and critical point experiments can only be performed with light beating techniques. Note that most of the scattering experiments discussed lead to Lorentzian optical spectra so that the photocurrent spectra consist of Lorentzians described by eq. (2.20) for heterodyne detection, or by eq. (2.14b) for homodyne detection, since the light scattered by thermodynamic fluctuations or diffusing macromolecules is almost always statistically Gaussian. In the case of light scattered from nonspherical macromolecules the optical spectrum was seen t o consist of a series of Lorentzians of different half-widths all centered a t w = w o . The heterodyne photo-
111,
4
41
COHERENCE A N D SIGNAL TO NOISE CONSIDERATIONS
163
current spectrum for this case will consist of a series of Lorentzians and is the same as the optical spectrum but centered at o = loo-wLo/. In the homodyne photocurrent spectrum, one finds a series of zerocentered Lorentzians, one for each Lorentzian component in the optical spectrum, and also a series of cross terms which arise from the beats between the optical Lorentzians (CUMMINSet al. [1968]). Finally, we point out that the light scattered by fluids very near the critical point need not be statistically Gaussian, so that the homodyne spectrum may depart from the form which was deduced under the assumption that E,(t) is a Gaussian random process (cf. CUMMINS [1968]). Additional complications arising from fluctuations in the light source which we have ignored here will be discussed in 3 4.
Q 4. Coherence and Signal to Noise Considerations In our development of the theory of light beating spectroscopy in 9 2 and of light scattering theory in 3 3, we made several simplifying assumptions which rarely apply in real experiments. These were: (1) the scattered radiation field is spatially coherent over the area of the photodetector; ( 2 ) the exciting light source provides a pure monochromatic field; (3) the measurement of the photocurrent spectrum P t ( o ) covers a sufficiently long time interval so that one actually measures the ensemble average value of P f ( w ) at each w . In this section we consider the effects of relaxing these assumptions. 4.1. SPATIAL COHERENCE
Consider the homodyne photocurrent spectrum predicted by eq. (2.14b) for a Gaussian optical field whose spectrum is a Lorentzian of halfwidth Am+ (optical) = y :
At o w 0, the light beating signal to shot noise ratio (ratio of the third term to the first) is:
which is just the mean photoelectron emission rate divided by the halfwidth of the optical spectrum. Now suppose we “break up” the photodetector surface ( A ) into
164
LIGHT BEATING SPECTROSCOPY
[In,
s
4
small area elements AA, each of which then produces the photocurrent spectrum of (2.14b), where (i} is the average photocurrent from one element. As we increase the size of the photodetector surface, the d.c. and signal terms both increase with the square of n = A/AA, while the shot noise term increases linearly with n. Thus the signal to shot noise ratio increases linearly with detector area. This result will hold as long as the optical field is spatially coherent over the entire detector. As the detector size increases, its dimensions will eventually surpass the distance over which the field is spatially coherent. Then the fluctuations in the photocurrent produced at different points on the detector will be uncorrelated, and the light beating currents from uncorrelated areas will be randomly phased with respect t o each other and will add up in random walk fashion. Thereafter, the signal term only increases linearly with the area A , so that no further increase in the signal to shot noise ratio is achieved by increasing the detector size. The effect of imperfect spatial coherence, which is one manifestation of the Hanbury Brown-Twiss effect, was discussed by FORRESTER et al. [1955] who considered the photodetector surface to be divided into N uncorrelated “coherence areas” ( A = NA,,,). If (i} in (2.14b) is taken as the current from the total photodetector surface, we must then divide the last (light beating) term by N so that the signal to shot noise ratio (4.1) becomes (4.2)
Since (i) and N both increase linearly with A , this is indeed independent of photodetector size. For the coherence area A,,, in (4.2), Forrester employed Acoh =
?’‘/on,
(4.3)
where SZ is the solid angle which the source subtends at the detector. The evaluation of Acoh has been discussed by BROWNand TWISS [1958] for planar sources. BENEDEK[1968a, b] cites the results of unpublished calculations by J. B. Lastovka for a three-dimensional source (a cube with edge a ) which suggest that the coherence area be defined as A,,h
= 2a2/Q.
Both of these equations for A,,, assume a spatially incoherent source. We briefly consider the evaluation of the area of coherence A,,
111,
5 41
COHERENCE A N D SJGNAL TO NOISE CONSIDERATIONS
165
"
Fig. 3. Van Cittert-Zernike theorem for spatial coherence.
for a quasi-monochromatic source. Let the source (which we will take as the two-dimensional spatially incoherent projection of the scattering volume on a plane (t,17) perpendicular to the direction of observation) have area b x b'. The detector (of area A = h x h') is at a distance R from the source (see Fig. 3 ) . Let A ~ , ( wbe ) the Fourier component of the fluctuating photocurrent from an incremental detector area AA at Pj. Then
Pi( w ) = 22 (Aij
(LO)
Ai,,( w )),
ji'
but since the field is statistically Gaussian (cf. MANDEL and WOLF [I9651 p. 275),
( A i , A i f } = (Aii}(Aijr} Jg'l)( Y ~ Y ~ , ) ) ~ , and thus PAW) =
22 ((4 (W)))21g(1) 5 'i
(Y,Yf)12
where ( 6 i ( w ) } is the mean current (at frequency W ) per .unit area, which is assumed to be constant over the detector. The normalized spatial coherence function g(l)(rlyP) for a uniform source is given by the Van Cittert-Zernike theorem (BORNand WOLF [1964] p. 508): (4.5)
166
IIII,
LIGHT B E A T I N G SPECTROSCOPY
9 4
where k = 2n/il, p = (X,-X,)/R, q = (Y,-Y,)/R, = h [ ( X : +Y,”) - ( X i +Yi)]/ZR. For the rectangular source (bxb’) of Fig. 3, eq. (4.5) is readily evaluated:
Whence
P<(W)=
sin2[ E ( X ,- X , ) ] (Si(0))Z [E
(xl-x,)1 (4.7)
where M = kb/2R and p = kb’/2R. If the detector is very small so that a(X,-X,) always << 1, the integrand of (4.7) FX 1, whence
Pi (m),oh
=
(si(W))2A2
and p(Y,-Y,)
[A (6i(W))]’,
are (4.8a)
which is just what we would expect for ideal spatial coherence. In the large detector limit where the maximum values of E(X,--X,) and @(Yl-Y2)are >>1, we can extend the limits of integration on (X,-X,) and (Y,-Y2) in (4.7) to fa, whence:
P,(W) = (Si(L0))ZA = [A(di(W))]2X
=
[A(Si(W))l2
1 22 -,
-
AQ
[i
-___
(bbyR2)l (4.8b)
where Q = bb‘/Rz is the solid angle which the source subtends at the detector. Comparison of (4.8b) with (4.8a) shows that in the large detector limit, the light beating signal is reduced from the value it would have if the field were perfectly spatially coherent by P,(W)
= Pi(W)COh/N.
(4.8~)
If we take the coherence area as Acoh = A2/Q, in agreement with eq. (4.3), then in the large detector limit N = A/(i12/Q)= A/A,,, and N is the “number of coherence areas”. Note, however, that the coherence area Acoh = A / N is really defined by eq. ( 4 . 8 ~ )from (4.7) and (4.8a). In terms of the number of coherence areas N we can rewrite eq. (2.14b) as
111,
s
41
COHERENCE A N D SIGNAL TO N O I S E CONSIDERATIONS
167
where
N=
(
1 A/(a2/.q
A << a2152 A >> 22/32.
In the intermediate region, A m A2/52, N must be evaluated by carrying out the integration of eq. (4.7) and will not be simply equal to A I (22/Q). BERGEet al. [1968b] have recently investigated the dependence of the ratio Pi(co),,,,,/(i)2 on the size of the detector. According to eq. (4.9) this ratio (for a constant total photocurrent (i)) should increase with decreasing detector area when A >> A2/52, and approach a constant value independent of A when A << A2/Q. This prediction is in good agreement with the results of the experiment. Note that the signal to shot noise ratio can be increased by reducing 52 which reduces N in eq. (4.9). Since Q = bb‘/R2, the scattering volume should be as small as possible. In practice the exciting beam is usually focused into the scattering volume for this purpose, although a fairly long focal length lens is used to avoid losing angular definition of KO. Finally, we note that the same factor N which reduces the signal to shot noise level in eq. (4.9) will appear in the heterodyne case. However, additional loss in spatial coherence can occur because of imperfect spatial matching of the local oscillator and signal fields. For this reason, the signal term in eq. (2.20) is frequently multiplied by a combined “heterodyne efficiency” E where E 5 l/N. 4.2. SOURCE SPECTRUM AND STATISTICS
In 9 3 we assumed that the exciting light source was perfectly monochromatic - i.e., an ideal single mode laser:
E L (t) = Eoe-iwot.
(4.10)
We now consider the effects of phase and amplitude variation in E,, and then take up the effect of utilizing multimode lasers. Whatever the form of E L ( t ) ,the scattered field at the detector, E s , will be given by
Es ( t ) = f ( 4 EL ( t ) >
(4.11)
168
LIGHT BEATING SPECTROSCOPY
[III,
s
4
where f ( t ) is a “scattering amplitude”. If the source is perfectly monochromatic (eq. (4.10)), then
E, ( t ) = Eoe-’*’otf(t), =
(18)
(lE81>2=
(4.12)
Io(lf(t)I2),
W
~ ( w )=
J-W
(1~/276)
ei(~-~o)7 (f* ( t )f (t+.)
w .
In 5 3.2, for example, f ( t ) is proportional to p,(t), the qth Fourier component of the density or concentration whose autocorrelation function uniquely determines the spectrum (cf. eqs. (3.16)-(3.20)). 4.2.1. Phase fluctuation
We have previously (5 2.2) utilized a model of a well-stabilized single mode laser in which the field is fixed in amplitude but subject to random phase fluctuation (GLAUBER[1965]) :
EL ( t ) = E0 e-iiwot+@(t)l,
(4.13)
with + ( t ) a stochastic variable. The spectrum of the scattered field
E , ( t ) = f ( t ) E0e-i[wot+4(t)l is then
I
I ( ~=) 2 276
I-” ”
ei(w--wo).r
(eiO(t)
f
x
(qe-i4(M7) f(t+.)
)dt.
Since f ( t ) and +(t) are presumably uncorrelated,
I b ) = GJ-” I0 *
,i(w-wo17
(ei4(t) e-i4(t+T)
>(f* (0f(t+.))dz.
By the convolution theorem of Fourier analysis, the transform of a product equals the convolution of the transforms, whence
where @ indicates convolution. Thus the spectrum of the ideal case (eq. (4.12)) must be convoluted with a term representing the spectrum of the laser. The observed optical spectrum will have a width in excess
III,
S
41
C O H E R E N C E A N D S I G N A L TO NOISE CONSIDERATIONS
169
of the relaxation rate y of f ( t ) . For sufficiently small y values, this broadening effect will dominate the optical spectrum. The light beating spectrum, however, is determined by G ( 2 () r )= (E:(t) E,(t) E,’(t+z) E,(t+t)). From eq. (4.13), ).(
~ ( 2 )
= 1 ~ ~ u1* (4 t ) f ( t )f*(t+.)
f(t+t)>,
(4.15)
which is the same result we would get with the ideal monochromatic exciting field, eq. (4.10). Thus we see that phase fluctuations of the exciting source have no effect on the homodyne spectrum. Similarly, it can be easily seen that phase fluctuations will not effect the heterodyne spectrum if the exciting source and the local oscillator are derived from the same laser. 4.2.2. Amplitude fluctuation Next consider an exciting source
E L (t) = E , ( t )erioot,
(4.16)
The convolution theorem again gives the optical spectrum as the convolution of the source spectrum with the spectrum of the scattering amplitude f ( t ) . For this case,
G“’ ).(
=
( E : ( t ) f * ( t )Eo ( t )f ( t )E; (t+.)
( E ; ( t ) Eo (t) E i (t+.) = Gg: (7)GY’ (z).
=
f * (t+.) Eo(t+t) t(t+z)> E , (t+t,)(f* (t) f ( t )f * ( t + t ) f ( t + z ) )
Now suppose that the exciting source is a narrow band Gaussian noise source (e.g., a 19*Hglamp). Then by the arguments of $ 2.2, since both E,(t) and f(t) are Gaussian random processes, the current autocorrelation function is
cz ).(
=
e(i)8(.)+(i)2
P+IgF; (.)l”l[l+lg:l)
If we let (.)Iz = exp (-2y,lzl), then (cf. eq. (2.14b))
e(i> Pa ( 0 )= -+(i)28’((3)+2(i)2X 7.c
and
I&) (.)Iz
(.)I”. =
(4.17) exp ( - 2 ~ ~ 1 4 ) ~
170
LIGHT BEATING SPECTROSCOPY
[III,
9 4
Two new terms appear in eq. (4.18) in addition to the first light beating term (which appears alone in eq. (2.14b) for excitation by a monochromatic source). However, if the width of the source y z is either very small or very large compared to yf, these new terms have no essential effect, and yf, the quantity of fundamental interest, can be extracted from the spectrum. Therefore, we could (in principle) utilize a rather broadband source and still measure very narrow light beating spectra. The reason that ordinary broadband light sources are not suitable for light beating spectroscopy lies in their limited brightness. The resultant large scattering volumes increase the number of coherence areas N in eq. (4.9) and thus reduce the signal intolerably. Extension of the preceding discussion to a single mode laser with small amplitude fluctuations is accomplished by combining eqs. (4.10) and (4.16):
EL( t ) = [E,+mE (t)]e-iuot,
(4.19)
where m is a “modulation depth” (m << 1) and E ( t ) is again a random Gaussian process. Again there are new terms in the photocurrent spectrum present similar to the last two terms in (4.18), but they are both multiplied by m2 and can usually be neglected for a reasonably stabilized laser source. 4.3.LIGHT BEATING WITH A MULTlMODE LASER SOURCE
We will now extend the results for homodyne and heterodyne detection, which were derived for a monochromatic source E ( t ) = E , exp ( - h o t ) (9 2 . 2 . and 9 2.3), and which were extended in the previous section to the case of a stabilized single mode laser E L (t )= E, exp {-i[w,t++(t)]}, to the more general case of a multimode laser source. I n practice light scattering experiments are usually performed with multimode lasers, since most commercially available high powered CW lasers operate multimode (e.g., the output of the widely-used Spectra-Physics Model 125 laser consists of approximately 17 modes spaced 83 MHz apart). The field EL(t)of a multimode laser operating well above threshold may be described by
EL ( t ) = 2EOj exp {-i[w,t++,
@)I}>
(4.20)
j
where the sum is over the modes of the laser, and E,,, wi and
+, ( t ) are
111,
$ 41
COHERENCE A N D SIGNAL T O NOISE CONSIDERATIONS
171
respectively, the amplitude (assumed to be constant), frequency, and the random phase factor for the jthmode. The optical spectrum of the multimode laser is given by the Fourier transform of
which is I(W) =
--I n _
[
~ j ~ E n [exp i ~ {i(w-wj)r}~ ~ ~ m ([exp {i+j (t111 [exp { - i ~ i (~+.)II) dr -cc
~
~
_
_
IEOjI
237,
z j
Thus the optical spectrum consists of a line a t the frequency of each mode with a width determined by the phase autocorrelation function for that mode, ([exp {i$j( t ) } ][exp {-i+j ( t + z ) } ] ) . The spectrum of the photocurrent produced by beating a multimode laser with itself is the Fourier transform of the photocurrent autocorrelation function (eq. (2.9)) :
C i(z)
=
e(i)d(t)
+ (i)"gC'
(t),
where
==
(2jIEnjI2)-' {<[zj,pE:jEop ~ X P{i[(~j-wp)t++j ( + 4 p
(t)I x
x I&,mE& Enmexp{i[( ~ T c - w T n ) (t+r) +b ( t + t ) -+,,, (t+r)I >I )>. Hence the photocurrent spectrum consists of a spectral line a t each of the intermode beat frequencies Iwk-wn,I. We consider now only the low frequency ( K = m ) component of the spectrum, since the intermode beats occur at frequencies of the order of 100 MHz, which is usually beyond the frequency range of interest in light scattering experiments. For k = m, the second double sum in gizl (t)collapses to a single sum and, since the cross terms in first double sum then average to zero, the first double sum is also reduced to a single sum. Thus (4.2la) g p (t)= 1, just as for a single mode laser (eq. ( 2. 16) ) .The photocurrent spectrum,
172
LIGHT BEATING SPECTROSCOPY
P t ( w ) = e(i)/n+(i)26’(w),
[IIL
§ 4
(4.21b)
consists of the shot noise and d.c. terms o d y , even though the optical spectrum has a finite width due to the random phase modulation. This result for the low frequency homodyne spectrum of a multimode laser is valid only under the assumption that the amplitudes Eoj of the laser modes are constant in time, while in a real multimode laser the Eojmay be slowly varying, thus giving rise to a low frequency component in the photocurrent spectrum. If this component in the laser photocurrent spectrum is not narrow compared to the spectrum of interest, then the laser must be stabilized before meaningful results can be obtained. The homodyne spectrum for light scattered using a multimode laser source can be calculated in the usual way from the photocurrent autocorrelation function
C i(7)= e206(z) (IEZ (t)12)+e2~2(IE,(t)121Es(t+~)12), where E, ( t ) is the optical field produced by the scattering of the laser beam by the sample and is given by (cf. eq. (4.11))
Es (4
= f ( t ) E L (t),
where the scattering amplitude f ( t ) is assumed to be a Gaussian random variable. Thus
ci (7)= e 2 0 ~ ( ~ ) ( l E(t)l2)(If(t)l2)+ L +e2a2(IE,
(ill2 lE, ( ~ + ~ ) l ” > ( l f ( ~ ) l 2 / f ( ~ f ~ l 2 ) ~
since E L ( t ) and f ( t ) are independent. The average current is
(i>
=
e 4 - G W) = eo(lf(t)12>(/EL. (t)l”>
so that
C i( r ) = e(i>s(z)+(i>2gC)(Z)g:2)(t), where gy) ( T ) = 1 + ( T ) l 2 since f ( t ) is a Gaussian random process. Since gp) (7)= 1 (eq. (4.21a)), we have
ci (7)= e(i>6(T)+(i)2
( W g p (T)I2),
(4.22)
which is identical to the current autocorrelation function derived for homodyne detection with a monochromatic exciting source (eq. (2.11)). Thus we conclude that the homodyne spectrum obtained for light scattering with an amplitude-stabilized multimode laser with an intensity (I) = ( / E L( t )12) is the same as the spectrum obtained using
111,
41
173
COHERENCE A N D SIGNAL TO NOISE CONSIDERATIONS
a monochromatic source or a single mode laser with the same intensity. Now we will consider heterodyne detection with a multimode laser source. (Heterodyne detection with a monochromatic source was discussed in 5 2.3.) The optical field incident on the photomultiplier is
E ( t ) = f ( t ) EL( t )+KEL (t)e-'OMZ,
(4.23)
where the first term arises from scattering by the sample and the second term is the local oscillator signal, which is just the laser signal (within a constant, K ) , upshifted by a frequency m M . (Schemes for producing the local oscillator signal will be discussed in § 5 . ) The photocurrent produced by either the local oscillator field KE, (t) exp ( -imo,t) or the scattered light fieldf(t) E L ( t )in the absence of the other is is or i, , where (is)
=
(iL01 =
ea(lf(t) 12)(lEL(t)
12>,
(4.24)
eolKI2 (lEL(t)12)
= iL0J
(4.25)
where the factoring of the correlation functions follows from the independence of the laser field and the scattering process, and the equality (iLo)= i, follows from the time independence of ]EL( t )l2 = xjIEOj12. The photocurrent autocorrelation function for the field E ( t ) is
C i(t)=e2a( 1 EL( t )12) ( If (tj+Ke-iWMti2)6 (t)+ + e 2 0 2 ( / E L ( t ) / I EL(t+t~~2)(~f(t)+Ke-'w~t(2~f(t+t)+Keiw~ Ten of the 16 terms in the last factor are zero, and by eq. (4.21a), (IEL(t)I21EL(t+T)l2) = (IEL(t)12)2, so we have
Ci )(.
= e((is)+iLO)S(Z)+
+e2a2 ( I E L
(4 12>2 {(lf(t)12I f ( ~ + ~ ) l 2 ) + 2 1 ~ I 2 < l f ( ~ ) l 2 > +
+2IKI2[eiWMt ( f * ( t ) f ( t + ~ ) > + e - ' ~(~ f ('t ) f*(t+.))I+IRI4) =
+(is)2g:2) +2i,,
+
(is) +2iLO (is) [eiWM'gp)( t ) + e - ' " ~ ' g ~ ) * ( t )+i$. ]
e( (is)+iL,)6
(t)
(t)
If i,
>> (is), we are left with
C i(7)
=
eiL,S(t)+i~,+2i,,
(is)[eiW~'gp) ( ~ ) + e - ~ ~ ~ (t)], ' g ~ ) *(4.26)
which is exactly the same current autocorrelation function that was obtained in 5 2.3 for heterodyning with a monochromatic source for which the current from either the local oscillator or sample in the absence of the other was i, or (is). I t therefore follows that the
174
LIGHT BEATING SPECTROSCOPY
[IIL
§ 4
photocurrent spectrum is the same for heterodyning with both the scattered field and the local oscillator signal derived from a common amplitude-stabilized multimode laser source as it is for heterodyning with a monochromatic source which produces the same scattered field intensity (or photocurrent (is)) and the same local oscillator intensity (or photocurrent iLo). Thus we conclude that for both the homodyne and heterodyne detection techniques the light beating spectrum obtained with a multimode laser source is the same as that obtained with a monochromatic source of the same intensity, except for the presence of the normally unimportant intermode beats. 4.4.SIGNAL
To NOISE
So far, we have computed the photocurrent spectrum Pf(co) in terms of the ensemble or time-averaged photocurrent (i). For the case of homodyne detection of a Lorentzian optical spectrum when there are N coherence areas on the detector, for example, we have from eq. (4.9):
Pf(co)=
e(i) ~
+ 2 (Ni ) 2 ~
7c
2y/n co2+(2y)2’
where we have dropped the d.c. term which is normally blocked in an experiment. The spectral power at frequency co consists of two terms: the signal term, and the shot noise term, e(i)/n. The shot noise term, being constant, poses no limitation in principle to the precision with which the signal term can be measured. (A method for accurately subtracting out the shot noise term is discussed in 3 5 . 1 . ) In practice, measurements of P t ( w ) are performed in finite time intervals, so that the observed spectrum is never identical to the prediction based on infinite time averages. The observed spectrum will exhibit fluctuations which will set the limit in practice on the precision with which the signal term can be measured. A meaningful index of the precision which can be expected in a given experiment is the ratio of the signal term to the r.m.s. uncertainty in the combined signal plus shot noise which we shall henceforth refer to as the signal to noise ratio S/N. HAUS[1968] has discussed the signal t o noise problem from the point of view of the statistics of the individual photoelectric emission events. Alternatively, the photocurrent can be considered as a continuous process, and the S/N ratio evaluated by standard methods of
III,
$ 41 PHOTO
COHERENCE AND SIGNAL TO NOISE CONSIDERATIONS
17.5
-
IF BANDPASS FILTER
INTEGRATOR
Fig. 4. Block diagram of an electronic spectrum analyzer.
electrical signal analysis (FORRESTER et al. [1955], BENEDEK [1968b]), which is the procedure we shall follow. A block diagram of an electronic spectrum analyzer is shown in Fig. 4. The optical signal I ( o ) is detected by a phototube which has an output current i, ( t ) with a corresponding power spectrum P, ( w ) . The photocurrent i, (t) passes through a narrow bandpass filter (filter A, the spectrum analyzer IF bandpass), is detected and squared and then integrated by the filter B, whose output current i, (t)is recorded. The squared signal i, ( t ) is a noise process and will contain fluctuations, no matter how large i3 ( I ) may be. The signal i, [t)corresponds to a finite time average of i 3 [ t ) ,unlike the infinite time average spectra derived for various cases in 5 2. The magnitude of the fluctuations in i 4 ( t )depend on the product 6T of the bandwidth 6 of the spectrum analyzer IF bandpass filter (filter A) and the time constant T of the averaging filter (filter B), as we shall now show using well-known results from linear circuit theory (see, for example, ASELTINE[1958]). We will calculate for the measured current i 4 ( t ) the ratio of the mean square fluctuation to the mean value squared,
by tracing the photomultiplier current i, ( t ) through the spectrum analyzer to the recorder. The photocurrent i, [t)is assumed to be a Gaussian random variable. If the photocathode area A is large compared to a coherence area kt,,h, then the photocurrent will have Gaussian statistics as a consequence of the central limit theorem. For very small photocathodes the photocurrent statistics will depart from Gaussian, going over to the exponential distribution W [ i )= (i)-l exp ( - i / ( i ) )in the limit A << ACoh.Thus the results of this section will be rigorously true only for the large detector limit, A >> Acoh. The photocurrent i l ( t ) is filtered by the bandpass filter A. The
176
LIGHT BSATING
SPECTROSCOPY
[III,
9: 4
spectral density P z ( w ) of the filter output is related to the input spectrum P, ( w ) by P2(w)
(w)12
= IyA
(4.28)
pl(w),
where Y,(w) is the admittance of filter A. (This theorem and other results from linear circuit theory which we will use are proved in ch. 15 of ASELTINE [1958].) In order to carry out the calculation explicitly we assume that the spectrum analyzer has a Lorentzianshaped IF bandpass with a center frequency p and full-width S. Therefore (4.29)
Further, we assume that the filter bandwidth 6 is small compared to the width of the signal spectrum P l ( w ) , so that P l ( w ) varies slowly over the width of the filter and hence to a good approximation (4.30)
The autocorrelation function C,(T) for the current i, (t) can now be calculated by the Wiener-Khintchine theorem:
I-, c, W
c2(T) =
P,(w)e-'"'dw
(4.31)
(o)eiB7e--iS7.
=
The effect of the detector and squaring circuit which follow filter A is to remove the carrier and square i, ( t ) ,so that i3 (t) = li, (t)1 2 , and thus we have, again applying the Wiener-Khintchine theorem,
P3(w)= -J 2n
(li,
(t)l2
li, (t+t)12)eiw7dt.
-W
If the input i l ( t ) of a linear system (such as filter A ) is a Gaussian random variable, so is the output i z ( t ) ,and therefore we have (cf. eq. (2.10)) (li2(t)lB
F2(t+T)l2)
= (li2(~)12Y+l(~;(~)
=
[C,
@)I2+
IC,
iZ(t+T))I2
l2
(T)
and (4.32)
111,
$ 41
177
COHERENCE A N D SIGNAL TO NOISE CONSIDERATIONS
The current i3 ( t ) is integrated by filter B, and the spectrum P, ( w ) of the output current i 4 ( t )is (as in eq. (4.28)) P4(w) =
IYB(w)12P3(W).
Filter B is an RC integrator with a time constant T = RC, so -7-1
Hence
To complete the calculation of eq. (4.27)
we need to find (i, ( t ) ) . The Fourier transform i, (0) of i, ( t ) is related to the Fourier transform of i3 ( t ) by
i,
(w, =
y, ( w ) i3 ( w ) ,
but 1
i3( w ) = 252
so that 1
(i4
(4) =2n
=
”
J-,
a7
,
J-, YB
C,(O),
Ii, (t)1 eciwtdt,
)
J-,
(]i,(t’)) , 1 eiwt’dt’dw (4.34)
(0 CiWt
since only i, (t’) in the integrand is affected by the ensemble average. We have finally the result (4.35)
The relation between the mean square fluctuation in the detected current and the mean current squared (eq. (4.35)) has been investigated in our laboratory by Mr. Ronald Reese and Mr. Donald Henry. The photocurrent produced by a white light bulb source was analyzed with a variable bandwidth spectrum analyzer (Panoramic SB-15) and averaged by a variable time constant RC integrator. The fluctuations in the observed current i 4 ( t )were monitored on an oscilloscope and
L I C H T R E A T I N G S PE C T K O S C O PY
LIII,
9 4
Fig. 5 . The spectrum analyzer output before averaging, for different bandwidths 6 1 2 ~ (a) : 7 . 5 Hz, (b) 18 Hz, (c) 33 Hz and (d) 68 Hz.
C O H E R E N C E A N D SIGNAL TO N O I S E C O N S I D E R A T I O N S
179
0
Fig. 6. The spectrum analyzer output after averaging, for different time constants: (a) T = 10 msec, (b) T = 27 msec, (c) T = 100 msec and (d) T = 270 msec. In each case the IF bandwidth 61272 = 33 Hz.
180
1111,
LIGHT BEATING SPECTROSCOPY
9
4
a digital record was obtained with a signal averager (Fabri-Tek Model 1052). Before presenting the quantitative results of the noise measurements let us examine the qualitative time dependence of the signal current on the bandwidth 6 and time constant T . Oscillograms of the detected spectrum analyzer output after detection but before averaging (current i 3 ( t ) in Fig. 4) are shown in Fig. 5 for different bandwidths 8/27c: (a) 7.5 Hz, (b) 18 Hz, (c) 33 Hz and (d) 68 Hz. (The oscilloscope time base is the same for each case, 50 msec/cm.) Note how the rate of fluctuation in the (Gaussian) signal increases with increasing bandwidth. Oscillograms of the signal after averaging (current i 4 ( t )in Fig. 4) are shown in Fig. 6 for different time constants, with an IF bandwidth 8/2z = 33 Hz in each case: (a) T = 10 msec, (b) T = 27 msec, (c) T = 100 msec and (d) T = 270 msec. Equation (4.35) predicts that the normalized mean square fluctuation in i 4 ( t )should be equal to (1 +dT)-l, and clearly the fluctuations decrease monotonically for (a) through (d), for which (1+6T)-1 = 0.33, 0.15, 0.046 and 0.018, respectively. The quantitative results of the noise measurements are summarized in Fig. 7, a graph of ((Ai4)2)/(i4)2 as a function of 6T. The solid line is the theoretical value (lfaT)-l and the points are the measured values. For wide spectrum analyzer bandwidths the fluctuations are
0.64
0
10
20
30
40
50
60
70
80
90
100
8T Fig. 7. Normalized fluctuations in the observed signal as a function of the product of the spectrum analyzer bandwidth 8 and the integration time T . The dots are the experimental points and the solid line is a plot of eq. (4.35).
111,
§ 41
COHERENCE A N D SIGNAL TO NOISE CONSIDERATIONS
181
small even for fairly short time constants, but the fluctuations increase markedly when the bandwidth is made very narrow, unless long time constants are used. Now we will apply the results we have obtained to the real problem of interest, the signal to noise ratio for light beating signals. We have from eqs. (4.30), (4.31), (4.34) and (4.35)
[( (Ai,)z)]'
=
BP, ( w ) (l+dT)-*,
(4.36)
(i4)
=
BP,(w),
(4.37)
where R is a constant, Pi(co) is the photocurrent spectrum, and w now denotes the center of the spectrum analyzer bandpass. The photocurrent spectrum in every case (@2.2-2.3) consists of a signal P, ( w ) plus a white shot noise background P,, = e(i)/x, so that (4.38) pi ( 0 ) = p s ( 0 ) + P S N Only the fraction P,/PsN of the observed current (i,) is the desired signal (eq. (4.37)),but both Ps(co)and PSNcontribute to the fluctua-
tions in i, . Thus we have finally that the true signal to noise ratio in the observed spectrum, the ratio of the signal amplitude at a frequency cr) to the r.m.s. fluctuation in the signal plus shot noise at that frequency, is (4.39)
(The same result is given in the review by BENEDEK [1968].) The signal to noise ratio as a function of frequency, (S/N),, attains its highest value when Ps(w)/PsNis a maximum, that is, at the center frequency coC of the photocurrent spectral line. Thus
The signal t o shot noise ratio at 0 = coC for the photocurrent spectra obtained in $9 2.2-2.3 for homodyne and heterodyne detection of a Gaussian random optical field with g(l)(z) = exp (-yIzl) is (4.40)
where N is the number of coherence areas subtended by the detector,
182
CIII, § 4
LIGHT BEATING SPECTROSCOPY
(is) is the photocurrent produced by the scattered field alone, and is a constant which has the values ,u = 1 for homodyning, ,u = 2 for heterodyning with wM 3 y (wlT is the local oscillator frequency), and ,u = 4 for heterodyning with wM = 0. In the usual case 6T >> 1, so that
,u
(4.41)
(If ST 5 1, then (S/N)max5 1, even if Ps/PsN>> 1, so that investigation of a spectral line would be extremely difficult.) If the signal to shot noise ratio is very large, (,u(&)/Ney) >> 1, then from eq. (4.41) we have
-I
W
(r
0 0
- - - - - - - -SH!!!I_NOISE_20
40
- - - - - - - - - -_----- 60
80
-- ---
100
FREQUENCY (kHz)
Fig. 8. Homodyne spectra for a xenon sample a t a temperature 0.30 C" above the critical temperature, illustrating the dependence of the signal to noise ratio on linewidth when Ps P S N .(a) 0 = 30". 2y/2n = 3.9 kHz, (b) 0 = 150", 2 y / 2 n = 57 kHz. I n both (a) and (b) the relative resolution ( s l y ) = 0.032 and the time constant T = 0.4sec.
>>
111,
§ 41
COHERENCE AND SIGNAL TO NOISE CONSIDERATIONS
(S/N)max= (ST)'.
183
(4.42)
Thus for a fixed relative resolution (Sly) of lines of varying widths y ,
(S/N)max = y+"(~/y)TI!
(4.43)
and the signal to noise ratio is proportional to the square root of the width of the line studied. This is a quite surprising result, one that is not intuitively obvious: for a fixed relative resolution (Sly), the signal to noise ratio decreases with decreasing linewidth y, even though the power per unit bandwidth in the signal spectrum P, (w) increases! ! ! The homodyne spectra in Fig. 8, recorded for a xenon sample a t a temperature 0.30C" above the critical temperature for two scattering angles, (a) 8 = 30" and (b) 8 = 150", illustrate the dependence of the signal to noise ratio on linewidth predicted by eq. (4.43).The scattering volume, scattered light signal (is), relative resolution (Sly), and time constant T are the same for the two spectra, and in both cases Ps >> P S N .Since the linewidth is given by y = xq2 (eq. (3.21))and q2(150")/q2(30") = 13.9,the spectral density P,(w) at w = 0 is 13.9 times greater for the 8 = 30" spectrum than it is for the 8 = 150" spectrum, yet eq. (4.43)predicts that the signal to noise ratio for the 8 = 150" spectrum should be (13.9)*= 3.7times that for the 8 = 30" spectrum. The spectra are seen to exhibit the predicted behavior. If the signal to shot noise ratio is small, then the signal to noise ratio is (4.44)
Thus in this case the signal to noise ratio for a fixed relative resolution (Sly) is inversely proportional to the square root of the linewidth y. Since the (S/N)maxis directly proportional to y* for large signal to shot noise ratios and inversely proportional to y* for small signal to shot noise ratios, it is of interest to find the optimum (S/N)maxwhen y can be varied:
where
(S/N)max = and the relative resolution (Sly) is held constant. The result is that the
184
LIGHT BEATING SPECTROSCOPY
[IIL
s
5
optimum signal to noise ratio is attained for PJP,, = 1. Thus if the physical quantity to be determined from a linewidth measurement can be determined equally well for different linewidths (e.g., by changing the scattering angle), then the proper choice of linewidth is the one that yields a signal to shot noise ratio of unity. I n three cases discussed in 9 3 the Rayleigh linewidth was proportional t o q2 : y = D,q2 for a dilute solution of spherical macromolecules, y = xq2 for a simple fluid, and y = Dq2 for a binary mixture. Thus in these cases the same information about the desired quantities D,, x and D can be obtained for different scattering angles (this is not true if y does not have a simple q2 dependence - see 9 3.1.2, 0 3.1.3 and 9 6.4). It follows from the preceding paragraph that the optimum signal to noise ratio is obtained when the scattering angle is chosen so that, if possible,
PsIPs,
= ,u(i,)/NeDq2 = 1,
(4.45)
where D denotes D,, x or D . Note that the laser beam power that can be effectively used in increasing the signal to noise ratio of a light beating experiment can be estimated from eq. (4.41). A laser beam power that yields a signal to shot noise ratio of unity results in a signal to noise ratio only a factor of two smaller than that produced with infinite beam power. Thus if the laser power were increased without limit from the level for which (is) = Ney/,u (i.e., Ps/P,, = l ) , the ultimate signal t o noise ratio would be the same as that obtainable (with (is) = Ney/,u) by increasing the integration time T by a factor of four. I n most experiments the small gain in signal to noise ratio obtained by increasing the laser power far beyond the level for which PsIPsN= 1 would be more than offset by the resultant detrimental sample heating.
5 5.
Apparatus and Procedure
The spectrum of light scattered by a sample can be measured by mixing the light with itself (homodyne technique) or with a local oscillator (heterodyne technique) in a square law detector (for example, a photomultiplier). The power spectrum of the photocurrent, P f ( o ) , is then normally measured with a spectrum analyzer which determines the Fourier transform of the current autocorrelation function ( i ( t ) i ( t + ~ ) )which , in turn is proportional to the second order correlation function of the optical field (9 2.1). Alternatively, the opti-
111,
9
5j
APPARATUS AND PROCEDURE
185
cal spectrum can be deduced from a direct measurement of the photocurrent autocorrelation function using an autocorrelation function computer. 5.1. HOMODYNE SPECTROSCOPY
The basic spectrometer used for homodyne measurements of an optical spectrum is diagrammed in Fig. 9 (FORDand BENEDEK [1965]). The output of a laser is focused on a sample and the light scattered through an angle 8, defined by apertures, is collected and focused onto the cathode of a photomultipler. The output photocurrent is Fourier analyzed with an electronic spectrum analyzer (various commercial spectrum analyzers are available which cover the spectrum from the subaudio region out to the microwave region). The output of spectrum analyzers is usually a voltage spectrum, proportional to [ P t ( w ) ] *which , must be squared to give the desired P f ( w ) . Typical spectra obtained with large and small signal to shot noise ratios are shown in Fig. 10. These spectra were obtained with a SingerPanoramic SB-15 spectrum analyzer for a carbon dioxide sample at the critical density and near the critical temperature (SWINNEY[1968]). After subtraction of the flat shot noise background, the spectra accurately follow the predicted Lorentzian line shape (eq. (2.14)), and the linewidth y = xq2 (eq. (3.2)) yields a value for the thermal diffusivity x in good agreement with thermodynamic measurements (SWINNEY and CUMMINS[1968]). The current autocorrelation function can be measured directly by substituting a correlation function computer for the spectrum analyzer and squaring circuit in Fig. 9. Figure 11 shows the correlation function obtained using a P.A.R. Model 100 Correlator, again for scattering from carbon dioxide near the critical point (CUMMINS[1968]). The correlation function is exponential as predicted (eq. (3.19)), and the correlation time is within a few percent of the value y-l obtained from linewidth measurements for the same sample conditions. Direct measurement of the correlation function is a far more efficient technique than the measurement of the spectrum with the usual swept-frequency spectrum analyzer (such as the Panoramic SB-15 or General Radio 1900 A), because these spectrum analyzers scan in frequency, measuring only one bandwidth at a time, whereas a correlation function computer (at least in principle) analyzes all of the signal all of the time. Thus the rate of data collection with a correlation function computer is faster than data collection with a swept-frequency
186
LIGHT BEATING
LENS
SPECTROSCOPY
SAMPLE
I\
Fig. 9. Homodyne spectrometer
Fig. 10. Typical spectra with large and small signal to shot noise ratios. The spectra were obtained for scattering from a carbon dioxide sample at the critical density and near the critical temperature.
111,
§ 51
APPARATUS AND PROCEDURE
187
< i ( t l i ( t + T ) ) vs T T increment: 5psec/channel
r-
Fig. 11. Photocurrent autocorrelation function for light scattered from carbon dioxide near the critical point.
spectrum analyzer by a factor which is of the order of the number of bandwidths swept. However, there exist commercial “real time” spectrum analyzers (e.g., Hewlett-Packard Model 8054, Federal Scientific Model PSD-g/AU) which simultaneously sample all frequency intervals. Such “real time” spectrum analyzers are in principle as efficient as correlation function computers.
Experimental considerations The signal to shot noise ratio for the photocurrent is inversely proportional to the number of coherence areas within the solid angle subtended by the collection optics (5 4.1). Since the number of coherence areas in a given solid angle decreases as the scattering volume decreases, it is desirable to make the scattering volume as small as possible. Therefore, the laser beam should have a small angular divergence and the focusing lens should have a short focal length. In order to have a well-defined scattering angle the collection aperture must be small. Also, the short focal length focusing lens requirement of the preceding paragraph must be relaxed for very small scattering angles, since the spread in angles in the focused incident beam can introduce a significant spread in the scattering angle. Multiple scattering is frequently a problem in scattering intensity measurements, particularly for highly turbid samples such as fluids near the critical point. The effect of multiple scattering on the spectral distribution of scattered light has been calculated for one case of interest: double scattering from thermal diffusion type density fluctuations in a fluid (FERRELL [1968]). Ferrell found that in the two limiting cases of forward and backward scattering the shape of the spectrum is insensitive to the presence of a double-scattering component.
188
LIGHT B E A T I N G SPECTROSCOPY
1111,
3
5
The spectrum of the photocurrent contains the desired signal spectrum plus a white shot noise background. The shot noise level can be determined by examining the spectrum at high frequencies, beyond the range in which the signal level is significant. An alternative technique for measuring the shot noise level is to substitute a light bulb (noise source) for the sample. The intensity of the light bulb is adjusted to produce an average photocurrent (i) equal to that which was produced by the sample. The signal t o shot noise ratio for the homodyne spectrum produced by the light bulb source is (9 2.2)
Ps (w
=
O)/PSN = (i)/Ney,
where N is the number of coherence areas subtended by the detector (9 4.1), and 1' is the linewidth of the light bulb spectrum (within a numerical factor, of the order of 1, which depends on the shape of the spectrum). Typical values are: phctocathode current (i) M A, N w 100, and for a 3000" K tungsten bulb, y M 1015sec-l. Thus Ps(w M O ) / P S N M lop9. Therefore, the signal spectrum for a light bulb source is entirely negligible compared to the shot noise, so that the light bulb effectively serves as a white noise generator. With a light bulb source the shot noise level can be determined at any frequency, so this technique imposes less stringent requirements on the frequency response of the spectrum analyzer than the technique of determining the shot noise level by examining the spectrum at frequencies much greater than the linewidths of the signals of interest. Furthermore, the white noise produced by the light bulb source can also be used to calibrate the frequency response of the entire spectrometer system, from the photodetector t o the recorder. Photomultiplier dark current is usually not a problem in light beating experiments because signals which are intense enough to yield usable signal to shot noise ratios result in a cathode photocurrent far greater than the dark current (see eq. (4.45)).Hence the principal criterion for selecting a photomultiplier tube for optical beating experiments is that it should have a high quantum efficiency in the optical region investigated. 5.2. H E T E R O D Y N E SPECTROSCOPY
Heterodyne spectroscopy differs from homodyning only in that the optical signal whose spectrum is to be determined is mixed with a monochromatic local oscillator signal at the photodetector. We will
111,
9 51
APPARATUS AND PROCEDURE
189
discuss the various techniques that have been developed for producing the local oscillator signal. Our discussion of heterodyne detection in 9 2.3 started with the assumption that the local oscillator is a monochromatic signal. In practice the local oscillator signal is always derived from the same laser which illuminates the scattering sample, since phase and frequency variations in the laser, which would otherwise distort the spectrum, are then automatically cancelled out. This fact has led to some confusion in terminology, since a “superheterodyne” spectrometer in which the local oscillator is derived from the same source as the signal (in contrast to an independent source) could be designated a homodyne (as opposed to heterodyne) spectrometer (cf. DELANGE[1968]). In the terminology which we are following in this review, a spectrometer employing a local oscillator (of any derivation) is designated a heterodyne Spectrometer, the term homodyne being reserved for the direct detection scheme of 9 2.2 in which only the signal field is present. Various techniques have been developed for obtaining the local oscillator signal. Most of them utilize the incident light at frequency wo directly as the local oscillator, so that wLo = w,,. From eqs. (2.19) -(2.20) we see that the spectrum will then be centered at w = 0, as in the homodyne case. The unshifted local oscillator can be obtained in several ways. LASTOVKA and BENEDEK[1966a] used some of the light transmitted through the cell, recombining it with the scattered light on a beam splitter. BERGE[I19671 put a piece of milky quartz in the scattering cell; by adjustment of the optics to locate the effective scattering volume in the sample very close to the surface of the milky quartz (which produces strong elastic scattering), the scattered signal and local oscillator signal can be made to originate within a single coherence volume which leads to a high mixing efficiency. LASTOVKA and BENEDEK [1966b] utilized the scattering from dust on the surface of the cell in the same way. A similar scheme which provides a more uniform distribution of the local oscillator power was employed in the TMV experiment of CUMMINSet al. [1968], who placed a Teflon wedge in the scattering volume with its edge partially intersecting the beam. Alternatively, some of the laser signal can be split off before the scattering volume, shifted in frequency, and recombined with the scattered light at a beam splitter. Since wLo # wo, the heterodyne current spectrum for this case is not centered at zero, but at lwLo-wol. Thus the entire optical spectrum is reproduced in the photocurrent spectrum, making it possible to study dissymmetry in the optical
190
LIGHT BEATING SPECTROSCOPY
[111,
s
6
spectrum which is, of course, lost if wLo = wo. Frequency shifting of the local oscillator from w,,to oLocan be achieved by Bragg reflection from travelling sound waves in an ultrasonic tank (cf. CUMMINSand KNABLE[1963]). Two “Bragg tanks” were used in the polystyrene diffusion broadening experiment of CUMMINSet al. [1964]. Despite the advantages of having a displaced local oscillator (wLo # wo), there is a particular difficulty with techniques in which the local oscillator beam traverses a different optical path than the scattered light, since the two signals must be recombined with perfect spatial cross-coherence in order to achieve the full signal of eq. (2.20). Any optical imperfection will tend to reduce the “mixing efficiency”, and thus reduce the light beating component of the photocurrent spectrum, leading to a reduction in the signal to noise level. The schemes discussed above, in which the signal and local oscillator are produced within the same volume, do not suffer from this difficulty since the signal and local oscillator light traverse identical optical paths. In that case, it is possible t o obtain a mixing efficiency close to unity and to achieve signal to noise levels comparable to those obtained with homodyne detection (ADAMet al. [1969]).
Q 6. Review of Rayleigh Linewidth Experiments In this section we review the light beating measurements of the linewidth of light scattered quasi-elastically by dilute solutions of macromolecules and by simple fluids and binary mixtures near the critical point. This technique for studying the dynamic properties of critical fluctuations can also in principle at least, be extended to solids. However the conditions under which solid state transitions may produce critical opalescence are rather restrictive, and no unambiguous case is yet known to exist (cf. SHAPIROand CUMMINS[1968]). For a discussion of other applications of the light beating technique to the study of distributed spectra, see BENEDEK[1969b]. 6.1. DILUTE SOLUTIONS O F MACROMOLECULES
The first experimental light beating measurement of a Rayleigh linewidth, reported in 1964, utilized dilute monodispersed solutions of polystyrene latex spheres (commercially available from the Dow Chemical Company). That experiment was performed with a heterodyne spectrometer, and the results exhibited the behavior predicted by eqs. (3.11 c)-(3.13) and (2.20) (CUMMINSet al. [1964]). Homodyne
111,
S 61
REVIEW O F RAYLEIGH LINEWIDTH EXPERIMENTS
191
measurements on the same material were reported by ARECCHI[1967]. Arecchi’s results verified the prediction of (2.14 b) for the optical spectrum (3.11 c), thus also demonstrating the random Gaussian nature of the scattered field. In 1967 Dubin, Lunacek and Benedek extended the technique to the study of several biological macromolecules (DUBINet al. [1967]). Their homodyne measurements of light scattered by bovine serum albumin, ovalbumin, lysozyme, tobacco mosaic virus (TMV), DNA and polystyrene latex spheres were all analyzed in accord with eq. (2.11 c), which is appropriate for spherical scatterers as we have seen. For the two largest macromolecules studied (TMV and DNA) the spectra were in fact found to give poor fits to single Lorentzians, as we should expect from the discussion of 3.1.2 for nonspherical scatterers. Recently, CUMMINSet al. [1968] have studied the spectrum of light scattered from TMV, utilizing the multi-Lorentzian spectrum predicted by PECORA [1964]. Both homodyne and heterodyne detection were utilized in this experiment which gave results for both the translational and rotational diffusion constants of TMV. With some additional sophistication in technique, the analysis of the spectrum of light scattered from complex macromolecules should provide additional parameters characterizing internal degrees of freedom in addition to the diffusion constants. Similarly, the spectral analysis of light scattered from self-propelled microorganisms can be used to study their movements, an effect which BERGEet al. [1967] utilized in a study of the salinity dependence of the motility of fish spermatozoa. 6.2. SIMPLE FLUIDS NEAR THE CRITICAL POINT
As a consequence of the ordinary linearized hydrodynamic equations, the Rayleigh linewidth for a simple fluid is given by the LandauPlaczek equation, y = x q 2 (eq. (3.21)), where x,the thermal diffusivity, equals A/pc,, where A, p and c, are the thermal conductivity, density, and specific heat at constant pressure, respectively. The specific heat c, diverges much more strongly than A in the critical region, so the Rayleigh linewidth should go to zero at the critical point. Thus measurements of the Rayleigh linewidth can be used to study the detailed temperature and density dependence of x in the critical region. Many of the thermodynamic properties of systems in the critical region are found to exhibit simple power law dependences on the reduced differential temperature E 3 IT-TcI/Tc (FISHER [1967],
192
L I G H T BEATING SPECTROSCOPY
[III,
9
6
HELLER[ 1967]), and experimental determinations of the numerical values for the exponents serve as crucial tests for the theories of critical phenomena. Above the critical temperature on the critical isochore c, (which diverges in the same way as the isothermal compressibility) is propord.Therefore x E Y - @ . Similarly, below the tional to E - Y , while A critical temperature on the coexistence curve, x cY’-*’. The exponent y is 1.00 for “classical” models (e.g., the van der Waals fluid) and is 1.25 for the 3-dimensional Ising model, which is mathematically identical to the lattice gas model; experiments indicate y = 1.3*0.1 (HELLER[1967]). Linewidth measurements combined with the known critical behavior of c, offered a way to determine the critical behavior of the thermal conductivity A , which was not known, although thermodynamic measurements of A for CO, by SENGERS [1965] did show that A has a critical anomaly. Linewidth measurements can be performed for an isothermal sample, unlike thermodynamic measurements of A, which require a temperature gradient in the sample and hence are limited to temperatures not too close to T,. The first measurements of the Rayleigh linewidth in a simple fluid were reported by FORDand BENEDEK[1965, 19661, for sulphur hexafluoride and shortly afterwards ALPERT et al. [1966] reported linewidth measurements for carbon dioxide near the critical point. These experiments showed that the Rayleigh linewidth follows the sin2 t 8 dependence predicted by the Landau-Placzek equation (where 0 is the scattering angle), and the linewidth was observed t o approach zero approximately linearly with T-T,, indicating that x goes to zero with an exponent y-y GX 1. However, these measurements were performed over small temperature ranges for samples whose mean density was not precisely known; therefore, the result y--y = 1 had a large uncertainty. Recent precision measurements of the Rayleigh linewidth in SF, (SAXMAN and BENEDEK[1968], BENEDEK [1968a, 1968bl) and in CO, (SWINNEY and CUMMINS[1968]) have shown that x for these two fluids does indeed go to zero as the critical point is approached, but the manner in which x approaches zero along the critical isochore is quite different for the two fluids, and in neither case does x depend linearly on T-T,. The results for the thermal diffusivity of CO, are shown in Fig. 12, which includes in addition to the data of Swinney and Cummins, light beating linewidth measurements of 15 very near N
- -
111,
9
61
193
REVIEW O F RAYLEIGH LINEWIDTH EXPERIMENTS
T , (SEIGELand WILCOX[1967]) and thermodynamic data for x far from T , (compiled by Dr. J. V. Sengers - cf. refs. 44-51 in MOUNTAIN [1966]). The exponent y - y for CO, (slope of Fig. 12), given by combining the three independent sets of data, is 0.73h0.02, while for SF, Saxman and Bendek found that y - y = 1.26&0.02. These different results for SF, and CO, are startling in view of the expected universality of the critical exponents. For both CO, and SF, below the critical temperature, it was found that y'-y' M 8, along both the gas and liquid sides of the coexistence curve. Thus the Rayleigh linewidth measurements indicate that the thermal conductivity of CO, has a strong critical point singularity, y m 0.6, both above and below T,, while SF, has this singularity below T , but is only weakly divergent if at all above T,. Recently KADANOFF and SWIFT[l96S], using scaling law techniques, have predicted that the thermal conductivity has a strong critical point singularity, A ~~5-1, where 5 is the two-body correlation N
0.001
0.01
LO
0.1
1-1,
10
w)o
(C")
Fig. 12. The thermal diffusivity of CO, along the critical isochore versus the difference and CUMMINS[1968]; m, thermodynamic data; A, temperature T-T,. 0 , SWINNEY SEIGELand WILCOX[1967]. The slope given by combining the three independent sets of data is 0.73&0.02.
194
[III,
LIGHT BEATING SPECTROSCOPY
-
S
6
-
length. On the critical isochore, t E-”, and on the coexistence curve, E N E-”’, where v M v’ M f . Thus A e-8 and x E*. The results for CO, are in good agreement with the prediction of Kadanoff and Swift, and are also in accord with the approximate equality expected for the exponents above and below T , (KADANOFF et al. [1967]). Preliminary results for xenon also indicate y-y M f (HENRYet al. [1969]). In contrast, extensive linewidth measurements of Saxman and Benedek for SF,, which were made for several isochores over a range of two orders of magnitude in E , clearly show a behavior above T , which is dramatically different from that observed for CO, and for binary mixtures (5 6.3). The cause of this difference is not presently understood. N
6.3. BINARY CRITICAL MIXTURES
The Rayleigh linewidth for a binary mixture is given by y = Dq2 (eq. (3.21)), where the binary diffusion coefficient D,which plays the role that x plays for simple fluids, goes to zero at the critical point. The first measurement of the Rayleigh linewidth for a binary mixture near the critical point was by Alpert and collaborators for the mixture aniline-cyclohexane (ALPERTet al. [1965], ALPERT[1966]). The Rayleigh linewidth was observed to vary approximately linearly with T-T, and with sin2 46 (cf. DEBYE[1965]). Similar behavior was observed by WHITEet al. [1966] for a critical mixture of polystyrene macromolecules in cyclohexane. Recent extensive measurements of the Rayleigh linewidth by Chu and collaborators (CHU [1967a], [1967b], CHU et al. [1968]) for the system isobutyric acid and water and by BERGEand VOLOCHINE [1968] for an aniline-cyclohexane mixture have revealed a y F% behavior for these mixtures rather than y F as indicated by the early experiments; hence the binary diffusion coefficient D E% in the critical region. SWIFT [1968] has shown that a binary mixture is the dynamic analog of a simple fluid, and, in particular, that the mass diffusion mode of a mixture corresponds to the heat conduction mode of a simple fluid. Thus Swift predicted that the binary diffusion coefficient D for a mixture and the thermal diffusivity x for simple fluid should exhibit the same critical behavior, which would be x D E ~ v, M $. The D F% behavior observed for two binary mixtures supports Swift’s prediction, as does the agreement between the exponents obtained for binary mixtures and carbon dioxide.
-
-
-
-
N
-
111,
61
REVIEW OF RAYLEIGH LINEWIDTH EXPERIMENTS
195
6.4.FIXMAN’S MODIFICATION
The equations of motion used in deriving the Landau-Placzek equation for the Rayleigh linewidth (eq. (3.21)) were the linearized [1960] has shown that these equahydrodynamic equations. FIXMAN tions must be modified slightly in the immediate neighborhood of the critical temperature in order to include the effects of long-range density correlations. BOTCH[ 19631 has incorporated Fixman’s modification in a derivation of the Rayleigh linewidth, and has obtained for the linewidth Y = xq2(1+q2E2). (6.1) Since the correlation length 6 diverges as T --f T,, the q 2 6 2 term should become significant very near T,, while far from the critical point qzE2 << 1 and the Landau-Placzek and Fixman equations become essentially equal. YEH [1967] was the first to observe a departure from the LandauPlaczek equation. I n linewidth measurements on xenon near the critical point he observed a very large Fixman modification term, equivalent to m 2300 A at E = 2 x Subsequent measurements in carbon dioxide (SWINNEY and CUMMINS[1968]), in an isobutyric acid and water mixture (CHU [1967a, 1967b], CHU et al. [1968], CHU and SCHOENES [ 1968]), and in an aniline-cyclohexane mixture (BERGE et al. [1968a]) revealed the presence of a much smaller Fixman niodification term for these systems: at F = 2 x 10-4, 6 = 150 k for CO,, 520 k for isobutyric acid and water, and 200 A for aniline-cyclohexane. Recent preliminary experiments on xenon by HENRYet al. [1969] also indicate a correlation length that is an order of magnitude smaller than that reported by Yeh*. It should be noted that the Fixman modification term in eq. (6.1) is a lowest-order correction to the Landau-Placzek equation, valid only when q is small (see HALPERIN and HOHENBERG [1967]). Furthermore, a recent study of the isobutyric acid and water system by CHU and SCHOENES [1968] indicates that the proper linewidth expression (for small q ) may have the form
Y where b
q2(1+bq26’2)>
(6.2)
+ 1 and 6‘is a correlation length different from the correlation
* Dr. Yeh has recently reanalyzed his xenon data, including in the analysis an aperture correction formerly omitted, and has obtained a correlation length an order of magnitude smaller than his earlier reported value (private communication).
196
L I G H T B E A TIN G S P E C T R O S C O P Y
[111,
§ 7
length t. Thus further experiments for different systems covering a range of temperatures very near T , and a large range of scattering angles are clearly needed. § 7. Conclusions
We have analyzed many of the technical problems that must be considered in any light beating experiment. For example, it has been shown that for both the homodyne and heterodyne detection techniques the light beating spectrum obtained with a multimode laser source is the same as that obtained with a monochromatic source or single mode laser with the same intensity, except for the presence of the normally unimportant intermode beats. It has also been shown that the signal to noise ratio for the light beating spectrum cannot be increased without limit by increasing the intensity of the exciting source; rather, only a small gain in S/N is realized by increasing the laser power beyond the level for which the signal to shot noise ratio is unity. The S/N (for a fixed relative resolution and a large signal to shot noise ratio) is proportional to the square root of the linewidth, so very precise measurements of extremely narrow spectral lines (of the order of Hz) will be difficult, no matter how powerful the laser source. On the other hand, if the linewidth can be broadened without decreasing the signal to shot noise ratio below unity by, for example, increasing the scattering angle, then the signal to noise ratio will be increased, even though the power per unit bandwidth in the signal spectrum is reduced. The applications of the technique of light beating spectroscopy in the past five years have demonstrated the value of the technique in the study of spectral lines far too narrow to be measured by conventional spectroscopic techniques. Light beating measurements of the Rayleigh linewidth in simple fluids and binary mixtures have yielded valuable new information on the dynamics of the critical region. For example, the critical behavior of both the thermal diffusivity of simple fluids and the diffusion coefficient of mixtures has been found to be described by a two-thirds power law, in agreement with the prediction of dynamical scaling. Although significant results have already been obtained with the light beating technique, the application of light beating spectroscopy to the study of critical systems and other systems must be considered to be in its infancy. Systematic light beating studies of the temperature, density and q dependence of the Ray-
1111
REFERENCES
197
leigh linewidth lie ahead; work is in progress in many laboratories and doubtlessly new results will soon be reported. Finally, the authors would like to acknowledge the help provided by members of the light scattering group at Johns Hopkins University, and to thank Professors E. Wolf, L. Mandel, and G. Benedek for reading the preliminary manuscript of this article and making many helpful suggestions which have been incorporated in the final version. We also wish to thank the Advanced Research Projects Agency and the Army Research Office (Durham) for financial support of our experimental light scattering spectroscopy program.
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IV MULTILAYER ANTIREFLECTION COATINGS BY
A. MUSSET and A. THELEN Optical Coating Laboratory, Inc., Santa Rosa, California, U S A
CONTENTS INTRODUCTION. . . . . . . . . . . . . . . .
.
203
SINGLE LAYER ANTIREFLECTION COATINGS . . 205 TWO-LAYER ANTIREFLECTION COATINGS ON GLASS. . . . . . . . . . . . . . . . . . . . . . 206
THE DESIGN METHOD O F EFFECTIVE INTERFACES. . . . . . . . . . . . . . . . . . . . . . 210 TWO-LAYER ANTIREFLECTION COATINGS ON HIGH-INDEX SUBSTRATE. . . . . . . . . . . .
212
THREE-LAYER ANTIREFLECTION COATINGS ON HIGH-INDEX SUBSTRATE. . . . . . . . . . . . 214
THREE- AND FOUR-LAYER ANTIREFLECTION COATINGS ON GLASS . . . . . . . . . . . . . . 217
. . . . . . . . . . . . .
222
THE COATING O F REFLECTION REDUCING MULTILAYERS . . . . . . . . . . . . . . . . . . .
224
ENVIRONMENTAL STABILITY O F MULTILAYER ANTIREFLECTION COATINGS . . . . . . . . . .
225
SYNTHESIZED LAYERS
OPTICAL PERFORMANCE O F COATINGS AND INCREASE I N TRANSMISSION TH’ROUGH AN OPTICAL SYSTEM. . . . . . . . . . . . . . . . . . . 225 PHOTOGRAPHIC APPLICATIONS. . . . .
. . . .
230
SUPPRESSION O F STRAY LIGHT I N AN OPTICAL S Y S T E M . . . . . . . . . . . . . . . . . . . . . 231 REFERENCES .
. . . . . . . . . . . . . . . . . . . .
236
Q 1. Introduction Advances in thin film technology and a continuing need for better performance of optical systems have created a strong interest in reducing the reflection of optical surfaces beyond that achievable by single layers. In Fig. 1 the reflectance of an uncoated glass surface of index 1.52 is compared with the reflectance of the same surface when overcoated with a single layer of index 1.38 [magnesium fluoride) and optical thickness 125 nm. Although the coating reduces the reflectance substantially there are two major deficiencies: 1. the residual reflectance at the minimum is not low enough for many applications; 2 . while the reflected light from the uncoated surface is neutral in color, that from the coated surface is not.
4%
I
0
400
500
600
700mp
WAVELENGTH
Fig. 1. Comparison of the reflectance of a single layer antireflection coating (solid curve) with the reflectance of the uncoated surface (dotted curve), ns = 1.52, ~ Z M= 1.0, n, = 1.38, 4n,d, = 510 nm. 203
204
MULTILAYER ANTIREFLECTION COATINGS
[IV, § 1
There are basically two ways one can improve the performance of a single layer, viz. 1. by varying the index of refraction of the film continuously with its thickness (inhomogeneous antireflection coatings) or 2. by constructing the antireflection coating of several homogeneous layers with different indices of refraction (multilayer antireflection coatings). Although the design, construction, and use of inhomogeneous antireflection coatings is still in the experimental state, multilayer antireflection coatings are widely used. It is for this reason that we shall restrict the discussions of this paper to homogeneous multilayer antireflection coatings. The literature on antireflection coatings is very extensive and scattered. No attempt will be made to trace the various designs historically. We shall restrict ourselves to the practically important designs and cover current design methods. Throughout the theoretical section of this paper we shall consider absorption-free materials, no dispersion, and normal light incidence. Another major restriction is imposed by the fact that only a few materials lend themselves to deposition as thin films. Table 1 contains a list of commonly used TABLE1 Coating materials Useful wavelength region
Material
Approximate refractive index a t 550 nm
Na,AlF,
1.35 1.38 1.45
(0.2 to 1 0 p m <0.2 to 5pm 0.2 to 8pm
SiO
1.55 1.6 1.65 1.85
0.3 to 8pm 0.3 to > 2 p m 0.2 to 7pm 7pm 0.6 to
Nd, 0, ZrO, CeO, ZnS TiO, Si Ge
2.0 2.1 2.30 2.30 2.30 3.40 (over useful range) 4.0 (over useful range)
0.4 t o > 2 p m 0.25 t o 7 p m 0.4 to 5 pm 0.4 to 1 5 p m 0.4 to 1 2 p m 8 pm 0.9 t o 1.3 to 3 5 p m
MgFz SiO,
Si,03 CeF, 3'
IV,
5
21
SINGLE LAYER ANTIREFLECTION COATINGS
205
TABLE2 Commonly used substrate materials Material
Approximate refractive index
Sapphire Synthetic quartz
1.7 1.45
Glass Si 1icon Germanium MgF, (Irtran 1) ZnS (Irtran 2)
1.5-1.7 3.45 4.0 1.38 2.2
Useful wavelength region (0.2 to 55 ,urn (0.2 to 4.0,urn and from 45 ,urn on 0.35 to 2.5,um 1.2 t o 50 pm 1.8 t o 25 prn 0.5 to 9 ,urn 0.7 t o 14.5,urn
materials. Table 2 lists some optical substrates for which antireflection coatings are desired. The process of designing multilayer filters is still an art. There are no methods yet of finding a certain design which can be proven to be optimum in its class. So the usual procedure is to use a design method which relates simple designs t o more complicated ones. Several methods are available and the selection for a particular problem is quite arbitrary and highly personal. The presentation in this paper is no exception. It is to be understood that the derivations are half-logical and half-intuitive. Some long formulas cannot be avoided. § 2. Single Layer Antireflection Coatings
The reflectance of a single homogeneous film on a glass surface is given by the formula (SMAKULA [1941])
Rfilrn
n2 (nM-ns)2- (nz-n2) (n2-ng) sin2 (2nndlil) = n2(nM+ns)2- (nif-n2) (n2-n;) sin2(2nnd/A) ’
with n : refractive index of the film, d : geometrical thickness of the film, n, : refractive index of the substrate, nM: refractive index of the surrounding medium, il : wavelength. Since the reflectance of the uncoated surface is
(1)
206
M U L T I L A Y E R A N TI R E F L E C T I 0 N C 0 A T I N G S
[IV,
§ 3
(3)
and particularly that
Rfilm= 0 for n
=
2nnd
d G Sand __ = hn,+n,etc. 2
(4)
Unfortunately there is no coating material with a refractive index low enough to fulfill condition (4) for normal glass surfaces (n rn 1.5). Magnesium fluoride with a refractive index of 1.38 is the best acceptable material. This leads to the limitations discussed in the introduction. For substrates with higher indices of refraction, especially n 2 1.3S2, single film antireflection coatings can be very effective. A significant advantage of a single layer antireflection coating is the fact expressed in relation (3) that its reflectance never exceeds the uncoated surface reflectance. This is not the case for multilayer antireflection coatings which contain materials with refractive indices higher than the substrate index.
5 3.
Two-Layer Antireflection Coatings on Glass
For normal glass surfaces (n rn 1.5) two layer antireflection coatings can eliminate one or the other deficiency of the single layer antireflection coatings as described in the introduction. Complete narrow band reflection reduction is accomplished by the so-called V-coating and a broadening of the minimum by the so-called W-coating. The reflectance of a two layer coating is given by (THELEN [1956]) . , x
R=-
n
l+X'
with
where n, : refractive index of the film next to the surrounding medium, p1 = 2nn,d,/J.,
IV,
9 31
ANTIREFLECTION COATINGS ON GLASS
201
n2: refractive index of the film next to the substrate, p2 = 2nn2d2/A.
For small R,
R In the V-type coating we set R
M
X.
=X =
0 which leads to
These relations allow a wide variety of solutions. Figs. 2 and 3 give some examples. From all the combinations of nl and n2 compatible with equations (8) and (9) the ones with the lowest values are preferred because they yield the broadest reflectance minimum (THELEN [ 19561). In the W-type coating we insert a half wave film (n2d2= +A) be4%
w V
5
3
0 lW _I
LL
w
m
2
I
0 WAVELENGTH
Fig. 2. Reflectance of various two-layer V-type ns = 1.52, nM = 1.0; solid curve: n, = 1.38 , n2 = 1.701, dotted curve 1 : +z1 = 1.38 , n 2 = 2 .3 0 , dotted curve 2: n, = 1.865, n2 = 2 . 3 ,
coatings on a substrate with index 4n,d, = 540 nm 4n,d, = 540 nm; 4n,d, = 698.5 nm 4nzd2= 113.3 nm: 4n,d, = 540
4n2d, = 540
nm nm.
208
MULTILAYER
ANTIREFLECTION
COATINGS
WAVELENGTH
Fig. 3. Reflectance of various two-layer V-type coatings on a substrate with indcx ns = 1.60, % ~ = f 1.0; solid curve: n, = 1.38 , 4n1d1 = 540 nm vz2 = 1.746, 4n,d, = 540 nm; dotted curve 1: n, = 1.38 , 4 n,d , = 671.5nm n 2 = 2.30 , 4n,d, = 113.2nm; dotted curve 2: n, = 1.818, 4vz,d, = 540 nm n2 = 2.30 , 4 n 2 d 2 = 540 nm.
12
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
"2
Fig. 4. Coefficientof cosz p in equation ( l o ) , A,, as a function of n,. n, = constant = 1.38, %M = 1, n~ = 1.52 (solid curve) and 1.60 (dotted curve).
IV,
§ 31
209
ANTIREFLECTION COATINGS ON GLASS
', \
,
_,I'
L . 4
..____. --
*r
%
with index
Fig. 5. nS = 1
4%
1
2
2
I 0 W
$ 3 W 0
6n 2
I
0 400
500
600
700mp
WAVELENGTH
Fig. 6. Reflectance of various two-layer W-type coatings on a substrate with index V Z = ~ 1.60, PZM = 1.0; solid curve: n, = 1.38 , 4n,d1 = 510 nm n, = 1.70 , 4n,d, = 1020nm; dotted curve 1: n, = 1.38 , 4n1d, = 510 nm n, = 2.10 , 4n,d, = 1020 nm; dotted curve 2: n, = 1.38 , 4n,d1 = 510 nm n, = 2.40 , 4vz,d, = 1020nm.
210
M U L T I L AY E R
A N T IR E FLE C T I 0N C0AT I N G S
[IV,
5
4
tween a quarter wave single layer antireflection coating and the substrate. By setting y , = 2y1 = 2 y equation (6) yields:
which is the reflectance of a single film with n = nl. But while for the single film this point is a minimum it is now a maximum as long as the coefficient of cos2y, A , < 0. Fig. 4 shows the coefficient A , as a function of n, for nl = 1.38, nbf= 1 and ns = 1.52 and 1.60. Fig. 5 gives some examples for a substrate of index 1.52 and Fig. 6 for a substrate of index = 1.60. While two layer antireflection coatings cannot eliminate both previously mentioned deficiencies of the single layer antireflection coating on normal glass surfaces they can provide excellent antireflection for substrates with higher indices of refraction as will be shown later.
Q 4. The Design Method of Effective Interfaces For the design of multilayer antireflection coatings the method of considering two effective interfaces inside the multilayer is a very effective method (DUFOUR[1954]; SMITH [1958]; THELEN[1960]; THETFORD [1970]). Two adjacent interfaces inside the multilayer system (Fig. 7 ) are selected and the Fabry-Perot formula is applied. One arrives at the following formula
IV, §
41
211
DESIGN METHOD O F EFFECTIVE INTERFACES
I
2
SPACERLAYER
7
?/ //
///////////.///////
JSUBSITTEM
SUBSTRATE
Fig. 7. Effective interfaces inside a multilayer.
with T (A) : Transmission through the total multilayer system, T,(A) : Transmission through subsystem I, T2(A): Transmission through subsystem 11, R(A) =d{R,(A) R,(A)} with R,, R, being the reflectivities of the subsystems I, I1 for light incidence from the spacer, r$, 4, : Phase changes upon reflection associated with R,, R, . = 4nnd/A with n, d being the index of refraction, physical B thickness of the spacer layer. The important feature of this formula is that the amplitude and phase relationships can be considered separately: T o @ )depends on the amplitudes of the reflectivities of the subsystems only and F(A) depends primarily on the phase upon reflection of the two subsystems and the thickness of the spacer. Both factors are always 5 1. For high transmittance (T FZ 1) the amplitudes have to be adjusted so that T o @ )M 1 and the phases SO that F(A) M 1. From the structure of equation(l1) it can be seen that
T o @ )= 1 only for Rl(A) = R2(A)
(12)
and
E(A) = 1 only for R(A) = 0 or sin2 +(r$,f#~~-B)
=
0.
(13)
The application of this method to two- and three-layer systems is relatively simple because the subsystems I and I1 reduce to single films.
212
MULTILAYEK ANTIREFLECTION COATINGS
[IV,
s
5
There are two groups of designs for which the amplitude condition (12) can be fulfilled for all wavelengths. It can beshown (Thelen [1969])
that the transmittance and reflectance of a multilayer are invariant to multiplying all indices of refraction (multilayer, substrate and medium) with a constant factor or replacing them with their reciprocal value while keeping the optical thicknesses constant. Group I : For the purpose of this discussion we consider subsystem I t o have a substrate index of nMand a medium index (spacer) of n,. We can shift one system into the other by multiplying all indices with a shift factor y: nx
=1
y n , and n, = yn,.
(14)
This leads to spacer index nx transformation
?%subII =
=
dn,nM,
Ynsub I = (nS/nM)+nsub I.
(15)
GroupII: Instead of shifting subsystem I into I1 by multiplying by a shiftfactor alone we now take the reciprocal values and then shift. The result is
nx = y/n, and ns = y/n,,
(16)
which leads to spacer index n, transformation
%z.,,b II =
=dnsnbf,
y/n,,,
I
= nRlns/nsub I.
(17)
Because of the previously discussed lack of coating materials with very low indices of refraction the direct application of these concepts is limited to substrates with indices of refraction well above 2. Antireflection coatings of this type were extensively discussed by MUCHMORE[1948] and YOUNG[1961]. They used electrical network synthesis for the derivation.
Q 5 . Two-Layer Antireflection Coatings on High-Index Substrate Group I-type coatings: By setting q.~= y1 = p2and n2 = n12/(ns/nL,) eq. (6) changes into
IV.
S 51
COATINGS ON HIGH-INDEX SUBSTRATE
213
Fig. 8. Reflectance of two-layer group I-type coatings on a high-index substrate with ns = 3.45, n M = 1; solid curve: n, = 1.56 , 4n,d1 = 1, n2 = 2.896, 4n,d, = A,; dotted curve: n, = 1.38 , 4n1d1 = 1, n2 = 2.56 , 4n,d, =A,.
x./
x
Fig. 9. Reflectance of two-layer group 11-type coatings on a high-index substrate with ns = 3.45, n M = 1; solid curve: n, = 1.56 , 4nld1 = A, nz = 2.21 , 4n,d, = A,; dotted curve: n, = 1.38, 4n,d1 = 10 n2 = 2.50, 4n,d, = A,.
214
M U L T I L A Y E R A N T I RE FLE CT I 0 N COAT1N GS
[IV,
I
ti
which yields a single broad minimum with zero reflectance at y = in. Fig. 8 shows two examples. Group 11-type coatings: With q~ = p1 = y 2 and n2 = nSnM/nl we arrive at (eq. (6))
This coating has a small maximum at y
= +z with
and two zero reflectance points at
where R, is the reflectance of the uncoated substrate. The total bandwidth for which R 5 R,,, is given by:
Fig. 9 shows two examples.
Q 6. Three-Layer Antireflection Coatings on High-Index Substrate In the previous paragraph the thickness of the spacer layer (with refractive index d ( n S n M )was ) zero. By giving it a finite thickness we can further reduce the residual reflectance and increase the bandwidth. The actual amount of the thickness depends on the phase factor and is difficult to optimize. Setting the thickness equal t o the thicknesses of the other two layers is an obvious choice, mostly because the designs are then more symmetrical and the analysis is simple. The reflectance of a three-layer coating is given by the formula (THELEN [1956])
IV, S 61
COATINGS ON HIGH-INDEX SUBSTRATE
21 5
with
x=
nS ~
(A2+B2)
4%
and cos 91 cos q2cos T3-
(%
nM n3 -
2)
cos pll sin p12 sin p3-
(2:: r:) ~
--
sin rplcos q2sin p13,
(::zl)
cos ql cos q2 sin v3+ - - - cos pll sin q2 cos v3
+
(2 :) ---
sin TI cos q2 cos y 3 -
(=-A nht n2
n1n3
sin y1 sin q2sin ps.
Group I-type coatings: By setting q.~ = v1 = q2 = p3 and n2 = d ( n S n M ) > n3 = n12/(nS/nAf) eq' (24) yields 4
Fig. 10. Reflectance of three-layer group I-type with nS = 3.45, n~ = 1; 1z1 = 1.56 , solid curve: n, = 1.8574, m3 = 2.896 , dotted curve: nl = 1.38 , n2 = 1.8574, n3 = 2.56 ,
coatings on a high-index substrate 4n1d1 = & 4n,d, = I , 4n3d, = 1,; 4nld1 = & 4n,d2 = & 4n3d3 = l o .
216
MULTILAYER
ANTIREFLECTION COATINGS
This coating produces two zero reflectance points at tan2 qo =
'd(nMnS)
+
nM/nlfnl/d(nSnM)
and the reflectance in the center at q
= $ 7 ~is
(27)
Fig. 10 shows two examples.
Group 11-type coatings: By setting = y1 = q2 = p3 and n2 = 2/(nsn,), n3 = nsnM/nl eq. (24) changes into
X./X
Fig. 11. Reflectance of three-layer group 11-type coatings on a high-index substrate with nS = 3.45, n M = 1 ; 4n,d, = I ,
solid curve:
n , = 1.56
dotted curve:
n 2 = 1.8574, 4n2d2 = I , n 3 = 2.21 , 4n,d3 = I , ; n, = 1.38 , 4n,d, = I ,
,
A,
n 2 = 1.8574, 4n2d,
=
n3 = 2.5
=I,.
,
4n,d,
IV, §
71
ANTIREFLECTION COATINGS O N GLASS
217
(28)
Now we have three zero reflectance points, one in the center a t in, and two at the points given by
pl =
Q 7. Three- and Four-Layer Antireflection Coatings on Glass Again, due to the lack of coating materials with a low enough index of refraction, it is not possible to design group I- and group 11-type coatings for glass substrates directly. We can, though, design antireflection coatings from air into a high index substrate and from this high index substrate into glass. We then can let the thickness of the high index substrate shrink to zero and we end up with a composite antireflection coating for glass. As a first combination let us use a two-layer group I-type coating to match from air into the dummy high-index medium with index n, and a single layer coating to match from the dummy medium to glass. This construction requires the following index relations (eqs. (15) and (4)): n2 = n,(n,/n,)+,
n3 = (nDns)+,
or n2/fil= n3/(fiMns)+.
(32)
The resulting three-layer coating has a broad minimum for 9 = in. In Fig. 12 the reflectance of three different designs are shown. It appears that the design with n2 = n, has the broadest minumum.
I
WAVELENGTH
Fig. 12. Reflectance of three-layer antireflection coatings on glass with n~ = 1.52. n M = 1; solid curve: n, = 1.38 , 4n1d1 = 510 um n, = 2.1 , 4n,d, = 510nm n, = 1.88 , 4n3d3 = 510nm; dotted curve 1: n, = 1.38 , 4n1d, = 510nm n2 = 1.9 , 4n,d, = 510nm n3 = 1.698, 4n3d3 = 510 n m ; dottcd curvc 2: nl = 1.38 , 4n1d1 = 510nm n2 = 2.30 , 4n,d, = 510 nm n, = 2.055. 4n,d, = 510 nm. 4%
I
0 WAVELENGTH
Fig. 13. Reflectance of four-layer group 11-typc coatings on glass with n~ = 1.52, '?%M= 1; solid curve: nl = 1.38 , 4n1d1 = 485 nm n, = 2.2 , 4n,d, = 485nm n, = 2.43 , 4n3d3 = 485nm n4 = 1.887, 4n,d, = 485 nm; dotted curve: n, = 1.38 , 4n1d, = 485 nm n, = 2.1 , 4n,d, = 485nm n3 = 2.322, 4n3d3 = 485 nm n4 = 1.887, 4n,d, = 485 nm.
IV.
S 71
219
ANTIREFLECTION COATINGS ON GLASS
WAVELENGTH
Fig. 14. Reflectance of three-layer group I-type coatings on glass with ns
=
1.52,
=
1.52,
n M = 1;
solid curve:
dotted curve:
n, = 1.38,
n, = 2.15,
4n,d, 4n,d,
n3 = 1.70,
4n,d, =
n, = 1.38, n, = 2.35,
4n,d, =
n3 =
4n,d, 1.70, 4n,d,
= =
= =
525 nm 1050 nm 535 nm; 525 nm 1070 nm 535nm.
WAVELENGTH
Fig. 15. Reflectance of a detuned version of one of the coatings of Fig. 14, ns n M = 1; solid curve: n, = 1.38, 4n,d, = 510 nm n, = 2.15, 4n,d, = 1020 nm n3 = 1.62, 4n3d, = 510 nm; dotted curve: n1 = 1.38, 4n,d, = 510 nm n2 = 2.15, 4n,d, = 1020nm n3 = 1.70, 4n,d, = 510nm.
220
MULTILAYER ANTIREFLECTION COATINGS
[IV, § 7
As a second example we use two group 11-type coatings. The resulting four layer coating has to satisfy the following two conditions (eq. (17)):
nln2 = nMnD,
n3n4= nDns,
(33)
which leaves the indices of three layers open. As an additional requirement we can specify that the reflectance in the center is zero. By a straightforward expansion of eq. (24) the reflectance of a four-layer coating for p = rpl = p2 = p3 = p4 = &n is found to be
R w nMn2n,/ns n1n3-n1n3/n2n 4 .
(34)
R is zero for n1
(35)
= (nM/nS)
n4
Combination of eqs. (33) and (35) leads to
ni = n:(ns/nM)8,
n2 2 = n:(nM/ns).).
(36)
The two examples of Fig. 13 prove that indeed very broad band antireflection coatings can be designed this way. As a third example we use two group I-type coatings. With eq. (15) we can establish the two index relations nz/nl = (nD/nM)+, n3/n4= (nD/ns)+,
(37 1
or n1 ( n S ) f / n 2
= n4 (nM)'/n3
>
(38)
which is the same as eq. (35). By setting n2 = n3 we can reduce the four layer coating to a three layer quarter-half-quarter coating. The index relation is now n4/n1=
(ns/nM)+.
(39)
The optimum bandwidth should be determined by varying n 2 . Fig. 14 indicates that the optimum occurs for n2 = 2.15. In order to further study the quarter-half-quarter coating let us set p1 = p, y 2 = Zp, and p3 = rp in eq. (24) and evaluate the reflectance for p = & + E with E + 0. We arrive at a relation of the following form:
Re,, =
((-- ):
5 4nM
nMnl nSn3
-
2
WAVELENGTH
Fig. 16. Reflectance of a detuned version of one of the coatings of Fig. 13, nS = 1.52, nM = 1; solid curve: n, = 1.38 , 4n,d, = 485 nm n 2 = 2.2 , 4n,d, = 485nm n3 = 2.435, 4n3d3 = 485 nm n4 = 1.837, 4n4d4 = 485 nm; dotted curve: n, = 1.38 , 4n,d, = 485 nm n, = 2.2 , 4n,d, = 485nm n3 = 2.435, 4n3d3 = 485 nm n4 = 1.887, 4n4d4= 485 nm.
WAVELENGTH
Fig. 17. Reflectance of a detuned four-layer coating on medium index glass, nS = 1.62,
nM = 1.0; solid curve:
dotted curve:
n, = 1.38 , 4n,d, = 485 nm n2 = 2.176, 4n,d, = 485 nm n3 = 2.45 , 4n3d3= 485nm n4 = 1.91 , n, = 1.38 , n, = 2.176, n3 = 2.45 , n4 = 1.98 ,
4n4d, = 485nm; 4n,d, = 485 nm 4n,d, = 485 nm 4n,d3 = 485nm 4n4d4= 485nm.
222
M U L T I L A Y E R A N T I R E F L E CT I O N C O A T I N G S
[IV,
8
WAVELENGTH
Fig. 18. Reflectance of two three-layer antireflection coatings for glass with odd thickness ratios, n s = 1.52, n M = 1.0; solid curve: n, = 1.38, 4n,d, = 567.2 nm n2 = 2.10, 4n,d2 = 212.3 nm n3 = 1.80, 4n3d, = 731.4 nm; dotted curve: n, = 1.38, 4n,d, = 500.4 nm n2 = 2.10, 4n,d, = 862.3 nm n3 = 1.8 , 4n,d, = 324.5nm.
which indicates that by slightly deviating from eq. (39) a broadening can be accomplished. The result of this detuning is shown in Fig. 15 for the practically important case of the quarter-half-quarter coating with n, = 1.38 (magnesium fluoride), n2 = 2.15 (zirconium oxide) and n2 = 1.62 (cerium fluoride) (THELEN [1965]; Cox et al. [1962]). Curves for detuned four-layer coatings are given in Figs. 16 and 17. Three-layer antireflection coatings can of course also be designed by applying the general method of effective interfaces directly (THELEN [1960]; THETFORD [1970]). An interesting design by Thetford is given in Fig. 18.
5 8.
Synthesized Layers
Most of the designs described so far require high indices of refraction which are still low enough to be possible but which are difficult to achieve. There are methods, though, to replace these less available materials with combinations of easily available ones. A graphical method was recently presented by Rock [1968]. OSTERBERG [ 19621 discussed a method based on admittance matching.
1 v.
5
SYNTHESIZED LAYERS
81
223
WAVELENGTH
Fig. 19. Reflectance of a synthesized four-layer antireflection coating on glass, H S = 1.52, n~ = 1; n, = 1.384, 4n1d, = 510nm n, = 2.35 , 4n,d, = 1020nm n3 = 1.55 , 4n3d3= 510nm n4 = 1.384, 4n4d, = 510nm.
4%
8 z
U
h 3
iw2 a
2
I
0 WAVELENGTH
Fig. 20. Reflectance of a synthesized four-layer vzs = 1.52, PZM 1.0; n, = 1.38, 4n,d1 = n2 = 2.30, 4n,d, = n3 = 1.38, 4n3d3 =
antireflection coating on glass.
5
510nm 1105nm 170nm n, = 2.30, 4n4d4= 85nm.
224
M U L T I L A Y E R ANTIREFLECTION COATINGS
[IV,
§ 9
A quarter-wave layer next to the substrate (index nF)can be replaced by two quarter-wave layers with indices nA and nB if the following relation is fulfilled: .AI%
=n
Fh'
Fig. 19 gives as an example a coating derived by OSTERBERG [1962] from one of the designs given in Fig. 15. A third method is based on the concept of Herpin equivalent layers (EPSTEIN[1952]). I n this way YOUNG [1965] derived the design shown in Fig. 20 again using one of the designs of Figure 15 as starting design. Since all synthesizing methods really only provide equivalent performance at the wavelength point of matching large deviations from the starting design can result. It is normally necessary to further optimize the thicknesses. Refining methods (BAUMEISTER [1958] ; THELEN [1969]) are generally used for this purpose (YOUNG [1965]).
5 9. The Coating of Reflection Reducing Multilayers The individual layers in a multilayer system are usually coated in succession by now-conventional thermal evaporation of the coating materials in a high vacuum chamber. An elevated substrate temperature of up to 300" C, and glow discharge cleaning contribute greatly to the adhesion and durability of the multilayer. Techniques for the controlled evaporation of some of the commonly used coating materials have been summarized by COX et al. [1962], Cox and HAS [1964] and BALL [1966]; certain mixed oxides have also been and PUTNER found to be suitable evaporants (THELEN [1965]). Proper control of the thickness of each layer may be afforded by measuring the reflectance of some suitably positioned monitor plate in the evaporation chamber and stopping the deposition when the reflectance has reached a predetermined level. A separate monitor plate area is recommended for each layer. To improve the control sensitivity for layers whose refractive indices are close to that of the monitor plate, the plate surface may be precoated with some suitable material in order t o simulate a relatively wide differencebetween the indices of the plate and film. Visual monitoring of layer thicknesses, though practised for single magnesium fluoride films and suggested for certain two-layer coatings (SAWAKIand KUBOTA[1953]), is inadequate for the control of multilayers.
OPTICAL PERFORMANCE O F COATINGS
225
Q 10. Environmental Stability of Multilayer Antireflection Coatings It is desirable for any optical coating to show good chemical and mechanical stability in addition to its optical performance; unless the coated surface is to be hermetically sealed, good coating stability is essential. Single layer coatings of magnesium fluoride only achieved widespread usefulness when, by glow discharge cleaning the substrate and depositing the magnesium fluoride at an elevated substrate temperature, good substrate adhesion and mechanically hard coatings could be produced. Multilayer coatings can also be made durable by employing these two basic techniques in the deposition process, together with proper choice of coating material. Environmental stability tests commonly applied include the (Scotch Tape) adhesion test and tests for abrasion resistance, exposure to humidity at elevated temperatures including temperature-humidity cycling, exposure to salt fog and salt spray and exposure to extreme temperatures (-260" to +300" C). Commercial multilayer coatings may be repeatedly cleaned without damage, exercising only the care usual in the handling of coated optics. This ability to remove greasy contaminants (thumb prints) from multilayer antireflection films is very important since a thumb print shows up vividly by reflection against the almost black background although on an uncoated glass surface it might pass unnoticed. Q 11. Optical Performance of Coatings and Increase in Transmission Through an Optical System Measured spectral reflectance curves of properly prepared films conform closely to computed data. Fig. 21 shows the measured spectral performance of a two-layer V-coating; the single surface reflectance has been reduced to the extremely low value of 0.02 % at the design wavelength. This figure may be contrasted t o the value of 1.3 % for a magnesium fluoride single layer on crown glass. Which of these two coatings is superior depends upon the application, for the reflectance of the V-coated surface rises steeply away from the minimum while the single layer reflectance curve is much flatter. Thus for white light applications the single layer will be more effective and the V-coating better for monochromatic light at its design wavelength. The V-coating of Fig. 21 was made for optics operating at the 633 nm helium-neon laser line.
226
M U L T I LAYE R
[IV,
AN T IREFLECTION COATINGS
I
11
nm WAVELENGTH
Fig. 21. Measured reflectance of a two-layer TiO,+MgF, reflectance a t the 633 nm helium-neon laser line.
V-coating giving 0.02
yo
WAVELENGTH
Fig. 22. Measured reflectancc of a three-layer antircflection coating consisting of MgF,+ZrO,+CeF, on crown glass. The coating was a quarter-half-quarter wave design a t 550 nm. (After Cox et al. [1962].)
s
111
OPTICAL PERFORMANCE O F COATINGS
227
WAVELENGTH
Fig. 33. Measured reflectance of a three-layer antireflection coating consisting of MgF, Nd,O, CeF, on crown glass. The coating was a quarter-half-quarter wave design a t 1100 nm. (After Cox e t al. [1962].)
+
+
Figs. 22 and 23 show the spectral performance of prepared threelayer films in the visible and near infrared wavelength regions; the reflectance is suppressed to lower than 4% over a span of several hundred nanometers. Fig. 24 is a scan of an experimental multilayer coating which shows suppression of the reflectance to below 0.8 yo over an extended spectral range of about 420 to 900 nm. Such wide-band coatings find application in instrumentation where both visual and near-infrared sighting is employed. In computing the effectiveness of systems in transmitting white light, rather than energy of a discrete wavelength, or the effectiveness of a coating in suppressing unwanted reflections, we need to integrate over the spectral region concerned. Thus the energy effectively reflected from a surface is given by
where R, is the reflectance of the surface a t wavelength A , E,l is the energy distribution of the incident light, S , is the spectral response of the detector, We thereby obtain the area below the spectral reflectance curve
228
M U L T I LAY E R A N T I R E F L E C T I 0N
I
400
500
600
700
LIV,
C0A T I N G S
800
900
5 11
nrn
WAVELENGTH
Fig. 24. Measured reflectance of a very wide band multilayer antireflection coating effective over the range 400-900 nm
weighted according to the characteristics of the source and detector. The fractional reduction of reflected light on coating a surface is thus
and it follows that the reflectance values for the coated surface can rise to quite high values in the spectral regions where E , and S , are small without impairing the usefulness of the coating. Thus, the coating of Fig. 22 is visually effective because the relatively high reflectivity at 400 nm occurs where the visual response is near-zero. Suppression of the reflectance of an optical surface leads to an increased transmission of light through that surface. For 2N surfaces the overall useful transmittance of an imaging system is T 2 X , T being the single surface transmittance. Fig. 25 shows how T 2 N varies with T for selected values of 2N. The improvement in transmittance can easily amount to one or more photographic stop numbers in a severalelement system. For a non-imaging system it is relevant to include the effect of intra-lens reflections, and Fig. 26 shows how the overall system transmittance then varies with T for selected values of 2N.
IV, § 111
OPTICAL PERFORMANCE OF COATINGS
PERCENT TRANSMITTANCE T FOR SINGLE SURFACE
Fig. 25. Overall useful transmittance of image-forming light for a system containing N elements in air as a function of the transmittance of a single surface.
It will be seen that for any selected value of the parameter 2N greater than unity the overall transmittance of the system is greater when the intra-surface reflections are considered. This transmittance increase is far from desirable for image-forming systems for it represents potential stray light in the image plane. Examination of Figs. 25 and 26 will show that the transmittance increase is greatest for large values of 2N and low values of the single surface transmittance. For any particular 2N value the increase is least when the single surface transmittance is greatest and it follows that the more effectively we
230
MULTILAYER ANTIREFLECTION COATINGS
PERCENT TRANSMITTANCE T FOR SINGLE SURFACE
90
95
100
Fig. 26. Overall transmittance of light for a system containing N elements in air as a function of the transmittance of a single surface. Both image-forming and stray light arising from intra surface reflections have been included.
can suppress surface reflections in a refracting systems the less stray light will arise. This is discussed more fully below.
12. Photographic Applications In the foregoing discussion of multilayer designs and their spectral characteristics the region of interest mainly considered was the visual, 400-700 nm region. If the multilayer coatings are to be used in visual
IV,
5
131
SUPPRESSION O F STRAY LIGHT
231
or pseudo-visual instrumentation it is appropriate to use a spectral reflectance curve which best fits this region. On the other hand, if the response of the system to be coated extends beyond these wavelength limits, close con. ideration must be given to the overall spectral response and the multilayer coating characteristics can critically affect this. With single layer magnesium fluoride coatings it is common practice to center the film at the wavelength of greatest interest, be this visual or photographic, and to stagger the coating thicknesses in a many-element system to avoid strong coloration in the transmitted light. The color effects likely to result when a many-element system is coated with single magnesium fluoride layers have been studied by MURRAY [1956]. SCHARF[1952] proposed a color index based on the total axial glass thickness and the difference between the transmission densities at 400 and 750nm. With multilayer coatings the proper selection of spectral response can be much more critical because, unlike single layers, the reflectivity can rise to values substantially in excess of the bare substrate values outside the low reflectivity zone. Thus, for example, a multilayer-coated lens reflecting a total of less than 1% of the incident light in the 400-790 nm region can reflect a total of 25% of the light at 1500 nm. The interference color by reflection has been calculated for 2- and 3-layer coatings by KUBOTA [ 19611. A lens manufacturer faced with the problem of coating a manyelement zoom lens for color photography must use a coating which produces acceptable color rendition in the final photograph. The spectral reflectance and absorbance of the coatings, the spectral transmittance of the lens elements and the response of the film all contribute to the system response and variations of the coating characteristics in the violet and near ultraviolet wavelengths can critically affect the color rendition. Some otherwise suitable coating materials become strongly absorbing at shorter wavelengths and those that remain transparent can not always be deposited controllably.
9 13. Suppression of Stray Light in an Optical System Although the advantages of reflection reducing coatings are nearly always explained in terms of the resulting increase of light transmission through the system in nearly every case the chief advantage lies in the reduction of unwanted light in the image plane (Fig. 27). This generali-
232
MULTILAYER ANTIREFLECTION COATINGS
[IV, §
13
Fig. 27. Intra lens reflections in an optical system. One possible path for stray light arriving at the image plane is depicted for a photographic triplet.
zation holds good whether we are considering a single lens (say an eyeglass lens) or a system containing very many lenses (say a flexible endoscope). The simplest case is perhaps that of the protective glass in front of an instrument dial. Sometimes proper optical design can lessen annoying reflections from the glass; for example, the glass can be formed concave t o the observer. The real solution though, lies in effectiv? use of surface coatings. Fig. 28 illustrates the increase in visibility of an instrument dial when the frontal glasses are multilayer-coated. For a sensitive instrument movcment whose pointer position may be influenced by the pressure of a static charge on the cover glass, the multilayer coating may be rendered conductive by incorporating a metal or conductive metal-salt layer. With proper design a resistivity of less than 350 000 12 per square may be realized with only imperceptibly increased reflectivi-
Fig. 28. The multilayer-coated instrument glass on the left greatly improves the visibility of the meter face by reducing unwanted reflections from the glass to negligible levels.
IV,
9
131
233
SUPPRESSION O F STRAY LIGHT
ty; the absorption of the conductive layer affects only the transmission of the coated surface and this typically falls to 30-90%. Intra-surface reflections in an eyeglass lens can give rise to irrelevant images when a bright source of light is in the scene, a situation habitually encountered under night-driving conditions. If the surface reflectivity of the lens is reduced to $yothese irrelevant images cease to be visible. Since the production of stray light by intra-surface reflections involves a double reflection (we do not normally consider any more than two reflections because the stray light intensity is then negligible), the effect of reducing the surface reflectivity from 5% to *yois to reduce the stray light inten;ity by two orders of magnitude. The successful suppression of stray light in the image by the use of multilayers stems from this fact. For a system involving many refracting elements the stray light increases because of the large number of surfaces that may form pairs. Moreover there is greater likelihood of a pair of surfaces causing the stray light to come to a focus in the region of the image plane. If this be the case the partially-focussed stray light may be of such brightness as to obliterate some wanted detail in the scene. We illustrate the increase of stray light with the number of elements-in-air of a refracting 201%
UNCOATED SYSTEM 16
12
!-
3
4
MUTILAYER- COATED SYSTEM
0
18
20
NUMBER OF ELEMENTS
Fig. 29. The stray light in the image plane as a function of the number of elements-in-air of a lens system. The single surface reflectivity has been taken as 4.0 yo for the uncoated system, 1.4 Yo for the magnesium fluoride-coated system, 0.35 yo for the multilayer-coated system. The stray light has been cxpressed as a percentage of the wanted light in the image.
244
MULTILAYER ANTIREFLECTION COATINGS
[IV,
$ 13
Fig. 30. Night time scene taken with a 15-element zoom lens coated throughout with single layers of magncsium fluoride. *4 150 watt P A R spotlamp in the center of the scene was aimed directly a t the camera lens.
Fig. 31. The same scene as in Fig. 30 has been taken under identical condition of scene, exposure and processing except that the zoom lens was now multilayer-coated.
Fig. 32. San Francisco’s Golden Gate Bridge a t night taken with a 15-element zoom lens coated throughout with single layers of magnesium fluoride. The driving lights of the parked car a t the foot of the picture were aimed directly a t the camera lens. Many of the irrelevant stray light images come to a focus in the region of the film plane.
Fig. 33. The identical bridge scene but with the zoom lens multilayer-coated.
236
MULTILAYER ANTIREFLECTION COATINGS
[IV, § 13
system in Fig. 29. We have here considered light passing axially through a centered system and computed the transmission factors for the system (a) when we ignore and (b) when we include the effect of intra lens reflections. The increase of the system transmission for case (b) represents the potential stray light in the image plane. Whereas a single layer of magnesium fluoride reduces the stray light by a factor of about 4 for a 10-element system for example, use of multilayers effect a further reduction by a factor of 10. Some of these effects are dramatically demonstrated by Figs. 30 through 33. The circular halo of stray light in Fig. 30 leads to an overall reduction of image contrast over a large area of the scene. The 15-element system was coated throughout with single magnesium fluoride layers and reference to Fig. 29 indicates that an uncoated system would produce a stray light intensity about three times as great. An identical zoom system was multilayer coated and Fig. 31 was secured under identical conditions of scene, exposure and processing. There is a major reduction in the level of stray light and contrast over much of the scene has been greatly improved. We should note here that the exposure times for the two photographs were identical but a more meaningful comparison would have been made had the exposure of Fig. 30 been increased (nearly doubled) to compensate for the lesser transmission of the magnesium fluoride-coated system. The contrast between the two photographs might then be expected to be even more striking. Figs. 32 and 33 were similarly taken with the same pair of zoom systems. With the zoom setting (focal length setting) chosen, several of the doubly-reflected stray light beams come to a best focus in the region of the image plane and generate an objectionable array of irrelevant images. Even quite minor images in a static scene become very obtrusive when the scene is panned; sunset scenes are classic cases where unwanted images intrude into the scene and it is not always feasible t o keep localized glare sources out of the object field. Fig. 33 shows how a high degree of stray light suppression can be achieved by using multilayers.
References BAUMETSTER, P., 1958, J. O p t . SOC.Am. 48, 955. Cox, J . T, G. HASSa n d A. THELEN, 1962, J . O p t . Soc. Am. 52, 965. Cox, J. T . and G. HASS,1964, Antireflection Coatings for Optical and Infrared
IV]
REFERENCES
237
Optical Materials, Vol. 2, in: Physics of Thin Films, eds. G. Hass and R. E. Thun (Academic Press, New York and London). DUFOUR, C . and A. HERPIN, 1954, Optica Acta 1, 1. EPSTEIN, L. I., 1952, J. Opt. SOC.Am. 42, 806. KUBOTA, H., 1961, Interference Color, in: Progress in Optics, Vol. 1 , ed. E. Wolf (North-Holland Publishing Co., Amsterdam). R. B., 1948, J . Opt. SOC.Am. 38, 20. MUCHMORE, MURRAU, A. E., 1956, J. Opt. SOC.Am. 46, 790. OSTERBERG, H., 1962, Military Standardization Handbook 141, Optical Design, Defence Supply Agency Washington D.C., Chapt. 21. PUTNER T. and R. BALL,1966, Vacnique 5, 3. ROCK,F., 1968, U S . patent pending. SAWAKI, T. and H . KUBOTA,1953, Sci. Light (Tokyo) 2, 128. SCHARF, P. T., 1952, J . SOC.Motion Picture Television Engineers 59, 191. SMAKULA, A., 1941, Glastech. Ber. 19, 377. SMITH,S. D., 1958, J . Opt. SOC.Am. 48, 43. THETFORD, A., 1970, A Method of Designing Three Layer Antireflection Coatings, Optica Acta, to be published. THELEN,A , , 1956, Optik 13, 537. THELEN,A , , 1960, J . Opt. SOC.Am. 5 0 , 509. THELEN, A , , 1965, U.S. Patent 3, 185, 020. THELENA , , 1969, Design of Multilayer Interference Filters, in: Physics of Thin Films, Vol. 5, eds. G. Hass and R. E. Thun (Academic Press, New York and London). YOUNG, L., 1961, J. Opt. SOC.Am. 51, 967. YOUNG, L., 1965, Appl. Opt. 4, 366.
This Page Intentionally Left Blank
V S T A T I S T I C A L P R O P E R T I E S O F L A S E R LIGHT* BY
H. R I S K E N I . Institut fur theoretische Physik der Universitat, Stuttgart, Gerrnany
* This article was written during the author’s stay at the Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota and was supported by Project Themis, Office of Naval Research.
CONTENTS
. . . . . . . . . . . . . . . . 3 2 . SEMICLASSICAL T H E O R Y . . . . . . . . . . . . 1.
$
INTRODUCTION .
244
3. SOLUTION O F THE LASER FOKKER-PLANCK
EQUATION . . . . . . . . . . . . . . . . . . . .
3 3
241
..
264
. . . .
272
4 . PHOTOELECTRON COUNTING DISTRIBUTION 5 . FULLY QUANTUM MECHANTCALTHEORY
252
REFERENCES . . . . . . . . . . . . . . . . . . . . .
291
§1. Introduction
In physics one may distinguish two developments. After the laws for certain quantities (e.g., the position of a small particle, intensity of an optical field, a current in a circuit) have been discovered, the fluctuation properties of these quantities (e.g., Brownian motion, fluctuations of light, noise in circuits) have been investigated several decades later on. The same development occurred in the laser field, only in a much shorter time period. After SCHAWLOW and TOWNES [1958] had shown how to extend the maser principle to the optical region, the first laser action was observed by MAIMAN [1960] and by COLLINS et al. [1960]. Then a large number of papers appeared dealing for instance with mode competition, frequency pulling and pushing and hole burning effects. A microscopic theory of these effects was given by several authors, the most complete investigations are the ones of HAKENand SAUERMANN [1963a, b] and LAMB[1964]. After the first laser was built, not much was known about the statistical properties of the laser light (eg., intensity fluctuation, linewidth) except that the linewidth was very small. One of the first attempts to derive the statistical properties of laser light was made by WAGNERand BIRNBAUM [1961] using a linear theory. As we know today the laser oscillation can only be described by nonlinear equations and therefore their results apply only to a light amplifier or a laser below threshold. The first microscopic theory of laser fluctuations seems to be the one of HAKEN[1964a, b] and soon afterwards several other papers appeared (HAKEN[1965, 19661, HAKENand WEIDLICH[1966], LAX [1966a, b], SAUERMANN [1965, 19661, RISKENet al. [1966a, b, c], PAUWELS [1966], KORENM A N N [1965]). In these investigations the nonlinear laser equations were solved by a linearization procedure, similar to the one used by BLACQUI~RE 1119531 and GRIVETand BLACQUI~RE 1719631 in the noise investigation of a maser oscillator. The results of the above laser theories agreed very well with noise measurements of FREED and HAUS [1965, 19661, ARECCHIet al. [1966], SMITHand ARMSTRONG [1966a], 241
243
STATISTICAL PROPERTIES O F LASER LIGHT
[v. § 1
and others. However, the linearized theories are not applicable near laser threshold. A more general solution of the nonlinear laser equation with noise, which was valid below, at and above threshold was derived by RISKEN[1965, 19661 and by LAXand HEMPSTEAD [1966], using a Fokker-Planck equation. The stationary solution of the laser FokkerPlanck equation was substantiated experimentally by SMITHand ARMSTRONG [ 1966b1, who showed that near threshold this solution is superior to those which followed from the linearized laser equations or from the signal plus noise model of LACHS[1965] and of GLAUBER [1966]. (For a review article pertaining to experimental studies of intensity fluctuation in lasers see ARMSTRONG and SMITH[1967].) After the appearance of these early papers, fluctuation phenomena in a laser were derived with more advanced methods. In the papers of WEIDLICH et a]. [1967a, b] and HAKENet al. [1967] the master equation of WEIDLICH and HAAKE [1965a, b] and the coherent state representation of GLAUBER [1963a, b] and SUDARSHAN 1119631 were used. SCULLYand LAMB[1966,1967a] and FLECK [1966a, b] derived and solved the master equation in the occupation number representation: MCCUMBER [1966], WILLIS [1967], GORDON[1967] and BRUNNER [1967] also developed a theory of laser noise. On the experimental side more detailed investigations were performed by ARECCHIet al. [1967a, b, c], CHANGet al. [1967] and DAVIDSON and MANDEL [1967]. Very good agreement was found not only with the stationary solution but also with the correlation function (derived from the Fokker-Planck equation by RISKENand VOLLMER[1967a] and HEMPSTEAD and LAX [1967]) andwith the transient behavior of the laser oscillation (calculated by RISKENand VOLLMER [1967b] and SCULLY and LAMB[1967b]). The main result of the theoretical and experimental investigation is that the statistical properties of laser light differ essentially from the properties of ordinary, i.e., thermal light, even if the linewidth of the thermal light would have been made as narrow as that of laser light ( e g , by use of filters) and if the intensity would have been made as high as that of laser light. For instance, the photoelectron counting probability distribution (for time intervals much shorter than the reciprocal linewidth) is a Bose-Einstein distribution for thermal light and for laser radiation far below threshold, whereas for laser light well above threshold the photoelectron counting distribution is a Poisson distribution. In the interesting threshold region one has a continuous transition between these two distributions. Because the laser threshold region is very small one may regard this transition as a phase transition.
v, 5 1;
I N TR 0 D U CTI 0N
243
For instance, below threshold the relative intensity fluctuations are large (phase a ) ; above threshold the relative intensity fluctuations are small (phase b). (For a more detailed discussion between phase transition and laser threshold see GRAHAM and HAKEN[1968].) The following two points make it difficult (and therefore interesting) to investigate the statistical properties of laser light. First, one has to treat a nonlinear equation with noise, preferably without a linearization procedure. Secondly, the noise originates from the quantum behavior of the light field (i.e., spontaneous emission). From the last point it follows that a rigorous description of the electromagnetic field should be given by quantum electrodynamics. Because of the large number of photons present in the laser resonator (number at threshold is of the order 4000), it turns out, however, that to a high accuracy (relative error of the order 1/4000) the light field can be treated classically provided a proper noise force is added, whereas the atoms are treated quantum mechanically (semiclassical theory). Near threshold the error due to this procedure is much smaller than the error due to the linearization of the laser equations. Although the accuracy of the semiclassical theory is high enough for a comparison with experiments, a more rigorous treatment is desirable from a theoretical point of view. The present review is divided into two main parts. In the first part (5s 2-4) the semiclassical theory is developed. The laser Fokker-Planck equation is derived in 5 2 ; this equation is solved in 3. Using a connection between photoelectron counting distribution and intensity distribution given by MANDEL [1958, 19631, the counting distribution is calculated and compared with measurements in 4. I n the second part (5 5 ) the fully quantum mechanical laser equations are derived using the master equation method of WEIDLICHand HAAKE[1965a, b]. I n 3 5.3 it is shown explicitly how the Fokker-Planck equation of the semiclassical theory can be obtained from the master equation without introducing any additional noise forces. Moreover first order quantum corrections are calculated in 3 5.3 for the purpose of supplementing the results of 3 3. It should be noted that the more logical approach would have been to start with 5 and then proceed with 3 3, where the laser Fokker-Planck equation is solved. Because the semiclassical theory is much simpler and because the small quantum correction to this theory has not been measurable up t o now, we have chosen the semiclassical laser theory as the starting point. In the whole article we have restricted ourselves to a homogeneously
644
STATISTICAL PROPERTIES O F LASER LIGHT
[v,
s
2
broadened two level laser system with one traveling mode operating not too far from threshold. The restrictions of homogeneously broadened, two level systems, and a running mode were made because of the simplification in deriving the laser Fokker-Planck equation. Without these restrictions one can derive the same Fokker-Planck equation as discussed in § 2. We have restricted ourselves to the region not too far from threshold for the following reasons: First, because of the low stabilization force near threshold, fluctuations have a large influence only near and below threshold, they have a negligible influence very high above threshold. Secondly, the theory is now well established not too far from threshold and it agrees with experiments very well. Third, for high pumping powers the one mode assumption is not valid even for a homogeneously broadened laser system with a traveling wave because in the high pumping region off resonance modes begin to oscillate too (see RISKENand NUMMEDAL [1968a, b, c], GRAHAM and HAKEN[1968]).
Q 2. Semiclassical Theory 2 . I . LASER EQUATION WITHOUT NOISE
In this paragraph we derive the semiclassical laser equations. Semiclassical means that we neglect the operator character of the light field and treat it as a classical variable but that the atoms are treated quantum mechanically. Since even at laser-threshold a large number (of the order 4000) photons are in the cavity the relative error due to neglecting the operator character is very small (of the order 0.1 yo) and has, up to now, no measurable influence on the photon statistics, see ARMSTRONG and SMITH[1967]. We postpone the more complicated fully quantum mechanically treatments to 3 5. I n order to derive these equations we confine ourselves to a running wave, single mode ring laser model with N two-level homogeneously broadened atoms. I t turns out that more general models (standing wave type resonator, inhomogeneous broadening, multi-level system:) lead, near threshold, t o the same equation; only the basic parameters of this equation are of a slightly different form. The equation of motion of the density matrix p(” for the p’th atom under the influence of an electric field E is given by
v,
5
21
245
S E M I CLASS1 CAI. T H E 0 R Y
where the Hamilton operator H
= H,-exE
(2.2)
consists of a free field part H , and an interaction part, -exE. The electric field is assumed to be polarized perpendicular to the laser axis z. Because we have assumed a two level system the density operator can be expanded in the two eigenstates 11) and 12) of the Hamilton operator H , Hol1)
(ilj)
= Elll), =
dij
H012)
=
%coo=
E212) &g-&1.
(2.3)
From (2.1) it follows that the equation of motion for the elements [&)I* = ( i l p ( @ ) l j )of the density operator p ( @ reads )
pi$) = &)
= iw, p g ) +i(e/%)x12E (pi$)-&))
-y2pi/”2) __
(2.4)
In deriving (2.4) and (2.5) we have assumed that there is no permanent dipole moment in the ground and excited state (11x1 1) = (21x12) = 0. The phase factors of the ground and excited states were chosen in such a manner that the matrix element x12= (1 1x12) is real. The underlined terms in (2.4) and (2.5) were added in order to describe damping of the off-diagonal elements (-y2pi$)) and of the diagonal elements (-y1(pi$)-&))) and in order to include pumping (ylo,/N). I n $ 5 we will see that these terms can be derived by adding reservoirs to the system. Eqs. (2.4) and (2.5) are the Bloch equations (y2 = l/T2, and BLOCH y1 = l/Tl) of the spin resonance theory (WANGSNESS [1953]) applied to the case of a dipole moment under the influence of an electric field. Eqs. (2.4) and (2.5) determine the density matrix elements under the influence of an electric field E. By the wave equation ( K describes the losses of the electric field)
the atoms react back on the electric field via the polarization 1 P(2, t ) = exla(fi$)(t)+f~)(t)). A (zP-z)eA
I:
(2.7)
The summation in (2.7) means that we have to sum up over a number of active atoms situated at zg which in turn are placed in a volume
246
STATISTICAL PROPERTIES OF LASER LIGHT
[v, 9 2
element A around z. This volume element may be so small that in it the electric field E ( z , t ) is practically constant, but so large that it contains a large number of active atoms. Since we are concerned with a one dimensional ring laser with one allowed direction of propagation, the electric field has the form of a traveling wave. Because of the relatively weak interaction with the active atoms, the amplitudes will change slowly. Therefore we can make the ansatz (V is the cavity volume) iw, (t-
v
2E0
where b ( t )is a slowly varying complex function with respect to the period 2n/w, (]&I << w,lbl) and where C.C.means that onehas to addthecomplex conjugate. The fact that b ( t ) does not depend on the space coordinate z restricts the field to one mode only. Furthermore it is implied in the ansatz (2.8) that we restrict ourselves to the tuned case (w,L/c = n 2 n ) , because the field must be periodic with the cavity length L. The normalization in (2.8) was chosen in such a manner, that b*b gives the intensity of the electric light field in photon numbers. Introducing
+
s ( t ) = 2 pi:) exp {iwo(t-zr/c)}
(2.9)
P
(2.10)
one obtains the following equations by inserting in (2.4)-(2.7) eqs. (2.8)-(2.10) and by neglecting antiresonant terms bf
Kb
= igs
S+y2s = -igbo &+yl(o-oo)
(2.11)
=
Zgi(s*b-sb*).
Here the coupling constant is defined by
where il = 2nc/w0 is the wavelength. The eq. (2.11) has a nonzero solution, if the pump parameter o,, is larger than the threshold value
> gthr
Ky2/g2,
(2.13)
" 3
9
21
247
SEMICLASSICAL THEORY
which is the SCHAWLOW-TOWNES [1958] formula. The steady state intensity is then given by (2.14)
Near threshold (oo M crthr) one can simplify (2.11)further. If the time variation of b and s is slow compared to the times l/y, and l/yz,the inversion is approximately (2.15)
Inserting the approximate inversion (2.15)leads, after neglecting the (small) derivative S, to the rotating-wave VAN DER POL [1927] or [ 18941 equation RAYLEIGH
b-(y-Bb*b)b
=
b-/?(d-b*b)b
=0
(2.16)
with
In deriving (2.16)and (2.17)we have assumed K << y z , which is usually fulfilled in a laser. The same eq. (2.16)is valid for a standing wave type resonator and for an inhomogeneously broadened line (see LAMB [1964],ARZTet al. [1966])and seems to be a general equation for every well behaved self-sustained oscillator near threshold. The form (2.16)is derived from the VANDER POL[1927] equation j-2j3(d-yZ)j+wZy
=
0
(2.18)
or from the RAYLEIGH [1894] equation j'-2/3(d-j2/3~2)j+w2y
=
0
(2.19)
in the following way. Inserting
y
=b
j
=
eciwt+c.c.
b e-iwt-iwbe-'wt+c.c.
y = i; e-iwt-
2iwbe-iwt-wz be-iwt+c.c.
into eqs. (2.18)and (2.19)leads, after neglecting the small underlined terms, to eq. (2.16)if higher harmonics are neglected.
248
STATISTICAL PROPERTIES OF LASER LIGHT
[v.
s
2
2.2. L A S E R EQUATION N E A R THR E S HOLD WITH N O ISE (LANGEVIN
METHOD)
Eq. (2.16) has a stationary solution of the form b = const. and therefore the electric field (2.8) has a pure sinusoidal time dependence without any linewidth and amplitude fluctuations. In order to obtain a finite linewidth or amplitude fluctuation one has, near threshold, to generalize the laser eq. (2.16). The simplest way of doing this is to include a noise term on the right-hand side of eq. (2.16), i.e., one has t o investigate
b-p(d-b*b)b
=
b--yb+pb*bb
= T(t),
(2.20)
where T(2)is a complex stochastic noise force (Langevin force) with zero average
( T ( t ) )= 0 .
(2.21)
This procedure is in close analogy to the theory of Brownian motion, where the velocity v of a particle obeys an equation of the form
i r f c r v=
r,
(2.22)
[1943]. The fluctuating force r see for instance CHANDRASEKHAR has to be added in (2.20) because the damping process (for instance, spontaneous emission) is not a continuous process as treated in 9 2.1 via a continuous damping term. If, for instance, a light quantum is emitted spontaneously, this is a discrete process. Therefore the final field is disturbed by such a process in a way similar to the way as the velocity of a small particle is changed by a collision process in the theory of Brownian motion. Since the process of spontaneous emission is much faster than the movement of the electric field b (t)the expectation value of two Langevin forces with a different time argument must be zero, i.e., they are &correlated. Since by the spontaneous emission no phase is produced, the expectation value of two Langevin forces is of the form
(2.23) (ri(t) r,(t')> = +es , j s ( t - t f ) , is the imaginary part of r. In where r, is the real part of r and complex notation (2.23) is equivalent t o
(r(t) r*(t')) = Q d(L-t');
(r(t) r(t')) = 0.
(2.23a)
The strength Q of the Langevin forces cannot be derived from a semiclassical equation, since spontaneous emission is a typical quantum
v, § 21
249
S E M I C L A S S I C A L T H E 0R Y
mechanical effect. For a rigorous method of deriving the laser equation see 3 5, where the fully quantum mechanical equations are given. In this section we determine the strength of the Langevin forces by the requirement that ( 2 . 2 0 ) and ( 2 . 2 3 ) lead to the correct spontaneous emission rate. Multiplying ( 2 . 2 0 ) with b* and adding the complex conjugate equation leads to (d/dt) (b*b ? =By( b* b 1-28( (b*b) )
+
+(T*(t)b ( t ) ) + ( b * ( t )
W?.
(2.24)
The expectation values containing the Langevin forces can be calculated using (2.20), (2.21) and ( 2 . 2 3 ) according t o
(r*(t) b ( t ) ) = (/'*(t)/'
6(z) dz)
--03
-J:,
-
(l'*(t) r ( z ) ) d z
Thus we finally obtain by inserting the inversion and the constant y (2 . 1 7 ) (dldt) (b*b)
=
g2
(2.25) =
+Q.
CT = N,--N,
2 - ((N2-Nl) b*b)-2~(b*b)+Q.
(2.15)
(2.26)
Y2
The first term describes the induced emission rate, the second the loss rate due to the cavity losses. In order to obtain the correct quantum mechanical spontaneous emission rate, we have to choose (2.27)
according to the Einstein theory of radiation. (For the sake of simplicit y we have neglected the number of thermal quanta, which is very small for laser light.) The number N , of atoms in the upper state depends on b*b because the inversion depends on it, i.e., N , = +,(N+o), however, near threshold N , and therefore Q may be regarded as a constant. 2.3. LASER EQUATION WITH NOISE (FOKKER-PLANCK EQUATION METHOD)
Although an equation for the moments ( ( b * b ) " ) can be directly derived from the Langevin eqts. (2.20), (2 . 2 3 ) and ( 2 . 2 7 ) , the method of introducing distribution functions is more convenient when treating
250
STATISTICAL PROPERTIES O F LASER LIGHT
[v. § 2
a nonlinear equation, as eq. (2.20). Since the rate of spontaneous emission Q is much larger than the rate due t o induced emission /Id and because the Langevin forces are &correlated and the light field is treated classically (continuous Markov process) the distribution function W ( b , t ) ( W ( b , t ) d 2 bis the probability that the amplitude b ( t ) lies in the range b < b ( t ) < b+db) can be calculated by the FokkerPlanck equation which reads in vector notation (b, = Re b, b, = Im b)
r
The drift coefficient
A i and the diffusion coefficient Qii are given by
1
A i = lim - ( b i ( t + A t ) - b i ( t ) ) l b ( t ) = b AT+O
(2.29)
AT
where Ib(t)=b means that at time t the expectation of b ( t ) has the sharp value b (see BHARUCHA-REID [1960], WANGand UHLENBECK [1945]). Integrating the Langevin eq. (2.20) and using vector notation, we have
b,(t+At)
=
bi(t)+ / t t + A r
[ p (d - Ib ( T ) I 2, bi ( T ) -Ti( T )]dt. (2.31 )
Inserting this expression into (2.29) and (2.30) leads, after using (2.21), (2.23) and (2.27), to
A i = p(d-b*b)bi
(2.32)
Thus the final Fokker-Planck eq. (2.28) reads in vector notation b = {b,, b,} (RISKEN[1965], LAX[1967c])
aw + /IV((d-~bl2)bW)= qAW, -
at
(2.34)
where V and the Laplace operator A act with respect to b. This final equation depends only on three parameters /I, d , q. Whereas the parameter /I and q are constants for each laser, the parameter d is variable and describes the intensity of the pumping (d < 0 below, d = 0 at
",
5 21
251
SEMICLASSICAL THEORY
and d > 0 above threshold). In the next section we will see that ,8 and q can be determined by measuring the intensity and the linewidth of the intensity fluctuation at threshold. The Langevin eq. (2.20) and the Fokker-Planck eq. (2.34) describe the noise of most self-sustained oscillators near threshold. Here it was derived for a homogeneously broadened two-level ring laser model, but it is also valid for a three-level system (HAKEN[1966]), inhomogeneous broadening (ARZTet al. [1966]), semiconductor laser (HAUG and HAKEN[1967]) and for a special cas? of a parametric oscillator (GRAHAM [1968]). In these papers the constants /3, d and q are given in terms of the fundamental constants. The Langevin eq. (2.20) with real parameters p, d, q describes noise near threshold in the absence of detuning. With detuning the parameters p, d are complex (see RISKEN and VOLLMER [1967b]). For numerical purposes it is convenient to introduce normalized variables
-b = (,4/q)*b; f = dpxI;
€
= d&t;
a
=
dGd.
(2.35)
Eq. (2.34) is then transformed into
aw
-
+v((a-~bl2)bW}= aw at
-
(2.36)
which only depends on the pump parameter u (a < 0 below, a = 0 at, and a > 0 above threshold). The bar over V and A indicates that one has to differentiate with respect to 6. In polar coordinates (6 = F e'q), this Fokker-Planck equation has the form
The scaling parameters 1/(q/p) and 1/2/(pq) can be determined experimentally by measuring the photon number in the cavity and the linewidth of the intensity fluctuation near threshold. Using measurements of ARECCHIet al. [1967c] we estimate 1/(q/p) = 4000, 1/1/(&) = 10-3 sec. From the distribution W(b,t ) expectation values containing one time only can be calculated. In order to obtain expectation values containing different times, one must know the joint distribution function W , [ W , ( b ( zt,; ) , b ( , - I ) , tz-l; . . .; b(1), t,)d2b(z)d2b(z-1) . . .d2b(l) is the joint probability that b(ti) lies in the interval b(i), b(i)+db(i)].
252
[V# §
STATISTICAL PROPERTIES O F LASER LIGHT
This joint distribution can be expressed by (t, > tz-l
3
> . . . > tl)
2
W,(b(”,t,; . . .; b(’), t l ) =
IT G(b‘i’, bci-l),
lti-ti-1l)W(b‘’), t l ) ,
(2.38)
i=2
where G is the Green’s function of the Fokker-Planck eq. (2.34), i.e., G has the initial value G(b(2),b(1), 0)
=
(j(b(2)-b(l)).
(2.39)
Q 3. Solution of the Laser Fokker-Planck Equation 3.1. STATIONARY SOLUTION AND ITS EXPECTATION VALUES
The stationary solution W ( f ,9,f) of the Fokker-Planck equation (2.37) surely does not depend on the time t. It does not depend on
the phase 9 either, because no direction of the b-plane is preferred. Hence the stationary solution W,,(f) has to obey the equations
Here S can be interpreted as a probability current in the f direction, which is a constant because of the first part of (3.1).This current must originate either from the origin or from infinity. In the present case, it has no physical meaning to assume a current S different from zero. Furthermore, it can be shown that for S # 0 the distribution function W s t ( f )does not go to zero sufficiently fast enough for 7 4 00. When S = 0 the stationary distribution function follows immediately from (3.1), viz., Jlr Wst(f) = -exp{-p+&zv2} 2n
1 - = /om
“4
exp{-p+&z
Jlr
= - efa2exp
2n
{-$(F-u)~) (3.2)
P}f df = F,(a) .
This stationary distribution function is shown in Fig. 1 for different pump parameters near threshold. It is a Gaussian distribution in the intensity I = f 2 with the mean value f = a truncated at f = 0. The moments M n and the generating function M ( s ) of the moments of the stationary distribution function are given by
Mn
=
(f.)= F,(a)/Fo(a)
~ ( s= ) <e18) = ~ , , ( a + 2 s ) / ~ ~ ( a ) ,
(3.3) (3.4)
v, S 31
THE LASER FOKKER-PLANCK EQUATION
-
I
I
2
4
6
1
8
253
1
1
0
Fig. 1. The stationary distribution (3.2) as a function of the normalized intensity
I=
P2.
where the integrals
can be reduced to the error integral 2
@(a) = -
1/n
Ia
e-"' dx
0
by the following recurrence relations
F,(a)
=
2(n-l)F,~,(a)+aF,_,(a);
F,(a) = 2+aF,(a) Fo(a) = &exp i (&~2))[l+@(+a)].
(n 2 2 ) (3.7)
The integrals F , are related to the parabolic cylinder function D , ( z ) by F,(a) = n ! 2:(n+l) $a2D - n-1(- 4 1 / 2 1 (3.8) (see GRADSHTEYN and RYZHIK[1965]). Because the generating function of the cumulants K , of the
254
STATISTICAL PROPERTIES O F LASER LIGHT
[vs
s
3
stationary distribution (3.2) can be expressed by (see for instance STRATONOVICH [1963] for a definition of cumulants)
the cumulants K,(a) are derivatives of each other K,+,(a)
dn d ds" ds
= --lnF,(a+2s)Js=,
d" da"
= 2"-KK,(a)
-10
-8
-6
-4
=
-2
= 2n-
d 2-Kn(a). da
0
2
4
d" d - InF,(a+2s)js=, da" ds (3.10)
6
8
10
Fig. 2. The first moment < f > = M , ( a ) = K , ( a ) (solid line) and the asymptotic expansions K,(a) = 2/lal--8/lals for a << --1 and K , ( a ) = a for a >> 1 (broken line) as a function of the pump parameter a. The unnormalized intensity ( I ) (in photon numbers) is only valid for threshold photon number = 4000 found by ARECCHIe t al. [1967c].
The first moment M,(a) = K,(a) is shown in Fig. 2 as a function of the pump parameter; in Fig. 3 the first four cumulants are plotted. The stationary distribution ( 3 . 2 ) was found by RISKEN[1965], LAX and HEMPSTEAD [1966], FLECK [1966a, b], SCULLY and LAMB[1966, 1967al WEIDLICH et al. [1967a]. The result of Scully and Lamb has a different form which, however, is practically identical to ( 3 . 2 ) . For unnormalized units the intensity is O ( a ) > = d41B (I(a)>. Thus ns = d a ( f ( 0 ) )= 4% 2 1 4 % is the number of photons at threshold, which ARECCHIet al. [1967c] have found for their laser to be 4000.
I
2.0 1.6
-
-
1.2
-
-
0.8 -
-
-
-0.4 -
-10
I -8
I -6
I -4
I -2
I 0
2
4
I
I
6
8
10
3.2. EXPANSION I N EIGENMODES
In order to obtain the joint distribution of Z'th order in eq. (2.38) or in order to give a transient solution of the Fokker-Planck eq. (2.37) it is sufficient to know the Green's function of the Fokker-Planck equation. This Green's function can be expressed in the following way (RISKENand VOLLMEK[1967a, b], HEMPSTEAD and LAX [1967])
where vnmare the eigenfunctions and A,, one dimensional Schroedinger equation ~:m+
[Anrn-vn(p)I~nrn
are the eigenvalues of the =
0
(3.12)
with the potential
Because of (3.2) and (3.11) the eigenfunction yo, belonging to the stationary eigenvalue I,, = 0 is
256
[v.
STATISTICAL PROPERTIES O F LASER LIGHT
s
3
-
yoo(f)= V‘FM exp{-+~*+$a~2).
(3.14)
The yrLnl(F) are assumed to be normalized. Thus, because of the orthogonality, we have (3.15)
The expression (3.11) is a Green’s function of the Fokker-Planck equation because, as one may verify by insertion, G is a solution of (2.37) and because of the completeness relations l o o
M
s(Y-Y’)
=
2 y n m ( f )y n m ( y ’ ) ; ~ ( v - q ~=’ ) 27G 2 -
m=O
n=-w
ein+P‘)
2
(3.16)
the initial conditions are given by G(Y, 31; F’, p’; 0)
1
= zS(f-F’) S(y-p’)
Y
=
S(6-6’).
(3.17)
I n order to obtain numerical results, one has to calculate the eigenvalues and eigenfunctions of the Schroedinger eq. (3.12). Only below (a << -1) and above (a >> 1 ) threshold this Schroedinger equation can be solved analytically (except of course for the stationary solution yoo).Thus one has to use either approximations (for instance a variational method, RISKEN[1966]) or a numerical integration of the Schroedinger equation (RISKENand VOLLMER [1967a, b], HEMPSTEAD and LAX[1967]). 3.3. CORRELATION FUNCTIONS
The two most important correlation functions for the light field are the correlation function of the amplitude (d2b= d(Re b) d(Im b ) )
g(a, t) = ( b * ( t + t ) b ( t ) > =JJb*b‘
W,(b, b‘; t)dZb d2b‘
(3.18)
and the correlation function of the intensity fluctuation
K ( 4 ).
=
< (Ib (t+.)
-,1
(lbl”)
(16 (4 12-
(Ibl”))
=JJ ( ~ b ~ 2 - ( ~ b ~ z ) ) ( ~ b ’ ~ z - (W,(b, ~ b ~ zb’; ) ) t)d2bd2b‘
(3.19)
(see BORNand WOLF[1964], MANDELand WOLF[196l]). In the stationary state, which will be treated here, these correlation functions and the joint distribution function of second order W ,
V,
s
31
257
THE L A S E R F O K K E R - P L A N C K E Q U A T I O N
depend only on the time difference t. Using (2.38), where W is now the stationary distribution (3.2), we obtain by inserting the explicit expression (3.11)
W 2 ( ffp; ,
f ' , CpI'> ).- = c
a
m
2 2
'no(') ~'nn"') y n r n ( fy)n r n ( f ein(p-m')exp{-AnrnI?I}. ') (3.20) 2~cf 2nf'
m=On=--m
Carrying out the integration leads to (in normalized units)
(3.21)
(3.22)
The correlation functions for
z=0
were already calculated in
3
3.1
Since all matrix elements V mare positive and their sum is one (3.24)
the Vm give the relative influence of the m'th order damping term. The Fourier transform of the correlation function gives the spectral profile. Because the correlation functions are sums of exponential functions, the spectral profile is a sum of Lorentzian functions. However, the influence of other Lorentzian lines beyond the lowest order is not very pronounced for the correlation function of the amplitude. Calculation of V t ) shows that l - V t ) = 2z=lV i ) is of the order of 2 % near threshold and smaller outside. Therefore the spectral profile is nearly a Lorentzian with a linewidth (in unnormalized units) -
AV = d p q Aln = u L q / ( I ) ; uL = Aln(I>.
(3.25)
The decay constant ill, is plotted in Fig. 4 as a function of the pump
258
STATISTICAL PROPERTIES O F LASER LIGHT
[v, § 3
Ii
'I
I: ' I
1
,'i I 1
!
Fig. 4. The eigenvalue ,Ilo (solid line) and the asymptotic expansionsl.,, = la1+4/lal for a << -1 and ,Il,= l / a for a >> 1 (broken line) as a function of the pump parameter a.
1.0
,It
I " ' " ( O
-10
i -8
-6
-4
-2
0
2
4
6
8
10
Fig. 5 . The linewidth factor U L = AlO(I) as a function of the pump parameter a.
parameter and in Fig. 5 we have plotted uL as a function of the pump parameter. The factor uL varies continuously from 2 to 1 by passing through the threshold region (RISKEN[1966], see also LAX 11967~1, LAXand LOUISELL119671); thus it connects the linearized laser, and the oscillator theories, below and above threshold (BLACQUI~RE [1953], GRIVETand BLACQUI~RE [1963]). The ratio of 2 to 1 occurs because above threshold the laser amplitude is stabilized and there-
Fig. 6. The first four matrix elements V E ) as functions of the pump parameter a (see (3.22)).
THE LASER FOKKER-PLANCK EQUATION
35
259
c
15
10
5 I
-10
-8
,
-6
I
-4
I
-2
I
,
0
2
I
4
I
6
8 a 10
Fig. 7. The first four non-zero eigenvalues A,, and the effective eigenvalue Aeff (see (3.26)) as functions of the pump parameter a. Very below ( a << - 1 ) and high above threshold ( a >> 1) the effective eigenvalue can be approximated by Jeff = 21al.
fore only half of the noise power (in yj direction) contributes to the linewidth. The correlation function of the intensity fluctuation does not contain only the lowest decay constant, as was approximately the case for the correlation function of the amplitude. One sees in Fig. 6 that approximately 4 terms of the series (3.22) have to be included for pump parameters slightly above threshold. A closer inspection of the potential V o ( f )of the Schroedinger equation shows (see RISKENand VOLLMER [1967a]) that for high pump parameters the eigenvalues are pairwise degenerate. (See also the plots of these eigenvalues in Fig. 7 and the eigenfunction and the potential in Fig. 8.) For large pump parameters only one decay constant prevails in agreement with the linearized laser theories (see HAKEN[1964a]). As was shown by RISKEN and VOLLMER[1967a], one may introduce an effective correlation function and decay constant, defined by
which agrees well with the exact expression (3.22) even slightly above threshold. The effective width Aeff is, however, 25 % larger than the lowest decay constant for a w 4.5. This deviation of Aeff from A,,, was substantiated experimentally by measurements of ARECCHIet al.
260
iv, § 3
STATISTICAL PROPERTIES O F LASER LIGHT
1
40
20
0
Fig, 8. The potential V , of tb.- :,chrocdinger eq. (3.12) and tlic first flvc and eigenfurctions for the pump parameter a = 10.
* ’ .:,
%’:!
uos
[1967b]. (A table of c l g vdiies,matrix elements and Aeff is contained in RISKEN[1968].) For unnormalized units the linewidth of the spectrum of the intensity fluctuation is thus given by 7
A%).( = 4%Aeff ( a ) . At threshold a typical value of Av,/(2n) is 1400 CIS (see ARRECHIet al. [1967c]). Below threshold (a << -1) the correlation function of the intensity fluctuation is the square of the correlation function of the amplitude, (3.27) This can be shown by using the asymptotic values of the cigenvalues and of K , and K,. It was proved by MANDELand WOLF[196l] that the relation (3.27) is valid provided the light field is a Gaussian process. Because (3.11) is a general Green’s function, higher order joint distribution functions (2.38) and correlation functions can be constructed. These, however, will not be given explicitly in this article. 3.4.TRANSIENT O F THE LASER OSCILLATION
Generally, the time dependent distribution function reads W(V,pl, f)
= /om/ozn
G(V, q ;f ’ , q’;i) W(V’,pl’, 0 ) f ’ df’ dpl’, (3.28)
v. 9 31
261
THE LASER FOKKER-PLANCK EQUATION
where W ( f ,v, 0) is the initial distribution. If we assume that the laser is switched on by suddenly changing the pump parameter from -a to a value near threshold (for instance by a sudden change of the resonator losses) the initial distribution is the distribution which contains no photons,
W ( f ,v, 0) = lim E+O
1
(3.29)
-d ( f - 8 ) .
2nf
Inserting the Green’s function (3.11) and (3.29) into (3.28) leads to
(3.31)
It can be shown by a power series expansion that y o I , L is ( ~proportion) al to 2/&. Therefore the constant (3.31) has a well defined finite value. For numerical calculations the series (3.30) must be truncated. Because of this truncation the expansion (3.30) cannot be applied to very small times i. For small times, however, the nonlinearity in the drift coefficient of the Fokker-Planck equation (2.36) can be neglected 1 -
-
__---
I
I
II*.H
F a = 8 ; ,Lo,= 14.651
0.8 -
0.6 -
\ 0.4 -
0 0
2
4
6
Fig. 9. The transient distribution function (3.33) for a
=
8.
262
[vt § 3
STATISTICAL PROPERTIES O F LASER LIGHT
I
I
‘ m ’
I
I
I
q30000
1
20000
I
0
O.ims
Fig. 10. The transient mean intensity of the distribution function (3.33) as a function of time for various pump parameters a. The unnormalized intensity (I> (in photon numbers) and t (in milliseconds) are valid for a threshold photon number 4000 and for a threshold linewidth of the intensity fluctuation 1400 CIS as found by ARECCHIet al. [1967a]. The three indicated regions are: I - region of spontaneous emission; I1 region in which spontaneous quanta are amplified by induced emission; I11 - saturation region. The solution I ( f )= a I ( 0 ) exp(2af)/[a- I(0)+I(O) exp ( 2 a f ) l of the normalized eq. (2.16) (B = 1; d = a ; I = 6.6) without noise is dotted in for a = 8. The initial value I(0)was chosen in such a manner that the solutions with and without noise agreed in the end of region 11.
and the distribution function can thus be approximated by the Gaussian function
The distribution (3.30) does not depend on the phase p because the initial distribution does not depend on it. The transient distribution of the intensity (3.33) Jo
is shown in Fig. 9. I n Fig. 10 and Fig. 11 the transient mean intensity
I
V,
31
263
THE LASER FOKKER-PLANCK EQUATION
( ( I ( f ) and ) the transient variance ( ( I ( f-)( I ( f ) ) ) 2 )are plotted. The moments ( I k ( f )obey ) the equations (Io = 1, K 2 1) d (I”(i)) = 2Ka(Ik (i)) -2K(f“+l (f) ) 4K2(I”’c-1 (f) ), df
+
-
(3.34)
which can be derived directly from the Langevin eq. (2.20) or the Fokker-Planck eq. (2.36). The initial condition that no photons are present for t = 0 is
(I”(0))= 0, k 2
(3.35)
1.
Using (3.34) and (3.35) the first two coefficients of the Taylor expansion for ( I ( i ) )are easily found to be
( I ( t ) )= 4f+4aP+ . . . .
(3.36)
Thus, in normalized units, all curves of the transient mean intensity start with the same finite slope. I n order to find the physical significance of this effect, we rewrite the expression in unnormalized quantities. We obtain (neglecting the number of thermal quanta)
( I ( t ) ) = AN,t+B(N,--N,),N2t2--B(N,--Nl),hrN2t~+ . . . (3.37) A
= I
2g2/y2; B = 2 g 4 / ( y J 2 ; I
I
I
I
W2--NJthr = Kyz/g2. I
I
I
I
(3.38) 1
-
5
-&to7
-
4
-
3
-
c\
-4.107
I
; 1
-
-.-
a
; 5
-
’?
I
I
L L
1 2
2
4
‘2
-2.107
b
1
:
-
o=O
I
I
0
Fig. 11. The transient variance of the distribution (3.33) as a function of time for various pump parameters. The unnormalized intensity is valid for the same threshold values as in Fig. 10.
264
S T A T I S T I C A L PROPERTIES O F L A S E R LIGHT
[v, § 4
We see that the term linear in t stems from the spontaneous emission rate. The second term stems from the spontaneous emission, which is amplified by induced emission. The third term describes the losses of the spontaneous quanta due to their finite lifetime in the cavity. Above threshold, the sum of both terms, which is quadratic in time, is positive. Higher expansion terms of (3.37) stem from induced emission and from the change of inversion. Thus we can mainly distinguish three regions: Spontaneous emission (region I ) , amplification of spontaneous emission by induced emisstion (region 11), saturation effects of the inversion (region 111). This is also shown in Fig. 10. The transient mean squared deviation or variance has the following interesting property above threshold (a > 0). It reaches a maximum value in the beginning of the saturation region I11 as shown in Fig. 11. This means of course that for this time, the spread of the distribution function is largest (see Fig. 9). For larger times the variance becomes smaller and finally reaches its stationary value. This can be physically interpreted as follows: Since in the switching on process the spontaneous photons are amplified, small fluctuations of the spontaneous photons lead, because of the large amplification, to large fluctuations in the beginning of the saturated region 111. Because of the large stabilization of the nonlinearity, these fluctuations are then diminished and thus reach finally their relatively low values. A t threshold (a = 0 ) no amplification occurs. Due to the low stabilization effect of the nonlinearity for a = 0, the variance reaches its largest value for t --f 00. No maximum occurs for finite 1 and a < 0. The transient calculations are in agreement with measurements performed by ARECCHIet al. [1967a].
5
4. Photoelectron Counting Distribution
4.1. GENERAL RELATIONSHIPS BETWEEN T H E PHOTOELECTRON
DISTRIBUTION AND INTENSITY DISTRIBUTION
The distribution function of the light field inside the laser cavity is not measured directly. Usually one measures the intensity outside the laser cavity with a photon detector. Because only a small fraction of the light intensity is transmitted by the mirror and finally is absorbed by the detector, one usually counts only a few photoelectrons in a given time interval T . For this reason one cannot directly apply the results of 3 3, where the intensity was treated as a continuous variable, but we must look for the discrete distribution of the photoelectrons.
V,
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41
PHOTOELECTRON COUNTING D I S T R I B U l I O N
265
The connection between the continuous intensity distribution and the discrete photoelectron distribution was given by MANDEL [ 1958, 19631. (See also KELLEYand KLEINER[1964], MANDEL and WOLF [1965] and 9 5.3 for a quantum mechanical derivation.) His result is that the probability $(%, T , t ) of finding TZ photoelectrons in the time interval t , t+T for a given intensity I ( t ) is given by p(rz, T , t ) = n ! [ G C ’ J ~ ‘ + dt’] ~ I ( ~exp ’)
{-
ct’Jt‘+TI(t’)dt’). (4.1)
In (4.1), x’ is the factor relating the average number of photoelectrons in the interval T and the time averaged intensity, i.e.,
rT
00
5=
2 np(n, T , t ) = GC’ a==0
I(t’)dt’.
(4.2)
The factor GC’ is determined by the mirror transmittance, by a geometrical factor giving the fraction of the intensity which falls on the photodetector and by the quantum efficiency of the photocounter. Since I ( t ) is not a given quantity but a stochastic variable, one must average (4.1) over the distribution of this stochastic variable in order to get the measured photon distribution, i.e.,
P(%
T ) = .
(4.3)
In (4.3) ( ) denotes the averaging process of the stochastic light intensity. The N-fold photoelectron counting distribution p (a1,T I ; n,, T,; . . .; a,, T,) of finding in a counter 1, n1 photoelectrons in the interval (tl, t,+T,), in counter 2, n, photoelectrons in the interval (t, , t,+ T,) and so on reads
P(%T,:
n2,
T,, . . .; a,, T , ) =
=
(P(%>
Tl, tl) P(%> T,, t z ) . . .P(%> T , , &)>> (4.4)
see BEDARD[1967b]. Generally, the calculation of the average in (4.3) and (4.4) is complicated mainly because one needs the joint distribution function for all I(t’) with Min ti 5 t’ 5 Max ti+Ti, i.e., a joint distribution of infinite order. 4.2. COUNTING DISTRIBUTION FOR SHORT INTERVALS
In this subsection we assume that the time interval T is short compared to the time in which the intensity changes its value appreciably
266
STATISTICAL PROPERTIES O F LASER LIGHT
TdPq = T
<< I/&.
[v, §
4
(4.5)
Because:the intensity does not change in the interval T one may write
r T I ( t ' )dt'
=
T I ( t ) = TdaI(t)
and average according to the one fold distribution w(1, i) which, in the stationary state, is
+
w ( I ) = z W s t ( d I )= JV exp {-$P++uI).
(4.7)
Therefore the photon counting distribution for short intervals is given by the Poisson transformation (for the inverse of this Poisson transformation see WOLFand MEHTA [1964], BBDARD[1967c])
9 (n,T ) =
lom $ (I) ecYfw
dI,
(4.8)
where v is an abbreviation for
Inserting (4.7) into (4.8) we obtain vn F,, (u -2v)
9(%T ) = -n ! Fo(u)
'
(4.10)
where the integrals F,(u) are defined in (3.5) or (3.7). The counting distribution (4.10) can be expressed in terms of parabolic cylinder functions, by (3.8) (see BBDARD[1967a]) or by the incomplete I' function (see SMITH and ARMSTRONG[1966b]). Typical counting distributions are shown in Fig. 12. For large average photon numbers (large v), the photoelectron counting distribution is nearly identical to the continuous intensity distribution
see Fig. 12c. The relations between the cumulants of the photocounting distribution (4.12)
reads in our notation (for a derivation see MANDEL [1958, 19631)
V.
I
41
267
PHOTOELECTRON COUNTING DISTRIBUTION
1
(n)= 1.12
0 1
1 0
2 1
3 1
6 1
5 2
6 1
7 3
I
1
a
-nV n
-
b n
0 1 2 3 4 5 6 7 s i 101 121 I41 161 1'81 1101 1/21 l l 4 l n
C
0
1
2
3
4
5
/d. dk' d$ db
40
6
7
'
11111111 11111111 11111111 l l l l " y ~ " l l ~ ' I I l l l l ~ ~
50
60 n
Fig. 12. The discrete photoelectron distribution (4.10) and the continuous distribution (4.11) for various v parameters (a: v = 0.5; b: v = 2; c: v = 8) but for the same pump parameter a = 2.
k,(a) =
= YK,(U)
(92)
k&) = ( ( a - ( a ) ) 2 ) = 'YK,(a)+~2K,(U) k, (a) = YK, (u) 3Y2K2(a)+Y31<, (a)
+
(4.13)
+
k , (u) = YK,(a) 7v2KZ( a )+6y3K, (a)+v4K4( a ) . The cumulants K n ( u ) of the stationary intensity distribution are plotted in Fig. 3. The k'th factorial moment of the pliotoelectron distribution (4.8) 00
(?PI)
=
2 a(%-1)
n=O
* *
. (n-k+l)p(a, T)
(4.14)
268
STATISTICAL PROPERTIES O F LASER LIGHT
[v. 3 4
is the k'th moment of the stationary distribution w(1)
(?PI)
= ((V1)k)
= vk
F,(u)/F,(u)
(4.15)
which, because of (3.8), can be expressed by the parabolic cylinder functions (ARECCHI et al. [1967c], BBDARD[1967a]). It should bc noted that the relationship between intensity cumulants viK, arid counting cumulants K,(u), i.e., the inverse of (4.13), is the same :IS that between factorial moments ( ~ [ ~ land ) moment:< (:ai)(see footnote, CHANGet al. [1967]).
' '
0.25t-
i i ........ ........ .__..
P in1
T--
"7 i
1
0.15 -
0.1
.__-
'-nonjinear oscillator
.. .....
0.05 -
0
"
0
'
1
2
4
6
8
n
Fig. 13. Counting distribution just above threshold (solid line), the Poisson distribution for the same + (dotted i lines) and the nonlinear oscillator distribution (4.10) giving the best f i t (dashed line). For n = 7, 8 and 9 the nonlinear oscillator and observed distributions are coincident. (Reproduced from SMITHand ARMSTRONG[1966b].)
The photoelectron counting distribution p (n, T ) near threshold was first measured by SMITHand ARMSTRONG[l966b]. (See also the review article of ARMSTRONG and SMITH[1967].) The excellent agreement between theory and measurements is shown in Fig. 13. One sees by inspection that the measured distribution is not a Bose-Einstein distribution (4.16) (a Bose-Einstein distribution has its largest probability always for n = 0), but that a Poisson distribution (n). exp { - ( n ) } / % !is a better approximation. The Eose-Einstein distribution (4.16) follows from (4.10) for pump parameters far below threshold (u << -1). This can be shown either by using an asymptotic expansion of the
v. § 41
P H 0 T O E L E C T R O N C 0 11N T I N G D I STRI B U T I 0 N
2ti9
0.4
I
0.2
? " ' O0-10
-8 -6
-4
I
-2
0
2
4
6
8
f0
Fig. 14. The parameter ~ ( aof) (4.18) and (4.19) (solid line), and the asymptotic expansions 7 = 1 -4/a2 for a << - 1 and 7 = 2ja2 for a >> 1 (broken linc) as a function of the pump parameter a. The experimental points are taken from ARECCHIe t al. [1967c]. (Thc abscissa K , ( a ) / K l(0) of Arecchi e t al. was rescaled and expressed as a function of a.)
integrals F,, ( a ) or hy substituting in (4.8) the intensity distribution far below threshold
~ ( 1=)+la1 exp (-+lalf>.
(4.17)
The transition from a Bose-Einstein to a Poisson distribution through the threshold region as obtained by changing the pump parameter, is best shown by looking at the variance < ( n - ( n ) ) z ) = (n)(1+?7(a)(n))J
(4.18)
where we have defined the parameter ~ ( aby )
r(a)=~ z ( a ) / W ) .
(4.19)
As seen in Fig. 14, there is a smooth transition from the variance of the Bose-Einstein distribution 17 = 1 to the variance of the Poisson distribution 7 = 0. The variance (4.18) consists of two parts; the first
part ( n ) stems from the discreteness of the photoelectrons whereas the second part ~ ( a (n>z ) stems from the fluctuation of the light beam. The last term gives rise to the BROWN-TWISS [1956, 1957a, b] effect. It should be noted that the last part ~ ( a (n)z ) remains finite as the pump parameter a goes to infinity because ( n ) increases proportional to a. However, if the light beam is attenuated in such a manner that ( 1 2 ) remains constant as the pump parameter a is increased, we finally obtain for a --t co a pure Poisson distribution with the variance ( n ) . In Fig. 14, the experimental points obtained by ARECCHIet d . [1967c] are also plotted. These authors were able to operate the laser at threshold a = 0 and measure the noise. A similar plot was also
270
STATISTICAL PROPERTIES O F LASER LIGHT
[v. § 4
obtained by DAVIDSON and MANDEL [1967]. By ~ ( u one ) can experimentally decide whether the laser is above (u > 0) or below threshold (a < 0 ) . One has only to measure the mean photon number and the variance. Above (below) threshold (4.20)
is smaller (larger) than ~ ( 0 == ) 'n-1 2
=
0.5708.
(4.21)
1
(4.22)
The first four reduced cumulants Qi
I
= Ki ( a ) [Ki ( a )
have been measured by CHANGet al. [1967] in good agreement with the values that follow from (3.10) or Fig. 3. A t the end of this section we briefly mention some further applications. I n the instationary case, the photoelectron counting distribution is given by (4.8) where now w(1) is the general instationary distribution (3.33). The two detector photoelectron counting distribution (4.4) reads in the stationary state for short intervals
where v i = a'da T iand where .T is the normalized time difference [i2-&l.Using (3.19) and (3.26) the cross correlation
c
m o o
(121122)
=
2 nln,p(.nl>% > 3
(4.24)
n,=O n,=O
now becomes (n1nz) =
(~2>+vlV2K(4 7)
= <.l>.
(4.25)
Therefore the correlation function K ( u , ?) respectively Aeff can be determined by measuring (n1n2) with a two detector method.
V.
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41
PHOTOELECTRON
271
COUNTING DISTRIBUTION
4 . 3 . EXPECTATION VALUES F O R ARBITRARY INTERVALS
In this subsection we drop the assumption (4.5) that the time interval T is short enough. In contrast to tj 4.2 we do not derive the counting distribution but merely discuss the expectation value 00
(n) =
2 .$(“it,
n=O
T )=
(I(t‘)>dt‘ = v K l ( a )
(4.26)
and the variance
(f(t’”(t’’))dt’ dt“- [/ff+T(l(l‘))dt’]
’). (4.27)
Using (3.19), (3.22) and (3.26) we obtain <(n-)2>
= vK1(a)+v2K2(a)
(4.28)
For AeffT<< 1 we have f(T) = 1 and thus obtain the old result (4.13). For Aeffp >> 1 we have f(T) = 2/(Aeffp).The part of the variance v2K2(a)f ( T ) which stems from the fluctuation of the light beam is therefore diminished by using a long time interval T >> &&’. Because AefP has a minimum near threshold, the part of the variance v3K2(a)f(T) is most important in the threshold region. 4.4. CONDENSATION EFFECT O F THE COUNTING DISTRIBUTION
We have seen in 3 4.2 and tj 4.3 that the variance ( ( n - ( n ) ) 2 ) is usually larger than the variance (n>of a Poisson distribution. Therefore the photoelectrons appear more or less in clusters. In order to investigate this condensation effect we ask for the conditional probabilit y Pc(?)df that we find in the infinitesimal interval f+?, f+?+df a photoelectron if we have already found a photoelectron at the earlier time i. This conditional probability is (see for instance MANDEL [1963])
272
STATISTICAL PROPERTIES OF LASER LIGHT
Iv9
s
5
Fig. 15. The ratio of the conditional probability density +,(.) to the unconditional probability p as a function of the normalized time 7 = z2/(&) for various pump parameters a.
Because of (3.19), (3.22), (3.26) and (4.19) we may write
where p is the probability density of finding a photoelectron. The excess of p,(z) over p , which is a measure of the condensation process, is shown in Fig. 15 as a function of the time 5. Below threshold (a << -1) and for small times t,p , = 2$, which is well known to be valid for Gaussian light, see MANDEL [1963]. Near threshold the curves decrease very slowly in time 7, but they still have an appreciable condensation effect for t = 0. High above threshold ;leff increases again and p c ( 0 ) m p , thus showing no more condensation effect in this region. The photoelectron distribution will therefore be a Poisson distribution.
Q 5. Fully Quantum Mechanical Theory 5.1. INTRODUCTORY REMARKS
In §§ 2, 3 and 4 we have treated the optical field E classically, i.e., we have neglected the operator character of b. I n order to describe the fluctuations, we introduced fluctuating Langevin forces whose strengths were determined by the requirement that they lead to the quantummechanically determined transition rate. The resulting Fokker-Planck equation near threshold described the experimental measurements very well quantitatively as shown in 4 4.However, it is clear that by neglecting the operator character of b, one can only obtain results with an error of the order of the commutator. For instance with the method of § 3 the expectation value (b+b) (in unnornialized units) can only be
v,
9 51
FULLY QUANTUM MECHANICAL THEORY
273
determined to the order unity. In order to calculate (b+b) with an error smaller than unity, fully quantum-mechanical equations must be used. Because in a laser the number of photons at threshold is of the order 4000 (see ARECCHIet al. [1967c]), the relative error is of the order 0.1 yo and far beyond the accuracy of the measurements. Although the shortcomings of the model which was used (eg., single mode) may introduce larger errors than the ones due to neglecting the operator character of 6 , it is interesting from a theoretical point of view to derive fully quantum-mechanical results. Two main methods exist for deriving laser fluctuations fully quantum-mechanically. One method (more connected with the Heisenberg picture) consists of adding Langevin operators to the Heisenberg equation of the field operator and of the atomic operators. These Langevin operators preserve the commutation relations, which otherwise are destroyed by the damping terms. This method was used for a damped harmonic oscillator by SENITZKY [l960, 19611, and for a laser system by HAKENand WEIDLICH[1966], HAKEN [1966], SAUERMANN [1965, 19661, and LAX [1966a]. (For a review article see HAKEN[1967], RISKEN[l968].) The other method (more connected with the Schroedinger picture) consists of deriving a density operator equation (master equation). This method was developed for a laser system by WEIDLICHand HAAKE[1965a, b]. The resulting equation and similar master equations were further investigated by et al. [1967], LAX [1967a, b], LAX WEIDLICHet al. [1967a, b], HAKEN and LOUISELL [1967], GORDON [1967], SCULLY and LAMB[1967a] and WILLIS [1967]. Although the two methods are equivalent (in the same manner as the Langevin method and the Fokker-Planck method are equivalent in the theory of Markov processes) the master equation approach is more appropriate for solving the nonlinear equations. In fact we show in 3 5.4 (see HAKEN et al. [1967]) that the solution of the master equation can be reduced to a solution of an ordinary (c-number) partial differential equation and thus numerical methods can be applied directly. The Langevin method requires analytical solutions of the equations and is therefore mainly restricted to linear or linearized equations. 5.2. MODEL AND DERIVATION O F THE LASER MASTER EQUATION
In Fig. 16 the block diagram of the model for the laser system is shown. Damping effects of the light field and damping and pumping effects of the atoms are described by the coupling to reservoirs R,
214
STATISTICAL P R O P E R T I E S O F LASER LIGHT
Iv. § 5
- -- - - - - - - - -1
Fig. 16. Block diagram of the laser system. L: light field, R: light field reservoir. A ( p ) :p’th atom, R ( p ) :reservoir belonging to the p’th atom, solid lines: interactions.
which will be assumed to consist of a large number of oscillators with different frequencies. Hence the total Hamilton operator H consists of the following parts:
H
= HL+HRSHL,
+CHL*
( p )*
P
(Because in a one mode traveling wave laser model the space dependence drops out (see 9 2), we have neglected space dependence in (5.1).) In (5.1) the Hamilton operator of the light field, the operator of the reservoir belonging t o the light field, and the operator of the interaction between light field and reservoir read
The Hamilton operator of the p’th atom, of the reservoir belonging to the p’th atom and the operator of the interaction between the p’th atom and the reservoir is given by
HA(p)= z f i ~ ~ A L ! f A!$ i
= LZ$(,U)a j ( p )
V,
I 51
FULLY QUANTUM
MECHANICAL THEORY
275
Finally, the operator of the interaction between field and atoms reads
Tn (5.2)-(5.4), b+ is the creation operator of the light field, a t ( p ) the creation operator of the i’th energy level of the p’th atom, cf is the creation operator of the p’th level of the light reservoir, cz(,) is the creation operator of the p (p)’th level of the reservoir belonging to the p’th atom; g&, g ~ ( p ) , p tand ( p ) g, are the (real) coupling constants between the ,u’’thatoms or light field and the corresponding reservoirs; g i j is the (real) coupling constant between atomic systems and the light field. As we will see later on, the first term in HA(P)R(,) describes real transitions whereas the second term describes virtual phase destroying transitions. In the interaction picture the creation and annihilation operators oscillate according to the free field frequencies
where the operators without time argument are the operators at t = 0. The information of the complete system (light and atoms and reservoirs) is given by the total density operator ptot, whose equation of motion reads in the interaction picture
with
276
STATISTICAL PROPERTIES O F LASER LIGHT
[v, 9 5
figD(c; exp {iS,t}+c, exp {-iSDt}) (b+eiwot+be-iwot) (5.9)
H i R ( t )= ?)
HL-\(pj (t) = zfigigzj (b+e'"ot+be-'"ot ) exp (i(cz-c3)t}AIy).
(6.10)
L, 3
The total density matrix ptot contains too much information and has too many variables (all the reservoir variables). Instead of using ptot one wants to describe the process by a reduced density operator P = TrR{Ptot}>
(5.11)
where all the reservoir variables are averaged over. The equation for p is the master equation and is derived by the following steps. First one assumes that the total density matrix factorizes at a certain time to in a density operator of the atoms and of the light field times a density operator containing the reservoir variables only ptot(tO)
= P(tO)pR(tO).
(5.12)
An iteration procedure for ptot(t) leads after two iteration steps t o
Pm)
-i
=
-i 2 t [HIP),Pto&)l+(X)
~ ~ l ~ ~ ~ ~ [ ~ l ( ' ~ ~ P (5.13) t O t ( ~ O ) l l ~
to
The master equation for p is obtained by taking the trace of (5.13) with respect to the reservoir system, then setting it equal to the time derivative of p and letting t go to t o , p(to) = lim Tr,{ p!:!(t)} t +to
(5.14)
The term containing the simple commutator in (5.14) vanishes for interaction operators containing reservoir variables and the double commutator vanishes for all terms except for those where both interaction operators contain variables of the same reservoir. This follows immediately from relations of the form Tr,(cip,) = TrR(cicp,pR) = 0 ($ f $'). Therefore the master equation (5.14) can be reduced to
v, g 51
FU LLY
277
Q U A N TU M M E CII A N I C A L T H E 0 K Y
Inserting (5.9) into (5.16) and neglecting antiresonant terms (i.e., terms with a time dependence of the form exp {i(wo+L?,)t}) and using
2 g:
exp{i(Q,-w,) ( t - z ) }
=
2~B(t--t)
(5.18)
9
TrR(C:C~PR(tO)}= % t h ; we finally end up with )L (;
=
K(%h+l){[b>
=
K{[bp,
Pb+l+LbP>
b+l$- Ib,
TrR{c~c:pR(tO)} =
b+I)+K%h{[b+,
p b + ] } f 2 K n t h L [ b ,P I ,
PbI+[b+P> bl}
bfl.
(5.19)
=
(5.20)
The &function in (5.18) appears because the frequencies 0, are assumed to be continuously distributed in the reservoir and therefore the summation over p can be replaced by an integration. The constant K is the damping constant of the resonator, nth = [exp (fiwo/KT) - 11-l is the occupation probability of photons with frequency L?, = coo of the reservoir of temperature T . Because the reservoirs are assumed to be infinitely large, nth does not change with time. The evaluation of (5.17) is made in a similar manner. Neglecting antiresonant terms, the equations corresponding to (5.18) and (5.19) read in this case TrR{Wi$’) ( t ) Wig)
(Z)
pR(tO)}= yij6ikdj,d(t-Z)*
(5.21)
In deriving (5.21), we have assumed that the energy levels are distributed irregularly in such a manner that the equation E , - - E ~ = E k - 8 1 is only solvable for i = k , j = L. The y j i are reservoir constants and they describe real transitions from level j + i for j # i and virtual, phase destroying transitions for j = i. After considering (5.21) and the corresponding expressions in which the factors in the trace are changed cyclically, we finally end up with
278
STATISTICAL PROPERTIES O F LASER LIGHT
[v. § 5
For a three-level system eq. (5.22) was derived by WEIDLICHand HAAKE[1965b]. The time change of the expectation value ( A i j ) = Tr{Aij p} due to the terms (5.22) is
= ~y7j(An.)Bij-+Z:(yi,+yjr)(Aij). r
(5.23)
r
Thus the y i j describe transitions from the level i to the level j . For i = j , (5.23) are the rate equations for the occupation numbers. Terms of the form yii only enter in the nondiagonal terms ( A i j ) ( j f i), where they describe damping by virtual processes additional to the real transitions. For the real transitions only outgoing processes are responsible for damping, as shown in Fig. 17. (See also SCHMID and RISKEN[1966].) For a two-level system (5.22) reads explicitly
Fig. 17. Transitions leading to a damping of the off-diagonal element A,, (solid lines) and transitions leading to no damping of the off-diagonal element A t , (broken lines).
++Y21{[4$)
P, Ai;’I+[A:$>
++yl,{[Ag’ p, Ag’I+[A:;’, ++y2,{[.1L$)
PAi;’l)
(5.24)
PAg’l}
p, AL$)I+[Ai& PAi$’l}.
For a two-level system it is convenient to introduce spin operators = sf;
Ag’+A;?’
A!$’ = sp; =
1,
+(A&A(r”’) 11
A&$’= ++szp,
= szp
A;?) = +-szp.
(5.25)
The product of two operators can be reduced to one operator according to (see also LAX [1963])
ApA g
z
A Lf) Sj,.
(5.26)
V,
s
51
FULLY QUANTUM MECHANICAL THEORY
279
I n a two-level system the reduction relations of the spin operators s-s+ = 1- s,;
s+s- = '2+ s
2
(5.27)
correspond to (5.26). Thus the total master equation for a two-level, resonant system ( e 2 - cl = mug)where anti-resonant terms are neglected, finally reads (5.28)
5.3. LASER MASTER EQUATION NEAR THRESHOLD
The purpose of this section is to derive a master equation containing only the light field. For this end we introduce the following expressions, which depend only on the light operators b, b+, (the index A indicates averaging over the atoms) : N
1 p = TrA{p},
p*
TrA
{ p=1 2
N
1
J
pz
=N
TrA{
2 2szpP)*
p=1
(5'32)
In (5.32) p is the density operator of the light field alone, p* and pz are the expectation values of the positive and negative frequency parts of the polarization and of the inversion. Using the master equation (5.28) for a two-level system, the following equations for p, pf, p, can be derived:
2 80
STATISTICAL PROPERTIES O F LASER LIGHT
[v, § 5
(5.37)
(b+Tr,(sLs; p}-TrA{sLs; p)b++ b Tr,(s;s;
p) -Tr,{sis;
p}b) (5.38)
PlfV
in (5.34) and similar terms in (5.35) and (5.36). This neglection is similar t o the procedure of WEIDLICHet al. [1967a], where the corresponding expansions of the density operator were neglected. However, the difference is that p, p*, pz are still operators in the light field domain and thus lead to results being equivalent for normal and antinormal distribution functions. A next step of the present method would be to include terms of the form (5.38) but neglect expectation values of the atomic system of spin operators belonging to three different atoms. For further considerations we restrict ourselves to the case of small K ( K << y z ) . Under this condition the terms containing K in (5.34)-(5.36) may be omitted and, near threshold, the time derivatives in q1 and gz may be neglected (adiabatic approximation). Eliminating p+ and p- in (5.33) and (5.36) by the eqs. (5.34) and (5.35) yields, after some calculations, the following master equation:
281
(5.40)
(5.41)
(5.42)
With the distribution functions Wn(u, u*)and W,(u, a*) every normally and antinormally ordered product of b+ and b can be calculated simply by using c-number integration over the distribution function Wn or W,: Tr{(b+)' bj p ( t ) }
=
I
( U * ) ~ UWn(u, ~ u*,t ) d 2 a
(5.43)
Tr{b'(b+)3 p ( t ) ) = j a ' ( u * ) j Wa(a,u*,t)d2a. The proof follows from the fact that the integration of (5.43) in u space corresponds to a differentiation in a space
The distribution function Wn(u,u*)is GLAUBERS[1963a, b] P-representation of the density operator p =
where) . 1
.r
l ~ J+'n(u, >
.*)
(GI d2u>
(5.45)
are the eigenstates of the annihilation operator b l u ) = ulu).
(5.46)
Note that W, is not necessarily positive. It therefore does not have the usual meaning of a probability distribution, but serves merely as a
282
STATISTICAL PROPERTIES O F LASER LIGHT
[v.
s
5
tool to calculate all quantum mechanical expectation values in the c-number domain. Continuous functions similar to (5.42) were first introduced by WIGNER[1932] and further investigated by MOYAL [1949]. The continuous functions used here were introduced and studied in detail by KLAUDER[1960], GLAUBER[1963a, b] and SUDARSHAN [1963]. (For a review article see the book of KLAUDER and SUDARSHAN [1968]; for recent investigations see the paper of AGARWALand WOLF [ 19681.) The next task is to find an equation of motion for the distribution functions W na from the master eq. (5.39) and (5.40). For this purpose we multiply (5.39) and (5.40) with 0 na and take the trace. By a proper cyclical permutation of the factors under the trace and by using [b, 0
= ia*O na;
[0na, b+] = ia 0 na
a2
a2
Tr{b+O,bp}
=
~
1,; Tr{bO,b+p} = aia a h * f a aia aiu* ~
we finally end up with the following equation
,g,
a
[. LU+iu* aiu* f i a ia*+2 1f,, (5.49) a a g2 ia-+iu*) +ia ia*]gna. [4 ‘2j aia aia* g2
= oAfna- __
iu-
__
YlYZ
a2
+l
__ Y1Y2
The equation for the distribution function W na follows from (5.48) and (5.49) by the replacement iu
--f
-a/&;
ict* -f
-a/a~*;
a/aia --f
Using real variables in vector notation
U;
a/aia* --f u*. (5.50)
v,
P
51
283
FULLY QUANTUM MECHANICAL THEORY
a1 = *(a+.*),
a2 = $(u-a*),
u = {a1,U z }
(5.51) the master equation for the distribution function has the form
+ V (urx2N G ,,-KW~,])
=
at
(5.53) The upper (lower) signs are valid for the normal (antinormal) distribution function W . Because of the denominator in ( 5 . 5 3 ) , the master equation contains derivatives up to infinite order. In the next step, we evaluate the denominator in (5.53). In order to see the magnitudes of the different terms, it is convenient to introduce the normalized coordinates (2.35), where ,I?, q and d are now defined by (u0 = uAN = (Ni-NY) is the total inversion)
B = 4 g 2 ~ / ( ~ 1 y 2 ) ;= g2N/(4y2), g2(Ni-NY)/y2 = 4q0, = K+,I?d,
-
u =dp/qd.
(5.54)
For the sake of simplicity we further put nth = 0. In the normalized equation, terms of the form U, Vii, d,JU/2 are all of the same ordernear threshold. Retaining only terms up to the order d ( p / q ) , we obtain
(a/ai)wna + ~ { u [ a - - l u l 2 ] ~ ~ , } - = d~~~ dPx{ { lu 1 '/ (4c+4)f ( -vu)/ (40,) Q ( f u [ (a I U I WnaI 1. - I U 1 */ (4uA) IWna}
-
-
)
(5.55)
d(q/P) is approximately the number of photons at threshold and is for a typical laser (ARECCHIet al. [1967c]), of the order 4000. Therefore the Fokker-Planck equation (2.36) is a very good approximation. The right-hand side of (5.55) are the first order quantum mechanical corrections of this Fokker-Planck equation. (By evaluating the denominator in (5.53) further, higher order quantum corrections can be found.) Some of these corrections have different signs for W , and W ,
284
STATISTICAL P R O P E R T I E S O F L A S E R L I G H T
iv, I 5
and thus lead to slightly different solutions for W , and W,, which the classical Fokker-Planck eq. ( 2 . 3 6 ) cannot describe. In the FokkerPlanck eq. ( 2 . 3 6 ) ,the q value was defined (nth= 0) by q = g2N2/( 2 y , ) . The differences between this definition and (5.54) lead to terms of the order 2 / ( p / q ) and thus are irrelevant in the classical Fokker-Planck eq. ( 2 . 3 6 ) .It is worthwhile to mention that in the quantum corrections a term of the form v{iilU14W} appears. This stems from the second expansion term of the denominator in (5.53). In unnormalized coordinates the distribution functions W , and W , are connected by (see GLAUBER[1963ab]; W,(u) = (ulpIu)/n) e-l01*W,(u+v)d2v,
(5.56)
which reduces, in first order quantum corrections and in normalized coordinates, to
W,(U)
=
Wn(U)++2/B/g2W,(U).
(5.57)
Because the difference between W , and W , is of the same order as the right-hand side of eq. (5.55), there is no need for distinguishing between W , and W , in the semiclassical limit, i.e., in the case where the right-hand side of eq. (5.55) is neglected. By a somewhat lengthy calculation it can be shown that the solution W, and W , of (5.55)obeys the relation (5.57) to first order corrections for all times. Eq. (5.55) can be solved by a similar manner as the Fokker-Planck eq. ( 2 . 3 6 ) . Perturbation procedure seems to be particularly appropriate for expressing the new eigenfunction and eigenvalues in terms of the old ones. In this work we want t o solve the more general eq. (5.55) only in the stationary state. The stationary distribution function depends only on the field amplitude f . As in 9 3 we can introduce a probability current S. The relation S = 0 reads now
(a-P )f -W’,,/ Wna-fd/rB/4{f
+ ($-$a)
W’/,, W },,
+fl9/(4c~~){-aT~+P~+PW’~,/W =, ,0. }
(5.58)
Because we want to calculate only first order quantum corrections, we may insert the classical result W’/W in the curly parenthesis. The coefficient of the term . \ / c / o A then disappears and we obtain
w ,,= ( 2 n ) - 1 , ~,{1h$v‘a P[~-+(P-U)~I)
exp ( - ~ P + + u P } . (5.59)
v, S 51
285
FULLY QUANTUM MECHANICAL THEORY
3
4
2
6
r.72
8
Fig. 18. Thc normal (W,) and antinormal (W,) ordered stationary distribution functions (5.59) of the laser master equation near threshold ( 5 . 3 9 ) , (5.40), for a pump parameter a = 3 and a threshold photon numbcr .\/(q/p) = 20. The solution WSt which follows from the semiclassical calculations is also shown. For a threshold photon e l al. [1967c], thc differences between numbcr 2/(q/b)= 4000 as measured by ARECCHI the three distribution functions are down by a factor of 200 and lie, in the drawing, completely inside the line thickness of Wst.
The normalization constant JV normalization constant of $ 3 Jlr a,
nL
can be expressed by the uncorrected
=N ( l . f i a d 6 ) .
(5.60)
In Fig. 18 the distribution functions W,,areplotted. Using the inlegral (3.5) and the recursion relation (3.7), we obtain 2n
s
f2W, ,fdf
=
(I)r$dfi
(5.61)
or in unnormalized coordinates (bfb)
=
d a ( I ) -4,
(bbf)
=
d4/p (I)++
(5.62)
where ( I ) is the uncorrected moment of $ 3. It should be recalled that only quantum corrections up to first order can be calculated with (5.59). For instance we obtain (b+b+bb) = (q/p) ( f 2 ) - 2 d q l B ( I ) ,
( b b b f b f ) = (q/P) (12)++dq//i(I) (5.63)
which are correct to first order but not to second order. Accidentally the commutation relation (bbbfb+-bfbfbb)
= 4(b+b)+2
is even fulfilled for the second order term 2.
(5.64)
288
STATISTICAL PROPERTIES O F LASER LIGHT
LV,
9 5
Because d(P/q) is of the order 1/4000, the difference between W , and W , is very small near threshold, i.e., where ( P ) is of the order one. For much larger pumping parameters, where for instance ( f z ) z / ( p / q ) is of the order one (the photon number is then the square of the photon number at threshold), quantum corrections may play an important role. For this case more expansion terms of the denominator in eq. ( 5 . 5 3 ) must be used.
Photon counting distribution Using the Glauber P-representation (5.45) of the density operator we obtain p(n)
==
(nldn)
==
s
~ e - 1 ~ ~- ~ 12, Wn(u)d2u. (5.65) I < ~ l ~ > I 2 W n (= ~)d2~
Since usually only a small fraction of the intensity inside the laser falls on the photon detector during a time interval T , the intensity is attenuated by a factor u‘T.Thus instead of W n ( u )Wn(u/y”(u’T)) , must be inserted (see also GLAUBER [1966], ARECCHI and DEGIORGIO [1968]). Then (5.65) is exactly the Mandel’s equation for the photoelectron counting distribution for short intervals (4.8). For normalized coordinates, we thus obtain by using (5.59), (5.65), (4.9) and (3.5) Vn
P (n) = n!F,o{ (1-%ad%)F , (a -2Y) +dfi [*-+a21 F,+l +%a4%iFn+z(a-W
-
&dB/qFv+,(a-2v)),
(a-2v) (5.66)
which differs from (4.10) by the small correction terms of the order d(P/q). SCULLY and LAMB[1967a] have derived a master equation in the occupation number representation for the light field alone. Writing their equation in operator form and applying the same technique, we find that without quantum corrections their equation agrees exactly with our Fokker-Planck eq. (2.36). The first order quantum corrections disagree with the ones derived here. This may be attributed to the fact that these authors use a slightly different pumping model (not an ideal two level system with a constant number of atoms as used here). 5.4. c-NUMBER EQUATION O F THE LASER MASTER EQUATION
In § 5.3 we have seen that the solution of the operator eqs. (5.39), (5.40), can be reduced to the solution of the c-number equation (5.52),
v,
I
51
FULLY QUANTUM MECHANICAL THEORY
287
(5.53). The operator p is then given by (5.45). Following a method of HAKENet al. [1967] (see also GORDON[1967], LAX(1967b], and LAX and LOUISELL [1967]), we derive in this section a c-number equation of the master eq. (5.28), which now contains all the atomic variables too. However, as we will see below, we do not need all N atomic variables, but only the ones which refer to macroscopic quantities, namely the total atomic dipole moment and the total inversion
Sf
=
Is;,
s- = 2s-,fi s, = ZS,.
P
P
(5.67)
P
We define the distribution function f by
f (u, u*,v, 7J*, I ; t)
= M'
J . . .J exp{-ivt-iv*t*-i~I-iua-iu*a*) x F(5, [*,
5,a, a*, t)d2Ed2a d t
(5.68)
where
F
= Tr{eiSS-efSszefS*S'eia'
e
b+ l a b
P(t)l = TWPl.
(5.69)
(For the sake of simplicity we only treat the normally ordered distribution function.) For the following analysis, it is convenient to use a decomposition of the operator 0 in the form
0
=
o,o,;
N
(5.70)
eiSs;eiSs.,eiS*s:. , 0L -- eia'b+eiab OA(fi)-
M' is a normalization constant, so that
S f d2ud2vd I = 1. u,u* are
the macroscopic variables associated with the lightfield operators b, b+, v, v* and I are quantities corresponding to the total complex dipole moments S - or Sf and the inversion S,, respectively. The definition (5.68) ensures that f is real provided p is a hermitian operator. Expectation values of the light operator can be obtained as in 5 5.3 by the first part of eq. (5.43). The same procedure may be applied to expectation values which are functions of the macroscopic variables Sf, S-, S,. If the resulting order of the operators does not agree with the order wanted, the usual commutation relations must be applied. Multiplying (5.28) with the operator 0 and taking the trace we have (5.71)
288
STATISTICAL PROPERTIES O F LASER LIGHT
[t-,
s
5
with (5.7%)
(5.73)
(5.74)
Using (5.30) we find
N
fYl2
[ 2 Tr~s,O*,spa,P) -Tr{s,sp*,oa,d
TrP*,s,spa,
-
+
(711+Y22)
['
PI1
Tr{s,,oA~cs~/~o~,p)~Tr{s~~UA~o'*~p>
-Tr{o.4pS~po'App)l
with 0
(5.75)
OA,O'*,;
oa,
=
JJ O,,O,.
1 (5.76)
v#a
Using the property of the trace that an operator product under it may be rearranged in a cyclic manner, we have brought the p in every term to the right-hand side. Because s-, s+ and s, obey eq. (5.27), we may express terms of the form s i s ; , s t s i , ,;s by the single operators. For our further analysis it is most important that the operators which occur in front of the density matrix under the trace can be expressed as linear combinations of derivatives of the operator O,, with respect to the variables 5, 5*, [ and of the operator OA,L;
o,,
=
eits;eicsz,eit*s;
(5.77)
v, 9 51
FULLY QUANTUM MECHANICAL THEORY
289
We exhibit these linear combinations explicitly by the following formulas:
s+O P .4P s-P
=
[+e-'c++eic(it)2(iE*)2-t
(it)(it*)]O,,
(5.78)
+ [2 ( e d -
(5.79)
We now calculate the term of eq. (5.73) which stems from the interac-
290
[v, § 5
STATISTICAL PROPERTIES O F LASER LIGHT
tion between the field and the atoms. Note that in this interaction only the operators S+, S- of the total atomic dipole moment occur so that we may immediately write =
+
+
-ig{Tr{ [OS+b OS-b+]p}--Tr{ [S+bO S-b+O]p}}.
(5.80)
Using the operator relations
(5.81)
we find
aF
(z)AL,
=
-ig
a a (-a(il*) a(itc) ~
-
~
a a($)
a ~
a(itc*)
(5.82)
The field term was already calculated in 9 5.3 (put g
=,);(
-“““;d-1 a
=
0 in (5.48))
F + ~ K ~ z ~ ~ (ia) ( ~ cFc. * (5.83) ) ) According to the definition of f this distribution function is a Fourier transform of F . Thus the expressions (5.79), (5.82) and (5.83) may be expressed by f and its derivatives. We write this resulting equation for f in the form +iu*
a(lR
vl
291
REFERENCES
af
-=
at
Lf
(5.84)
where the Liouville operator L consists of the atomic part, the atomfield interaction and the field part
L = LA+L,,+L,.
(5.85)
The different contributions are defined as follows:
(5.86) +fryl2
[ N(e-a/aI-
+ ava v+ av*a v*
1)
-
~
-
I
2 (e-a/aI- 1)I
(5.87)
(5.88)
Taking into account only first and second derivatives the leading terms of this equation agree with the Fokker-Planck equation derived and solved below and above threshold by RISKENet al. [1966a, b, c]. For a complete solution of the two-level laser problem (without linearization of the coefficients, adiabatic approximation), however, the full eq. (5.84) must be investigated.
References AGARWAL, G. S . and E. WOLF,1968, Phys. Rev. Letters 21, 180. ARECCHI,F. T., A. BERNEand P. BURLMACCHI, 1966, Phys. Rev. Letters 16, 32. ARECCHI,F. T. and V. DEGIORGIO, 1968, Phys. Letters 27A,429.
292
STATISTICAL PROPERTIES O F LASER LIGHT
[v
ARECCHI, I;.T., V. DEGIORGIO and B. QUERZOLA, 1967a, Phys. Rev. Letters 1 9 , 1168. ARECCHI, F. T., M. GIGLIOand A. SONA,1967b, Phys. Letters 2.54, 341. ARECCHI,F. T., G. S . RODARI and A. SONA,1967c, Phys. Letters 2 5 A , 59. AKMSTRONG, J . A. and A. W. SMITH,1967, in: Progress in Optics, Vol. 6, ed. E. Wolf (North-Holland Publishing Co., Amsterdam; John Wiley and Sons, New York) p. 211. ARZT,V., H. HAKEN, H. RISKEN, H. SAUERMANN, C. SCHMID and W. WEIDLICH, 1966, Z. Physik 1 9 7 , 207. B~DARD G.,, 1967a, Phys. Letters 2 4 A , 613. BEDARD,G., 1967b, Phys. Rev. 1 6 1 , 1304. B~DARD G.,, 1967c, J . Opt. Soc. Am. 5 7 , 120. BHAKUCHA-REID, A. T., 1960, Elements of the Theory of Markov Processes and Their Applications (New York-Toronto-London, McGraw-Hill Book Company, Inc.). B L A Q U I ~ RA,, E , 1953, Ann. Radio Elect. 8 , 36. BORN, M. and E . WOLF, 1964, Principles of Optics (Pergamon Press, London). BROWN, 13. Hanbury and R . Q. TWISS,1956, Nature 1 7 7 , 27. BROWN,13. Hanbury and R . Q. TWISS,1957a, Proc. Roy. Soc. London A 2 4 2 , 300. BROWN, R . Hanbury and K. Q. T w ~ s s ,3957b, Proc. Roy. Soc. London A 2 4 3 , 291. BRUNNER, W., 1967, Ann. Physik 2 0 , 53. CHANDRASEKHAR, S., 1943, Rev. Mod. Phys. 15, 1; this article is contained in: Selected Papers on Noise and Stochastic Processes, ed. Nelson Wax (Dover Publication, Inc., New York, 1954). V. KORENMANN, C. 0. ALLEYand U. HOCHULI, CHANG, R . F., R. W. DETENBECK, 1967, Phys. Letters 2 5 A , 272. COLLINS,R. J.. D. F . NELSON,A. D. SCHAWLOW, W. BOND,C. G. B. GARRET und Ti'. KAISER,1960, Phys. Rev. Letters 5, 303. UAVIUSON, F . and L. MANDEL,1967, Phys. Letters 2 5 A , 700. FLECK, J . A,, 1966a, Phys. Rev. 1 4 9 , 309. FLECK, J. A,, 1966b, Phys. Rev. 1 4 9 , 322. FREED, C. and H. A. HAUS,1965, Appl. Phys. Letters 6 , 85. FREED, C. and H. A. HAUS,1966, Phys. Rev. 1 4 1 , 287. GLAUBER, R. J., 1963a, Phys. Rev. 1 3 0 , 2529. I<.J.. 1963b, Phys. Iicv. 1 3 1 , 2766. GLAUBER, GLAUBER, R. J., 1966, Proc. Puerto Rico Conf. on Physics of Quantum Electronics, eds. P . Kelley et al. (McGraw Hill Book Co., Inc., New York). GORDON, J . P., 1967, Phys. Rev. 1 6 1 , 367. GRADSHTEYN, I. S. and I. M. RYZHIK,1965, Table of Integral, Series and Products (Academic Press, New York). GRAHAM,R., 1968, Z. Physik 2 1 1 , 469. GRAHAM,K. and H. HAKEN, 1968, 2. Physik 2 1 3 , 420. GRIVET,P. and A. B L A Q U I ~ R1963, E , Proc. Symp. on Optical Masers, ed. Jerome Fox (Polytechnic Press, New York) p. 69. HAKEN, H., 1964a, Z. Physik 1 8 1 , 96. HAICEN, H., 1964b, Phys. Rev. Letters 1 3 , 329.
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REFERENCES
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HAKEN, H., 1966, 2 . Physik 190, 327. HAKEN,H., 1967, Dynamical Processes in Solid State Optics, in: 1966 Tokyo Summer Lectures in Theoretical Physics, eds. R. Kubo and H. Kamimura (Tokyo). HAKEN, H., H. RISKENand W. WEIDLICH, 1967, Z. Physik 2 0 6 , 355. HAKEN, H . and H. SAUERMANN, 1963a, 2 . Physik 173, 261. HAKEN, H. and H. SAUERMANN, 1963b, Z. Physik 176, 47. HAKEN, H. and W. WEIDLICH, 1966, 2. Physik 189, 1 . HAUG,H. and H. HAKEN,1967, 2. Physik 2 0 4 , 262. HEMPSTEAD, R. D. andM. LAX,1967, Phys. Rev. 161, 350. KELLEY,P. L. and W. H. KLEINER, 1964, Phys. Rev. 136, A316. KLAUDEK, J . R., 1960, Ann. Phys. 11, 123. KLAUDER, J. K. and E. C. G. SUDARSHAN, 1968, Fundamentals of Quantum Optics (W. A. Benjamin, Inc., New York, Amsterdam). V., 1965, Phys. Rev. Letters 14, 293. KOKENMAN, ,_ LACHS, G., 1965, Phys. Rev. 138, B1012. LAMB, W. E., 1964, Phys. Rev. 134, 1429. LAX,M., 1963, (Q 11) Phys. Rev. 129, 2342. LAX,M., 1966a, (Q IV) Phys. Rev. 145, 110. LAX.M., 1966b, (Q V) in: Physics of Quantum Electronics, eds. P. L. Kelley, B. Lax and P. E. Tannenwald (McGraw-Hill Book Co., Inc., NewYork) p. 735. LAX,M., 1967a, (Q VII) I E E E J . Quantum Electron. QE-3, 37. LAX,M., 1967b, (Q X ) Phys. Rev. 157, 213. LAX,M., 1967c, Phys. Rev. 160, 290. LAX,M. and R. D. HEMPSTEAD, 1966, Bull. Am. Phys. Soc. 1 1 , 111. LAX,M. and W. H. LOUISELL, 1967, (Q I X ) IEEE J . Quantum Electron. QE-3, 47. MAIMAN,T. H., 1960, Nature 187, 493. MANDEL,L.,1958, Proc. Phys. Soc. 7 2 , 1037. MANDEL,L., 1963, Fluctuations of Light Beams, in: Progress in Optics, Vol. lT, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) p. 181 MANDEL,L. and E. WOLF,1961, Phys. Rev. 124, 1696. MANDEL,L. and E. WOLF,1965, Rev. Mod. Phys. 37, 231. MCCUMBEK,D. E., 1966, I’hys. Rev. 141, 306, MOYAL, J . E., 1949, Proc. Cambridge Phil. Soc. 45, 99. PAUWELS, H . J., 1966, I E E E J . Quantum Electron. Q E - 2 , 54. RAYLEIGH, 1894, Theory of Sound, Vol. 1 (London; Dover Publications, Inc., New Y-ork, 1945) p. 81. RISKEN,H., 1965, Z. Physik 186, 85. RISKEN,II., 1966, Z. Physik 191, 302. RISKEN,H., 1968, Fortschr. Physik 16, 261. KISKEN,H. and K. NUMMBDAL, 1968a, Phys. Letters 2 6 A , 275. RISKEN,H. and K. NUMMEDAL, 196813, Phys. Letters 2 7 A , 445. RISKEN,H. and K. NUMMEDAL, 1968c, J . Appl. Phys. 39, 4662. RISKEN, H., C. SCHMID and W. WEIDLICH, 1966a, Phys. Letters 2 0 , 489. KISKEN,H., C. SCHMID and W. WEIDLICH, 196613, 2. Physik 193, 37.
294
STATISTICAL PROPERTIES OF LASER LIGHT
rv
KISKEN,H., C. SCHMID and W. WEIDLICH, 1966c, 2. Physik 1 9 4 , 337. RISKEN,H . and H. D. VOLLMER, 1967a, 2. Physik 2 0 1 , 323. RISKEN,H . and H. D. VOLLMER, 1967b, 2 . Physik 2 0 4 , 240. SAUERMANN, H., 1965, Z.Physik 1 8 8 , 480. SAUERMANN, H., 1966, 2 . Physik 1 8 9 , 312. SCHAWLOW, A. L. and C. H . TOWNES,1958, Phys. Rev. 1 1 2 , 1940. SCHMID, C. and H. RISKEN,1966, 2. Physik 1 8 9 , 365. SCULLY, M. and W. E. LAMB, 1966, Phys. Rev. Letters 1 6 , 853. SCULLY, M. and W. E. LAMB,1967a, Phys. Rev. 1 5 9 , 208. SCULLY, M. and W. E. LAMB,1967b, in: Proc. Intern. School on Quantum Optics, “Enrico Fermi”, Varenna, Italy, to be published. SENITZKY, J . R., 1960, Phys. Rev. 1 1 9 , 670. SENITZKY, J . R., 1961, Phys. Rev. 1 2 4 , 642. SMITH,A. W. and J . A. ARMSTRONG, 1966a, Phys. Letters 1 9 , 650. SMITH,A. W. and J . A. ARMSTRONG, 1966b, Phys. Rev. Letters 1 6 , 1169. STRATONOVICH, K. L., 1963, Topics in the Theory of Random Noise, Vol. 1 (Gordon and Breach, New York). S U D A R S H A N , E. C . G., 1963, Phys. Rev. Letters 1 0 , 277. VAN DEK POL,B., 1927, Phil. Mag. 3 , 65. WAGNER, W. G. and G. RIRNBAUM, 1961, J . Appl. Phys. 3 2 , 1185. WANG,M. C. and G. E. UHLENBECK, 1945, Rev. Mod. Phys. 1 7 , 323; this article is contained in: Selected Papers on Noise and Stochastic Processes, ed. Nelson Wax (Dover Publications, Inc., New York, 1954). WANGSNESS, R. I<. and F. Bloch, 1953, Phys. Rev. 8 9 , 728. WEIDLICH, M’.and F. HAAKE,1965a, 2 . Physik 1 8 5 , 30. WEIDLICH, W. and F. HAAKE,1965b, Z. Physik 186, 203. WEIDLICH, W., H . RISKENand H. HAKEN,1967a, 2 . Physik 2 0 1 , 396. WEIDLICH, W., H. RISKENand H. HAKEN, 1967b, 2. Physik 2 0 4 , 2 2 3 . WIGNEII,E., 1932, Phys. Rev. 4 0 , 749. WILLIS,C . R., 1967, Phys. Rev. 1 5 6 , 320. WOLF,E . and C. L. METHA, 1964, Phys. Rev. Letters 1 3 , 705.
hTote added i n proof Some important references, wich have appeared after the completion of this manuscript: H. HAKEN,1970, Laser Theory, in: Encyclopedia of Physics, Vol. XXV/2c, ed. S. Fliigge (Springer Verlag, Berlin-Heidelberg-Pew York). Here a complete and extended laser theory is presented. This work also contains a review of more recent investigations The Lecture notes of the 1967 Quantum Optics Course in Varenna, Italy, are now available: Proc. Intern, School of Physics ”Enrico Fermi”, Course XLIl ”Quantum Optics”, ed. R . J . Glauber (Academic Press, New York and London, 1969). R . L . STRATONOVICH, 1967, Topics in the theory of Random Noise, Vol. 11 (Gordon and Breach, New York). I n part 2 of this book ”Nonlinear Self-Excited Oscillations in the Presence of (Classical) Noise” some results of 9 3 are also derived.
VI
COHERENCETHEORY OF SOURCE-SIZE COMPENSATION I N INTERFERENCE MICROSCOPY BY
T. YAMAMOTO Research Laboratory, Nipporn K o g a k u K . K . , Japavz
CONTENTS INTRODUCTION.
. . . . . . . . . . . . . . . .
.297
GENERAL THEORY O F TWO-BEAM INTERFERENCE MICROSCOPES . . . . . . . . . . . . . . . . 300 COHERENCE DIFFRACTION THEORY O F IMAGE FORMATION AND TWO-BEAM INTERFERENCE
306
LOCALIZATION O F FRINGES W I T H PARTIALLY COHERENT LIGHT. . . . . . . . . . . . . . . .
315
. . . . . . . . .
317
SOURCE-SIZE COMPENSATION
PRACTICAL METHODS O F SOURCE-SIZE COMPENSATION I N SHEARING INTERFERENCE MICROSCOPE W I T H POLARIZED LIGHT . . . . . 321 IMAGES O F SOURCE-SIZE COMPENSATED INTERF E R E N C E MICROSCOPES . . . . . . . . . . . .
331
.
338
REFERENCES . . . . . . . . . . . . . . . . . . . . .
338
CONCLUSION . . . . . . . . . . . . . . . . .
Q 1. Introduction An interference microscope combines the two functions of interferometer and microscope into a single instrument. In the early stages of the development of the interference microscope, the interferometer and the microscope were integrated by placing in front of the microscope objective a miniature interferometer called a micro-interferometer. The micro-Jamin due to SIRKS (1893), the micro-MachZehnder due to PRINGSHEIM (1899), the micro-Michelson and -cyclic form due to SAGNAC (1911) and the polarization version of a microJamin due to LEBEDEFF[1930] are described by KRUG,RIENITZand SCHULZ[1964]. Since it is not easy to include these micro-interferometers in the working distance of the objective and since they introduce considerable aberrations, the numerical aperture obtainable and hence the useful magnification were severely reduced. As a matter of course, it is desirable that the optical path difference produced by the microscopic object can be measured through objectives of high numerical aperture with unrestricted illumination aperture. LINNIK [1934] was the first to bring the microscope objectives into each arm of the Michelson interferometer. Because of its symmetrical form, there was, in principle, no limitation t o aperture. In contrast t o it, in a two-beam interferometer with dissimilar paths, the light source should normally be a small point of monochromatic light, to obtain interference fringes of high contrast. Recently, many efforts have been made to find systems for a microscope which enable an extended light source to be used *. These are called “source-size compensated” by STEEL[1962], or simply “compensated” by FRANCON [1956]. The importance of source-size compensation in interference microscopy should also be stressed for other * I t IS to be noted that the use of an extended source IS possible in a two beam Interferometer but not In a multiple-beam interferometer. So only two-beam systems will be considered here The discussion concerning the so called tilt compensation (STEEL [I9673 p 90) IS also exclnded In this work 297
298
SOURCE-SIZE
COMPENSATION
[VI,
ci 1
reasons than the highest resolution and image quality, that is, -
-
a luminous image (high irradiance of the image) can be obtained, the fringes are localized, i.e. superposed in the image plane so that any other interference fringes, for example those arising from reflections within the interferometer, are out of focus, optical sectioning is possible because of the very short depth of field, as ALLEN and BRAULT[1966] indicated.
The principles of source-size compensation may be classified by forms of interferometer into three groups: -
When the Michelson, Mach-Zehnder interferometer, or cyclic interferometer is to be utilized, it suffices to arrange identical optics in each path with perfect symmetry *, as is the case of the forementioned Linnik system.
-
In common-path interferometers such as shearing or double-focusing interferometers, one may give the two beams just the same shearing or defocusing distances as that t o be given when observation is made, before they illuminate the object. This case will be fully discussed below.
-
In the Michelson interferometer with dissimilar arms such as the Twyman-Green interferometer, both the optical paths and the apparent distances of the image of the end mirror in the two arms of the interferometer must be equal, as HANSEN[1942, 1954, 19551 showed. This is true in the case of a lens-testing interferometer or a Michelson interferometer used for interference spectroscopy (MERTZ [1959], CONNES [1956], STEEL [1962]), but is rare in an interference microscope (HANSEN[1942, 1954, 19551, GERHARDT and LENK[1960, 19621 and GERHARDT [1967]). (In this case, the term “field-widening” is sometimes used instead of source-size compensation. )
A variety of methods of source-size compensation were reported by many authors. Each contains some of following features: 1. The interferometer can adapt itself t o the ordinary (conventionally
designed) microscope without introducing appreciable aberrations or a change of illumination conditions;
* The condition can always be satisfied when the interferometer is used in double passage by an auto-stigmatizing element placed a t one end.
VI,
s
11
INTRODUCTION
299
2. It imposes no additional limitation on the use of the microscope,
e.g. the working distance of objective or condenser; 3. The second-order effects of the aberrations of the optical components
in the interferometer which limit the usable source-size will be kept sufficiently small; 4. The compensation system can be adjustable for the condition of the various objects, such as a large range of curvatures of the reflecting object to be observed; 5 . The interferometer may be simple to manipulate as well as to con-
struct. The theory of compensation is closely related to that oi localization of fringes (BORNand WOLF [1959]) and very recently has been based on the theory of coherence by DYSON1119501, HANSEN[1955], HANSEN and KINDER[1958], HOPKINS [1957b, 19671, STEEL [1959, 1965, 19671, FRANCON and SLANSKY[1965], FRANCON [1966] and YAMAMOTO [ 1965, 19681. Aberrations in the interference microscope were studied only by a few authors (SLEVOGT [1954] and KINDand SCHULZ [1959]) from the geometrical-optical point of view. Although the effects of diffraction caused by finite apertures within the interferometer are traditionally ignored in interference theory, the coherence theory may provide a unified treatment applicable to interference phenomena entailing diffraction and aberrations (STEEL [1959]). As STEEL [1959] and YAMAMOTO [1965] showed, the image forming properties of a microscope incorporated with the compensated interferometer under illumination with any degree of coherence can be treated as a linear system by multidimensional Fourier analysis techniques. The coherence theory of compensation by Fourier analysis will be useful because the compensation can be described more analytically than otherwise and because incomplete compensation due to diffraction, aberrations of optical components or other reasons can be analyzed. It is the outline of this theory that the author intends to give. This review article first gives the general theory of two-beam interferometer in § 2 and next the unified theory of image-formation and interference in terms of coherence in S 3. Then, its applications to the theory of localized fringes and to the theory of source-size compensation are described in 3 4 and 5 . In 6 we present the classification of practical methods of source-size compensation which have been recently achieved. In 7, a description of the images produced by a compensated interference microscope is given.
300
SOURCE-SIZE COMPENSATION
[VI,
§ 2
Q 2. General Theory of Two-Beam Interference Microscopes * 2.1, GENERAL THEORY O F TWO-BEAM INTERFEROMETERS
A general treatment of the influence of source-size on the visibility of the fringes in two-beam interferometers is described by BORNand WOLF [1959], HOPKINS[1967], SCHULZ1719641 and STEEL [1965, 19671.
Suppose that the source is a quasi-monochromatic incoherent source extelzded about C (Fig. 1 ) . Let 3 denote a two-beam interferometer,
U
I
Fig. 1. General description of the two-beam interferometer.
0 the image plane, passing the axial point 0, common t o both paths. If a typical point P in the plane O is viewed from the source side, two images are seen at P, and P, from each beam in two planes passing the image points 0, and 0, of 0, 9, and O,,inclined t o each other a t an angle (angle of shear) and separated by the distance h along the axis. Similarly, a typical point S of the source, viewed from the plane of observation 0, appears at S, and S, in two planes passing the image points C, and C, of C, ul and u,, inclined to each other at an angle (angle of tilt) and separated by the distance h along the axis. When the source u is considered as the entrance pupil, its images u1 and u, are taken as the exit pupils for each beam. The difference p in optical path along the two rays reaching P from S by two different paths may be expressed as
P *
=
[sp,s,pl-"sp,s,pl
= P,+P,+P,,
(2.1)
Thc considerations of this section are based in part on the work of STEEL[1967].
30 1
TWO-BEAM I N T E R F E R E N C E M I C R O S C O P E S
When the distances ?%=u and @ = x are small compared with and K1respectively, we may have approximately (see BORNand WOLF [I9591 p. 293, STEEL[1967]) two symmetrical relations,
sp,
P, p,
= -S =
*
U/R-$hiU12/R2,
(2.5)
-2 . x/R-@jxl”R2,
(2.6)
__
where s is the projection normal to CO, of the vector separation between P, and P,, h the scalar separation parallel t o C 0 , between P, and P, , and R the distance from C to 0, in the source space. s”, h and R are corresponding quantities in the image space. s, 3, h and h are called the shear, tilt, shift and lead respectively by STEEL[1967]. It is to be noted that, by the Malus-Dupin theorem of geometrical optics (BORNand WOLF [1959], p. 130), [$,+$,) is independent of the position of S, and [$,+Po) is independent of that of P. Suppose that the normalized radiance of the source is E ( U , v) at the frequency v, and that 4. is uniform for all points in the source. Then, since 9, and Po are also independent of the position S or u, the intensity distribution at P in the plane of observation is represented ~
bY I ( p ) = 11+I2+21/[1112)
X 2[y(pl,
p,, -P,/c) exp{i(Snv/c)$,)
(2.7)
where I,, I , are the intensities at P through the system due to each beam alone, and
y ( P , , P,, - P o / c )
=
exp{i (~ZV/C)P~}J E ( U , v)R-, exp{i(2nvlc)Pu}du. U (2.8)
y is called the degree of coherence between two points P, and P, at two instants t and t-(p,/c), and -(Po/c) is called the delay by STEEL [1967]. Thus, the fringe position in the plane of observation varies as the sum pa+$, varies as a function of x, while the fringe visibility as defined by Michelson is given by IyI x 21,12[1,+12). If a circular source (angular radius m) of uniform radiance is assumed, the degree of coherence y can be expressed in terms of Lommel functions of the
304
SOURCE-SIZE
COMPENSATION
[VI,
s
d
first kind, as in STEEL[1965], SCHULZ and MINKWITZ[1961], SCHULZ [1964], MINKWITZand SCHULZ [1964]: y (PI, P, ,
-p,/c)
exp{ - i d } / d x x [ U I ( 2 d , 2nll)+iU,(27d, 2nrl)l
= exp{i (2nv/c)po}
(2.9)
where
5
= UZh/A,
(2.10)
17
=Xl4/4
(2.11)
a = Clv. These results show that high-contrast fringes are obtained only near the point whose two images are situated a t 5 = q = 0. 2.2. LOCALIZED FRINGES
From equations (2.8) and (2.5), the maximum visibility is obtained in two ways: one is reducing the source to a point, the other reducing p , to zero, and in the latter there are three cases. That is, (a) cr
--f
(b) s
=
0; 0 and
h
=
0;
(c) Is1 # co, K = co and h = 0; (d) h # co,R = co and s = 0. Otherwise the visibility falls off. Condition (a) means that the source approximates a point so that the fringes are non-localized as the radiation field is coherent everywhere. Condition (b) shows that the two images PI and P, of the observed point P must be exactly coincided three-dimensionally. * Sometimes, s is called shearing distance, and h defocusing distance. Only over a region of small s and h, the fringes will have good visibility as the degree of coherence attains to unity; the fringes are localized. When s = 0 and h w 0 (not vanishing except on a line, but sufficiently small) are satisfied, the fringes are called Fizeau fringes or fringes of equal thickness. Either condition (c) or (d) indicates that the fringes are localized in the focal plane of the whole optical system, which always exists; they are called fringes (localized) at infinity or fringes of equal inclination. Then, if s" = 0 and b # 0, the fringes are circular and * This conclusion is valid even when the shear is not uniform but arbitrarily takes place. See, for example, the rotatory shearing by ARMITAGEand LOHMANN [1964].
VI,
s
21
T W O - B E AM 1 N T E R F E R E N C E M 1 C R 0 S C 0 PE S
303
termed Haidinger rings; while if s" # 0 but h = 0, they are straight and equidistant and they are termed Brewster's fringes (BORNand WOLF[1959], p. 308). In general, these conditions are satisfied imperfectly in the image plane and it is necessary to reduce the source size for sufficient visibility. This would severely limit the resolution and the luminosity of the interferometer. 2.3. I N T E R F E R E N C E MICROSCOPE AND SOURCE-SIZE COMPENSA-
TION
An interference microscope is the integration of an interferometer and a microscope, one being incorporated with the other. A microscopic object to be observed is introduced in either one of the two intermediate planes (say conjugate with the plane of observation 0,as shown in Fig. 2. The interferometer is generally represented as divided into two parts, that is, one before the object as 4 and the other after it as 4'. The rest is the same as in Fig. 1.
&)
Fig. 2. General description of the two-beam interference microscope.
F,
E,
Let S be the vector separation between and normal to the optical axis, where F, and F2are conjugate with P in the plane of observation. P, and P, are similarly defined as conjugate with P in the source space. Suppose that w , ( E ) and w , ( c ) denote the variations of optical path due to the object in either beam, at F, and p2respectively. The difference of these two variations, denoted by w,will be added to the optical path-difference in place of 9, in eq. (2.1). Then we have
I(P) = I , + I 2 + 2 2 / ( ~ 1 ~ 2 X ) ~ " Y ( P 1p,, , -P,/c) exp{iyx}l, where
(2.12)
304
SOURCE-SIZE
COMPENSATION
[VI
5 2
and y(P,, P,, - p , / c ) should be found for the two images P, and P, to be located by retracing the system which includes the object. Hence, the difference between changes of optical path due to the object in either beam w can be rendered visible in its magnified image by the interference between the two beams. When the symmetric interferometer with separate arms such as a Michelson or a Mach-Zehnder is used, w, is normally taken as zero: zw, 3 0. I n the case of the shearing interferometer where the axial separation & between the two planes 8,and 8,is zero, but the same variation (say wl)at points separated by the distance s in the object plane is used instead of w,:
w
= w,(X+S)-w,(x),
(2.15)
a,.
where X is the position vector of P, in the object plane I n the double-focus interferometer applied to interference microscopy, the axial separation is usually taken so large that the disturbance due to the object placed in the plane 8,for the path 2 may be lumped in the (aperture) plane G , or its conjugate (source) plane u and represented by &,(u).* This affects the visibility of the fringes but not their position. Hence, in this case, we can put
w
= -w,(Z).
(2.16)
I n these common-path interferometers, it is generally desirable that the shear S or the shift k in the intermediate space (object space) be as large as possible. Therefore, to have a good visibility with a large source, another interferometer Y must be placed between the source and the object so as to compensate B or h, that is, to reduce s or h to zero in the source space. Such a special pair of interferometers Y and 9'are called source-size compensated. Source-size compensation being carried out, two kinds of sharp fringes can appear in the field of view: straight-line fringes when h = 0 and circular fringes when s" = 0. Here the field is called fringed field. The phase difference w due to the object may be measured by reading a local displacement of the fringes with respect to those which are to be formed in the absence of the object. So these fringes are called reference fringes in this work. If both h = 0 and 4. = 0 are assumed, the fringes expand out indefinitely and the intensity becomes uniform across the field. This is often called u n i f o r m field. The phase-difference appears as a
*
Practically, it can be considered independent of the position of
Pz.
VI,
9 21
TWO-BEAM INTERFERENCE MICROSCOPES
305
change of the interference color in white light, or as the intensity change in monochromatic light. For a sensitive observation, the background for zero phase difference of the object should be made as black as possible. The degree of blackness, or extinction factor as it is termed by INOUE and HYDE [1957], is limited by the fall of degree of coherence due to aberration, unequal division of amplitude, depolarization, scattered light and so on. 2.4. D E L A Y COMPENSATION
If a light source is not quasi-monochromatic, the intensity I ( P ) in eq. (2.12) should be integrated with respect to Y over the spectral bandwidth. Because exp{i(Znv/c)$,} is a rapidly changing function with respect to Po, the larger the bandwidth of the light, the more rapidly the visibiIity drops with increasing 9,. Thus, all the interference microscopes using thermal light are designed to compensate the delay - (2nv/c)$, by three methods: 1. optical-path compensators, 2. symmetric or cyclic construction, 3. self-compensating elements such as a Savart plate, a Wollaston prism, etc, which are composed of two identical crystal elements having an opposite birefringence. The delay introduced by the microscopic object and its dispersion varying with Y is usually so small that the visibility will not be affected practically. But, the difference in dispersion between the two separate microscopes for two beams presents a serious problem because of their complicated optical construction (HANSEN[ 19301, RANTSCH [1949]). This is one of the reasons why the common-path interference microscope may be adopted advantageously. In the discussion below, we will assume the interferometer to be delay-compensated. 2.5. USE O F L A S E R AS A S OUR C E FOR I N T E R F E R E N C E MICROSCOPES
If a single-mode gas laser is used as the source, neither the source size-compensation nor the delay compensation is necessary because of its extremely high spatial and temporal coherence. Moreover, holographic interference microscopes have been described very recently (GABOR and Goss [1966], VAN LIGTENand OSTERBERG[1966], ELLIS[1966], SNOW and VANDEWARDER [1968], MAGILL and WILSON [1968]). But, the conventional light source of an interferometer cannot be replaced totally by a laser, and the compensation techniques have
306
SOURCE-SIZE COMPENSATION
[VI.
9 3
still a constant importance in interference microscopy. The reason is as follows. As several authors indicated (MURTY [1964], MURTYand MALACARAHERNANDEZ [1965], STEEL [1965], POLZE[1965]), a lot of beams reflected from numerous surfaces of a microscope objective interfere near the image plane with the main beams to produce unwanted fringes with a very good visibility. It is impossible t o distinguish the true variation in intensity due to the two main beams from the spurious ones, for a microscopic object is usually intricate in structure. The unwanted fringes can be eliminated only by diffusing the laser light to give an enlarged extent and by establishing a source-size compensation t o get localized fringes. In addition to that, even the delay compensation is needed unless one has a priori knowledge of the approximate path-difference due to the object measured in the wave-length unit, whereas the white fringe can provide a fiduciary mark for measurement (FRANCON [196l], p. 177).
Q 3. Coherence Diffraction Theory of Image Formation and Two-beam Interference So far we have neglected the effects of diffraction and aberrations attendant upon the optics. There are two defects in the ray optical theory which was given in 9 2 . Firstly, in the presence of these effects, it is impossible to exactly retrace the system in order to find the conjugate points of the point of observation, especially for a large source. Secondly, when the incoherent source plane CT as shown in Fig. 1 is imaged onto the plane of observation, there would be zero visibility of fringes, unless the two points PI and P, coincide perfectly. This is not the case in practice, as HOPKINS [ 1957bl indicated. * 3.1. STEEL’S UNIFIED THEORY
STEEL[1959] developed a new formulation of the diffraction theory of image formation and two-beam interference, which is linear for sources of any degree of coherence. Consider the analytical signal V(P, t ) at a point of the image plane of a two-beam interference microscope, which is expressed as the sum of the two analytic signals reaching the point by each path:
* We can only apply the treatment given in § 2 by replacing approximately with an incoherent source the partially coherently illuminated aperture plane which lies a t a sufficiently great distance from the source (HOPKINS [1967]).
VI,
g 31
C O H E R E N C E D I F F R A C T I O N T H E 0 RY
V(P, t ) = Vl(P, t)+V,(P, t ) .
307
(3.1)
The mutual coherence function T(P, Q, t) of WOLF [1954, 19551 between the radiation at two points P and Q in the image plane for a time difference t is represented by the sum of the mutual coherence functions Tij(P,Q, t) between the radiation from the beam i at the point P and that from the beam j at the point Q for the time difference t:
W', Q, t) = W(P, t ) . V*(Q,t+.)> =
2 Tij(P,Q, t);
i, j
= 1,2,
ij
where
rij(P>Q, t) = (v,(P,t ) *
vT(Q,t+t)>,
(3.2) (3.3)
and the angular bracket denotes a time average and the asterisk the complex conjugate. If the Fourier inverse transform with respect to z is taken, we have a fundamental expression of two-beam interference in terms of mutual spectral density (BORNand WOLF[1959], p. 501),
where
I-, 03
fij(P, Q, v)
=
Tij(P,Q,
t)exp(i2nvt)dt
(3.5)
and the circumflex denotes the time Fourier transform. On the other hand, as ZERNIKE[1938] has shown, each mutual spectral density is propagated according to the generalized HuygensFresnel principle (BORNand WOLF [1959], pp. 514, 531). With the approximation that reduces the Huygens principle for amplitudes to a two-dimensional transform, the mutual spectral density on one of the reference surfaces is reIated to that on the subsequent reference surface by a four-dimensional Fourier transform, since it is a function of the four spatial coordinates that denote the position in a plane of the two points (BLANC-LAPIERRE and DUMONTET [1955]). In Fig. 3, a general system of microscope with Kohler illumination system is considered. In most two-beam interference microscopes, two microscopes are assumed to be so arranged in parallel for two paths that they possess the same optical axis coincided virtually. Consider now, for each path, the field stop plane, the reference sphere touching the entrance pupil plane at its center, the object plane, the referencesphere touching the exit pupil plane at its center, and the image plane
308
SOURCE-SIZE COMPENSATION
[VI,
§ 3
as shown in Fig. 3. We take these surfaces as the reference surfaces. We have assumed here that the separate optical system is aligned with the same optical axis * and has the same number of stops. Each optical system need not be identical but may possess partly or totally a different magnification. So the double-focusing interferometer can be included in the theory.
*’
_ - T,----+
I I
I
I
-
--t--T2---i
I
I I I
I
I
Lc I
- _ _ T2’
---
I I
I
To condense the notation, points are specified by a position vector normal to the optic axis and we use the following reduced coordinates (see Fig. 3) for the path i (i = 1, 2) , instead of geometrical coordinates (in capitals) :
* A slight misalignment can be interpreted as a difference in aberration between the two systems.
“1,
§ 31
u.= a
309
COHERENCE DIFFRACTION THEORY
1 1 1 ui, iii = ui, u: = Ti sin ui T i sin Gi Tl sin ui I
u;,
(3.7)
where the subscript i denotes a quantity relating to the path i, a i , Fii, Fii, ni are the refractive indices of the spaces as shown in Fig. 3, u i , G i ,Gi,a; the angular radii of the circular pupils * in conformity with the usual sign convention, Ti, p i , T: the distances measured from left to right, from the centre of the field stop, of the object and of the image to their corresponding reference spheres respectively. For systems obeying the isoplanatic condition or corrected for the transversal aberrations on the reference sphere (HOPKINS [1951, 1965, 1966]), the Gaussian image of a point at xi in the object plane is at xi = xi in the image plane and similarly ui = ui.If there is no difference in magnification between the systems for the two paths, nor in the angle the pupil subtends at the object (image) plane, x = x1 = x2,u = u1 = u2 etc. hold. Thus, the scheme of the unified theory may be expressed as follows: For the illumination system: 00
f i j ( i i i ,Ti,v) =
, v) exp{i2n(iiixi-Vjyj)}dxidyj, (3.8) [[ f i j ( x i yj, JJ-m
f i j ( i i i 3,, , v) = f i ( Z i , v ) f ; ( T j , v ) * f i j ( B i ,T,, v),
(3.9)
For the observation system:
T i j ( U i ,v;, v) = //-:Pij A
,
(Fi, yj,v) exp(i2n ( u ; X i - v~yj)}dxidyj, (3.12) A
P;j(u;,v;, v) = fi(UL,v)f;*(v;, v) * r i j ( u :v;, , v),
(3.13)
where mutual spectral density through one of the reference surfaces is denoted by a plain symbol before the plane and by the same symbol with a prime after the plane. L i ( X i ,v) denotes the spectral amplitude transmission of the object * The aperture stops are assumed to coincide with the pupils in position, but not always in size. They may be sometimes larger than the pupils.
31 0
[VI,
SOURCE-SIZE COMPENSATION
s
3
placed in the path i . fi(ui, v) or fi(ui, v) is the spectral amplitude transmission or pupil function of the condenser or objective for the path i. For example, if the condenser has no absorption and a wave aberration W i ( u i )for a circular pupil, it is f i ( U i , ).
5 pi; 1 4>Pi-
= “xP{i(24c)Wi(ui)}, =
luil
0,
(3.15)
We assume again that the image planes for both paths are so closely situated that the one (say x;) may be taken as the image plane common to both beams by considering the focal shift of the other as a defocusing aberration. Hence, we have the reduced co-ordinates for the point X’ in the image plane
, n’ * sin & x1 = x; x ,2 = n‘ - sin a;
ii
1
x.
(3.16)
Putting p’ = sin dJsin a;,
(3.17)
x; = p’xi.
(3.18)
we get By using eqs. (3.8)-(3.14), we can calculate all of P,,(x:, y i , v) = 1, 2) in the image plane, from the knowledge of P,,(x,, y,,v) on the source. Then, by summing them up to get P(P, Q, v ) given in (3.4), and taking its Fourier transform, r ( P , Q, z) is obtained:
(i,j
r ( P , Q, r ) = r11(4, Y ;, r)+rZz(p’xl, P’Y;, z) +rl2(4> P’Yl, ~)+rzl(p.’x;, Y L .I. When the two points P and Q coincide for z the point P will be obtained as
I ( P ) = T ( P >p, 0) = f&;,
= 0,
(3.19)
the intensity at
x;,0 ) + ~ 2 2 ( p ’ x ; ,E L K ; , 0)
+2%{G&;,
p’&
0)).
(3.20)
It is worthwhile to note that for quasi-monochromatic light of frequency Y&Av, mutual coherence function or mutual spectral density can be approximately expressed by the mutual intensity function: T(P, Q, 0 ) ~exp{-i2z+r} or f’(P, Q, O)d(v-$), providing 1x1 << l / A v (BERAN and PARRENT [1964]). This is Steel’s unified theory of image formation and interference (STEEL[1959]) with slight modifications; the object or the imaging
VI,
s
31
COHERENCE DIFFRACTION THEORY
311
system plays the role of linear filter for mutual spectral density. Since the interference effect in the image plane is expressed by T,z(xi, p'xi, v), the function T,,propagating through the system plays the basic role in two-beam interference theory. We will call this simply the cross-coherenee. The mismatched aberrations of the two systems can be taken into account by the transmissions for the cross-coherence, which are given by
The scalar theory for large aperture seems to be valid for a semiaperture up to about 30" (HOPKINS [1943]). The validity with respect to image extent was fully discussed by DUMONTET [1955]. 3.2. EXTENSION TO THE POLARIZATION INTERFEROMETER
In the polarization interference microscope, the two beams traverse the same optical system. Two paths are distinguished by the direction of vibration perpendicular to each other, which is to be designated with subscript 1 and 2 . The directions of (1) and (2) are so chosen that they are parallel with the principal plane of crystal elements. In order to make radiation of different direction of vibration interfere, an analyzer is needed. If we assume that the direction of the analyzer makes an angle 8 with the axis (l),we obtain the mutual coherence function at two points and at two instants in the image plane:
T(P, Q, t) = ((cos 8 - Vl(P, t)+sin 8 . Vz(P, t ) ) . (COS 8 - V,(Q, t+t)+sin 8 V,(Q, t + t ) ) * ) .
-
(3.21)
Defining T i j ( P ,Q, t)as the mutual function between two components i and j as WOLF [1954, 19561 introduced, we have a fundamental expression of polarization interference, which corresponds to eq. (3.4):
T ( P , P, 0) = cos2 8 - T,,(P, P, 0) +sin2 6 - T,,(P, P, 0) + 2 sin 8 - cos 8 . 9{T12(P,P, 0)).
(3.22)
I t is evident that the same propagation law will still hold for Tij(P, Q, t) as before only if the direction of vibration never changes (nointermingling of two components occurs). If T12(P,P, 0) is normalized with d(Tll(P,P, 0) . TZ2(P,P, 0)), it equals the degree of polarization, provided that T,,= I',, (BORNand WOLF [1959], p. 547). For maximum visibility, it is necessary that a polarizer be introduced, and that the analyzer be set at 8 = &in. Under the under-lined
312
SOURCE-SIZE
COMPENSATION
[VI,
§ 3
condition, depolarization due to the rotation of the plane of polarization cannot be contained in the system. 3.3. SHEARING ELEMENTS AND TILTING ELEMENTS
Consider a shearing element and a tilting element which are introduced in each path to give rise to a lateral displacement to the image of the field stop (“window”) and to that of the aperture stop (“pupil”), as shown in Fig. 4. They may be beam-splitters, mirrors, plane parallel plates, prisms, Savart plates, or Wollaston prisms in polarization interferometers. Taking the example in the image space, the integrand in eq. (3.14) can be replaced, if the quadratic and higher order terms in the variables may be neglected, by & ( u ; , vj, v) exp{-~2n((u~+s^~)(x~-s~)-(v~+s”~)(y~-s~))}, (3.23)
where 3; is the displacement of the beam i at the pupil surface, normal to the axis, s: that at the image plane, S ; , si.similarly defined, and they are measured in reduced coordinates. That is, generally, fij(ui, vj, v) Qij(ui,v;; xi,y;) e x p { ~ i 2 n ( u ~ x l - v ; y : ) ~ , (3.24) with
Q 23.. = exp{--i2n(3;xl--s;u;)+i2n(d~y;-d~ v;)} x x exp{*i2z (Si si-di s ; ) } ,
(3.25)
where the positive sign indicates that the direct Fourier transform should be taken in the space where the element is placed, and the negative sign indicates that the inverse transform should be taken. In other respects, the expression of Qij’s never change, irrespective of the space where the element is placed. Hence, these elements can be interpreted as a pupillary filter * and an objective filter for coherence (mutual spectral density), the transmission of which is Qij for the beam i and j . YAMAMOTO [1965, 19681 showed that the coherence transmission of both a Savart plate and a Wollaston prism is in the form of eq. (3.25). In such a commonpath interferometer, the distinction between i and j as subscript of coordinates disappears.
* If the shear is kept very small, the pupillary phase filter equivalent to the shearing element has the transmission such as l-i2nsiui for the beam i. Apart from the constant 1, this filter produces an image amplitude proportional t o the modulus of the gradient (in the direction s i ) of the phase due to the object (IWATA [1949], BREMMER [1951]).
VI, §
31
COHERENCE DIFFRACTION THEORY
---_-_-_-
T‘- -
-_
313
1
----
(b) Fig. 4. Illustrating the shearing and the tilting element; (a) a Wollaston prism (for simplicity, the plane of localization of fringes L is taken t o be normal t o the optical system axis), (b) a Savart plate.
For a Savart plate which is so placed that its principal plane is parallel to the x1axis or the x2 axis and displaces each component beam by the distances S in the direction parallel to the axis, the transmission is given by (3.26) i, j = 1, 2 , Q 23. . = q,(x,u ) q:(y, v); with (3.27) q,(x,u ) = exp{i2ns”xi}.exp{-i27ts.u,},
-
314
[VI, §
SOURCE-SIZE COMPENSATION
s=-
n sin M
a s,
$=-
1
T sin GI
s,
3
(3.28)
where1/2S denotes the shearing distance * and T the distance between the aperture and the field (objective or image). For a Wollaston prism which deviates each component beam by the angle of a in the positive or negative direction of the x1 axis, the coherence transmission is given by Q11= Q21=
q(x>u) q * ( ~ v)> , q*(x>u) 4 * ( ~ v), ,
Q12 Q22
= q(x, u)q
( ~v), , = 4 * b 3 U ) q(yJv),
(3.29)
with
q(x, u) = exp{i2nlxl} * exp{-i2nsul},
(3.20) (3.31)
where 20. denotes the splitting angle t and 5 the distance from the object plane t o the Wollaston prism. So far we have neglected the following causes which affect the coherence of polarized light: variation of path-difference denoted by Q::), depending on the second power of the incidence angle: the curved fringes at infinity by a Savart plate and the hyperbolic fringes at infinity by a Wollaston prism when it is illuminated just behind by a slit [1952]; FRANCON and parallel to its localized fringes (FRANGON SERGENT [1955]); depolarization due to the rotation of the plane of vibration inside the crystal element cut at a large angle such as 45' with the optic axis. Both restrict the usable source size and should be examined to the second approximation, after some practical method of source-size compensation is devised. It is shown that the influence on the coherence is approximately given by the integral of QI",' (u',v', x',y ' ) , taken over the pupil, and it is equivalent to the one due to the astigmatic aberration of the optical system (YAMAMOTO [1968]).
*
-
.
S = d2 e(nz-n$)/(nj++zg)where 2e is the thickness of the Savart plate, no and 12, the ordinary and extraordinary indices of the crystal and n the index of refraction of the surrounding medium. t d = { ( n e - n o ) / n }. tg E , where E is the wedge angle.
VI. § 41
316
LOCALIZATION O F FRINGES
§ 4. Localization of Fringes with Partially Coherent Light In a general arrangement which is illustrated in Fig. 5 in the case of a Savart plate, the object plane X is supposed to be an extended source M of arbitrary coherence Fij(xi, yi,0) between the two beams i and i. The optical system 0 forms of the image of M at M'. S is generally considered as a shearing element, whose coherence transmission is given by eq. (3.25). By using the method given in the preceding section, the mutual intensity in the plane X is represented by *
r(x',Y',0) = c ij
ss
FZi(Xi, yi,0) * @&;.-si-xi, y ; - s i - y i ) x
x exp(-i2n (dixi-diyj) where
'f9'(xiJ
Yj)
- exp{-i2n = gi(xi)
(disi-dj sj))dxi dyi (4.1)
gi*(Yj)'
(4.2)
Here, Qii is the coherence spread function for the beams i and j , gi(xi) being the amplitude spread function of the system for the beam i. M'
M
I
X,
(b) Fig. 5. Calculation of the coherence of a radiation field passing through (a) a shearing element and an image-forming system, or (b) an image-forming system and a shearing element, in the order named.
*
Hereafter, the integral sign without limit will stand for
s-",
.
316
SOURCE-SIZE COMPENSATION
[VI,
§ 4
The intensity at the point X’ becomes, if the system is assumed stigmatic, i.e. g i ( x i )= 6(xi),
I ( X ’ ) = ql(x;-s1, x;-sl, o)+T;z(p’x;-s2, p k - s 2 , 0) + 2 9 [q2(x;-Sl, p ’ x i - s z , 0) x x exp{-i2n ((Sl-p’ B,)x;-Sl sl+ S , s,)}], where p = sin =,/sin ul, p’ = sin uk/sin u;,
(4.3)
and FFj(P, Q, 0) denotes the value of rij(xi, yj, 0) in the plane X when x iis replaced by P, yj by Q. The visibility is determined by the modulus of cross-coherence between such two points in the source as
x1 = x;-sl,
yz = p ’ x ; - s z .
(4.4)
Incoherent source being assumed, it follows that the unit visibility is obtained only if the (overall) magnification of the system is equal for the two paths: p = p‘ and psl-sz
if p
=
= 0,
1, sl-sz
=
0.
Eq. (4.6) is equivalent to the classical rule of Raveau (COTTON [1934]) which states that the plane of localization lies at the intersection of two rays derived from a single incident ray. In the case of a Savart plate, eq. (4.6) means that u1 = up = 0; for each shear is perpendicular with each other: s1* s2 = 0. The fringes are localized in the focal plane. When we place a Savart plate in front of an aberrant system with a finite aperture, it is shown that the visibility of the fringes in the focal plane is equal to the modulus of the optical transfer function of the system at the normalized spatial frequency (Sl-Sz) with respect to infinity, while the variation in fringe position indicates its phase variation. Similarly, in the analogous arrangement as shown in Fig. 5(b), we find the mutual intensity across the image plane as
r ( X ’ , Y ’ , 0) = 2 e x p ( - i 2 n ( 8 ~ ~ ~ - 8 ~ y* ~exp(i2n(Sisi-Sisi)} )} x ij
VI, §
51
SOURCE-SIZE
COMPENSATION
317
Although the form is quite similar with eq. (4.1), the essential difference is the definition of shears. In eq. (4.11, si and g i are a reduced shear and a reduced tilt in the object space, whereas in eq. (4.7) s: and Si are the ones in the image space. It is impossible for si of a Savart plate to tend to null, unless the image plane goes to an infinite distance. The fringes are localized at infinity.
Fig. 6. Localized fringes by a Wollaston prism.
For a Wollaston prism as shown in Fig. 6, the condition for localization eq. (4.6) is expressed as sin u * 5 = 0, where 5 is the distance from the source (object) to the Wollaston; it is impossible to make u equal to zero while keeping 5 definite. Consequently, the localization is obtained in the image plane only when 5 = 0. Q 5. Source-Size Compensation The combination of the two arrangements as shown in Figs. 5(a) and (b) enables us to constitute a source-size compensated interference microscope. There are four possible modes according to which one is placed before the other, the image plane and the exit pupil of the precedent system being coincided with the object plane and the entrance pupil of the subsequent system respectively. We call the coincided planes the intermediate planes, denoted by XI o r x 2 for the beam 1 or 2, and place the object to be observed in either plane.
318
SOURCE-SIZE COMPENSATION
[VI,
s
5
Fig. 7. Source-size compensation in the field plane: illustration of the case when two shearing elements are placed outside a microscope (condenser-objective system).
It is to be noted that the source plane X and the (final) image plane X are common t o both beams in the sense explained before (9 3). If we take as an example the case shown in Fig. 7, the one (Fig. 5(a)) is placed before the other (Fig. 5(b)) with each pupil superposed, similar operations as before lead to the mutual intensity in the image plane being represented by
qx',Y',0) =
ss
z: ij
Tij(Xi,
Y j , 0)
x & & - (si+s'i)-xi,y : - ( s j + s ; ) - y j ) x exp{-i2n( (d,x,+sixi) - (djyj+d;yj))}dx,dyj xe~p{-i2n(8~s~-~,s,-d~s~+d~s~)),
(5.1)
Y j ) = &(Xi) . g"?(Yj),
(5.2)
where &(Xi,
and gi(xi)denotes the overall spread function of the composite system (condenser-objective system), with &,j the overall transmission of coherence. When the approximation that the whole system is stigmatic is made, one finds easily
r(x',Y', 0) = 2 lyj(x;-(s,+s;),y ; -
(Sj+Si),
0)
ij
x exp{-i2n((di+d;) xi- (Sj+Si) yj)} x exp(i2n (si(di+di)
s; (dj+ S i ) ) } .
-
(5.3)
Thus it is concluded that the one system can compensate the loss of cross-coherence due to the other, if the following two conditions are both fulfilled:
(I) the overall magnification between the planes X and X' must be equal for the two paths:
VI,
5 51
319
SOURCE-SIZE COMPENSATION
m, = m2 = m
where
p = sin a,/sin u,,
or
p =p’,
p’ = sin ailsin
(5.4) M;;
(5.5)
(11) the ratio of the sum of the reduced shears for each path must be equal to the ratio of the apertures subtended a t the source plane : P(Sl+S&
=
( s 2 + 4
0.
(5.6)
“ I f p u l , (sz--s,)+
or
(.;-s;) = 0,
(5.7) (5.8)
(Sl+s;) = ( s 2 + 4 ) .
Using condition (I), this is also expressed in terms of the relative shear with respect to one path which is measured in geometrical coordinates, (S,-S,)+m(S;-s;)
=
0.
(5.9)
(111) In addition to these conditions, the condition for the zero phase of cross coherence:
* If
(8,fs”;) -p’(8,+8k)
=
0.
(5.10)
p’ = 1,
(s^,-SA1)+
(S‘;-s”;)
=
0,
(5.11)
or ($1+$;)
=
(8,tQ
(5.12)
is satisfied, the fringes disappear. Using the condition (I),this is also represented as (5.13) where T,, T ; , T,, Tb are the aperture distances for each path, as defined in 0 3. STEEL[I9591 derived these conditions in a slightly different way. I t is shown that they never change even when the finite aperture and aberrations of the system are taken into consideration or when any other mode of combination of the two is adopted. From only the two conditions (I) and (II), source-size compensated interference microscopes presenting a frilzged field can be derived. All the conditions
* In a common-path interferometer, p = p’ = 1. The same condition must be valid for maximum contrast in interferometers of a different type.
320
SOURCE-SIZE COMPENSATION
[VI,
s5
(I), (11) and (111) being satisfied, the ones showing a uniform field will result. In order to find polarization interferometers belonging to the latter, F R A N ~ [1956, O N 19661 showed the following compensation principle equivalent to eqs. (5.7) and (5.11). We may call this the principle of localized fringe transfer: (a) the two fringe systems of the birefringent elements, which may be either fringes at infinity or localized fringes, must be exactly superposed in one plane (we will call it the plane of superposition); (b) the two birefringent systems must be so oriented that the splitting produced by one is exactly cancelled by the other. When eqs. (5.7) and (5.11) are perfectly established as well as the condition p = p’ = 1, the two exit pupils and the two exit windows are exactly coincided. Each of the rays emitting from the source proceeds as if the shears and tilts never occurred. Therefore, the validity of the conditions (5.7) and (5.11) is not confined to the planes of the source and the image. They are virtually valid for any other pairs of conjugate planes; that is, the interference phenomena are “non-localized”. It is this fact that justifies the equivalence of FranCon’s conditions to ours. U’
!
I c
I
I
Fig. 8. Source-size compensation on the pupil surface.
To complete the study of the compensation, suppose that another objective is placed in the image plane and assume that it forms the image of the exit pupil plane U‘ at U“, and that p = p’ equals to unity. In Fig. 8, the complex system is represented by a simple objective 0. By taking the Fourier transform of the mutual intensity r ( X ’ , Y , 0 ) , we get eq. (5.11) as the condition for recovering the modulus of cross-coherence and eq. (5.7) as the condition for zero phase,
VI,
§ 61
PRACTICAL METHODS
32 1
so that the two conditions are interchanged. The fringes to appear in the pupil are just what is called “source fringes”, while the fringes to appear in the image plane are called “test fringes” by STEEL[1965, 19671. These are specified as complementary. These considerations not only give general grounds for reasoning the principle of the practical methods of source-size compensation in any type of two-beam interferometer, some of which will be given in the next section, but they provide a useful tool for approaches to new designs of interference microscopes. It is last but not least that the analysis based on eqs. (5.1) and (5.2) will be used to study the influence of imperfect compensation upon image-formation by an interference microscope (§ 7 ) .
8 6. Practical Methods of Source-Size Compensation in Shearing Interference Microscope with Polarized Light In this section, we will confine ourselves to presenting all the practical methods of source-size compensation in a shearing interference microscope. Most of them were described by KRUG,RIENITZ and SCHULZ[1964], FRANCON [1961, 1966, 19671, BRYNGDAHL [1965], but the purpose of our study is somewhat different from theirs. We will consider them as theoretical consequences of the discussion made in the previous section, and will classify them systematically into several modes of construction, add some which have been derived analytically, and finally make a simple comparison from a theoretical point of view. As stated earlier (9 2.4), in interference microscopes the interferometer with dissimilar arms is rare, because matching of the dispersion in the two arms is difficult. In interferometers of symmetrical form, there is no particular problem about source-size compensation except matching aberrations and adjustment of the optics which was discussed partly by HOPKINS[1957b, 19671. Interference microscopes in common-path form can be well realized with polarization interferometers. The double refracting systems such as Savart plates or Wollaston prisms can divide the light beam into two beams polarized at right angles with exactly equal amplitude, independent of the wave-length and nearly of incidence-angle. At the same time, they produce shearing (lateral displacement) or double focusing (longitudinal displacement) * between the two polarized beams.
*
The best known form is described by SMITH[ 19551.
322
SOURCE-SIZE
COMPENSATION
[VI,
5
6
Ordinarily, the double refracting system is self-compensated satisfactorily. The delay is not worth due consideration. In the present status of technique, the double-focus method is not so suitable for interference microscopy as the shearing one, because the possible defocusing cannot be so large as to give an undisturbed reference wave-front (KRUG,RIENITZand SCHULZ [1964], p. 136); moreover, as STEEL[1967] pointed out, it is also difficult to form reference fringes without affecting the visibility. This is why it is excluded in this section.
Fig. 9. Practical methods for obtaining the fringed field: (a) NOMARSKI and WEIL [1955], (b) NOMARSKI and WEIL [1955], (c) GONTIER[1957], GUILD[1957],
(d) GONTIER[1957], GUILD[1957], (e) UHLIG [1965];
VI,
s
PRACTICAL METHODS
61
323
6. 1, METHODS F O R ORTATNING FRINGED F I E L D O F VIEW
6.1.1. I n the object plane
When only eq. (5.7) is valid, the fringes of spacing inversely proportional to I (d,--$,)+ (&--$;)I appear across the field. The optical-path difference due to the object can be measured by reading the deformation of the fringes which play the role of reference fringes. When each of the two Savart plates or the two Wollaston prisms is placed outside
(b')
(d')
2
, , (e')
I I
, , I
(a') YAMAMOTO [1968], (b') YAMAMOTO [1968], (c') FRANFON and PRAT [1964],
(d') YAMAMOTO [1968], (e') YAMAMOTO [1968].
324
S O U R C E - S I Z E C 0 M P E N S A T 10 N
[VI, §
6
of a microscope (condenser-objective system), one before and the other after, eq. (5.7) is rewritten as s+s‘
=
0
(6.1)
in terms of scalar reduced shears defined by eq. (3.28) or (3.31). From the relation (6.1), we get S .m
=
-S’,
(6.2)
for Wollaston prisms 05 . m
=
-a‘(’,
(6.3)
for Savart plates
where m is the magnification of the condenser-objective system between the planes x and x‘. A most simple solution is that m is unity. Then, the illuminating system and the image-forming system must be identical and arranged symmetrically with respect to the intermediate (object) plane, and the two birefringent systems have to be also identical and placed in an appropriate orientation. The system with transmitted light can immediately be converted to a reflection system. All the methods as shown in Figs. 8 and 9 follow this principle and can use a large light source. NOMARSKI and WEIL [1955] is the first to have described this kind of system with his modified Wollaston (NOMARSKI [1955]), as shown in Figs. 9 [a), (b). The fringe spacing i is given by
i
=
@/a)* T / ( T - ( ) ,
where 20 is the splitting angle of the Wollaston. The equivalent Savart systems are shown in Figs. 9 [a’), (b’). 6.1.2. In the pupil surface
As illustrated earlier, the complementary fringes of spacing inversely proportional to 1 [sz-sI)+ (d-s;)] appear across the pupil when eq. (5.11) is established. The fringes represent also the wave-front aberration, with respect to a chosen position of the object plane, which will affect the visibility of the fringes that must appear in the image plane when the same optical system is used as a part of an interference microscope. The corresponding equation to eq. (6.1) is given by 3+gr = 0
(6.5)
in terms of scaler reduced tilts defined by eq. (3.28) or (3.31). From this, we find the relations quite similar to eqs. (6.2) and (6.3):
VI.
§ 61
PRACTICAL METHODS
for Savart plates
s
4L = s’,
325
(6.6)
-o’(T’-t’), (6.7) where 4L is the magnification for the entrance and pupil planes of the whole system. If 4L is assumed to be unity, the pupils lie in the principal planes of [1957] and GUILD[1957] described independently the system. GONTIER the system shown in Figs. 9 (c), (d) for the study of the irregularity on the pupil surface. FRANCON and PRAT[1964] described its equivalent shown in Fig. 9 (c’), where two modified Savart plates by Franqon are used. It may be converted to the reflection system as shown in Fig. 9 (d’). In these cases, ordinary Savarts are used with the light source of more limited extent because of the curvature of fringes. For the same reason, Franqon’s modified Savart may be used in Figs. 9 (a’), @’). FranCon and Prat applied the system to an interferential focometer for the lens L, because the fringe spacing i is proportional to the focal length f of L: for Wollaston prisms o(T--5) A
=
i = (A/ZS)f.
(6.8)
For optical testing of the microscope objective, the systems as shown in Fig. 9 (e) and Fig. 9 (e’) can be used. UHLIG[1965] tested the mechanical tube length by the former system, because the fringe spacing varies proportionally with 5-l. YAMAMOTO [ 19681 utilized the latter to test the aberration of an objective with respect to the image plane conjugate to the reflecting object plane. This is no more than a polarization version of the classical Waetzman’s interferometer (WAETZMAN [ 19 121 ) . 6.1.3. Comparison between the uses of Savart plates and Wollaston prisms
The position of a Savart plate along the optical axis is completely free, but the fringe spacing is constant, unless it is split into two wedged ones which slide with respect to each other (TSURUTA [1963]) or it is replaced with two counter rotating Savart plates (STEEL[1964]). On the contrary, the spacing of the fringes by the Wollaston varies as we move it along the axis. 6.2. METHODS FOR OBTAINING UNIFORM FIELD O F VIEW
When both the conditions indicated by eqs. (5.7) and (5.11) are satisfied, the two beams become coherent and the intensity distribution turns uniform everywhere, with a large light source. Since, both
326
SOURCE-SIZE COMPENSATION
[VI,
a
6
eqs. (6.1) and (6.5) are satisfied, eqs. (6.2) and (6.6) or eqs. (6.3) and (6.7) hold. In order that eqs. (6.2) and (6.6) are satisfied for two Savarts, the optical system placed between them should be afocal, namely, telescopic, and the ratio of the shearing distances S , S’ of the Savart plates should be equal to the constant magnification of the afocal system,
yjs
1
-m
=
-&,
(6.9)
For two Wollastons to be used, they should be placed in the conjugate planes of the optical system which are placed between the Wollastons, and the ratio of the splitting angles 0, 0‘ of the Wollastons should be connected with the magnification rji for the two planes where each Wollaston is placed, O’/O =
-1jrji.
(6.10)
In Fig. 10, the methods for obtaining a uniform field can be classified in four modes of combination of the two fundamental forms as shown in Figs. 5 (a) and (b), where the exit pupil of the first system and the entrance pupil of the second are usually taken a t infinity: mode (1): two birefringent systems are both outside the condenserobjective system (Figs. 10 (a)-(e)); - mode (2): two birefringent systems are both between the condenser and the objective (Figs. 10 (f), (g)); - modes (3) and (4):one birefringent system is outside the system and the other inside it (Figs. 10 (h), (i)). -
The uniform field can be converted into the fringed one without spoiling compensation, when we place some additional interferometer which forms its localized fringes, real or virtual, in the image plane. 6.2.1. Mode (1) The most simple method which is shown in Fig. 10 (a) was described by FRANCON and YAMAMOTO [1962]. (a) and @) which FRANCON and CATALAN[1960] described follow the principle given by eq. (6.9). The adjustment for obtaining afocality was made by the movement of the condenser. (c) is the well-known Smith’s method (SMITH[1947]), where the two Wollastons are placed in the focal planes of the condenser and objective (T = 5‘ and T‘ = Since the focus of high aperture objective or condenser is not accessible, Nomarski realized this principle with his modified Wollaston prism. FRANCON and YAMAMOTO [1962]
c’).
VI,
S 61
PRACTICAL METHODS
(i)
7/-!
327
m t
uJ+
Fig. 10. Practical methods for obtaining the uniform field (transmission system) : (a) FRANFON and YAMAMOTO [1962], (f) LEBEDEFF [1930], (b) FRANFON and CATALAN [1960], (9) LEBEDEFF [1930], (h) FRANFON [1957], (c) SMITH[1947], (a) FRANFON and YAMAMOTO [1962], (i) LINDBERG [1952]. [1957], (e) FRANFON
328
SOURCE-SIZE COMPENSATION
[VL
I
6
extended Smith's method t o the case where T--5 # 0 and T'-(' = 0 are valid, which is shown in (d). (c) and (d) follow eq. (6.10). A mixed system consisting of a Savart plate and Wollaston prism was described by FRANCON [1957], which is shown as (e). In this case, a modified Wollaston prism should be used so that both the direction of vibration and the direction of shear may be parallel to those of the Savart plate. 6.2.2. M o d e (2) LEBEDEFF[1930] used the well-known interferometer by JAMIN [1868]. This is the first system which was source-size compensated
for a uniform field. It consists of two identical crystal plates cut at 45" to the optical axis, with a half-wave plate between them. The object is also introduced between them. We can suppress the halfwave plate by replacing the two identical plates with two identical Savart plates, but the non-straightness of their fringes reduces the usable aperture (LEBEDEFF [1930]), see Figs. 10 (f), (g). 6.2.3. M o d e s (3) and (4)
FRANCON [1957] used two Savart plates, one under the object and the other behind the afocal system consisting of the objective and an auxiliary lens (Fig. 10 01)).This system enables one to convert the well-known Franqon's interferential ocular ( FRANCON [ 1952,19541) into the source-size compensated one, just as the system shown in Fig. 10 (e). LINDBERG[1952] described a mixed form as shown in Fig. 10 (i). 6.2.4. Reflection system
Since some of the above mentioned systems are symmetrical with respect to the object plane, they can be adapted immediately to a reflection system as shown in Figs. 11 (a) and (c) (SMITH[1947], NOMARSKI and WEIL [1955], FRANCON [1953]). Later, DYSON[1963] made the beam reflect back and traverse the optical system twice so that the interferometer became an extremely stable one (Fig. 11 (b)). More recently, FRANFON and YAMAMOTO [1962] proposed the system shown in Fig. 11 (d), where an auxiliary lens L, is added to make the whole catadioptric system (L,+L,+L,) afocal. So far, reflection systems have been described which are to observe the p2ane reflecting object. To observe the curved object, YAMAMOTO [I9681 described the system which is shown in Figs. 11 (e) and (f). A varifocal system is introduced, which consists of two lenses L, and
VI,
5
329
PRACTICAL METHODS
61
M'
M
M
M'
t M'
5'
(e1
Fig. 11. Practical methods for obtaining the uniform field (reflection system): (a) SMITH[1947],
(b) DYSON[1963], (c) FRANFON [1953],
(d) FRANFON and YAMAMOTO [1962], (e) YAMAMOTO [1968], (f) YAMAMOTO [1968].
330
SOURCE-SIZE COMPENSATION
[VI,
J
6
L, both of which have the power, equal but contrary in sign. As the distance between them varies, the focal length of the composite system varies inversely proportionally to it. Its principal planes H and H’, however, stay always at the focal plane of L, and L, respectively. I n the system shown in Fig. 11 (e), if the principal plane H lies in the plane M, conjugate with the image plane M’ of the microscope, the ensemble system including the reflecting surface can be made afocal with unit magnification * by coinciding the focus of the varifocal system (L,+L,) with that of the catadioptric system. In order to do it, one may move the diverging lens relative to the converging one while keeping the latter immobile. Consequently, by placing two identical Savart plates, one before the varifocal system, the other after the objective, source-size compensation will be established, irrespective of the magnification of the objective and nearly of the curvature of the object. I n the system shown in Fig. 11 (f), by placing the first Wollaston a t an appropriate distance from the diverging lens and moving them as a body, the first Wollaston W may be imaged on the second W‘ with a constant magnification independent of the magnification of the objective for a large extent of range of curvature of the object. 6.3. OR J E C T I V E COMPENSATION AND P U P I L L A R Y COMPENSATION
I n the source-size compensation for obtaining a uniform field of view, two different types of compensation may be distinguished by the place where two rays derived from a single ray intersect with an angle, i.e. the position of the plane of superposition (9 5 ) . The case when the position is in the exit pupil of the objective may be called the pupillary compensation, while the case when it is close to the object plane, the objective compensation (FRANCON and YAMAMOTO [1962]) t. The important thing is that the aberrations of the objective affect the interference in quite different manners in these two cases. Consider the two methods which were shown in Figs. 10 (a) and (c). We illustrate them again in Fig. 12, the objective compensation by solid lines and the pupillary compensation by dotted lines. To obtain the same shear PIP, in the object plane, the two rays are separated in the pupil plane by the distance m i n the former, but *
Because M and M’ are the principal plane pair of the catadioptric system.
t The first objective compensation was described for double focusing interference [1951]. niicroscopes by PHILPOT
VI,
§ 71
331
IMAGES OF INTERFERENCE MICROSCOPES
M
M'
Fig. 12. Comparison between the objective compensation and the pupillary compensation.
null in the latter. In other words, the two images of the exit pupil are sheared by to the pupil in the former. Since the fraction of diameter corresponds to a normalized spatial frequency s",--s^, , it is shown that the visibility reduces to the value of the optical transfer function at the frequency &,-& with respect to the object P, or P, (YAMAMOTO[1965, 19681). Even when the condenser-objective system is corrected perfectly, the visibility never attains to unity in the case of objective compensation. The more the shear amounts, the more serious loss will be involved. Aberrations accelerate the deterioration. On the other hand, in the pupillary compensation, the surface of the exit pupil is really curved and astigmatic; moreover, large amounts of distortion are present in the pupil plane. Consequently, the superposition of the localized fringes becomes imperfect. And SO the pupillary compensation will be difficult with a high aperture.
5 7.
Images of Source-Size Compensated Interference Microscopes
If the source-size compensation is attained, the object can be observed through a microscope with a large light source which is usually imaged in the entrance pupil of the condenser. For simplicity, we assume that the optical system satisfies p = p' (eq. (5.5)) and p equals unity. Then, it is not necessary to distinguish the variables for beam 1 from those for beam 2. As stated earlier, if an object is introduced which is characterized by amplitude transmission Li(B)for the path i, the coherence transmission is given by eq. (3.11),
Then the mutual intensity in the image plane is represented by the
332
SOURCE-SIZE
COMPENSATION
[VI.
§ 7
following double convolution integral, using eqs. (4.1) and (4.7):
r&d,y‘, 0) =
/!
ri.(x, y, o)@zj(B-si-x, y-sj-y) I
X A i j ( X , p)@ i j ( X ’ - s ; - X ,
-
y’-sj-y) x exp{-i2z((dix-diy) (Bix’-$y’))} x exp{-i2n ( (siBi-sjdj)- (s:B ~ - . s ~ i ~ ) ) } d x d y d ~ d ~ .
+
T,,, Tlz, After calculating this integral for all pairs of i, j , i.e. T,,,
rZ1, putting x’= y‘, we obtain the intensity distribution in the image
plane, I ( x ’ ) = T ( x ’ ,x’,O), following the scheme of the unified theory. Since the object is partially coherently illuminated by each beam alone, rii’s (i = 1, 2) agree with the intensities which were given with the transfer factors or transmission cross-coefficients (HOPKINS [1953], BORNandWoLF [1959] p. 523) by HOPKINS[1953,1957b], with the exception of a bodily displacement due to si . But, the cross-coherence such as T,,slightly differs from them, as we will discuss below. If the object is sufficiently low contrast in the sense that
Li@)= \Li(X)l exp{iyi(n)}
(7.3) (7.4)
llLi(%)l-ll << 1, Pi(%) << 1, as MENZEL [1958, 19601 and SLANSKY [1959, 1960, 19621 described, all Tij’s can be so simplified that their Fourier transforms can be separated into two parts, the one related to the spatial Fourier transform of yi(X)and the other to the spatial Fourier transform of lLi(X)l-l, i = 1 or 2. 7 . 1 . IMAGE INTENSITY
If the assumption is made that the mutual intensity in the entrance pupil of the condenser is approximated by Dirac’s delta function* rij(u, v) = G(u-v--d), (7.5) where B is the sheared distance produced by the first shearing element, we find the intensity in monochromatic light at the point x’ in the image plane: Fij(x’, x’,0) = C . exp{-i2n(d+$’) x’}
x / / A i j ( r n + n , n)tij(&, 0; m + n + d , n ; s+s’) x exp{-i2nrn(x’-s;)
*
+i2zns‘}dmdn,
(7.6)
This assumption means that a perfect collector lens images a light source onto the entrance pupil of the condenser through the first shearing element.
VI, §
71
333
IMAGES O F INTERFERENCE MICROSCOPES
where s
C
= s.-s. 2
=
3’
g
= d a. - - $ .3 7
s’ = s a. - s . 3 ’ 8’ = s,.-s.9 ) ’
I
n‘
”I
exp{i2n [( s i + s ; 8) - (sig i-sj Sj-si $ +sj .() ) >.
(7‘7) (7.8)
Here, 3iij(u,v) denotes the Fourier transform of &(R, p) in eq. (7.l ) , the product of the two-dimensional inverse Fourier transforms
of Li[%)and L,[y): 3iii(U,
v)
= &(u)*
Zj*(v),
L
(7.9)
m
Zi(u)=
L t ( X ) exp(i2nu2)d%,
(7.10)
and t i j is defined as follows:
tij[a, b ; a‘,6‘;c ) =/fdv+a)fXv+b)
fi(v+a)fi*[v+b’) . exp{i2ncv}dv,
(7.11)
where f i , fi are the pupil functions of condenser, objective for the beam i. The tij’s may be called the transfer coefficients of the two-beam interference microscope. The region of integration is shown in Fig. 13. For tii [i = 1, 2), the two circles of the condenser apertures denoted by f i and f: coincide with each other, for 8 = di-di = 0. Then tii’s become equal t o the transfer factors. In the absence of the object or in the presence of the object uniform in transmission (L,[X) = L,(%) = l ) , &(m+n, n) = d(m+n) * d(n).
(7.12)
Then, one finds
rij(x’, x’, 0)
x
J
=
exp{-i2n[8+d’)x’} x
fi(v+s^)fj(v+s^) fj*(v)fj*(v)* exp{i2n(s+s’)v}dv.
(7.13)
On the one hand, the influence of the imperfect compensation s+s’ # 0 can be analyzed by evaluating the integral given by eq. (7.13). This leads to a tolerable deviation from the compensation conditions. On the other hand, if the compensation conditions are satisfied, the integral is no more than the optical transfer functions of the whole system at the normalized spatial frequency d . This is the expected intensity in the absence of objects. When ap-
334
SOURCE-SIZE COMPENSATION
Fig. 13. Integration domain for the transfer coefficients of the two-beam interfercnce microscope when fi, f j denote t h e pupil of the condenser for the beams i and j respectively, similarly f ; , /; t h a t of t h e objective, measured in reduced co-ordinates. The area enclosed by a thick line is the integration domain.
plying the HOPKINS’Sformulae which yield aberration tolerances (HOPKINS[1957a]), we can determine the tolerable aberrations for an interference microscope, so as to give no appreciable amount of intensity in the black background. YAMAMOTO [1965, 19681 discussed these problems in the case of his interference microscope with very small shear, which is called differential interference in the German [1967]). literature (eg. FRANGON 7.2. HARMONIC ANALYSIS O F IMAGE INTENSITY
When we take the two-dimensional Fourier transform of eq. (7.6) for each pair (i, j ) , the compensation conditions being satisfied, we find
bij(m)= C exp(i2nms~)
x j t i j ( S , 0; m + n + $ , n ; 0) A,,(m+n, n ) exp(i2nns’}dn,
(7.14)
VI,
s
71
IMAGES O F INTERFERENCE
MICROSCOPES
335
where C is given by eq. (7.8). Summing up all bii’s, we obtain b(m),the two-dimensional spatial frequency component of the image intensity. bii (i = 1, 2) represents the image formed by each beam alone, and the image contrast depends mainly on the bij’s (i # j ) . STEEL [1959] described the b ( m ) in the interference microscope with separate arms, where an object is placed in one path and a clear object in the other path and no shear is present. Then, it is by the following expression that the aberration balancing between the two optical systems for the beam i and j can be analyzed: t i j ( O , 0; m ,0, 0) =
s
f i ( v ) f j * ( v ) f ~ ( m + ~ ) f ~ * ( v )(7.15) dv.
For the low-contrast object, which is called the Zernike-Gabor object ~ ~ L O H M A N N from eqs. (7.3), (7.4) and (7.10),theFourier [1956], transform of the object is expressed as the following sum:
Z(m) = d(m)+Z’(m).
(7.16)
If we designate the Fourier component of the amplitude and the phase of the object by a ( m ) a n d p ( m ) , * i.e.
s-,
W
a(m)=
{ I L ( i ? - l } exp{i2nm2j&,
(7.17)
W
p(m)= S _ q ( . i ) exp{i2zmdjd.i,
(7.18)
the interference microscope using a large source, that is, with partially coherent illumination, plays the role of a linear filter for a ( m ) and p (m). The transmissions for these are called “amplitude contrast function” and “phase contrast function” by MENZEL [1958, 19601 and denoted by d ( m )and S ( m ) (MAR~CHAL and FRANGON [1960]).+ Menzel also showed the d ( m ) and 9 ( m ) for the interference microscope with separate arms, where an object is placed in one path only. YAMAMOTO [1965, 19681 described these functions for the interference microscope of differential shearing when a circular aperture is assumed. These results are roughly illustrated in Fig. 14, where y is the phase angle introduced by the compensator. The dotted line *
We have the following relations: Z’(m)+l’*(-m)= 2 a ( m ) , Z’(m)-Z’*(-m) = i2p(m). t d( m ) and Q(m ) are defined as the coefficients of a(m ) and p ( m ) in the expression of b ( m ) ,i.e. ( m# 0 ) . b ( m )= C b i i ( m )= d ( m ) . a , m ) $- 9 ( m ). p ( m ) $9
336
SOURCE-SIZE COMPENSATION
ivI, § 7
Fig. 14. Representation of the amplitude contrast function d ( m ) and the phase contrast function 9 ( m ) in the case of (a) interference microscope of Michelson type, (b) interference microscope of very small shear (the case when the shear in the object plane is given by @/(N.A.), where (N.A.) is the numerical aperture of the objective). Both the objective and the condenser are assumed to be free of aberrations and have thc same numerical aperture.
indicates the optical transfer function of the microscope objective which is defined with incoherent illumination. 7 . 3 . E F F E C T S O F THICKNESS O F THE OBJECT U N D E R OBSERVATION
It was stressed earlier that. interference is non-localized, if the sourcesize compensation has been achieved for a uniform field and an object is assumed to be infinitesimal in thickness. The actual object, however, may have locally varying, appreciable thickness and hence varying inclination.
---I?--Fig. 15. Illustrating the localization of interference by a wedge-shaped object.
If a wedge-shaped object is placed in one path as shown in Fig. 15, a shear s due to the inclination of the upper surface takes place, and the virtual image of the intermediate plane 2 conjugate with the image is shifted toward the object by the distance h:
VI,
§ 71
IMAGES OF INTERFERENCE MICROSCOPES
s = -Z(l-(no/n))
h
=
-
8 n sin u/A,
337
(7.19) (7.20)
{ ( ~ o - n ) / f i o }=e p/no,
where I is as indicated in Fig. 15, h the shift at the center as defined in 3, 8 the wedge-angle, no and n refractive indices of the object and the surrounding medium, e the mechanical thickness, cp the optical thickness of the object, i.e. p = (no-n)e.
(7.21)
For maximum contrast, s should evidently be null. If the inclination 8 is not zero, the interference becomes localized a t I = 0. One must focus to the upper surface of the object within the depth of focus range of the objective. When the reference fringes are to be introduced, it is on this surface that they should be also localized. The shift h (defocusing) by eq. (7.20), however, still remains. * For actual objects, e, p, s, h and 8 are all considered as functions of location; 8 may be replaced with (grade ( X ) ( . The defocusing can be included in the amplitude transmission defined as (l-@iP)} for ii2 _I p2 L (2,23) = exp(-i(2zv/c)q(~)
for d 2 > p2
(7.22)
where /?= - (nsin u)2/n,,and p is the reduced radius of the condenser aperture. Consider the image produced by a shearing interference microscope. Substituting eq. (7.22) for L i ( X ) in eq. (7.1) and calculating with appropriate approximation, the image intensity is obtained as follows:
I&’)
= T(x’,x’,0)
=
2 exp(i(2nvlc)(1-iflp2)
(p(x’-s~)-p(x’-ss;))}. (7.23)
ij
The factor (1-@p2) is the “obliquity effect” of illumination aperture size, for which INGELSTAM [1960], INGE LsTAMand JOHANSSON [1958], TOLMAN and WOOD[1956], GATES[1956], BRUCE[1955, 19571 and THORNTON [1957] have derived the correction formulae. With reflected light microscopy, we may put no = -n in eq. (7.20).
* The tilt will also take place in the exit pupiI. This effect was described recently by GUILLARD[1963-19641, and was ignored here.
338
SOURCE-SIZE COMPENSATION
[VI
Q 8. Conclusion An outline of a general theory of source-size compensation in twobeam interference microscopy has been presented. It is formulated in terms of four coherence functions rijbetween the radiation from the beams i and j at two points. The representation of these functions was investigated on any one of the reference surfaces (pupil planes or window planes) in the image-forming system including several shearing and/or tilting elements. This representation applies to both the localization of fringes and the source-size compensation in any plane of interest, whereas the latter was described formerly as the superposition of two systems of localized fringes. These studies have yielded simple formulae which serve as an analytic tool for devising new designs, some of which were described above. It was also deduced that the transfer properties of source-size compensated interference microscopes can be expressed by the newly defined transfer coefficients of the interference microscope, characteristics somewhat different from those of ordinary microscopes. They can be utilized for examining the influence of imperfect compensation and fixing permissible aberrations for high visibility. This is indispensable information for an attempt to produce any kind of interf erence microscope. The explained theory which we owe principally to Steel unifies the theory of interferometer and that of image-formation by a microscope. It has been proven that the theory throws a new light on the difficult theoretical and practical problems encountered in interference microscopy.
References ALLEN, R. a n d J . W. BRAULT, 1966, I m a g e Contrast a n d Phase-Modulated Light Methods i n Polarization a n d Interference Microscopy, in: Advances i n Optical and Electron Microscopy, eds. R. E. Barer and V. E. Cosslet (Academic Press, New York) p . 77. J . D. a n d A. LOHMANN, 1964, Optica Acta 12, 185. ARMITAGE, BEKAN, M. and G. B. P A R R E N T , 1964, Theory of Partial Coherence (PrenticeHall, Englewood Cliffs, New Jersey) p. 111. BLANC-LAPIEKKE, A. a n d P. DUMONTET, 1955, Rev. O p t . 3 4 , 1 . BORN, M. a n d E. WOLF,1959, Principles of Optics (Pergamon, London). B R E M M E R , H., 1951, Physica 17, 63. BRUCE, C. F., 1955, Aust. J. Phys. 8, 224. BRUCE, C. F., 1957, Optica Acta 4 , 127.
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BRYNGDAHL, O., 1965, Applications of Shearing Interferometry, in: Progress in Optics, Vol. 4, ed. E. WOLF(North-Holland Publishing Co., Amsterdam) p. 37. CONNES,P., 1956, Rev. Opt. 35, 37. COTTON,A., 1934, Rev. Opt. 13, 153. DUMONTET, P., 1955, Optica Acta 2, 53. DYSON,J., 1950, Proc. Roy. SOC.204, 170. DYSON,J., 1963, J. Opt. SOC.Am. 53, 690. ELLIS,G. W., 1966, Science 154, 119s. FRANGON, M., 1952, Rev. Opt. 31, 65. F R A N ~ M., O N1953, , Rev. Opt. 32, 349. FRANFON, M., 1954, Optica Acta 1, 53. FRANFON, M., 1956, Interferences, diffraction et polarisation, in: Handbuch der Physik, Vol 24, ed. S. Flugge (Springer, Berlin) p. 171. FRANFON, M., 1957, J. Opt. SOC.Am. 47, 528. FRANFON, M., 1961, Progress in Microscopy (Pergamon, Oxford). M., 1966, Optical Interferometry (Academic Press, London). FRANFON, M., 1967, Einfuhrung in die neueren Methoden der Lichtmikroskopie FRANFON, (Verlag G. Braun, Karlsruhe) p. 102. F R A N ~ M. ON and , L. CATALAN,1960, Rev. Opt. 39, 1 . FRANFON, M. and R. PRAT, 1964, Optica Acta 11, 252. FRANFON, M. and B. SERGENT, 1955, Compt. Rend. Acad. Sci. 241, 27. FRAN~O M.Nand , S. SLANSKY, 1965, Coherence en Optique (C.N.R.S., Paris). FRANFON, M. and T. YAMAMOTO, 1962, Optica Acta 9, 395. GABOR, D. and W. P. Goss, 1966, J. Opt. SOC.Am. 66, 849. GATES,J . W., 1956, J. Sci. Instr. 33, 507. GERHARDT, U. and H. LENK,1960, Feingeratetechnik 9, 529. GERHARDT, U. and H. LENK,1962, Feingeratetechnik 11,208. GERHARDT, U., 1967, Feingeratetechnik 16, 505. GONTIER, M. G., 1957, Compt. Rend. Acad. Sci. Paris 244, 1019. GUILD,F., 1957, Phys. SOC.Year Book, p. 30. GUILLARD, M., 1963-1964, Rev. Opt. 42, 463; 43, 27, 64, 349. HANSEN, G., 1930, 2. Instrumentenk. 5 0 , 460. HANSEN, G., 1942, Zeiss Nachr. 4, 109. HANSEN, G.,1954, 2. Angew. Phys. 6,203. HANSEN, G.,1955, Optik 12, 5. HANSEN, G.and W. KINDER,1958, Optik 15, 560. HOPKINS, H. H., 1943, Proc. Phys. SOC.55, 116. HOPKINS,H. H., 1951, Proc. Roy. SOC.A208, 263. HOPKINS, H. H., 1953, Proc. Roy. SOC.A217, 408. HOPKINS,H.H., 1957a, Proc. Phys. SOC.B70, 449, 1162. HOPKINS, H. H., 1957b, J. Opt. SOC.Am. 47, 508. HOPKINS, H.H., 1965, Japan J. Appl. Phys. 4, suppl. 1, 31. HOPKINS, H. H., 1966, Optica Acta 13, 343. HOPKINS,H. H., 1967, The Theory of Coherence and Its Applications, in: Advanced Techniques of Optics (Van Nostrand, London) p. 189. INGELSTAM, E. a n d L . P. JOHANSSON, 1958, J. Sci. Instr. 35, 15. INGELSTAM, E., 1960, Problems Related to the Accurate Interpretation of Microinterferograms, in: Interferometry (H.M.S.O.,London) P. 137.
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SOURCE-SIZE COMPENSATION
INOUE, S. and W. L. HYDE,1957, J . Biophys. Biochem. Cytol. 3, 831. IWATA, G., 1949, Proc. Phys. SOC.Japan 4 , 195 (in Japanese). J , , 1868, Comp. Rend. Acad. Sci. Paris 6 7 , 814. KIND,E. G. and G. SCHUIZ, 1959, Optik 1 6 , 2. KRUG,W., J . RIENITZand G. SCHULZ,1964, Contributions to Interference Microscopy (Hilger and Watts, London). LEBEDEFF,A,, 1930, Rev. Opt. 9, 385. LINDRERG, O . , 1952, 2 . Physik 1 3 1 , 231. LINNIK, W., 1934, 2 . Instrumentenk. 5 4 , 462. LOHMANN, A., 1956, Optica Acta 3, 97. MAGILL, P. J . and A. D. WILSON,1968, J. Appl. Phys. 3 9 , 4717. M A K ~ C H AA.L ,and M. FRANCON, 1960, Diffraction (Rev. Opt., Paris) p. 97, MENZEL,E., 1958, Optik 1 5 , 460. MENZEL,E., 1960, Die Abbildung von Phasenobjekten in der optischen Ubertragungs-theorie, in: Optics in Metrology, ed. P. Mollet, p. 283. MERTZ,L., 1959, J . Opt. SOC.Am. 4 9 , No. 12, p. iv. G. and G. SCHULZ, 1964, Optica Acta 1 1 , 89. MINKWITZ, MURTY,M. V. R. K., 1964, J. Opt. SOC.Am. 5 4 , 1187. MUKTY, M. V. R. K. and D. MALACARA-HERNANDEZ, 1965, Japan J . Appl. Phys. 4 , suppl. 1, 106. G., 1955, J . Phys. Radium 1 6 , 9 (S). NOMARSKI, NOMARSKI, G. and A. R. WEIL, 1955, Rev. Metallurgie 5 2 , 121. PHILPOT, J. St., 1951, Some New Form of Interference Microscope, in: Le Contraste de Phase et le Contraste par Interfkrences, ed. M. FranCon (Rev. Opt., Paris) p. 45. POLZE, S., 1965, Monatsberichte der Deutschen Akademie der Wissenschaften zu Berlin 7 , 631. RANTSCH,K., 1949, Die Optik in der Feinmesstechnik (Carl Hanser, Munchen). G. and G. MINKWITZ, 1961, Ann. Physik (7) 7 , 371. SCHULZ, SCHULZ, G., 1964, Optica Acta, 1 1 , 43, 131. SLANSKY, S., 1959, J . Phys. Radium 2 0 , 13 (S). SLANSKY, S., 1960, Rev. Opt. 3 9 , 555. SLANSKY, S., 1962, Optica Acta 9 , 277. SLEVOGT, H., 1954, Optik 8, 366. SMITH,F. H., 1947, Brit. Pat. 639014, U.S.P. 2, 601, 175. SMITH,F. H., 1955, Microscopic Interferometry, in: Modern Methods of Microscopy, ed. A. E. J . Vickers (Butterworth Scientific Publications, London) p. 76. SNOW, I<. and R.VANDEWARDEK, 1968, Appl. Opt. 7 , 549. STEEL,W. H., 1959, Proc. Roy. SOC.2 4 9 , 574. STEEL,W. H., 1962, Optica Acta 9, 111. STEEL,W. H., 1964, Optica Acta 1 1 , 9. STEEL,W . H., 1965, Two-Beam Interferometry, in: Progress in Optics, Vol. 5, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) p. 145. STEEL,W. H., 1967, Interferometry (Cambridge University Press). THOKNTON, B. S., 1957, Optica Acta 4 , 147. TOLMAN, F. R. and J. G. WOOD,1956, J. Sci. Instr. 3 3 , 236. TSURUTA, T., 1963, Appl. Opt. 2 , 371. UHLIG,M., 1965, Mikroskopie 2 0 , 247. JAMIN,
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VANLIGTEN,R . F. and H. OSTERBERG, 1966, Nature 2 1 1 , 282. WAETZMAN, E., 1912, Ann. Physik (4) 39, 1042. WOLF,E., 1954, Nuovo Cimento 12, 884. WOLF,E., 1955, Proc. Roy. SOC.A 2 3 0 , 246. WOLF,E., 1956, The Coherence Properties of Optical Fields, in: Astronomical Optics and Related Subjects, ed. Z. Kopal (North-Holland Publishing Co., Amsterdam) p. 177. YAMAMOTO, T., 1965, Optica Acta 12, 229. YAMAMOTO, T., 1968, Thbse d’Etat (Paris). ZERNIKE, F., 1938, Physica 5 , 785.
This Page Intentionally Left Blank
VII VISION I N COMMUNICATION BY
L. LEV1 Deflt. of Physics, T h e City College, City UniversitJJof New Y o r k , New Y o r k , N . Y . , U S A
CONTENTS
9
1.
9 3 9 9
2 . BRIGHTNESS FUNCTION
BASIC CONCEPTS . . . . . . . . . . . . . . . .
345
. . . . . . . . . . . .
347
3 . SPATIAL FREQUENCY RESPONSE .
. . . . . . .
351
4.
NOISE I N T H E VISUAL SYSTEM . . . . . . . . .
359
5.
SHAPE O F MTF, LINEARITY AND STATIONARITY
365
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . .
368
APPENDIX: METHODS O F MEASURING THE MTF O F THE TOTAL VISUAL SYSTEM ABOVE THRESHOLD
368
REFERENCES . . . . . . . . . . . . . . . . . . . . .
370
Q 1. Basic Concepts A communication system” mediates between an input and an output signal, both of which are usually measurable quantities. They may be voltages in an electrical system or luminances in an optical system. In many systems, however, this output is supplied to a human observer making him, strictly speaking, the final stage of the system. It is then impossible to predict over-all performance without consideration of the characteristics of the last stage in the system, the human sensory detection system. This fact often tends to be camouflaged by the intuitive knowledge we have of the human sensory performance; but this is no more than a camouflage, and detailed knowledge of sensory functioning is prerequisite to an optimum design of such a system. It is the purpose of this article to discuss the relevant characteristics of human vision in terms which permit their consideration in communication system analysis. 1 . 1 . PROBLEMS OF PSYCHOPHYSICAL INVESTIGATIONS
The main obstacle to such a discussion is the difficulty of measuring the output, which is the purely subjective sensation of the observer - or his formulation of this sensation. Such measurements, attempting to relate psychological output to physical input, are termed “psychophysical” and techniques developed for them are often severely limited in the absolute accuracy attainable. A number of useful techniques have been developed permitting comparison of sensations and, with due care, meaningful results can often be obtained. We take here as the output the observer’s sensation. In a “detection” situation, this will be his feeling of whether a certain stimulus was present or absent in his field of view. In an “estimation” situation it will be the value of luminance, shape, etc. that the observer ascribes to the stimulus.
* By communication system we refer t o any arrangement which 1s to convey information. When applied t o optics, this excludes, for Instance, images formed for their esthetic values. 345
346
VISION I N COMMUNICATION
[VII.
§ 1
1.2. THE FUNDAMENTAL CHARACTERISTICS
Three major factors determine the performance of any communication system. 1. Relationship of output to input level. This is generally characterized by the transfer characteristic; in the visual system it is called brightness function, because the sensation caused by luminance is called brightness. 2. Frequency dependence. This is called spectral response; in the visual system it will be the modulation transfer function (mtf), defined below. 3. Random effects. Unpredictable effects entering system performance are called noise. In optics spatial noise is frequently called granularity. 1. The transfer characteristic controls primarily the range of input levels over which the system will respond usefully. 2 . The spectral response limits the rate at which information can be transmitted or the rate of signal variation to which the system will respond effectively. I n the usual sequential communication channel, the rate referred to is temporal (bits per unit time). When dealing with spatial images, however, the rate may also be spatial (bits per unit distance, area, or volume). 3. Noise is the factor which controls the ability of the above characteristics to limit system performance. The concept “noise” covers indeterminate additions to the signal, limiting the accuracy and reliability with which the input can be determined from the output or vice versa. In an electrical communication channel, this may be random voltage fluctuations generated thermally in a resistor; in a photograph, it may be the granularity of the emulsion. Always, the presence of noise changes the output in a way which is not predictable. The significance of the unpredictability lies in that, while the degradations due to transfer characteristic and spectral response limitations can usually be fully accounted and compensated for - at least in principle,* this is not true for noise, so that it is ultimately the presence of noise which limits system performance on both counts of input range and frequency. Thus our task is to find these three psychophysical characteristics for the visual system, and in this review one paragraph is devoted to
* Exceptions occur when the slope of the brightness function or the spectral response vanish absolutely. Then limitations exist independent of noise.
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each of them: 5 2 to the brightness function; 9 3 to the modulation transfer function; and 5 4 to the noise characteristics of the visual system; 3 5 is devoted to a discussion of linearity and stationarity considerations. Both temporal and spatial frequency responses are relevant, but only spatial frequencies are treated here. The spatial spectral response may be measured as the ratio of output to input modulation when a sinusoidal luminance pattern is imaged. In this form the spectral response is called modulation transfer function (mtf)*. In this connection, the object modulation is defined in terms of maximum and minimum luminances, LmaXand Lmin,respectively:
M , = Lmax -Lmin Lmax+Lmin
and the image modulation in terms of the corresponding illumination values (Emax and Emin):
The mtf value is, then
T
= MJMo.
In our psychophysical situation, the sensed brightness values, and Bmin,may be substituted for the illumination values. I t is occasionally easier to measure the spatial distribution of the output resulting from a point input. This is called point spread function and, if symmetrical, is related to the mtf by the mathematical process known as the (Fourier-) Bessei transform (WATSON [1962] pp. 453-454, O’NEILL [1963]). When a line source is used, the line spread function is obtained in the image plane. When symmetrical, this is the ordinary Fourier transform of the mtf.
B,,
Q 2. Brightness Function 2.1. PSYCHOPHYSICAL MEASUREMENT TECHNIQUE (GREEN AND SWETS [1966])
Methods for measuring the sensed brightness can be divided into threshold and supra-threshold. In the threshold techniques, the
* The concept of mtf is sometimes restricted t o stationary systems with a linear transfer characteristic, but it may also be defined in the more general terms used here, though its usefulness is somewhat restricted in nonlinear systems.
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stimulus is decreased until it is no longer sensed (or increased until it becomes just noticeable). The stimulus is then said to have its threshold value. We may use this method to find the threshold luminance change (AL) at any luminance level. If we then assume that the sensed brightness difference at the threshold is some fixed value AB, this permits us to build up a brightness curve, a small step at a time, and to investigate such matters as the effect of adaptation and stimulus area on the brightness function (GROSSKOPF [1963]). However, to overcome the highly questionable assumption of a constant AB, suprathreshold methods must be used. The most obvious of these is magnitude estimation (STEVENS [1956]). Here the observer is shown a reference stimulus to which an arbitrary value, say 10, is assigned. He is then shown test stimuli and asked to assign to them values relative to that. Surprisingly, this method yields some useful and relatively consistent results, despite its apparent crudeness. Closely related to it is the method of ratio scaling (EKMAN [1958]), where the observer must bisect the gap between two reference stimuli, or divide it in some other ratio. Interocular comparison is another supra-threshold measuring technique important in studying adaptation and other field effects. This method is based on the observation that the two eyes operate fairly independently, so that adaptation and other field effects in one eye, which may have an important effect on its brightness function, have relatively little effect on the other eye. Some interaction has, however, been noted (see e.g. SAUNDERS [1968]). In summary, the major weakness of the threshold method lies in the assumed constancy of the just noticeable brightness difference, independent of luminance level. This is overcome by the direct estimation methods, but these suffer from large inherent inaccuracies. Another problem common t o all methods is due to the strong dependence of the brightness function on the adaptation level. But, when the spot luminance differs significantly from the background luminance, it must affect the adaptation level, so that it becomes very difficult to obtain a brightness function independent of adaptation level (GROSSKOPF [1963]). Small test spots and short exposures can minimize, but not eliminate, these effects. 2.2. INSTANTANEOUS B R I GHT NES S FUNCTION
Due to the important effects of adaptation, the change in perceived
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brightness with a change in the luminance of the field of view can be measured under two basically different conditions, yielding greatly different results. The instantaneous brightness function is obtained when we measure the perceived brightness change corresponding to a luminance change when the stimulus is presented very briefly, in a manner such that adaptation is not affected. The steady-state brightness function is obtained when the brightness of a uniformly luminous field is measured for various luminances, but always after full adaptation has taken place. The steady-state brightness function thus includes adaptation effects and differs significantly from the instantaneous brightness function. * Classically, the brightness function has been taken to be logarithmic. This “Fechner’s law” was based on Weber’s law - the observation that, in certains regions, the just noticeable difference (jnd) was a fixed fraction of the stimulus luminance. Assuming that the jnd corresponds to a fixed brightness difference, this implies a logarithmic response. This assumption of a logarithmic response seems to be rather arbitrary and has, recently come under severe attacks (STEVENS [ 19611). Magnitude estimation measurements indicate that the instantaneous brightness function has the form (STEVENS and STEVENS[1963]):
B
=
k(L-Lo)B,
(3)
where L is the stimulus luminance and k , Lo and j3 are constants; k is the gain factor of the visual system, Lo may be considered a luminance threshold, and p is a measure of the response non-linearity. For a dark-adapted eye, p = 0.333, Lo = 0. If k is set equal to ten, B is obtained in brils, which are units proposed for brightness (STEVENS and STEVENS [1963]). Plots of j3,Lo and k as functions of adaptation level are shown in Fig. 1. These values were obtained with test targets of 0.1 radians diameter, exhibited for 2 seconds but the results are quite insensitive to changes in target size. A reduction in flash duration to bring it into the region of Bloch’s law (less than about 0.1 seconds), however, does increase the exponent (STEVENS [1966b]). * By dealing only with uniform fields we avoid discussion of the closely related phenomenon of induction which describes the effect of the luminance of one area element on the perceived brightness of another area element, when both are presented simultaneously (STEVENS [1966a]).
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I
0
2
3
Log Adoptotion Level
4
§ f
5
(Log Td)
Fig. 1. “Constants” in the expression for the instantaneous brightness function: and STEVENS[1963].) variation with adaptation. (After STEVENS
2.3. STEADY-STATE BRIGHTNESS FUNCTION
The steady-state brightness function follows an entirely different course - a course which can be approximated by an “automatic gain control” type equation (LEVI[1969]), the gain factor being
B,/(K+Et)
(4a)
kEt = B,Et/(K+E!).
(4b1
=
and the adaptation brightness
B,
=
Equations (4a) and (4b) imply that at very low levels of adaptation illumination, E,, the gain factor is constant (at B,/K) and that at very high levels the sensed adaptation brightness, B,, is constant at B,. The fact that, in the adapted eye, the perceived brightness approaches a fixed value asymptotically, helps explain the phenomenon of “brightness constancy” - objects retaining their apparent brightness, even though the illumination level changes. It also explains Weber’s law very simply: at high luminance levels the gain factor is inversely proportional to Et, and therefore a fixed fractional change in luminance will result in a fixed difference in perceived brightness. If the just noticeable illumination fraction is 6 = AEIE,,
(5)
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then the threshold brightness difference is
which is independent of the actual adaptation luminance level E, .
Q 3. Spatial Frequency Response 3.1. SUBSYSTEMS OF THE VISUAL SYSTEM
The formation of the final perceived image can be analyzed into a number of distinct stages, each of which modifies the image; thus each of these stages could have an mtf associated with it. These stages are 1. The optical media (cornea, lens, etc.), which form an optical image of the viewed object on the retina. 2. The retina, which diffuses the light passing through it on its way to the photoreceptors - the rods and cones. 3. The pattern of photoreceptors, which spatially quantize the image illumination by summing the sensed illumination over small regions. 4. The neural interconnections in the two plexiform layers in the retina, which combine the outputs of the various photoreceptors in various ways before these combinations are fedinto the optic nerve. 5. The optic nerve fibers in the lateral geniculate body in the thalamus portion of the brain, which connect to new fibers leading to the cortex, the outer layer of the brain. 6. The cortex, where the signals arriving via the afferent nerves are translated into a conscious image. Although the point spread function is probably different at each of these stages, it cannot presently be measured except at the output stages (1) and (6). The first stage is usually called the optical subsystem and stages 2-6 the “retina-brain” subsystem of the total visual system. The characteristics of these subsystems have been measured individually and in combination. 3.2. OPTICAL SUBSYSTEM
The mtf of the optical subsystem has been measured extensively by observing the image of a slit as formed on the retina (FLAMANT [1955], WESTHEIMER and CAMPBELL [1962], KRAUSKOPF[1962], ROHLER[1962], CAMPBELLand GUBISCH[1966]). Acting on the light reflected from the retina, the eye media reimage the retinal image
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in the neighborhood of the original slit, and this secondary image can be analyzed photometrically to yield the line spread function corresponding to a double passage through the ocular media. By Fourier transform, this yields the mtf corresponding to a double passage. This, in turn, is the square of the single-passage mtf of the ocular media - assuming perfectly diffuse reflection at the retina. This assumption has been confirmed by careful measurements (CAMPBELL and GUBISCH [1966]) and contradicted by others (ROHLERet al. [1969]). The mtf has also been measured by this method, using a sinusoidal object luminance (WESTHEIMER [1963]). The mtf of the optical subsystem can be considered as consisting of two factors - diffraction effects and aberrations. Diffraction effects of a circular aperture produce an mtf of the form:
where
y
= Afv/D
A is the wavelength of the light used, f is the effective focal length of the eye, Y is the spatial frequency, and D is the pupil diameter. It implies that there can be no image modulation for v > vo where * vo = D/Af m 1130, cycles/mm
(9)
and D, is the apparent pupil diameter in mm. Aberrations in the eye's optical system lower the mtf below the value given by eq. (7) (WESTHEIMER 1119831). This effect is quite small when the pupil diameter is below 2 mm and increases in significance as the pupil diameter increases. Thus, instead of increasing continually with pupil diameter, as implied by eq. ( 7 ) , the mtf begins to decrease beyond a pupil diameter of about 2.4 mm (CAMPBELLand GUBISCH [1966]). * This estimate is based on Gullstrands schematic eye (SOUTHALL [1937] pp. 56-59), which places the iris about 21 mm from the fundus. Data given there for the lens imply that this enlarges the pupil about 5%. On the other hand, the corneal curvature enlarges the pupil diameter by about 6.5%. The value of v,, in the text is based on 0 . 5 5 5 ~ light and a refractive index of 1.336 for the image space.
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S P ATIAL FREQUENCY RESPONSE
A representative set of curves, obtained with white light from a high-pressure mercury lamp is shown in Fig. 2.
Pupil Diameters
+ 2mm A
3mm 3.8mm 4.9mm 5.8mm 6.6mm
LL
IE
0
50 Ret.
Spatial
100 150 Freq. (cycles/mm)
200
Fig. 2. Mtf of eye optics for various pupil diameters. (From CAMPBELL and GUBISCH[1966].)
In addition to the techniques just described, more direct measurements have been made using excised steers’ and pigs’ eyes (ROHLER [1962], DE MOTT [1959]). These showed poorer results, but would not seem to be immediately applicable to a live, human eye. Another method (ARNULF and DUPUY[1960]) used to measure the mtf of the optical portion employs an ingenious technique to form on the retina a sinusoidal pattern of known modulation. Using a splitfield technique, the observer views a sinusoidal target of unity modulation, restricted to one half of his field of view. The other half is covered by a sinusoidal pattern formed directly on his retina. This latter pattern is adjustable to match the modulation of the former. When a match is established, the known modulation of the pattern will give directly the mtf of the optical portion. This adjustable pattern may be formed as follows (ARNULFand DUPUY[ 19601, BERGER-L’HEUREUX-ROBARDEY [ 19651, WESTHEIMER [ 19601, CAMPBELL[ 19681). A pair of narrow, parallel strips of coherent, monochromatic light is imaged in the pupil. This results in the appearance of regularlyspaced (Young’s) interference fringes on the retina of the observer.
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The fringe spacing is controlled by the spacing of the strips and, at any one spacing, the contrast is adjusted until a match is obtained. The contrast may be varied by changing the amount of flux entering through one strip relative to that through the other (e.g. by shortening one slit) or by adding a controlled amount of incoherent light to the incident radiation. Extensive results obtained with this method have been published (BERGER-L’HEUREUX-ROBARDEY [19651). 3.3. RETINA-BRAIN PORTION
The method just described has also been used to find, indirectly, the mtf (TRB) of the retina-brain portion. On the assumption that, at threshold, the sensed modulation has some fixed value ( K ) ,independent of spatial frequency, we need measure only the values of the modulation on the retina (M,(v)) at threshold at various spatial frequencies. Clearly K = M,(v)T,,(v), and hence, M,(v) = K/ T RB ( v ) T&. This technique has indeed been used (WESTHEIMER [1960], CAMPBELLand GREEN [1965], CAMPBELL[1968]). Representative results for the retina-brain portion are shown in Fig. N
100
-
L
.s c
-a 0
-0
0
5
210
-
r 0
e t, r
I
3.4. TOTAL VISUAL SYSTEM
The mtf of the total visual system may be obtained as the product: of the mtf’s of the subsystems just discussed. On the other hand, a
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variety of methods have been used to measure the mtf of the total visual system directly. Here again the most popular method seems to be threshold measurements. A sinusoidal luminance variation at some fixed spatial frequency is presented to the observer and the modulation is reduced until it is no longer detectable. The required luminance patterns have been obtained by modulating a crt raster (CAMPBELL[1968], SCHADE [1956], PATEL[1966]), by generating an interference pattern (ARNULF and DUPUY[1960], BERGER-L'HEUREUX-ROBARDEY [1965]), by means and of photographic transparencies (ROSENBRUCH [1959], DE PALMA LOWRY[1962], VAN NES and BOUMAN [1967], FRY[1969]), and by using the moir6 effect (MENZEL [1959]). Three supra-threshold methods have been used to obtain the visual mtf. Two of these use split-field photometry; that is, the region whose luminance is to be measured (the test pattern) is placed adjacent to a region of controllable and known luminance, and the luminance of the latter is adjusted until it appears to match the test pattern luminance. This technique has been used to measure the mft of the visual system directly (BRYNGDAHL [1964, 19661) and also to measure its step-function response (LOWRYand DE PALMA [1961]), which, upon differentiation and subsequent Fourier transform, yields the line spread function and the mtf, respectively. (Cf. also MENZEL [1959].)
1.0 -
0.5 -
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The third method (DAVIDSON [1968], WATANABE et al. [1968]) is more direct and similar to the magnitude estimation method. It is based on an observer rating the relative contrast of two sinusoidal patterns of different spatial frequencies. Results for the split-field method a t high luminance (BRYNGDAHL [1966]) are shown in Fig. 4. These methods are described in more detail in Appendix 1; some of their shortcomings, too, are described there. 3.5. COMPARISON O F RESULTS
The mtf’s for the two fractions of the visual system can be combined and compared with those found for the total visual system. Such a comparison does not seem to have been made. * It is not attempted here in view of the large discrepancies between the data obtained by the various workers for the total visual system. These discrepancies make such a comparison too uncertain to be of much value. I n attempting to compare the results published by the numerous investigators, it must be noted that many factors affect the performance of the visual system in general (in addition to the special factors associated with threshold measurements). Among these are pupil diameter, the state of accommodation, and retinal illumination and adaptation level. The effect of pupil diameter must, a t least in part, be due to the changed optical mtf - a reduction in the amount of longitudinal spherical aberration, and in the effect of both this and chromatic aberration, with reduction in pupil diameter. The effect of accommodation, too, has been investigated (SCHOBERand HILZ [1965]): a significant deterioration of the performance has been noted as the eye accommodates to shorter viewing distances. This may be due to increased aberrations contributed by the eye’s lens, whose anterior curvature increases significantly with such accommodation. The effect of the retinal illumination level on the mtf is less obvious. We would, of course, expect this level to affect the mtf via the noise level for those data which were obtained by the threshold method. On the other hand, if this noise has a generally uniform spectrum, the * One author evaluated the optical portion directly (CAMPBELL and GUBISCH [1966]) and also as the ratio between the mtf’s obtained for the total system and for the retina-brain portion (CAMPBELL[1968]). However, a comparison of these was not included and, indeed, the discrepancies seem quite large, especially a t the higher spatial frequencies.
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detected noise should have a spectrum matching the visual mtf - at all levels. The change in noise level should, therefore, affect the level of the threshold but not its relative spectral pattern. The effect of the illumination level on the mtf shape as measured by supra-threshold methods is even more difficult to explain. These variations imply that the visual spread function changes, perhaps in a manner analogous to changes in the spectral sensitivity curve, in the Purkinje effect (SOUTHALL [1937] pp. 274-275), which is explained in terms of a gradual transfer from rod t o cone vision with increasing illumination. In 9 5 it is shown that a major part of these changes may be due to the non-linearity in effects responsible for the shape of the mtf. On the whole, data of visual system mtf published agree qualitatively: at low spatial frequencies the mtf rises with increasing frequency until a maximum is attained; as the spatial frequency increases further, the mtf levels off and begins to drop. * (Some workers report a secondary maximum beyond the primary peak (PATEL[1966], LOWRYand DE PALMA [1961].) At still higher frequencies, the mtf drops at an approximately exponential rate. Quantitatively, however, agreement between the various measurements is extremely poor, even though these can be compared only on a relative scale; each is known only within a proportionality factor which can be chosen independently for each set of data. Even the choice of a reference point, where all the curves could be made to agree, is not obvious. The techniques of two authors (MENZEL [1959], BRYNGDAHL [1964, 19661) ensure that the mtf at zero spatial frequency is unity. The data of other workers cannot, however, be extrapolated reliably to zero frequency, and this can therefore not be used as a reference point. Another reasonable choice of reference point would seem to be the mtf peak. But even the location of the peak varies widely among the reports. Indeed, rather than treating it as a reference point, the location of the peak may be used as an index for illustrating the extent of the discrepancies. Figure 5 shows the spatial frequency at the mtf peak, as a function of the mean retinal illurnination, as reported by nine different workers. Only those who reported results with sinusoidal, rather than “square* I t has been stated (LOWRY and DE PALMA [196l], RONCHIand VAN NES [1966]) that some observers (WESTNEIMER [1960], ROSENBRUCH [1959]) have failed t o confirm the existence of a peak away from the origin. These statements are irrelevant, however, since the observers cited did not investigate the low spatial frequencies a t which the low-frequency depression occurs.
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wave” test patterns, are included. Dependence on retinal illumination was used as the independent variable, because this dependence seems t o have been investigated more fully than dependence on pupil diameter and accommodation. The values of these latter parameters, for each plot, are listed in the box insert in the figure. To note the magnitude of the inconsistencies, compare, for instance, the results of PATEL [1966] with those of VANNES and BOUMAN [1967] obtained under almost identical conditions and yet differing by a factor of almost two over most of the range. An extreme discrepancy appears when the results of SCHADE [1956] are compared with those of BRYNGDAHL [1964]; they differ by a factor of four throughout and at low light levels by a factor of eight. c
pupil Ref. diam
E E
a
view dist
-a
0 0 a
-01 0.2
0.5
I
2
5 10 20 50 100 200 05k Ik Retinal Illumination (Td)
2k
5k IOk
Fig. 5. Location of the peak of the visual mtf as a function of mean retinal illumination, as reported by a number of workers. Pupil diameters and viewing distances used are listed in the insert. 1. SCHADE[1956], 2. PATEL [1966], 3. VAN NES and BOUMAN [1967], 4. BRYNGDAHL [1964, 19661, 5. DE PALMA and LOWRY[1962], 6. LOWRY and DE PALMA [1961], 7. CAMPBELL[1968], 8. DAVIDSON[1968], 9. FRY[1969].
Such discrepancies are unusual even for psychophysical measurements, especially in view of the obviously great care taken in obtaining the data. Undoubtedly, part of the explanation lies in the great variations between individuals - but the available data do not indicate that such variations can account completely for the discrepancies. Perhaps there are other variables, not yet considered, which influence
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the results so drastically, and a search for such variables would seem to be in order. One factor, the orientation of the test pattern, has already been shown to affect the threshold. Visual sensitivity is highest for vertical and horizontal lines and significantly lower for lines at 45” to these (WATANABE et al. [1968]).
5 4.
Noise in the Visual System
The third fundamental characteristic of the visual system is its noise. We recall that this includes any factor which causes an unpredictable deviation of the detected value from the actual one. This includes not only measurable physiological quantities, such as spurious neural pulses, but also random factors occurring at the higher, cortical level. Thus we might say that noise varies inversely with attention and that the experience of an observer in an experiment reduces the noise level in his visual system, for the particular task involved in that experiment. 4.1. T H R E S H O L D MEASUREMENTS
Since the observer does not sense this noise directly, we must use indirect methods for evaluating it. Although estimation experiments could be used to determine noise levels, detection experiments, determining the threshold signal levels, are easier to apply. In its simplest form, this approach equates the just noticeable difference (jnd) with the noise level as mapped into the stimulus domain. Much work has been done to establish the luminance jnd. The results of one major investigation (BLACKWELL [1946]) covering the threshold contrast, C, for circular discs over a large range of illumination levels is shown in Fig. 6. Contrast is defined as C = AE/E (10) where E is the retinal illumination due to the background and A E is an illumination increment. For threshold contrast, A E is the just noticeable increase in illumination. The retinal illumination is given in troland. * These data correspond to a 50 yo detection probability.
* The illumination in troland is the object luminance (in cd/m2)multiplied by the pupil area (in mmz). The original report presented the data in terms of luminance. Published data (DE GROOTand GEBHARD[1952]) concerning pupil diameter were used t o convert to retinal “illumination”. The troland is popularly referred to as a unit of illumination, though, dimensionally, it is a unit of intensity and represents the intensity at the observer’s pupil.
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-4
-3
-I 0 I Log Retinal Illumination
-2
2
3
4
4
( Log Td)
Fig. 6. Threshold contrast for uniformly luminous circular discs of various diameters, as a function of retinal illumination. (From BLACKWELL [1964].)
The absolute thresholds, i.e. the illumination required for detection against an absolutely dark background, are listed in Table 1. TABLE1 Retinal illumination at absolute threshold Target diameter (minutes of arc)
3.6 9.68 18.2 55.2 121
Illumination
77.6 9.42 3.66 0.518 0.210
The fact that the threshold contrast varies significantly with luminance level and target size demonstrates that no single number can describe the noise characteristics in any useful way. If the target had not been circular, or not been uniform, what would the threshold contrast have been? What is the threshold contrast for 99% detection probability? To answer such questions, the spatial spectrum of the noise and its dependence on the luminance level must be investigated,
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and it is not enough to obtain an “effective” noise level - the distribution of noise levels must be determined. Once all this is known, we can calculate the detection probability for any target assuming an optimum detection strategy - or any other specific detection strategy. But, what strategy does the visual system use? Clearly, much more must be known about the higher-level functioning of vision before detailed noise characteristics can be extracted from threshold data. It is therefore with a severely limited amount of experimental data that we approach the investigation of noise characteristics. 4.2. N O I S E SOURCES A N D LUMINANCE D E P E N D E N C E
On the basis of physical theory, we must ascribe to the visual system at least three noise sources. 1. Detector noise
By whatever mechanism the eye converts radiation into a neural impulse, we must expect this mechanism to operate occasionally even in the absence of incident radiation, and, the more sensitive the detector, the more likely it is to be triggered spontaneously. This prediction has been confirmed experimentally and the effect has been called “dark light” (RUSHTON [1963]), l z o . 2 . Sensation and neural noise
At the other end of the neural pathway, we must expect spontaneous stimulation of sensation, even in the absence of neural activity in the optic nerve - the process which translates neural impulses into sensation must be expected to take place occasionally even in the absence of neural impulses (LEVI [1969]), resulting in sensation noise, n s . In addition to this “sensation noise” spurious pulses must be expected to occur at all neural terminals, such as in the plexiform layers in the retina and in the lateral geniculate body. We shall not discuss these further, however, because at the present state of knowledge they may be treated together with either of the preceding two noise sources. 3. Radiation noise In addition to the noise sources internal to the visual system, there is noise superimposed on the entering radiation at the time it enters the eye. This noise is due to the quantum nature of light the fact that light arrives in individual quanta. This noise has
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been analyzed most extensively and is often referred to as “quantum noise” (SCHADE[1956], ROSE[1957], BOUMAN [1961], MORGAN [1965]).
These three types of noise differ significantly in their dependence on luminance level. The sensation noise may be expected to remain constant -independent of luminance. The other noise terms are affected by the “gain” of the visual system. Since the gain factor ( k in eq. (3)) depends strongly on the adaptation level and, therefore, on the retinal illumination, these terms, too, must be expected to vary with illumination. In addition, whereas the detector noise should remain constant, the radiation noise varies directly with the square root of the illumination, as shown by statistical considerations. Thus there are at least three terms in the description of the illumination - dependence of noise, N y , in the sensation domain:
N y ( E ) = k(E)n,B+k(E)(adE)B+n,, (11) where a is a constant and no, n, are the noise levels defined above. We drop the Lo-term in eq. (3), because it is negligible when operating near the adaptation luminance level. It is usually more convenient t o express the dependence in the stimulus domain, and this is readily written by dividing each term by k and raising it to the 8-l power: N,(E) = n,+adE+“,/k(E)I1/fi. (12) If we substitute for k its value as given in eq. (4),we find, finally:
In the limits, then, we have for the illumination-to-noise ratio:
E - N
E/cl
for E << K1lB,(.n,/a)2
Ns ( E l and
E lim -E+W N,(E)
c2,
where c1 = n,+[n,K/Bo]l’B
and
c,
=
(B0/n,)l/fi
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are constants. In the intermediate range, the middle term of eq. (13) seems to control (ROSE[1957], BOUMAN [1961], MORGAN[1965]):
EIN, m 1/Ela.
(15)
4.3. SPATIAL SPECTRUM O F NOISE
No data concerning the spatial spectrum of noise in the visual system seem, as yet, to have been published. The following approach could, in principle, be used to find it. In the threshold method of finding the visual mtf (T,), it is taken to be proportional to the reciprocal of the modulation ( M )at threshold. This is based on the assumption of a fixed signal-to-noise ratio (p) at threshold: P = M(y)Tv(y)/n
and hence
M-l(v) = T,(v)/pn. This is proportional to T,(v) if the noise n, too, is fixed. If n varies with frequency, however, the frequency dependent quantity which is found, will be
T X y ) = Tv(.)/n(v).
(17)
(Here we have assumed that only the portion of the noise in the neighborhood of Y is effective in masking the signal; the other portions of the noise can, presumably, be filtered out.) On the other hand, the supra-threshold methods do yield T,(v). Thus we can find the noise spectrum from
4.) = T&)/T:(V),
(18)
the ratio of the mtf values obtained by the two methods. Unfortunately, the various results published for both T , and T$ are so inconsistent, that no meaningful ratio could be taken. 4.4. AMPLITUDE DISTRIBUTION O F NOISE
Since visual noise levels are not directly accessible to measurement, indirect methods must be used to find them. The principle underlying one method is illustrated in Fig. 7, which depicts the statistics of an experiment in which the observer is to decide whether luminance L A or LB was presented. The perceived brightness values B, and BB corresponding to these are determined by the brightness function and
364
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[VIL
§ 4
are marked on the abscissae-axis. Because of the random fluctuations in the visual system, B, and B, represent only mean values of the perceived brightness. The value of the brightness actually perceived will deviate from B, and B, with a probability density shown by the
BI
BA Bo BB Bz Brightness
Fig. 7. Detection probabilities with hypothetical noise distribution.
hypothetical bell-shaped curves. The solid curve represents the probability density when luminance L , is viewed and the broken curve the probability density when luminance L , is viewed. The range of abscissae B , t o B,, which spans the region of overlap, represents, then, the range of observed brightness which will leave the observer in doubt as t o which luminance was viewed. If the observer is forced to make a decision, his method may be presumed * to involve the choice of a partition point, B, ; if the observed brightness is less than B,, he will decide for L,, otherwise for L,. The exact choice of B, can be shown to be influenced by the relative frequency of L , and L , and by the rewards and penalties involved in making the right and wrong decisions (GREENand SWETS[1966]). In any event, the shaded area represents the error probability when in fact L A is viewed and the crosshatched area when L , is viewed. With a fixed L A , as L , is increased, the region of overlap will shrink at a rate determined by the shape of the curves. Thus it should be possible to determine the form of the bell-shaped curves by compiling statistics of the error rates for variously spaced L, and L,. In practice, the determination of the probability densities is hindered by the very large number of detection-decisions that must be made before the * His partition point itself could be presumed to be randomly distributed around a mean value, such as B,, but we shall not consider this more complicated assumption here.
VII,
9 51
SHAPE OF MTF, LINEARITY A N D STATIONARITY
365
shape of the curves can be determined with any degree of certainty. One effort along this line (BLACKWELL [1953]) failed to find a clearcut decision on whether the noise or itslogarithm is normally distributed. 4.5. NOISE ON THE OBJECT
When the object enters the visual system already masked by noise, the detection problem must be treated on a system basis. Such investigations have been made (MORGAN[1965], COLTMANand ANDERSON [1960]) but are outside the scope of the present review.
5 5.
Shape of MTF, Linearity and Stationarity
5.1. LINEARITY O F BRIGHTNESS FUNCTION
Fourier transform and, therefore, the concept of mtf will lead to useful results only in systems that are linear, or almost so. If the mtf depends on the amplitude of the signal, the response at any spatial frequency will be influenced by the signal level at other frequencies and by their relative phases. In the visual system it has been found that the brightness function obeys a power law with an exponent, @ (0.33 < @ < 0.5). It is therefore quite non-linear. For very small modulations, however, we may approximate the brightness variations by @ times the illumination variations so that the techniques of linear systems can be used for such limited modulation amplitudes (DAVIDSON [1968], WATANABE et al. [1968]). Note that these non-linearities will not show up in split-field measurements. The test field and the comparison field are affected equally by the non-linearity; the comparison is made in the sensation domain and the measurements of both in the stimulus domain. The only mtf measurements which could, in principle, reveal this nonlinearity are those based on direct magnitude estimation (DAVIDSON [1968], WATANABE et al. [1968]), and these, as mentioned, suffer from such a great scatter that it is difficult to derive absolute modulation values from them. 5 . 2 . SHAPE O F THE VISUAL M T F
Besides the brightness function, the main determinants of the mtf are the area effects - brightness sensation at one point due to illumination at another. Their linearity, too, must be investigated. As a first step, these area effects themselves must be identified and we attempt this on the basis of the mtf shape.
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9 5
The mtf of ordinary passive optical systems must have its maximum value at the origin (V = 0 ) . If the mtf at any other spatial frequency (v # 0 ) exceeds that at the origin, this implies a spread function which is negative in some regions. * This, in turn, implies a negative illumination which is impossible with “passive” systems. The observation of a maximum away from the origin therefore implies a negative, or inhibitory effect in the visual system. Specifically, it seems to point to an effect whereby any stimulated element radiates a far-reaching inhibitory influence into its environment (SCHADE[1956], DE PALMA and LOWRY[1962], FRY[1969]), in addition to a short-range positive spread due, perhaps, to scattering [1967]). Ator similar effects in the retina (VANNES and BOUMAN tempts at fitting such two-term spread functions to the observed effects have been made in terms of two Gaussian functions (SCHADE [1956]) and using an exponentially decaying curve in analogy t o the scattering in photographic emulsions (VANNES and BOUMAN [1967]). 5.3. NON-LINEARITIES O F AREA EFFECTS
When considering a hypothesis of a spread function which is negative over part of its range, the question immediately arises: has anyone actually observed a negative brightness all by itself? Is there a sensation of “blacker than black”? In the absence of reports of such a sensation, we must assume that the inhibitory mechanism is just that, that it can only inhibit existing stimulation but not produce negative stimulation in isolation. This implies a saturation effect in the inhibitory mechanism - its effectiveness drops in low luminance regions. Such a saturation effect can be observed in the asymmetry of the Mach-bands (MARIMONT [1963]) and in the appearance of sinusoidal luminance patterns (BRYNGDAHL [ 1966]), to be discussed in more detail below. The non-linearity just considered is an “effect non-linearity” since it is controlled by the illumination level at the effect site. There is evidence also of “cause non-linearity” controlled by the illumination a t the cause site or by the adaptation level.** It seems that the in-
* Since the mtf is (the absolute value of) the Fourier transform of the line spread function, L ( x ) ,this implies :I J-L(x)exp(i2nvx)dxl > I JL(x)dxl. The triangle inequality shows that this is impossible with L ( x ) real and positive: IJL(x)exp(i2nvx)dxl5 J IL. (x)exp (i2nvx)Idx = S L ( x )lexp (i2nvx)Idx SL (x)dx. ** These have been called “finite-spread’’ non-linearities in contrast to the “zerospread” non-linearities treated in the preceding section (INGELSTAM [1965]).
VII,
s
51
367
SHAPE O F MTF, LINEARITY A N D STATIONARITY
hibitory effect grows super-linearly with the illumination at the point causing the inhibition or with the adaptation level. The inhibitory effect seems to decrease with mean illumination to such a degree that one worker (PATEL[1966]) has found the low-frequency depression of the mtf t o disappear entirely at 3 Td. Such a non-linearity would also account for the broadening of the low-frequency depression as the luminance rises, as reported by many who have studied the effects of mean luminance on mtf, regardless of the method used (SCHADE [1956], PATEL[1966], DE PALMA and LOWRY[1962], VAN NES and BOUMAN [1967], BRYNGDAHL [1966]). In Fig. 5 this finds expression in an upward shift of the peak frequency with rising luminance. Only one worker (BRYNGDAHL [1966]) seems to have reported separate brightness data for peaks and troughs of sinusoidal test patterns, permitting a more detailed study of non-linearities. He shows the apparent luminance of peaks and troughs as a function of modulation; one typical example is reproduced in Fig. 8.
0.25
0.5 Modulation
0.75
I.o
Fig. 8. Apparent (solid lines) and actual (broken lines) peak and trough luminance in a sinusoidal test pattern of 15 cycles/mm on the retina. (From BRYNGDAHL [1966].)
At photopic brightness levels, the curves for the apparent troughluminance show a descent rate which is much higher than that for the actual trough-luminance (shown in a broken line in Fig. 8). But when the modulation exceeds about 0.3, the descent begins to slow down progressively. (For modulations of 0.5 and larger, the apparent trough luminance remains almost constant - a t zero.) This is the behavior expected on the basis of the earlier hypothesis of “effect
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APP.
sub-linearity”. The initial high descent rate is simply a manifestation of the inhibition effect due to the neighboring luminance peaks, as these become more and more pronounced. The subsequent flattening is the result of the saturation effect. The super-linearity of the apparent peak-luminance is less obvious. It may be explained, however, on considering the effect of the comparison field on adaptation. When this field is less luminous than the test pattern, as it is during trough measurements, it will not have much effect on adaptation. When it is brighter, however, as it is during peak measurements, it will affect adaptation significantly (STEVENS [1966]) and, hence, enhance the inhibition effect and, with it, the apparent relative peak luminance, due to the “cause super-linearity’’ hypothesized earlier. 5.4. STATIONARITY, HOMOGENEITY, OR ISOPLANATISM
Fourier transform analysis is applicable only if the system characteristics are spatially constant. Clearly, if the mtf changes in the distance covered by one signal cycle, it cannot be very meaningful to speak of an mtf. I n vision this may mean that such analysis must always be restricted to rather small area elements. If this restriction is observed, however, the mtf concept can still be very useful (DAVIDSON [1968]). This requirement, for mtf invariant with location, is analogous to the stationarity requirement in statistical analyses. It has also been called homogeneity (DAVIDSON[ 19681) and isoplanatism (BORNand WOLF[1965]).
Acknowledgments The work for this review was supported by the Office of Naval Research. The encouragement by Dr. G. C. Tolhurst, Chief, Physiological Psychology Branch, ONR, and Prof. H. Lustig, Chairman, Dpt. of Physics, City College, is gratefully acknowledged. APPENDIX
Methods of Measuring the MTF of the Total Visual System above Threshold Three methods of measuring the visual mtf above threshold were mentioned. Here these are described in some more detaii.
VII, APP.]
M E T H O D S O F MEASURING THE M T F
369
In the first method (BRYNGDAHL [1964, 19661) the field is split down the middle. In one half of the field the observer sees a sinusoidally varying luminance pattern and in the other half a uniformly illuminated field. The observer then adjusts the luminance of the uniform field until it matches the peaks of the sinusoidal luminance pattern, and the known luminance Lo,,, of the uniform field is recorded. This is then repeated for a match to the troughs, yielding the minimum apprehended luminance equivalent (Lomin). These methods yield the luminance-equivalent perceived modulation:
Mo
= (Lomax--lomin)/(Lomax+Lomin).
When this is compared with the objectively measured modulation, M , in the test pattern, the value of the luminance-equivalent mtf at that frequency is obtained: T(Y) =
M,/M,
where the modulation
are, respectively, the actual luminances at the peaks and LmaX,L, and troughs of the test pattern. In the other split field approach the observed luminance variations are measured across an edge which, objectively, represents a rapid transition from light to dark. When such a transition is viewed, an “overshoot” phenomenon is observed, i.e. an even lighter band appears in the light region near the transition and an even darker band in the dark region. These bands are referred to as Mach bands (GRAHAM [ 19651). Such resulting apparent luminances were measured by means of a controllable comparison field and the results of these measurements yield the integral of the line spread function of the visual system. The Fourier transforms were found for the derivatives of both the apparent and the actual luminance functions. Their ratio yields the mtf (LOWRY and DE PALMA [1961]). In the direct method, two sinusoidal patterns are presented t o the observer successively, and he is asked to indicate which has the greater modulation. One of these is at the “standard” spatial frequency, which is picked arbitrarily, and the other is at the frequency at which the mtf is to be found. This method yields, then, the mtf relative to that at the “standard” frequency (DAVIDSON [1968], WATANABE et al. [ 19681).
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I n this context a brief criticism of the various techniques may be appropriate. The threshold techniques suffer from a weakness analogous to the one mentioned in connection with the brightness function. In these techniques, the actual quantity measured is the threshold contrast. The reciprocal of this contrast - as a function of spatial frequency is then taken to be identical with the mtf, within a proportionality factor. This conclusion is based on the tacit assumption that the “noise” in the visual system affects the detection of all spatial frequencies equally. The validity of this assumption is the subject of Section 4.3. In the supra-threshold split-field methods, luminance is compared to luminance so that the non-linearities of the brightness function (Section 2 . 2 ) are canceled out of the results. Therefore we should not expect the mtf-values obtained to correspond to the perceived brightness mtf. (For small modulations, the perceived brightness mtf should be /Itimes the measured luminance mtf.) I n the split-field method employing the sinusoidal pattern, the experiment is clearly set up so that it must yield unity mtf at the origin (v = 0 ) , so that the absolute values of the mtf obtained there have no physical significance. Thus, the modulation “enhancement” apparent at intermediate frequencies may not be an enhancement in the absolute sense. In view of the heavy dependence of mtf on adaptation level (cf. Fig. 5 ) , the major weakness of this method would seem to be its interference with adaptation. Changes in adaptation level, as measurements are made, would seem to be inevitable and this, in turn, must distort the results obtained. This is discussed further in Section 5.3.
The supra-threshold method in which modulations at different frequencies are compared successively would seem to suffer from the same weakness as the magnitude estimation methods - inherent inaccuracies and large scatter.
References AKNULF,A. and 0. DUPUY,1960, Compt. Rend. Acad. Sci. Paris 2 5 0 , 2757. BERGER-L’HEUREUX-ROBARDEY, S., 1965, Rev. Opt. 4 4 , 294. BLACKWELL, H. R., 1946, J . Opt. SOC.Am. 36, 624. BLACKWELL, H. R., 1953, J. Opt. SOC.Am. 43,456. BORN, M. and E. WOLF,1965, Principles of Optics, 3rd ed. (Pergamon, London) p. 482.
VII]
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BOUMAN, M. A., 1961, History and Present Status of Quantum Theory in Vision, in: Sensory Communication, ed. W. A. Rosenblith (MIT, Cambridge) p. 377. BRYNGDAHL, O., 1964, J. Opt. Soc. Am. 5 4 , 1152. BRYNGDAHL, O . , 1966, J . Opt. Soc. Am. 5 6 , 811. CAMPBELL, F. W., 1968, Proc. I E E E 5 6 , 1009. CAMPBELL, F. W. and D. G. GREEN,1965, J. Physiol. 1 8 1 , 576. CAMPBELL,F. W. and R. W. GUBISCH,1966, J. Physiol. 1 8 6 , 558. COLTMAN,J . W. and A. E. ANDERSON, 1960, I.E.E.E. Proc. 4 8 , 858. DAVIDSON, M., 1968, J. Opt. Soc. Am. 5 8 , 1300. DE GROOT,S.G. and J. W. GEBHARD, 1952, J . Opt. SOC.Am. 4 2 , 492. DEMOTT,D. W., 1959, J . Opt. SOC.Am. 4 9 , 571. DEPALMA, J. J . a n d E . M. LOWRY, 1962, J . Opt. SOC.Am. 5 2 , 328. EKMAN, G., 1958, J . Psychol. 4 5 , 287. FLAIMANT, F., 1955, Rev. d’Opt. 3 4 , 433. FRY, G. A , , 1969, J . Opt. SOC.Am., 5 9 , 610. GRAHAM, C. H., 1965, Vision and Visual Perception (Wiley, New York) p. 549. GREEN,D. M. and J . A. SWETS, 1966, Signal Detection Theory and Psychophysics (Wiley, New York) Ch. 4. H., 1963, Vision Res. 3 , 457. GROSSKOPF, INGELSTAM, E., 1965, Japan J. Appl. Phys. 4, Suppl. 1, 15. KRAUSKOPF, J., 1962, J . Opt. SOC.Am. 5 2 , 1046. LEVI,L., 1969, Nature 2 2 3 , 396. LOWRY, E. M. and J . J . DE PALMA, 1961, J . Opt. SOC.Am. 5 1 , 740. MARIMONT, R. B., 1963, J . Opt. Soc. Am. 5 3 , 400. MENZEL,E., 1959, Naturwissenschaften 4 6 , 316. MORGAN, R. H., 1965, Am. J. Roentgenol. Radium Therapy Nucl. Med. 9 3 , 982. O’NEILL,E. L., 1963, Introduction to Statistical Optics ( Addison-Wesley, Reading, Mass.) Appendix A-3. PATEL, A. S.,1966, J . Opt. SOC.Am. 5 6 , 689. ROHLER,R., 1962, Vision Res. 2 , 391. ROHLER,R., U. MILLERand M. ABERL,1969, Vision Res. 9, 407. RONCHI,L. and F. L. VANNES, 1966, Atti Fond. Giorgio Ronchi Contrib. 1st. Nazl. Ottica 2 1 , 218. ROSE,A , , 1967, Advan. Biol. Med. Phys. 5 , 211. ROSENBRUCH, I<. J . , 1959, Optik 1 6 , 135. RUSHTON, W. A. H., 1963, J . Opt. Soc. Am. 5 3 , 104. SAUNUERS, J . E., 1968, Vision Res. 8, 451. SCHADE Sr., 0. H., 1956, J . Opt. Sot. Am. 4 6 , 721. SCHOBER, H. A. W. and R . HILZ, 1965, J . Opt. Soc. Am. 5 5 , 1086. SOUTHALL, J. P., 1937, Physiological Optics (Oxford; Dover, New York 1961). STEVENS, J . C. and S.S. STEVENS, 1963, J. Opt. SOC.Am. 5 3 , 375. STEVENS, S. S., 1956, Am. J. Psychol. 6 9 , 1 . STEVENS,S.S.,1961, Science 1 3 3 , 80. STEVENS, S . S., 1966a, J . Acoust. SOC.Am. 3 9 , 725. STEVENS, S. S., 1966b, Perception and Psychophys. I , 96. VANNES, F. L. and M. A. BOUMAN, 1967, J . Opt. Soc. Am. 5 7 , 401. WATANABE, A,, T. MORI,S.NAGATA and K . HIWATASHI, 1968, Vision Res. 8, 1245.
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WATSON, G. N., 1962, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University Press, London). WESTHEIMER, G., 1960, J. Physiol. 1 5 2 , 67. WESTHEIMER, G., 1963, J. Opt. SOC.Am. 53, 86. WESTHEIMER, G. and F. W. CAMPBELL,1962, J. Opt. SOC.Am. 52, 1040.
VIII
THEORY O F PHOTOELECTRON COUNTING * BY
C. L. MEHTA University of Rochester, Rochester, N . Y . , U S A
*
Work supported by the U.S. Army Research Office (Durham).
CONTENTS
3
1.
INTRODUCTION . . . . . . . . . . . . . . . . .
375
9 2 . PHOTOELECTRON COUNTING FORMULA . . . . 377 9 3. INTENSITY FLUCTUATIONS . . . . . . . . . . . 4 . PHOTOCOUNTING DISTRIBUTION .
384
. . . . . . .
399
9
5.
DEAD TIME EFFECTS . . . . . . . . . . . . . .
411
9 3 9
6.
MULTIPLE CORRELATIONS . . . . . . . . . . .
418
7.
INVERSION PROBLEM . . . . . . . . . . . . .
423
8. TWO PHOTON ABSORPTION
. . . . . . . . . .
429
APPENDIX A . SOME PROPERTIES OF COMPLEX GAUSSIAN DISTRIBUTIONS . . . . . . . . . . . . .
431
A P P E N D I X B . SOLUTION O F AN INTEGRAL EQUA434 TION . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . .
. . . 437
5 1.
Introduction
Fluctuations of light beams have been, for a long time, the subject of extensive investigation. Since the work of EINSTEIN [1909a, b] on fluctuations of blackbody radiation, several workers (EINSTEIN and HOPF [1910]; SMEKAL[1926]; BOTHE [1927]; FURTH[1928a, b]; MANDEL [1958]; WOLF [1960]; BBDARD[1966a]; GLAUBER[1966]; JAKEMAN and PIKE[1968]) have studied the radiation fluctuations in light beams of arbitrary spectral profile, on the assumption that the light amplitude can be described by a Gaussian random process. The development of highly coherent sources like lasers, where it is well known that the wave amplitude is not described by a Gaussian random process, stimulated the interest in fluctuations of light beams obtained from non-thermal sources. One of the most practical methods of studying fluctuations of light beams including the determination of their spectral and statistical properties is by means of photoelectric detectors. Widespread interest in developing the photoelectric detection techniques in recent years has led to several theoretical as well as experimental investigations to determine the exact relationship between the photocounting data and the statistics of the incident light radiation (BROWNand TWISS[1956, 1957a, b]; MANDEL, [1958, 1959, 1963a, b]; MANDEL et al. [1964]; KELLYand KLEINER[1964]; BELLISIOet al. [1964]; WOLF and MEHTA [1964]; GLAUBER[1966]; ARMSTRONGand SMITH [1967]; KORENMAN[1967]; LEHMBERG [1968]). Several review articles have discussed this problem in one form or another. Statistical description of the optical fields, particularly the introduction and properties of the coherence functions may be found in articles by MANDEL and WOLF [1965] and by GLAUBER [1965]. Many of the applications of fluctuation measurements are discussed by MANDEL[1963a] and MANDELand WOLF [1965]. An account of the experimental investigations on intensity fluctuations of and SMITH[1967]. Some statistical laser fields is given by ARMSTRONG 375
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THEORY O F PHOTOELECTRON COUNTING
[VIII,
5
1
properties relating to photoelectron counting and, in particular, the distribution of the counting interval between two photocounts has [ 19691. Theory of laser fluctuations is been considered by ROUSSEAU discussed by RISKEN[1970] in another article in this volume. A particularly interesting application of fluctuation measurements to light [ 19701 in beating spectroscopy is discussed by CUMMINSand SWINNEY another article in this volume. References to most of the original articles may be found in these review articles. In the present article, we will mainly consider the relationship between the statistics of the photoelectron counting and that of the light falling on it. More specifically, we will study the probability distribution p ( a )for the emission of n. photoelectrons and some of the statistical constants associated with it (e.g. the mean, the variance, the factorial moments etc.) as functions of the detecting time, the coherence time, and other statistical and spectral properties of the light beam falling on it. From the experimental measurements on the photoelectron counting, one can draw conclusions on the statistical properties of light which in turn have wide applications to fields such as spectroscopy and stellar interferometry. The basic quantity of interest which enters in the formula for p ( n ) is the probability density P ( W ) of the light intensityintegrated over a fixed time interval. Even when one knows the statistical distribution of the wave amplitude function, it is quite difficult to determine the probability density function P ( W ) of the integrated light intensity. Similar problems also appear in dealing with noise in electric circuits, where one is interested in the statistics of the output power, when that of the input current is given. RICE[1944, 19451 has discussed this problem a t length. Further development was carried out, and in fact solutions in certain specific cases were given by SLEPIAN[ 19581; KACand SIEGERT [ 19471; MIDDLETON[ 19571. In these problems, the input signal is described by a real random process. However, in the problems treated in this article, it is advantageous to use the complex analytic signals associated with the real fields (GABOR[ 19461; MANDELand WOLF[ 19651). This generalization from real to complex random processes is rather straightforward. The use of analytic signals come more naturally in discussion of the interaction of radiation with atomic systems (MANDELet al. [ 19641). We begin in 9 2 with a brief account of the counting distribution formula, which was first obtained by MANDEL [ 19581 on purely classical arguments. A corresponding relation has later been derived quantum
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§ 21
PHOTOELECTRON COUNTING FORMULA
377
mechanically (KELLEYand KLEINER[1964]; GLAUBER [1966]; LEHMBERG [1968]; MOLLOW [1968]; KORENMAN [1967]). If use is made of the Sudarshan diagonal representation of the density operator (SUDARSHAN [ 19631; GLAUBER[ 19631; MEHTA and SUDARSHAN [1965]; KLAUDER et al. [1965]; MEHTA [1967]; KLAUDERand SuDARSHAN [1968]), it is possible to rewrite this relationship in a form identical with Mandel's original formula. The use of c-number functions rather than operators in discussing these problems is therefore quite adequate. In singular cases the probability density P ( W ) , appearing in this manner need not satisfy the properties of a classical probability density, and has to be interpreted as a generalized function or as a distribution. However, for the typical cases considered in this article, P ( W ) is always a well behaved function. In the next two sections, we will discuss the form of the probability density P ( W ) and the counting distribution p ( n ) in several typical cases. In these discussions, it is assumed that the different photoelectric counts may be taken to be statistically independent. However, this is not so in general. The corrections due to the presence of dead time effects (DELOTTOet al. [1964a, b]; JOHNSON et al. [1966]; BBDARD[1967a]) is considered in § 5 . In 6, we consider the correlations in the output of two or more photodetectors. The existence of such correlations was clearly demonstrated by the early experiments of BROWNand TWISS [ 1954, 1957a, b]. These correlation measurements have applications to stellar interferometry (BROWN and TWISS[1954, 1957a, b]; PURCELL [1956]; MANDEL [1963a]; MANDEL and WOLF [1965]; WOLF [1966]; BROWN[1964, 19681). From such measurements, one can in principle also determine some of the higher order coherence functions of the light. The determination of the complete statistical distribution of the integrated light intensity from photocounting measurements is considered in 9 7 . Lastly in 5 8, we consider the counting statistics when the photoelectric emission takes place with simultaneous absorption of two photons. This is the simplest nonlinear interaction of radiation with atoms in the photodetector.
Q 2. Photoelectron Counting Formula The probability distribution p(n, t, T ) of registering n photoelectrons by an ideal detector in a time interval t , t+T is given by a Poisson transform relation
p ( n ,t , T ) =
n!
e-awP(W)dW,
(2.1)
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THEORY OF PHOTOELECTRON
COUNTING
[VIII,
g 2
where ct is the quantum efficiency of the detector and W is the integrated light intensity ( I )
W
=
I(t’)dt’.
(2.2)
P ( W ) is the probability density of the random variable W . The formula (2.1) was first derived by MANDEL [1958, 19591 by classical arguments. The two basic assumptions made in his derivation are: (1) the probability of registering a photoelectric count in a very short time interval At is directly proportional to At and to the instantaneous intensity of the light falling on the detector. (2) Different photo-electric counts are statistically independent events. However, since the photoelectric effect is itself a quantum phenomenon, the two assumptions made above on classical grounds are not completely satisfactory. MANDELet al. [ 19641 used first order perturbation theory to rederive formula (2.1). The method was semiclassical, in that the radiation field was considered to be a classical stochastic process and its interaction with the electrons of the detector was treated quantum-mechanically. They considered first a single realization of the electromagnetic field interacting with one atom detector. Then assuming that in practice, a photoelectric detector could be considered as a group of independent atoms interacting with the radiation field, they were able to show that the probability of photoemission of an electron in a time interval t, t+At is proportional to the classical measure of the light intensity defined in terms of the complex analytic signal: P(t)At = c t l ( t ) At. (2.3) Here I ( t ) = V*(t) . V(t), (2.4) V(t) being the analytic signal (GABOR[1946]; MANDEL and WOLF [1965]) associated with the real vector potential A ( t ) of the electromagnetic field, and cc is the quantum efficiency of the detector. cc depends on the geometry and various other parameters of the detector. It was also assumed in this derivation that the light falling on the detector is a quasi-monochromatic plane wave and that the time interval At is much smaller than the coherence time of the light but much larger than the period of the light. From eq. (2.3) and on the assumption that different photoelectric emissions are statistically independent events, it follows as shown below (see also MANDEL [1963a]) that the probability that there be
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g
21
PHOTOELECTRON COUNTING FORMULA
379
n photoelectric emissions in a finite time interval t , t+T is a Poisson distribution W" e-W, (2.5) n! ~
where W is the time integrated light intensity defined by eq. ( 2 . 2 ) . Eq. ( 2 . 5 ) refers to the counting distribution appropriate to a single realization of the intensity ensemble. The probability that would normally be derived from counting experiments is obtained by taking average of the expression (2.5) over this ensemble and one then obtains the result expressed by eq. ( 2 . 1 ) . To see that ( 2 . 5 ) follows from ( 2 . 3 ) , we divide the time interval t , t f T in a large number N of sub-intervals each of length AT = T / N . The number N is so chosen that AT satisfies all the requirements of At used to derive ( 2 . 3 ) . Let zk be a random variable which takes on the values 1 or 0 respectively, depending on whether or not there has been a photo-emission in the interval t + ( k - l ) A t , t+kAt; (k = 1 , . . ., N ) . The total number .n of photoemissions is then given by the relation N n= 'k'
c
k=l
Let Gr(A; t, T ) be the generating function of the random variable n, which is defined by the relation *
G,(A, t, T ) = Z: n
Assuming that all zk are independent, we find that N
Gr(A>t , T ) =
Cn
(1-AjZr)
k=l
N
((1-A)Z.).
=
(2.8)
k=l
Since zk is either 0 or 1, (l-n)..
=
l-Azk
and we therefore obtain
N
Gr(A, t, T ) = 17 { 1 - A $ ( z k k=l
1))
N =
JJ { l - h d ( t + K A t ) A t } k=l
--f
exp [-a2
SI"
I(t')dt'] as N - t co.
(2.9)
* Subscript r is used here to emphasize that we are considering a particular realization r of the intensitv ensemble.
380
THEORY O F PHOTOELECTRON COUNTING
The probability
pr(n,t , T ) is now given by
[VIII,
p
2
(2.10)
in agreement with ( 2 . 5 ) .It may be noted that the intensity fluctuations may also be taken into account by taking ensemble average of the generating function
G(1, t , T ) = (G,(A, t , T ) r =/om
ecawhP(W)dW.
(2.11)
The generating function is thus the Laplace transform of the probability density P ( W ) .If, as is usually the case, the incident field may be described as a stationary process, the generating function as well as the probability distribution are both independent of the initial counting time t . Our discussion shows that the fluctuations in the photoelectric emission may be regarded as due to two causes: ( 1 ) Intrinsic fluctuations in the detection process. This is due to the random ejection of the photoelectrons, even when there are no fluctuations in the intensity of the light beam falling on the detector and results in a Poisson distribution of the photo-electric counts. (2) Fluctuations in the intensity of light falling on the detector. In general, the resultant formula (2.1) is, of course, not a Poisson distribution. A complete quantum mechanical derivation of the photoelectron counting formula was first derived by KELLEYand KLEINER[1964] (see also GLAUBER[1966]). More recently KORENMAN [1967] and LEHMBERG [1968] have also given a quantum mechanical derivation of the counting distribution with less restrictive assumptions. These derivations are fairly involved. The final result may be expressed in a form similar to eq. (2.1). I t is found (KELLEYand KLEINER[1964]; GLAUBER [1966]; LEHMBERG [1968]) that the probability of counting n photoelectrons in a time interval t , t+T is given by
p ( n ,t , T ) =
(:9 :- .) e-aw.
,
(2.12)
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where u is again a constant, the quantum efficiency of the detector, and W is given by
'$l
=
IC
d s S I T dt' A"t(r,t ' ) . A^(r,f).
(2.13)
A
In eq. (2.13), A (r,t ) is the positive frequency part of the field operator (vector potential) and the integration ds is performed over the surface of the detector. The colons in (2.12) denote normal ordering, and the angular brackets denote the quantum expectation value (0)= Tr(38) where j3 is the density operator of the radiation field. Let us expand the field operators A ( r , t ) and A t ( r , t ) in the form h
A
A .
A(r,t )
=
2h a",u,t(r,q ,
(2.14a) (2.14b)
The operators a",, a"1 are the annihilation and creation operators respectively of the photon with mode label A, and satisfy the comLet I(un>) = JJ,Iu,) and Iuh) be mutation relations [a,, u1.1 = an eigenstate of a", : &,I%> = uAluA). (2.15) n
(2.16) (2.17)
If we make use of the diagonal representation of the density operator (SUDARSHAN [1963]; GLAUBER[l963]; MEHTA [1967]; KLAUDER and SUDARSHAN [1965]) (2.18) we may express the expectation value on the right-hand side of (2.12) as an integral
(2.19)
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where
W’ = W’(t,1‘)= I c d r / t t + T d t ~ v * ( t’) r , V ( r ,t’).
(2.20)
In obtaining (2.19), we also used eq. (2.16) and its Hermitean adjoint. Further, if we define
we may rewrite eq. (2.19) in the form (2.22)
The functional +({v,}) is properly normalized, but, in general, it is not a positive definite functional. Consequently in general P ( W ) is not a positive definite function. However, for radiation fields produced from most of the available sources one may interpret P ( W ) as a probability function. In general, of course, one must regard it as a generalized function. We thus find that the basic formula of photoelectron counting is essentially unaltered by field quantization. Let us consider some general consequences of the photoelectron counting formula (2.1) or (2.22). The average and the mean square of the number of photoelectrons is given by (n) =
2 .P(%
t , T ) = .(W(t, TI)
(2.23)
n
(n2)=
znn2p(n, t, T ) = cx<W)+a2(W2).
(2.24)
n
From (2.23) and (2.24), we obtain the following expression for the variance, first given by MANDELet al. (1964), ( ( A n ) 2 )= =
(n2)-
(n)2
(.>+.Y(AW)z>,
(2.25)
where
((AW)’}
=
(W2)-(W)2.
(2.26)
Formula (2.25) shows that the variance of the fluctuations in photoelectrons may be regarded as consisting of two parts: (1) The fluctuations in the number of classical particles obeying Poisson distribution (term ( n ) ) ; (2) The fluctuations in the classical wave field (the wave
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interference term c ~ ~ ( ( A W ) ~This ) ) . result is strictly analogous to a celebrated result of EINSTEIN [ 1909a, b] relating to energy fluctuations in an enclosed blackbody radiation.* A similar formula was later derived by FURTH[1928b] for energy fluctuations in a thermal radiation field of arbitrary spectral profile. More recently GHIELMETTI [1964] derived an expression analogous to eq. (2.22),withn representing the total number of photon (rather than the number of photoelectrons emitted in a detector) in a radiation field. Thus we find that strictly analogous formulae hold for both the energy fluctuations in a closed radiation field and those in the photoelectric counts registered in a photo-detector. It was noted earlier in the quantum mechanical derivation of formula ( 2 . 2 2 ) that P ( W ) is, in general, not a positive definite function. Thus the result that ((AW)z) 2 0 which holds for all classical probability distributions P ( W ) is not necessarily true in general. One may therefore expect to find cases where the variance of the fluctuations in the number of photoelectric emissions becomes smaller than that expected from classical particle statistics. For example, when the radiation field has a well defined number of photons (ie. when the density operator corresponds to an eigenstate of the number operator), ( ( A n ) 2 )= 0, and from ( 2 . 2 5 ) we see that ( (AW)z) is then negative. For radiation fields from a well stabilized laser, the intensity is essentially constant and ( (AW)z) = 0. The variance ( (Am)2) is then seen to be equal to ( m ) i.e. same as that for a system of classical particles. For fields obtained from thermal sources, P ( W ) is always positive so that for such fields 2 0 and hence the variance of the fluctuation in the number of photoelectrons is always greater than ( n ) . One may also relate various other moments of n to those of W . Thus in particular, one finds that the Kth factorial moment of n is simply proportional to the Kth moment of W : (dk’)
3
(n(n-1) . . .(%-A+
1)) = M k ( W ’ k ) .
(2.27)
* Einstein gave a purely dimensional argument to interpret the second tcrm in his formula for the variance of the energy fluctuations. LORENTZ [1916] later verified the correctness of this interpretation (see also the footnote on p. 434 of this article). Einstein’s forniula may thus be regarded as a reflection of the wave particle dualism of the radiation field. For a lucid account of the significance of Einstein’s result, see BORN [ 19491.
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Q 3. Intensity Fluctuations The basic quantity which enters in the formula for the photoelectron counting distribution is the probability density of the integrated light intensity W . In this section we will investigate the form of this probability density for few of the typical cases of interest. 3.1. POLARIZED THERMAL LIGHT
Let us assume that the light beam is completely polarized. In this case we can describe the wave field by a scalar random process V ( t ) in the form of an analytic signal. * We also assume that the light beam falling on the detector originates from a thermal source. The random function V ( t )may then be represented as a stationary complex Gaussian process. The instantaneous intensity I ( t ) is given by
I ( t ) = V*(t) V(t).
(3.1)
Since V is distributed according to a Gaussian distribution, the probability density of I is an exponential function
P(I)=
1 ~
exp(-I/(I)).
(0
(3.2)
We are interested in the statistical properties of the integrated light intensitv (3.3)
From (3.1) and (3.3) it follows immediately that the mean and variance of W are given by
( W >= ( O T ,
=
where
I+)
J, J
(3.4)
jr(t-t’) 12 dt dt’,
(3.5)
is the correlation function r(T) =
(V*(t)V(t+t)).
(3.6)
* For an introduction to the statistical description of wavefields see MANDELand WOLF[I9651 $ 3 .
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I n deriving eq. ( 3 . 5 ) we have made use of the moment theorem relating to complex Gaussian random process [cf. eq. (A.5b) of the appendix]. It is shown later in this section (eq. (3.31)) that the nth cumulant of W can be expressed as rT
rT
A result similar to this was conjectured for a real Gaussian random process by RICE [1945] and was later proved by MIDDLETON [1957] and SLEPIAN[1958]. Eqs. (3.4) and ( 3 . 5 ) are special cases of (3.7). From the knowledge of the cumulants, one can determine the moments of W . Thus in particular ( W > = K1, ( W 2 ) = .2+.:, (W3)
=
K3+
3 K 2 K1+
K:,
(W4)= K q + 4 K 3 K 1 + 3 K ~ + 6 K : K 2 + K ~ ,
where the summation on the right-hand side of the last expression includes all possible positive integers nl, n 2 , . . ., nk such that n,+n2+
. . . +nk
=
n.
(3.9)
So far we have considered only the statistical constants of W . It is very difficult to derive an exact expression for the probability density of W in which T is arbitrary. In fact no simple expression for P ( W ) is known for any case of direct physical interest. However, it is not difficult to derive asymptotic expressions for P ( W ) when the parameter T is either very small or very large. When T is very small compared to the coherence time T,, the intensity I ( t ) may be considered to be constant in the time interval of duration T and W is then, approximately, equal to I T . Hence from eq. ( 3 . 2 ) we may write (3.10)
where ( W ) = T ( I ) .
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On the other hand, when T is very large compared t o the coherence time, we may divide this interval into a large number of sub-intervals each of which is greater than or of the order of T,. The contributions to W from each of these sub-intervals are random variables and may be considered as statistically independent. From the central limit theorem, one may then conclude that W is approximately normally distributed (cf. RICE [1945]):
P ( W ) = (232((AW)2)}-* exp {-+(W-(W))2/((AW)2)}.
(3.11)
The mean ( W ) and the variance ((AW)2) may be determined from eqs. (3.4) and (3.5) respectively. In the limit when T becomes very large, all the fluctuations in the intensity may be expected to be smoothed out on integration. W may then be regarded as a constant corresponding to a &function distribution: P(W)= 6(W-(W)). (3.12) Such a distribution is expected, to be appropriate for light from an incandescent lamp, for example, for which even the fastest available detectors will average out all the fluctuations present in the light beam. One may also obtain an approximate expression for P ( W ) when T is arbitrary. Let us divide the time interval T into say N subintervals each of length 6t, (T = N 6 t ) . Let us further assume that it is possible to choose 6t so small that there are no appreciable fluctuations in the wave field during this time and also so large that the wavefields belonging to different time intervals are uncorrelated. Thus 6t is of the order of the coherence time. We may then regard V(n6t);n = 0, 1, . . ., N-1, as N independent complex Gaussian random variables with equal variance and we may write N
wm 2
V" (n6 t ) V ( n6 t ) 6t.
(3.13)
n=O
From eq. (A.16) of the appendix, it then readily follows that aN
P(W)= -
WN-1
Z N (N-l)!
e-+aW
(3.14)
The constants a and N may be determined from the requirement that the mean and the variance of W should agree with those given by
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INTENSITY FLUCTUATIONS
eqs. (3.4) and (3.5). Thus one obtains
N
=
[
T2
/oT/
Jy(t-t’)
J2
dt dt’]
-’,
(3.15)
and a = 2N/(W),
(3.16)
where y ( 7 ) is the normalized correlation function
Y(.)
=
r(t)/r(o).
(3.17)
The distribution (3.14) of the integrated intensity W which is seen to be a Gamma distribution was first suggested by RICE [1945] as an appropriate approximate distribution. It agrees very well with the exact distribution in the two extreme limits when either T is very small or when it is very large. When T is very small, we see from (3.15) that N is nearly unity and P ( W )then reduces to an exponential function in agreement with eq. (3.10). When T is very large, N is also very large and in this limit P ( W )may be approximated by a Gaussian distribution and eventually tends to a &function in agreement with eqs. (3.11) and (3.12). The time interval 6t chosen in deriving the gamma-distribution (3.14) has the physical interpretation of being of the order of coherence time. From (3.15) we find on simplification that (3.18)
When T is large, (3.18) reduces to Mandel’s definition of the coherence time (MANDEL [ 1959]), (3.19)
and appears to be a reasonable measure of coherence time for at least thermal radiation (see also WOLF [1958]; MANDELand WOLF[1962]; MEHTA [1963]). The problem of determining the probability density P ( W ) may be reduced to solving an associated integral equation (SLEPIAN [1958]; KACand SIEGERT[1947]). Let us assume that A,, A,, . . . are the eigenvalues of the integral equation (3.20)
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arranged in decreasing order A, 2 A, 2 . . .. The kernel r(t-t') is Hermitian and positive definite (MEHTA et al. [1966]) and hence the eigenfunctions can be chosen to form an orthonormal set JOT +*(lC)
(t)+tL) (t)dt = 8,,
(3.21)
and we may write (3.22) k
Let us also express the random function V ( t )as
v(t)= 2
(3.23)
Ck+(k)(t)>
k
where ck are random coefficients
*. We may then write
r(t-t')= (V*(t')V ( t ) )= 2 (c : c , > # ~ * ( ~ ) ( t ' )+("(t).
(3.24)
On comparing (3.22) and (3.24) we obtain (3.25)
= 'k'kt'
(c:cI)
Further, since V ( t ) is a Gaussian random process, the coefficients ck which are linear functions of V ( t )are distributed according to a multivariate Gaussian distribution and it follows from (3.25) that
$(iC))
=
expi-
(n'k)-l
2
A,11Ck12).
(3.26)
k
Now from eqs. (3.23), (3.3) and (3.21) it follows that (3.27)
From eq. (A.14) of the appendix we therefore obtain the following expression for the characteristic function of W : c ( h ) = (,Ihw)
(1-iAkh)-1.
=
(3.28)
k
We see that the problem of finding the characteristic function associated with the random variable W is reduced to finding the eigenvalues Ak of the integral equation (3.20) and then evaluating the product (3.28). From (3.28) we may also write down for the cumulant gener-
* Representation (3.23) of the random function V ( t )is known as Karhunen-LoBve expansion (see for example DAVENPORT and ROOT[1958] p. 96).
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FLUCTUATIONS
ating function of W :
(3.29)
Hence the rtth cumulant of W is given by == (9'-
K,
1)!
2 A:,
(3.30)
k
from which it follows that K,
=
1)!
(yt-
soT
r(")(t, t ) dt,
(3.31)
T(t-t'),
(3.32)
where
P ) ( t ,t ' )
=
and
r(,)(t, t ' ) =
P - 1 )
(t, t " ) P )(t", t') dt",
rt
2 2.
(3.33)
The evaluation of the eigenvalues 1, of the integral equation (3.20) can be carried out explicitly for the interesting case when the spectral [1958]), i.e. when profile of the light is Lorentzian" (SLEPIAN (3.34)
In eq. (3.34), Y,,is themid-frequency and cr is half-width of the spectral line. In this case the correlation function r(t)is given by r(Z) =
r(o)e-.l7l e-Zvivo7
(3.35)
The integral equation with this kernel ** is solved in Appendix B. From eqs. (B.12) and (B.19) of that appendix we obtain the following expression for the characteristic function: C ( h ) = euT [cosh 2+8 sinh z
(3.36)
\
* Strictly speaking g ( v ) must be zero when v < 0. However for the quasi-monochromatic case y o >> 0,we may take (3.34) to approximately hold for all frequencies. ** Actually the solution t o the integral equation with kernel I'(~-~')exp(2niv,(t- -T')} is given in that appendix. However the eigenvalues 1, are the same in both cases.
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where 2 =
{$T2-2io(
W)h}*.
(3.37)
In Fig. 1 the moduli of the characteristic functions calculated from 1.0
I
I
h-1.0
IC(h)l
0.5 - h = 2.0
-_h = 4.0 h.10.0
---_
_---
------__
_ _ ---_ I
0
/
--===- -------
/
I
t P(w/iv)
1.5
1.0
0.5
0
0.5
1.0
1.5
2.0
Fig. 2. The probability density of the normalized integrated intensity of completely polarized (9= 1 ) thermal light. (After JAISWAL and MEHTA [1969].) The spectral profile is Lorentzian.
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INTENSITY FLUCTUATIONS
the approximate expression (3.la), viz.
C,(h)
=
(1-2ih/~)-~
(3.38)
are compared with the exact values calculated from (3.36). As expected the two curves coincide in the two extreme limits when aT << 1 and aT >> 1. The probability density P ( W ) may be obtained by taking the Fourier inverse of (3.36). The results of numerical computations are shown in Fig. 2. It is found that even for aT 1 the values of P ( W ) given by the approximate expression (3.14) agree very well with those given in Fig. 2. For a Lorentzian spectral profile, one may also evaluate the first few cumulants of W explicitly. From (3.7) one obtains after straight forward but lengthy calculations the following expressions for the first five cumulants (MIDDLETON[ 19571): N
K1
==
K~
=
(W), (W)2(28-1+ep2fl)/(282),
K3
=
3(W)3{
K~
Kg
=
=
(B-
+ (B+
1)e-2fl}//33> 10e-2fl$-5 28e-28+ep4fl-29
1)
[- B2 + + 1. 8ec2fl 36ec2fl 3 ( 2 0 e ~ ~ f l + e - ~ j + 7 ) + 5 ( W ) 5 [+ B2 P3 B4 6(W)4
2e-2P
___--
8B4
2P3
f
_.
~
~
13) 1, +-3(12e-2fl+e-4flB5
(3.39)
where aT.
(3.40)
If one uses (3.8) and (3.39) one may also evaluate the first few moments of W . 3.2. PARTIALLY POLARIZED THERMAL LIGHT
Next let us consider a plane stationary, partially polarized wave traveling in the z-direction, and let V(t)be the vector analytic signal describing the field at a fixed point in space at time t. The integrated light intensity is then given by
W ( T )=
loT
V*(t)* V ( t )dt.
(3.41)
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The Cartesian components V,(t) and V,(t) of V are assumed to be distributed according to a complex Gaussian random process. The statistical properties of V are then completely characterized by the 2 x 2 coherence matrix (3.42) where
Tij(.) = (V ?(t)Vj(t+.)). Let
(3.43)
+im) be the eigenfunctions of the matrix integral equation
IoT 2
rij(tl-tz)
(i = X , Y),
+im)(tz)dt2= v m + j m ) ( t l ) J
(3.44)
i=o, y
corresponding to the eigenvalues v, . The functions 41"' may be chosen to satisfy the orthonormality condition JOT
4:'") ( t )
( t ) dt = 6,,
.
(3.45)
Following a method strictly analogous to that used in deriving eq. (3.28) we may express the characteristic function C ( h ) of W in the form C ( h ) 3 (eihw) = (l-iv&-l. (3.46) m
The cumulants of W are now given by the formula K
~
=
( k - l ) ! I T d t 1 . . . J o T d t k x . .. ~ F i l i z ( t l - t z x) 0
il
is
XFi,i,(t2-t2) . . . P ( k i l ( t k - t l ) , (3.47)
= rji(.). where fij(.) When T is very small, in comparison with the coherence time, we may write eq. (3.44) in the form
(3.48) Since the two eigenvalues of the matrix J ( 0 ) are &(l+cY)(I); $(l-Y)
v2 =
$(l-S)(W).
(3.49)
From eqs. (3.46) and (3.49) we then obtain (see also MANDEL [1963b]) C (h) = (1 -$ (1+Y) (W)k}-l (1-$i (1-9) (W)h}-l,
(3.50)
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and
P ( W )=
1 ~
P ( W ) (exp
)I.
2 w
(- (1+1m > exp (- V - P ) ( W ) 2w
-
(3.51)
Fig. 3 shows the behavior of this distribution for various values of the degree of polarization P.
I
1.0
T <
-
0.8
-
0.6
-
0.4
-
0.2
0
I
I
I
2
3
Wfi
Fig. 3. The probability density of the normalized integrated intensity of partially polarized thermal light; T << T,.(After MANDEL[1963b].)
When T is very large compared to the coherence time, one may use the central limit theorem, as before, to show that P ( W ) is Gaussian and tends to a delta function d ( W - ( W ) ) for extremely large values of T . In the special case when the second order cross-spectral density (MEHTA and WOLF [1967]; MANDEL and MEHTA [1969]) is approximately Lorentzian, we have M
I'ij(0)e-ulrl.
(3.52)
The eigenvalues v, of the matrix integral equations (3.42) may now
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be determined and the product on the right-hand side of (3.46) may be evaluated. We find that (HELSTROM [1964]; JAISWAL and MEHTA [19691) C ( h ) = C+(h)C-(h), (3.53) where cosh z*+$ sinh z*
and B again denotes the degree of polarization. The probability density P ( W ) may be obtained numerically by taking the Fourier inverse (3.56) of (3.53). By contour integration, it is possible to express this integral as a rapidly convergent series (JAISWAL and MEHTA [1969])
where
Q(y)
= el
(cosh y + t sinh y
(3.58)
B
Y
and (3.59)
(3.60) In eq. (3.58), = oT and a,,are the roots of eq. (B.9) arranged in an increasing order, i.e. wk < wo being the first positive root. A similar series expansion was obtained by JAKEMAN and PIKE[1968] for the completely polarized case (P = 1). Fig. 4 shows the behavior of P ( W )for various values of the parameter ,h' for completely unpolarized radiation (-9 = 0).
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395
t
Fig. 4 . The probability density of the normalized integrated intensity of the unpolxized (9’ = 0) thermal light. (After JAISWAL and MEHTA[1969].) The spectral profile is T,orentzian.
3.3. LASER LIGHT
An ideal laser emits light of a well stabilized intensity. This implies that even though there may be fluctuations in the phase, the amplitude of the wave field remains constant. The probability density of the instantaneous intensity may now be approximated by a delta function: P ( 1 ) = 6(1-(1)). (3.61) However, the probability density P ( I ) given above is not completely consistent with experimental observations. A close analysis of available photocount experimental data by ARECCHIet al. [1966a] has indicated that the moments of the photocounts observed experimentally are consistently larger than those expected from (3.61). A better experimental fit can be obtained if one assumes that the probability density P ( I ) is a Gaussian distribution, with variance much smaller than its mean ( R ~ ~ D A R[1966b]). D This modification is consistent with most of the recent theoretical models for laser noise (RISKEN[1965, 1968, 19701; LAXand HEMSTEAD [1967]; MCCUMBER [1966]; SCULLYand LAMB[1967]). We give below the probability
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density of the laser output intensity as derived by Risken, in his treatment of the laser as a Van der Pohl oscillator: 2
1
P(I)= exp &Io l+erf w
{-
(i
-w)2] .
(3.62)
The parameter I , is related to the average intensity ( I ) , (3.63) and the parameter w varies from large negative values to large positive values as the laser is brought from well below threshold to well above threshold. It is to be noted that, since the coherence time of the laser light is usually very long compared to the counting interval T , the probability density of the integrated intensity W is simply proportional to that of the instantaneous intensity I . In another model used for predicting the statistical properties of laser light, one considers the laser output to be an amplitude stabilized field on which a small amount of thermal noise is superimposed (LACHS[1965]; GLAUBER[1966]; MAGILL and SONI[1966]). Lachs and Glauber considered a single mode case and assumed the amplitude stabilized part of the field to be in a coherent state
6,
=
Iv,>(v,~
=
s
S ( 2 ) ( ~ - ~ c ) ( ~ )d2v, (d
(3.64)
whereas the thermal part has the usual Gaussian diagonal representation (3.65) The density operator of the superposed field is then given by
Assuming that the counting interval is small compared to the coherence time and using eqs. (3.66) and (2.1) one finds that the counting distribution may be expressed in terms of the Laguerre polynomial
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INTENSITY FLUCTUATIONS
Here (n,) = U C T I V , / ~ and (nT) = uT(IT) are the average counts obtained from the coherent and from the thermal fields respectively. As may be verified, the average and the variance of n are given by
< ( A n ) 2 )=
(nC>+(lZT);
(3*68)
The problem of determining the statistics of a harmonic signal field mixed with a thermal noise field has been extensively studied in recent years (MAGILL and SONI[1966]; LACHS[ 1965,19671; FILLMORE [1969]; JAKEMAN and PIKE[1969]; JAISWAL and MEHTA [1970]; MEHTA and JAISWAL [1970]) and is considered in the next section. 3.4. HARMONIC SIGNAL M I X E D WITH T H E R M A L F I E L D
Let us assume that the field V ( t )is obtained on superposition of a harmonic signal plus a thermal field, both of which are completely polarized. I t is further assumed that the states of polarization of the two fields are identical. We may then write
V ( t )= Vc(t)+VT(t).
(3.69)
The harmonic signal
V , ( t ) = j V ,lei(+@
(3.70)
has a constant amplitude and a random phase y . The thermal field V T ( t )is considered to be a Gaussian random process with an autocorrelation function
r(z)
=
(V:(t) v T ( t + t ) ) .
(3.71)
Let us expand the fields V,(t) and VT(t)in the form (3.72) (3.73) where
+(k)
are the orthonormal eigenfunctions of the integral equation
IOT
~ ( z - T ’ ) +(k) (t’)dt‘
=
(t).
(3.74)
Following a method analogous to that used in deriving eq. (3.28), we find that the characteristic function C (h) of the integrated intensity P T
(3.75)
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THEORY O F PHOTOELECTRON COUNTING
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$ 3
is given by
The product on the right-hand side of eq. (3.76) has been evaluated explicitly by JAKEMAN and PIKE[1969] for the special case when the spectral profile of the thermal field is Lorentzian. The probability density P ( W ) is, of course, obtained by taking the Fourier inverse of the above expression. When T is small compared with the coherence time of the thermal field, we find that only one eigenvalue namely A = T r ( 0 ) contributes significantly to the product on the right-hand side of eq. (3.76). It may be shown that this approximation leads to a counting distribution identical to that given by eq. (3.67) (see also MAGILL and SONI[1966]; JAISWAL and MEHTA [ 1 9 7 0 ] ) . For arbitrary values of T , an expression similar to (3.31) may be obtained for the cumulants of W . We have from (3.76)
The rtth cumulant of W is therefore given by K,
=
(n-1) !
2 A,”+%! 2 luk12A;--1 k
=
k
( n - 1 ) !soTI.(n’(l. l)dt+n!/oT/dt d l ’ ~ ~ ( l ) ~ ( n - l ) ( l , l ’ ) ~ ~ ( ~ ’ ) , (3.78)
where P ) is the kth iterated kernel as defined by eqs. (3.32) and (3.33). Eqs. (3.8) and (3.78) may be used to evaluate the moments of W . In particular we find that
<w>= (W,>+(WT> ( W 2 )= (W)2+
soTs
II’(r-d)12drdz’+
The results of this section may readily be extended to include the case when the incident light is in any particular state of polarization.
VIII,
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P H O T O C O U N T I N G DISTRIBUTION
399
The nth cumulant of the integrated intensity may in this case be shown to be given by
S, 7I‘;,) t ) dt + T
K,
=
(n- 1 ) !
(t,
(3.80)
where
I - p (t, t’) = (V&(t’)V.&)),
(3.81)
Q 4. Photocounting Distribution As pointed out earlier, the light fluctuations are not measured directly, but are usually inferred from the photoelectric measurements. According t o Mandel’s formula, the probability p (n) of detecting n photoelectrons in a time interval T is given by
p ( n )=
IOm n!
ecaWP( W )dW.
In 9 3 we evaluated the probability density P ( W ) in a few typical cases, and for various values of the counting time T . In this section we will study the corresponding behavior of p (n). 4.1. POLARIZED THERMAL LIGHT
Let us first consider a plane wave beam of completely polarized thermal light. When T << T,, we note from eq. (3.10) that the probability density P ( W )is exponential. The corresponding photocounting distribution p(n) is readily seen to be given by the Bose-Einstein formula * (4.2)
On the other hand, if T is very large compared to the coherence time T,, so that the conditions of eq. (3.12) are satisfied, the counting dis-
* I t is of interest to observe that the Bose-Einstein distribution is also the most likely distribution, if the only information given is the average number of counts .This can be seen by maximizing the entropy C p ( n )l o g p ( n ) with the constraints X p ( n ) = 1 and C n p ( n ) = .
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THEORY O F PHOTOELECTRON COUNTING
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f 4
tribution is Poissonian. For moderately large values of T (- 10 T,), the probability density P ( W ) is approximately Gaussian [eq. (3.11)]. If we retain terms to the first order in ( ( A W ) z ) / ( W ) zwe , obtain the following expression for the counting distribution (MCLEANand PIKE[1965], see also BBDARD[1966b]):
where ( n ) = ct(W). For a polarized thermal light beam with Lorentzian spectral profile, GLAUBER [1965, 19661 has suggested another asymptotic formula valid for T >> T,, namely
where
c = {(l/T~)+2(n)/(TT,)):,
(4.5)
S,(x)= (2x/n):e"K,_+(x),
(4.6)
and K,-; is the modified Hankel function of order n-$. This formula is in good agreement with the exact expression for p ( n ) only when T 2 lOT,. Since no explicit expression is available for P ( W ) for arbitrary values of T, it is not possible to give the corresponding expression for * ( a ) . However, if we use the Rice's approximate expression for P ( W ) [eq. (3.14)], we obtain from (4.1) the following formula for p ( n ) (MANDEL [1959]):
ll(n+N) +(%)
=
1
n!r(N)( l + ( n ) / N ) N
1
(l+N/(n))"'
(4.7)
The parameter N is given by eq. (3.15). Formula (4.7) is encountered in statistical mechanics in connection with the fluctuation of bosons in N-cells of phase space; in the present context N is, however not necessarily an integer. The expression (4.7) is valid in the two extreme limits T << T, and T >> T,, and appears to be a fairly good approximation for the intermediate values of T also, as is evident from our earlier discussion in connection with the corresponding probability density P ( W ) . For the special case when the spectral profile of the light beam is either Gaussian or rectangular, BBDARDet al. [1967] have evaluated the first few factorial moments using eq. (4.7) and
10
I
10-1
I
I
I
I
16'
lOl
I
10
10-2
PT
I
I
I
10'
I
10
oT
Fig. 5. A comparison of the factorial moments calculated from the approximate probability [eq. (4.7)] (broken curves), with the exact values calculated from the cumulants [eq. (3.7)]: (a) Rectangular spectral profile; (b) Gaussian spectral profile. (After BBDARDet al. [1967].)
w
z
405
THEORY OF PHOTOELECTRON COUNTING
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§ 4
compared them with the exact values obtained from eqs. (2.27), (3.7) and (3.8). Their results are reproduced in Fig. 5. It is to be noted that the generating function
is related to the characteristic function
C (12) =
eihwP(W) dW /om
of W by the formula
G(s) = C(itcs).
(4.9)
From eq. (3.28) we may therefore write
G ( s ) = JJ ( 1 + t c d k ) ~ l ,
(4.10)
lc
where 3Lk are the eigenvalues of the integral equation (3.20). The product on the right-hand side of eq. (4.10) is evaluated explicitly in Appendix B for light with Lorentzian spectral profile. From eqs. (4.9) and (3.36), we find that coshz+&sinhz
(",' + iT) )-', -
-
(4.11)
where z = .[0~T~+2oT(n)s}~.
The counting distribution p ( k ) and the factorial moments may be obtained from G (s) by repeated differentiation: dk
k!
dsk
(4.12) (firr1>
(4.13) s=l
(4.14)
The factorial moments in this case may, of course, be directly obtained from eqs. (2.27), (3.8) and (3.39). In Fig. 6 we compare the first few counting distributions and the factorial moments obtained from eqs. (4.4) and (4.7) with those obtained from the exact formulas (4.13) and (4.14).
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DISTRIBUTION
N
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4
4.2. PARTIALLY POLARIZED THERMAL LIGHT
When the counting time T is small compared with the coherence time, the probability density P ( W ) for partially polarized thermal light of degree of polarization B is given by eq. (3.51). From (3.51) and (4.1) we then obtain the following expression for the counting distribution $(a) (MANDEL [1963b]):
(4.15)
One may interpret this result as a distribution of n bosons in two cells of phase space with occupation numbers in the ratio ( 1 - 9 ) to ( 1 +P). The variance ( ( A n ) 2 )is given by ( (An)
> = (n>{1+4 (1 +g2) <.>>.
(4.16)
When T is large compared to the coherence time, P ( W ) is approximately Gaussian and+(%)is therefore of the form given by eq. (4.3). It is possible to obtain an approximate expression for p (n)analogous to (4.7). The coherence matrix J ( 0 ) of eq. (3.42) may be diagonalized by a unitary transformation. The intensity I ( t ) may thus be divided into two parts I l ( t )and 12(t),which, on account of the Gaussian nature of the wavefield, are statistically independent random variables. Their averages are the eigenvalues of J ( 0 ) :
I ( t ) = Il(t)+I,(t),
(4.17)
(1,) = + ( l + q < o >(1,) = + ( 1 - 9 ) ( I ) .
(4.18)
The contribution to the photocounting distribution pl(K) and p , ( k ) from I l ( t )and12(t)separately may be approximated by (4.7). We may therefore write (see also HELSTROM [1964]) n
P(n) =
2 $1(K)92(n-J4)
1c=o
2N ]-n+k.
(n>(1 -9)
(4.19)
VIII, 5 41
405
PHOTOCOUNTING DISTRIBUTION
It is to be noted that this expression holds for light beams with arbitrary spectral profile and that it leads to the correct distribution in the two extreme limits T >> T , and T << T,. From (4.19) we find that the mean square fluctuation of the number of counts is (4.20) in agreement with the formula of WOLF[l960]. The moment generating function G(s) = ( ( l - ~ ) ~may ) be expressed in terms of the eigenvalues of the matrix integral eq. (3.44): G(s) = (~+c~s,u~)-~. (4.21 m 100.0
,
,
I
I
4
t-
-1 ......
___----....... .......... ................
O ' ........................... ' l
0
lo./.--
1
........... 10 .__..( ...... ............. __.. .......
.....
0.5
9-
10
-
406
THEORY O F PHOTOELECTRON COUNTING
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4
-----__ -----__
--__ ...._..
'
.
Fig. 8. Variation of the normalized factorial moments with nT. Different curves for factorial moments of the same order correspond to B = 0, 0.25, 0.5, 0.75 and 1.0 respectively in the increasing order. (After JAISWAL and MEHTA [1969].)
For the special case when the spectral profile of the light is Lorentzian, i.e. when eq. (3.35) holds, the product on the right-hand side may be evaluated explicitly (cf. eqs. (4.9) and (3.53)). The first few factorial moments may also be evaluated in this case by using eqs. (2.27), (3.8), (3.47) and (3.52). Results of numerical evaluation are shown in Figs. 7 and 8. 4.3. LASER LIGHT
Light from an ideal laser would not exhibit any intensity fluctuations [cf. eq. (3.6l)l and the counting distribution in this case would be Poissonian: (4.22)
VIII,
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PHOTOCOUNTING DISTRIBUTION
407
However as mentioned already in 9 3.3, laser fields do exhibit some intensity fluctuations and the fcrmula (4.22) needs modification. If we use Risken’s result (eq. (3.62)) for the probability density of the intensity, and also assume that the counting interval is small compared to the coherence time, we obtain from eq. (4.1), the following expression for the counting distribution (RISKEN[1965, 19681; BBDARD[1967c]): 2 Dnexp(-wD+aD2)
p ( n )= - +i n!
lferf w
I-,
e-22(x+c)n dx,
(4.23)
where D = crI,T, c = w - i D , and the other parameters are the same as defined earlier in 5 3.3. The integral on the right-hand side of (4.23) may be expressed in terms of incomplete gamma functions or as parabolic cylinder functions. The counting distribution obtained on the assumption that the field may be approximated by a harmonic signal plus Gaussian noise is given by eq. (3.67). There is no simple expression for the counting distribution when T is arbitrary. However for the model in which the output is considered as a mixture of a harmonic signal and Gaussian noise, one may use eqs. (2.27), (3.8) and (3.78) to obtain the factorial moments of n (JAISWAL and MEHTA [1970]; see also JAKEMAN and PIKE[1969]). 4.4. BUNCHING EFFECTS
We have seen earlier [eq. (2.25)J that the variance of the photoelectric counts is in general different from, and is usually greater than, that expected from a Poisson distribution. This is a reflection of the fact that photons do not arrive at random, but they possess a certain bunching property characteristic of Boson particles (MANDEL [1963a]; UHLENBECK and GROPPER[1932]; LONDON[1938, 19431). Bunching effects may more clearly be understood in terms of the conditional probability p , (t1z)dz that a photoelectric count be registered in a time interval dz at t+t, given that one count has been registered at time t. Assuming that the light is stationary, it may readily be shown (MANDEL [1963a]; MANDEL and WOLF [1965]) that this conditional probability density is given by the formula
$,(tit)
=
.(I(t) I ( t + r ) > / ( I ( t ) > .
(4.24)
From eqs. (2.27) and (4.24) it follows that the second factorial moment
408
THEORY O F PHOTOELECTRON COUNTING
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s
4
of the counts registered in a time interval T may be expressed in the form (4.25)
and on differentiating twice with respect t o T , we find that (4.26)
Eq. (4.24) may be used to evaluate the conditional probability density @,(tlz) for the typical light beams:
A. Intensity stabilized field In this case the instantaneous intensity is constant and we find that Pc(tl.) = (4.27) which is independent of z. The separate counts are therefore statistically independent as expected.
B. Polarized thermal light In this case the wave amplitude is a Gaussian scalar random process, and we find on using the moment theorem for Gaussian distribution that Pc(tlz) = a (4.28)
T
Fig. 9. The normalized conditional probability V ( T ) = p,(tlt)/p,(tIm) for a thermal light with Lorentzian spectral profile.
VIII,
§ 41
409
PHOTOCOUNTING DISTRIBUTION
where y ( z ) is the normalized coherence function. The quantity q ( z ) = @,(tjz)/p,(tjm) is plotted in Fig. 9 for the special case of Lorentzian spectral profile.
C . Partially polarized thermal light Let r i j ( z ) = ( V y ( t ) V j ( t + z ) ) be the second order coherence tensor, where V ( t ) is the field perpendicular to the direction of propagation. In this case
Pc(tlz) = . ( 0 [ 1 +
c z: l ~ i ~ ( ~ ) l Z / ( W 1 . i
(4.29)
j
If we further assume that the light is cross-spectrally pure ( MANDEL [1961]; MANDEL and MEHTA [1969]), namely that its cross-correlation function obeys the condition (4.30)
z: 2 i
where
l r % j ( z ) l z= 3 ( 1 + ~ ~ 2 ) ( ~ ) 2 1 Y ( ~ ~ 1 2 >
(4.31)
j
B is the degree of polarization (WOLF [1959]). From eqs.
(4.29) and (4.31) we then find that Pc(tb)= 4 ) [ 1 + 3 ( 1 + W
lv(z) I”.
(4.32)
D. Harmonic signal p l u s thermal field Consider next the case when the field is given by eq. (3.69). We find in this case, that
[
Pc(tlz) = 40 1 +
IW)l2+2(rC>Ir(z) 1 cos [+b)+ (Wc-Wl-)TI
<w
1
7
(4.33)
where o C / 2 n is the frequency of the harmonic signal and r ( z ) = Ir(z)I exp [i(+(z)-wz)]. This equation may readily be generalized to partially polarized light. Finally by using the results of RISKEN[196S], one may evaluate the conditional probability$, ( t i t )for a Van der Pohl oscillator (RISKEN [1968]). In Fig. 10 the quantity
.l(r) = P c ( t / t ) / $ c ( t l a ) = l + W ( t ) Ar(t+z)>/(w is plotted for various values of the parameter w.
(4.34)
410
[VIII,
THEORY O F PHOTOELECTRON C O U N T I N G
20 1
I
1
I
I
I
I
4
1
I
1 1O
0
k
Fig. 10. The normalized conditional probability q ( ~for ) a Van der Pohl oscillator. (After RISKEN[1968].)
The enhancement of the counting rate (excess of V(Z) over unity) has been demonstrated by recent experiments of MARTIENSSENand SPILLER [1964]; ARECCHIet al. [1966b]; MORGAN and MANDEL [1966]; DAVIDSON and MANDEL [1967]; MANDEL[1970]. The analogous
01 0 0
o
A-D
I
I 500
0
300
c
200
x
1000 600 400 (y sec)
1 1500 900 600
I 2000 1200 8 00
Fig. 11. The normalized conditional probability q ( z ) . -4 ( A - V = 1.25 cmjsec), (0- V = 2.09 cmjsec), C ( + - V = 3.14 cmjsec): with light from pseudothermal sources, D ( - Laser) : with single mode of a He-Ne laser. The values of V refer t o the linear velocities of the ground glass plate and are related t o the coherence times of the light. (After ARECCHIe t al. [196Gb].)
B
VIII,
51
DEAD T I M E E F h E C T S
r
(a I
84F.;'i
.-
'... \,
I
411
I
1
LORENTZIAN SPECTRUM w i t h A v = 200Mc/r
I I 1 2 3 4 5 6 INTERVAL r, in n see
Fig. 12. An illustration of the phenomena of photon bunching. (a):with light from a 198 Hg source; (b): with light from a tungsten lamp. The ordinate represents essentially a quantity which is proportional to the integral J:;T(z)dr, where z2 is a constant. (After MORGANand MANDEL[1967].)
enhancement in the rate of coincident counts recorded by two photo detectors illuminated by two partially coherent thermal light beams was first observed by TWISSet al. [1967] (see also TWISSand LITTLE [1959]; REBKAand POUND [1957]; BRANNEN et al. [1958]; JANOSSY et al. [196l]). The correlation of counts using two or more detectors will be considered in 9 6. The results of ARECCIIIet al. [1966b] and MORGANand MANDEL [1966] are reproduced in Figs. 11 and 12.
5 5.
Dead Time Effects
5.1. GENERAL THEORY
In the derivation of Mandel's formula [eq. (2.1)], it has been assumed that the photoelectric emissions in a photodetector are statistically independent of each other. However, in practice, this condition is not strictly satisfied. Most of the detectors require some characteristic time, called the recovery or dead time, after each registration of a count, during which the detector does not respond to any external field. In this section we study the effect of finite dead time on the statistics of photoelectrons. For the purpose of making a precise analysis, we will assume that the detector has a constant inoperative period t,i.e. after each registration of a count by the detector, there exists a time interval t during which no further photo-emission may be registered. KOLIN [1934] appears to be the first person to establish experimentally the effect of dead time on the counting statistics in photoelectric emissions of ultraviolet radiation. The first theoretical in-
412
THEORY OF PHOTOELECTRON COUNTING
[VIW
5 5
vestigation of the dead time effects appears to be due to DELOTTO et al. [1964]. Following the method of FELLER [1948] they were able to obtain an expression for $ ( a ) with dead time corrections up to the order of ( Z / T ) More ~ . recently BBDARD[1967a] has found an exact expression for $(n) taking into account the dead time effects. We follow essentially his treatment. It is to be noted that the dead time effects are only appreciable when t / T is not too small. In practice z is usually small, we will therefore assume that T is also small compared with the coherence time, The integrated intensity W may therefore be taken to be proportional to the instantaneous intensity
:J
I ( t )dt w I T .
W ( T )=
I n the absence of the dead time effects, the probability distribution @ ( a of ) the photo counts is given by the formula
Let us take a particular rth realization of the stationary ensemble of the fluctuating intensity. For this realization, the counting distribution in the absence of dead time effects is Poissonian
(aIT)"
A(%T ) = -__
n!
exp ( - a I T ) .
(5.3)
Let S, be the interval of time extending from the initial time up to the (Kf1)th count. Obviously the probability that S , be less than or equal to t is identical with the probability that there be at least ( K + l ) counts in the interval t. Let us denote this cumulative probability distribution by F,(K, t ) , i.e., let F,(k, t )
= Prob.
(S,
5 t) (5.4)
It is then readily seen that
p,(k t ) = F , ( k - L t)--F,(K,
t).
(5.5)
We also introduce the probability f , ( k , .u)d.u that S, lies in the in-
VIII,
5 51
413
DEAD TIME EFFECTS
terval u,u+du, so that
Also let T , be the sub-interval of time extending from the Kth to the (k+l)th count; Tobeing the sub-interval between initial time and the first count. Then S , is the sum of these sub-intervals s k =
To+T,+
. . . +T,.
(5.7)
All the T , are independent random variables and all of them except T o have the same probability density which we shall denote by +(t). ( T ois being singled out because there is no dead time effect up to the first count.) Since To = S o , the probability density of To is f,(O, T o ) and can be evaluated from eqs. (5.3), (5.4) and (5.6). We find that Jo'f,(O,
.u) d.u = F,(O, =
4
l-P,(O, t ) ,
(5.8)
t 2 0.
(5.9)
and hence f,(O,
t ) = ale-"",
To evaluate +(t), we note that the requirement that T , be greater than t is identical with the requirement that there be no count in the interval (Skp1+z, S,-,+t), where z is the dead time of the counter. Hence we find that Prob. (T, 2 t )
=
ItW
+(t) dt
= P,(O,
t-z),
(5.10)
and on using (5.3) we obtain the result
+[t)= tlIe-ar(t--7), t 1z =
t
0,
(5.11)
Let @(s) and @,(s) denote the Fourier transforms of +(t) and f , ( K , t ) respectively, i.e. @ ( s ) ==
and Q k ( s )=
Jam
+(t)eistdt
s," f , ( K ,
t)eistdt.
(5.12)
(5.13)
414
THEORY O F PHOTOELECTRON
COUNTING
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s
5
Since T o ,TI, . . ., T , are all independent random variables and since S, = To+T,+ . . . + T k , we obtain the following relation:
From eqs. (5.9)-(5.13) we find that Q 0 ( s )= ccl/(aILis),
(5.15)
a' uI-is
(5.16)
and @(s) =
eisr.
~
On substituting from (5.15) and (5.16) in (5.14) and on taking the Fourier inverse, we obtain
Hence the cumulative distribution function F,(k, t ) is given by
(5.18) where (5.19) is the incomplete gamma function (ABROMOWITZ and STEGUN [1965]) and Pk = B k ( ~ = ) cclT(l--K~/T). (5.20) We have up till now considered only a particular realization r of the ensemble of fluctuating intensity. However the photoelectric measurements will provide us with the average over this ensemble,
F ( k T , ).
= =
(F,(k, T ) )
I
F,(k, T ) P ( I )d l .
(5.21)
If we make use of the series expansion (5.22)
VIII,
$ 51
415
D E A D TIME EFFECTS
of the incomplete gamma function, we may readily express F(K, T , t ) in the form (-l)’[~(l--Kz/T)]”+”+~
1
F(K, T , t) = k ! 2-
s ! ( K + ~ + 1)
h+s+l~
(5.23)
where p l is the lth moment of the probability density P ( W ) of the integrated light intensity:
W ’ P ( WdW, )
W =IT.
(5.24)
The photocount distribution p(n, T ) is obtained from eq. (5.5) and is given by $(%,
T , t) = F(n-1, T , t ) - F ( a , T , t) 1
= - “ ( Y h Pn-l)>-(Y(n+l,
(5.25)
n!
I t may readily be verified that in absence of dead time effects we obtain the usual relation
(t =
0),
(5.26)
and the Zth moment ,ul of P ( W )in this case is equal to the Zth factorial moment of the photocount distribution. Equality (5.26) no longer holds when dead time effects are present. The first order correction due to dead time effects may be obtained by retaining terms up to order t / T . From (5.23) and (5.25) we find that
P ( % ,T , t) =
( ~
e-aw [ l + n ( ~ W - n + l ) 21) T +o
(;I.
(5.27)
Let us now apply these results to laser light and to thermal light. 5.2. INTENSITY STABILIZED LASER LIGHT
For a well stabilized single-mode laser, the probability density of the light intensity is given by
P ( I ) = 8(1-(1)).
(5.28)
For counting time intervals of duration T short compared with the
416
[VIII,
T H E 0RY 0F P H O T O E L E C T R O N COUNT I N G
s
5
coherence time, we obtain from eq. (5.25) the following expression for
where
Pk Pk(%)== ~ ( 1 T(l--Kt/T). )
(5.30)
From the remarks of 9 3 on p. 386, it is evident that this result also holds for white light; for such light the coherence time is extremely short compared to any of the characteristic times appearing in the detecting process. JOHNSON et al. [ 19661 have observed dead time effects using tungsten lamp. In Table 1 below, we compare the PoisTABLE1 Typical photo-count distributions for light from a tungsten lamp
~
0 1 2 3 4 5 6 7 8 9 10
~~~
0.266069 0.352275 0.233206 0.102922 0.034067 0.009021 0.001992 0.000376 0.000063 0.000009 0.000000
0.266069 0.361202 0.235906 0.098721 0.029743 0.006873 0.001267 0.000192 0.000025 0.000002 0.000000
0.266072 0.361124 0.235805 0.098750 0.029819 0.006926 0.001280 0.000196 0.000025 0.000003 0.000000
The values of the parameters fia and t / T providing the best fit to the experimental data are 1.324 and 0.01908 respectively. The subscripts refer t o the Poissonian distribution (P),the calculated distribution (c) and the experimental~D ly observed distribution by JOHNSON et al. [1966] (E). (After B ~ D A I[1967a].)
son distribution and the distribution obtained from (5.29) with the experimentally observed distribution. The existence of the dead time effect is evident from the excellent agreement between the values in the last two columns of Table 1. To the first order in z / T , we find from (5.27) that the quantity f ( n ) = (~+l)P(~+ll/P(nyL)
(5.31)
VIII.
$ 51
417
DEAD TIME EFFECTS
where Go = a ( I ) T , is linear with slope --2E0z/T. The results of experimental observations due to Johnson et al. is presented in Fig. 13.
t
0
1
1
2
3
4
5
6
7
8
n
Fig. 13. Variation of the quantity f ( n ) = ( n + l ) p ( n + I ) / p ( n with ) n . The crosses represent the experimentally observed values; the dashed line represents a Poisson distribution for a mean of 1.324 counts per microsecond and the full line represents the dead time corrections [eq. (5.31)] corresponding to a dead time of 19.09 nsec. (After JOHNSON e t al. [1966].)
5.3. THERMAL LIGHT
Consider partially polarized thermal light. We have seen earlier in 9 2 that the probability density for the integrated intensity in a partially polarized thermal light is given by 1
pP(w>[exp (-
P ( W ) = ___
2
w
(1+.5P)(W)
1
-exp
(-
(1--B)(W) 2w
1
1
9
(5.32)
418
THEORY OF PHOTOELECTRON
COUNTlNG
[VIII,
$ 6
where 9 is the degree of polarization. It is being assumed that the counting interval T is much smaller than the coherence time of the light beam. From eqs. (5.32), (5.24) and (5.25) we obtain the following expression for F (k T , z) :
F(k, T ,T )
1 =-
293
(5.33)
where
W,
=
(5.34)
a ( W ) +(l+S) (l-kt/T),
and
W _= .(W)
+(l-9) ( l - k ~ / T ) .
For completely polarized light that
(5.35)
(.P= 1) we readily find from 1-k t / T
I"',
(5.33)
(5.36)
where 5, = a ( W ) is the average number of counts in the absence of dead time effects. For t = 0, eqs. (5.36) and (5.5) give the Bosenamely Einstein distribution with parameter +in, $ ( k , T , 0)
n,'C =
~-
-
(1+ii,)"fl
(5.37)
For usual thermal sources, the condition T << T , cannot be satisfied in practice. However, one may use the socalled pseudo-thermal sources (MARTIENSSEN and SPILLER[1964]) to verify the validity of eq. (5.33).
Q 6. Multiple Correlations In this section, we consider correlations in the photoelectron counting with two or more detectors. Let us consider N detectors situated at different points in a radiation field of a plane quasi-monochromatic stationary light beam. Let n, be the number of counts registered by the kth detector in a time interval (tk,tk+Tk), k = 1, 2, . . ., N . Then, assuming a single realization of the intensity ensemble, it follows readily from eq. (2.5) that the joint probability of registering n, counts by the first detector, n2 counts by the second
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MULTIPLE CORRELATIONS
419
detector etc., is given by
where uB is the quantum efficiency of the kth detector,
and I k ( t )is the instantaneous intensity at time t of the light incident on the Kth detector. The probability that would normaIly be obtained from counting experiments is the average of (6.1) over the intensity ensemble:
(6.3)
The multi-dimensional generating function G(sl, s 2 , . . ., S N ) may be defined by analogy with (4.8) G(s1, s2, . . ., S N )
=
( ( I - S ~ ) ~ ~ ( ~ - .- .S( ~1 )- ~ s ~~ ). ~ ~ ) . (6.4)
From (6.3) and (6.4) we find that
From eq. (6.3) it follows that the multiple correlation of the counts is simply proportional to the multiple correlation of the integrated intensities : ( n l n 2 . .. .n,iy> = A ( W I W , . . . W N ) ,
(6.6)
where A
= U1U%. .
. UN.
(6.7 1
We may also write
( n l n 2 .. . % N )
=A
tN+TN
dz, . . . Jt
,
dtN X
xP'"N)(tl,
where
F
N
Y
N
)
. . ., znr; tl,. . ., ZN), (6.8)
is the (2N)th order coherence function
p ( N * N ) ( .t ., l ,t .N , t i , . . ., tN) z=
(v*(tl). . . v*(tN)V(t,) . . . v ( t N ) ) .
(6.9)
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THSORY OF PHOTOELECTRON COUNTING
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If T I , T,, . . ., TN are all small compared with the coherence time, we find from (6.8) that (nln2 . . . nN)
N
(AT,T, . . . T N ) I ' ( N , N ) ( ~ , ,t,, . . ., t N ; t,, . . ., t ~ ) . (6.10)
The multiple correlation of the photoelectron counts in this case gives information about the higher order coherence functions of the field. Eq. (6.6) may also be expressed as a relation between correlations of the fluctuations An, = n,-(n,), and AW, = W,-(W,). By making a multinomial expansion of the product An, An,. . . AnN and using (6.6) repeatedly, we find that
(An,An,. . .An,>
= A(AW,AW,.
. . AWN).
(6.11)
The values of these correlations will of course depend on the type of light illuminating the detectors. If one assumes that the light originates in a laser with a well stabilized amplitude, these correlations vanish. The counting distribution $(n,,n,, . . ., nN) in this case as obtained from eq. (6.3) is a product of independent Poisson distributions: (6.12)
We consider now in some detail the case when the light originates in a thermal source. Consider first the correlation in two detectors. For the sake of simplicity, we will assume that the light is completely polarized, so that the wave fields at the two detectors may be characterized by complex scalar stochastic variables V,(t) and V,(t) respectively. From eq. (6.11) it follows that (AnlAn,)
Jt;Tl
= uluz
dt J.;TZ
dt' ~I'l,(t-t'~~2,
(6.13)
where T12b)
=
cv:
( t )V,(t+T)>
(6.14)
and where we have made use of the moment theorem of the Gaussian random processes [see eqs. (A.5) of the Appendix A]. By making the change of variables x = t-t', y = t+t' and integrating over y,we may rewrite eq. (6.13) in the form
VIII,
S 61
421
MULTIPLE CORRELATIONS
(AnlAn2)
= alu2
-Io
TrTz
(T,7.,n)~I'(tl-t2+x)l2dx]
.
(6.15)
When the counting intervals of the two detectors are equal (i.e. when T,= T , = T ) ,we obtain T
(AnlA.n,)
= X,~z
ST
( T - l ~ l ) l ~ 1 2 ( ~ + ~ ) l 2 d x , (6.16)
where t is the time delay t,-t,. The existence of this correlation was first observed by BROWN and TWISS[1956, 1957a, b]. A more detailed discussion and its applications to stellar interferometry is given by MANDEL [1963a] (see also BROWN[1964]; MANDELand WOLF[1965]; BROWN[1968]). In deriving eq. (6.15) [or (6.16)], it was assumed that the incident light is completely polarized. To consider the unpolarized orpartially by polarized thermal light also, one only needs to replace lT12(x)12
2 lrij(rl>
r29
-x)12-
i, j
Here Tij(rl, r,, x) is the second order coherence tensor I'ij(r1, r2,
x) = CVP(.,>
t ) Vj(r2, t+x)>,
(6.17)
and r,, r2specify the spatial locations of the two detectors. Next let us consider the correlations in three detectors. From (6.11) it follows that
(An, An, An,)
=
a, a, a, (A W ,A W ,A W3),
(6.18)
where
AW, = I t r T k V ? ( t )V,(t)dt-TT,(Vz(t) V,(t)).
(6.19)
For simplicity, it is again assumed that the light is completely polarized. Making use of the moment theorem (eqs. (A.5)), we find from (6.18) after some calculation that
432
THEORY OF PHOTOELECTRON COUNTING
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5 6
By taking the integrating times T,, T,, T , small compared with the coherence time and also by suitably adjusting the time delays between t,, t , and t,, it is possible to use this correlation to obtain the information about the phase of r,,(T). This method of obtaining the phase of r from triple correlation measurements was suggested by GAMO [1963]. The problem of determining the phase of r from the knowledge of its modulus has been considered by a number of authors in recent years (WOLF[1962]; WALTER[1963]; ROMAN and MARATHAY [1963]; MEHTA [1965a, 19681; DIALETIS1119671; NUSSENZVEIG [1967]), as it is of considerable interest, particularly for stellar interferometry and interference spectroscopy. Alternative methods for determining the phase of from correlation measurements has been considered by MEHTA [1965a, 19681. I n the special case when the integrating times T I ,T,, . . ., T N are small compared with the coherence time, it is possible to evaluate the multi-dimensional generating function G(s,, s,, . . ., SN) ( B ~ D A R D [1967b]; see also CANTKELL[ 1970al). I n this special case we may write
r
W , = T,VZV,,
k
= 1, 2,
. . ., N
(no summation)
(6.21)
where V , are distributed according to the Gaussian probability distribution
Here the matrix A,, is proportional to the moment matrix 2(A-l),,
=r z l c =
(VV,>,
(6.23)
and 1A I denotes the determinant of A . From eqs. (6.4) and (6.21)-(6.23) we find that (6.24) G(s1, s,, . . ., S N ) = l&v-l, where l d ~isl the determinant of the N x N matrix whose elements are given by ( d ~ )= , ~8,,+cc,T,s,I',,
The probability torial moments G(s,, s,, . . ., S N ) If we further
(no summation over k ) .
(6.25)
distribution p(nl,n,, . . ., n N ) and the various facmay be evaluated by repeated differentiation of ( B ~ D A R[1967b]), D but the process is very laborious. assume that all the detectors lie within a single
VIII,
B
71
423
INVERSION PROBLEM
coherence area * on a plane normal to the direction of propagation we may write lrk,lz = r,,r,,.In this case eq. (6.24) simplifies to
G(s1,
Szj
..
. p
SN) =
{1+(~1)s1+(n,)sz+
. . . +(~~N)sN}-'-
(6.26)
On making use of the relation
(6.27)
we find that
Recently FILMORE [1969] has considered the problem of multidetector counting statistics for a mixed thermal and coherent radiation, again assuming that the counting intervals Tkare small compared to the coherence time. The general case of arbitrary T , is considered by DIALETIS[1969], CANTRELL[ 1970bI and MEHTA and JAISWAL [1970].
Q 7. Inversion Problem In previous sections, we have mainly considered the effect of the statistics of light beams on the statistics of the photoelectron counting. The problem of determining the probability density P ( W ) from a given $ ( a ) will be referred to as the inversion problem. A solution of this problem provides a direct method of determining P ( W ) and provides also some information about the statistics of the radiation field. We show below that in principle, it is possible to determine the complete probability density of the light intensity from the knowledge of the distribution of the photoelectrons (WOLFand MEHTA [1964]). The probability that there be n photoelectric emissions in a time interval t , t+T is given by the Poisson transform (see eq. (2.1)) f i ( n ) = /om
0 n" ! e-OrWP ( W )dW,
* This implies that for points r,, r z within such an area, the normalized coherence function ynl T,l/{T',.T'Lz}* is unimodular.
424
THEORY O F PHOTOELECTRON COUNTING
[VIII,
5
7
where
From ( 7 . 1 ) , one may readily connect the statistical constants, such as the moments, the cumulants etc., of Wwith those of n (MANDEL [ 1 9 5 9 ] ) . Let M ( w ) ( x )and M ( " ) ( x ) be the moment generating functions of W and n respectively, (7.3)
and (7.4)
From eqs. (7.1)-(7.4) we find that
M ( " ) ( x )= M ( W ) [ ~ ( e 2 - l ) ] ,
(7.5)
and (7.5b) Identical functional relations hold for the cumulant generating functions K (x)= log M (x)of W and n: (7.6a)
K ( " ) ( x )= K ( W ) [ c r ( e n - l ) ] ,
(7.6b)
On expanding eq. (7.6b) [eq. (7.5b)l in powers of x and on comparing the coefficients, we may express the cumulants (moments) of W in terms of the cumulants (moments) of n. If K~ and K iare the cumulants of W and n respectively, we find that MK,
=K,,
tc2tc2
= K,-K,,
U
= K K3-3K2+2K,, ~
~
M4K4
= K4-
6K3+ 1 1 K2- 6K,,
U 5 K 5 = K,-10K4+35K3-50K2+24Kl,
etc.
(7.7)
Relations similar to ( 7 . 7 ) between the moments of W and n could also be obtained directly on observing that according to ( 7 . 1 ) the moments of W are proportional to the factorial moments of n :
dC(Wk= ) (ark'>= ( n ( n - 1 ) . . . ( n - - K + l ) ) .
(7.8)
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5
71
I NVERSION PROBLEM
425
'From (7.8) we readily find that
.(W> = (n), u2(W2) = (n2)-(n), u3(W3)= (n3)-3(n2)+2(n),
etc.
(7.9)
To obtain the complete probability density P(W) we set F(x) =
IOw
eirnwP(W)ecaW dW.
(7.10)
Then by the Fourier inversion formula, 1
I-, co
P(W)= 2n eaw
F(x)ePizWdx.
(7.11)
Now from (7.10) and (7.1), we have, formally (7.12)
Thus the required probability density P ( W ) may be obtained from the knowledge of p(n), by first evaluating F ( x ) from (7.12) and then evaluating (7.11). If the counting time can be adjusted so that it is much shorter than the coherence time of the light, then
W =IT.
(7.13)
Eq. (7.11) then yields the probability density of the instantaneous intensity. When the counting interval is small the dead time effects of the counter become appreciable and one must take these corrections into account. We will, however, ignore them in the present discussion. Further, if one assumes that the light beam falling on the detector is plane polarized, stationary and quasi-monochromatic, the probability density of the scalar wavefield V may be assumed to be independent of the phase of k' (KANO [1964]; DIALETISand MEHTA [1968]; MANDEL and MEHTA [1969]). Using the fact that I ( t ) = = V* ( t ) V ( twe ) readily find the following expression for the probability density of the complex wavefield V : 1
P ( V ) = - P(1). ?L
(7.14)
426
THEORY O F PHOTOELECTRON COUNTING
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f 7
The probability density of the real part V(r) of V may be obtained by integrating (7.14) with respect to Yci). By proper change of variables, this leads to the expression, (7.15)
To illustrate our method, let us consider a few typical examples: (1) Let us suppose that p (n)is given by the Bose-Einstein distribution (7.16)
Then (7.12) becomes
F ( x ) = ((n)+l-i(n)x/a)-',
(7.17)
and (7.11) gives
P ( W ) = (W>-l exp ( - W / ( W > ) ,
(7.18)
where ( W ) = ( n ) / u . When T << T,, this leads to an exponential distribution for the probability density of the instantaneous intensity. Moreover, if the light is linearly polarized, we obtain from (7.14) a Gaussian distribution for the complex wavefield V . It may readily be verified that, if p ( n ) is distributed according to a Bose-Einstein distribution between N cells of phase space, i.e., if (7.19)
the probability density P ( W ) is a gamma distribution
where ( W ) = (.)/a. (2) As a second example, consider the case when p ( n ) is given by the Poisson distribution (7.21)
In this case eq. ( 7 . 1 2 ) gives
F ( x ) = exp {(n)[(ix/a)-l])
(7.22)
VIII,
s
7j
INVERSION PROBLEM
427
and hence from (7.11) we obtain
P(W)= 6(W-(W))
(7.23)
where ( W ) = (.)/a. Thus we find that the photocount distribution is Poissonian, if and only if there are no fluctuations in the integrated intensity. The method just outlined is only useful when a closed analytic expression for p ( n ) is known. In practice, however, we can determine p ( n ) experimentally, only for a few small values of n. Even, if p ( n ) is small for larger values of n, we cannot neglect p ( n ) ,since F ( n ) in that case becomes a polynomial and the Fourier inversion then gives a singular function for P ( W ) . In such cases one must use some approximation techniques. Let us assume for example, that the experimental photocount distribution can be approximated as a sum of some smooth analytic functions, say exponentials, of n, il4
p(n)=
C aje-bjn. -
(7.24)
j=1
Eq. (7.12) then gives (7.25)
and from (7.11) we then find that M
P(W)=
2 a j exp (bj-
(ebj-l)W}.
(7.26)
f= 1
Another approximate formula for P ( W ) may be obtained by using the orthogonal properties of the Laguerre polynomials L k ( x ) . If we write m
(7.27)
then the coefficients C, are given by the relation
C,
=
a/om e-“”P(W)L,(aW) dW.
(7.28)
Here I,,(%) denotes the Laguerre polynomial (7.29)
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THEORY O F PHOTOELECTRON COUNTING
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5
7
From eqs. (7.1), (7.27)-(7.29) we then find that ( B ~ D A R[1967b]) D
The techniques for the inversion that we just outlined require the direct use of the counting distribution $ ( a ) . However, if one knows only the first few moments of n, eq. (7.9) gives the corresponding moments of W . From these moments, one can determine some general features and approximate expression for P (W) (KENDALLand STUART[1958] p. 148; ELDERTON [1938]; see also BBDARD[1967b]). So far we considered only one detector and we saw that we can obtain the information about intensity distribution a t points on the surface of the detector. One can, in principle, extend the argument using several detectors and obtain information about joint probability distribution of the intensities at several points. Consider for example two detectors. Let p(n,, a,) be joint probability that the first detector registers VL, photons in time interval t,, t,+T, and the second detector registers n2 photons in the interval t,, t,+T,. Then, we have, from (6.3) (~2W2)n2
n,!
x e-alW1--azW2 P(W,, W,) dW,dW2. The joint moments of W,, U’, factorial moments of nl, n,:
(7.31)
are simply proportional to the joint
&1 ~ 2k ~ ( W ~ ~ W= l ; j(zn) ~ 1 l ~ ~ z l ) .
(7.32)
Relations similar to (7.7) connecting joint cummulants of W,, W , and of n, , n2 may also be readily derived. One can obtain the complete joint probability distribution P (W,,W,) from the knowledge of p (n,, n,) by first evaluating
and then taking the double Fourier inverse
(7.34)
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3 81
T W O PHOTON ABSORPTION
429
In cases when a closed form of $(a,, n2) is not available, one has to use approximation techniques similar to those described for a single detector.
5 8.
Two Photon Absorption
Throughout our discussion on the photoelectric detection, we have considered the interaction of light with the photo-cathode as being linear in the intensity. This is true whenever the photoionization occurs with the absorption of a single photon. The simplest non-linear interaction of radiation with the photo-cathode occurs when the photoionizing transition takes place through the simultaneous absorption of two photons. Each of these two photons may have energy smaller than the first ionization potential of the atoms in the cathode, but their combined energy exceeds this threshold. Obviously the transition probability for two photon absorption is much smaller than that for one photon absorption. However, such processes cannot be ignored if the light intensity is sufficiently large as may be in the case of a laser beam. GOEPERT-MAYER [1931] appears to have been the first to consider the theory of two photon transition by an atomic system. More recently a large amount of experimental as well as theoretical work on this subject has been carried out. (See for example BRAUNSTEIN and OCKMAN [1964]; YATISVet al. [1965]; KIELICH[1966]; LOGOTHETICS and HARTMAN[1967]; MOLLOW [1968]; TEICHand DIAMENT[1969]; JAISWAL and AGARWAL [1969]). In this section we briefly consider the effect of the statistical features of the radiation field on the statistics of photoelectron counting. The rate of two photon absorption may be evaluated much the same way as for a single photon absorption, except, that one now takes into account the second order terms of the perturbation expansion. One finds (JAISWAL and AGARWAL [1969]) that the probability of detecting n photoelectrons in a time interval T (in the case where each electron is emitted via two photon absorption) is given by
where W , is now the time integral of the square of the intensity
w2= J-tt+TP(t) at,
430
TH E 0 R Y
0 F P H 0 T0E LE C TR 0 N C 0 U N TI N G
[VIII,
8
and t12 is some constant. When the light incident on the detector is stationary, $ ( n )does not depend on t. When T is large compared to the coherence time, one may conclude from an argument based on the central limit theorem of probability theory that P(W,) is a Gaussian distribution, which tends t o a delta function in the limit of extremely large values of T . The distribution p ( n )is then Poissonian. This is also the case for light from a n amplitude stabilized laser, with T much smaller than the coherence time. If the light is completely polarized and originates in a thermal source, the intensity distribution is exponential [eq. (3.2)]. Hence, if T is also small compared with the coherence time, P(W,) is given by
P(W,)
=
{2W2(W,)}-3 exp {- (2W2/(W,))3}.
(8.3)
Substituting (8.3) in (8.1), we obtain the following expression for (TEICHand DIAMENT [1969]; JAISWAL and AGAKWAL [1969])
$(%)
and where U ( k ,x) is the parabolic cylinder function (ABRAMOWITZ STEGUN [ 19651)
U(K-+, x) =
W)
tL-l exp (-+t2-tx-&x2}
dt.
(8.5)
A similar result may also be obtained for a partially polarized light (JAISWALand AGARWAL [1969]) for which P(W,) is given by [cf . eq. (3.51)]
For completely unpolarized light, we obtain from (8.6), on taking the limit B 3 0
The corresponding counting distributions may be obtained by substituting (8.6) or (8.7) in (8.1). The result may again be expressed in terms of parabolic cylinder functions.
VIII, APP. A]
43 1
COMPLEX GAUSSIAN DISTRIBUTIONS
Further development in this direction such as correlation between photoelectron counts using two or more detectors may be carried out and will be useful, when more experimental data become available. One may also consider cases when simultaneous absorption of three or more photons take place. However, transition rates for such processes are extremely small. APPENDIXA
Some Properties of Complex Gaussian Distributions z
Let us define the probability density p ( z ) of a complex variable (x,y real) in the following manner: The expression
= xfiy
p ( z ) d2z E p (x,y) dx dy
P.1)
denotes the joint probability that the real and imaginary parts of z take on values in the intervals x,x+dx and y , y+dy respectively. The multivariate probability densities are defined in a similar way. The complex variables zl,z 2 , . . ., z, are said t o be distributed as a multivariate (n-dimensional) Gaussian distribution, with zero mean, if the joint probability density p(zl, z 2 , . . ., z,) is given by
whereA is a Hermitian, positive definite matrix and IA I its determinant. In this appendix we discuss some of the properties of this distribution. Since A is a Hermitian matrix ( A t , = A;), it can readily be shown that the distribution (A.2) is equivalent to a Gaussian distribution in 2n real variables xl,y l ; x2,y 2 ; . . ., x,, y, with the following properties: (xzx,) = (YtY,), (A.3a) <XzY,> =
-<X,Y,>.
(A.3b)
Most of the properties of the distribution (A.2) may therefore be derived from those of a Gaussian distribution in real variables (MEHTA [1965b]; MILLER [1964]). We list below few of the main properties of complex Gaussian distributions which have been used in the text: (1) The moment matrix and the matrix A are related by the formula
(z,*z,)
=
2(A-l),t.
(A.4)
432
[VIII,
THEORY O F PHOTOELECTRON COUNTING
App. A
(2a) All odd order moments vanish (23;
. . . ztnzjlzjz. . . Zj,,)
= 0,
m f m'.
(Ah)
(2b) All even order moments ( 2 : . . . Z : ~ Z , .~ . . z,,) may be expressed in terms of the second order moments by means of the formula
where 1,denotes summation over all m ! permutations p , q, . . ., Y of 1 , 2, . . . ) m. (3) The characteristic function A(tl, t z ,. . ., t n ) defined by the formula z
is given by the relation
(4) Let w,,wz,. . ., wk ( k 5 n) be h linear combinations of z j , I I
where c ( l j are constants. Then w,are distributed according to the kdimensional Gaussian distribution
where
(B- l ) Ll= ?
2 %;z%L,A;l; I , 1'
=
1, 2 , .
. h. .)
(A.lO)
2, J
(5) Let U be the sum of the squared moduli of the z,'s, n
(A.11) Then U is distributed according to the probability density n
where a j ( j
= 1, 2, .
n
. , n ) , are the eigenvalues of the matrix A . Also
VIII, APP. A]
COMPLEX GAUSSIAN DISTRIBUTIONS
433
the cumulants of U may be expressed as
)'LK
Tr ( A - l ) k
=
2"(K-l)!
=
(A-l)! Tr (Adk),
(A.13a)
(A. 13b)
where M is the moment matrix whose elements are given by = (ZfZi).
Properties (1)-(3) may be derived by direct integration (MEHTA [1965b]). The properties (1) and ( 2 ) may also be derived by repeated differentiation of (A.7) (REED[1962]). Property (4)followsimmediately from (3) if we first derive the characteristic function of the distribution w,,w 2 ,. . ., w k . Property ( 5 ) is derived below. From (A.2) and (A.ll), we obtain the following expression for the characteristic function of U : (exp (ihU) )
IA l/{det(Aij-2ihdij)} n
[ AI
(A. 14)
(ai-2ih)-l, j=1
where ai are the eigenvalues of the matrix A . On taking the inverse Fourier transform and evaluating the integral by contour integration we obtain the following expression for p ( U ):
-
$IAj
2 e-+kU r-J (ai-ak)-l. k=l
(A.15)
j#k
It was assumed in this derivation that the eigenvalues a j are all distinct. If there is any degeneracy, the expression has to be modified (or is to be taken in a limiting sense as the eigenvalues approach each other). When all of the eigenvalues aicoincide (i.e. when A is a multiple of the unit matrix) we may evaluate $ ( U ) directly. In this case one finds that
(A.16)
434
jvm, .4pp. B
THEORY OF PHOTOELECTRON COUNTING
The variance of U is given by
* = na2 =
w2
~.
(A.17)
n
As an illustration of eq. (A.15), we consider the case when n From (A.15), we may write
=
2.
(A. 18a) When a,
=
a2 = a we take the limit a,
--f
a2 and find that
p ( U ) = $a2Uedau.
(A.18b)
To obtain the cumulants of U , we note from (A.14) that the cumulant generating function K ( U ) = log ( e x p ( h U ) ) (A.19) may be written as
(A.20)
From (A.20) it follows that the cumulants of U are those as given by eq. (A.13).
APPENDIXB Solution of an Integral Equation In this appendix, we solve the integral equation
* It is worth mentioning t h a t the second term alone in tion formula for blackbody radiation
EINSTEIN'S
[1909a, b] fluctua-
( (AdE,)z> = hv(dE,>+ ( ~ ~ / X n v ~ d v B ) < d E , ) ~
follows from cq. (A.17). Here one assumes t h a t the radiation field is thermal and t h a t the diffcrcnt frequency components of the field i n the interval dv have the same variance. of eq. (A.17) then corresponds to and n the number of modes in the frequency interval dv is given by the expression 8nv2dvB/c3.The second term on the right-hand side of the above formula for energy fluctuations thus corresponds t o t h e contribution from classical wave aspect of the radiation field. The first term on t h e right-hand side, namely hv(dE,> is the contribution from the particle aspect of t h e radiation field. (See also footnote on p. 383 of the present article.)
435
SOLUTION O F AN INTEGRAL E Q U A T I O N
V I I I , APP. B]
which occurs in the discussion of thermal light with Lorentzian spectral profile. Let us differentiate (B.l) twice with respect to t and use the relations d - lz-z’l = &(T--t‘), dz and d - E ( Z - - Z ’ ) = 2d(z-z’), 03.3) dz where ~ ( x=) f1 according as x > or < 0 and S(x) is the Dirac delta function. We then find that (z) satisfies the following differential equation
+
where w2 =
(20(1)/1) -$.
P.5)
Hence +(z) must necessarily be of the form
+ (z)
=
,4 eiw7+BeciwT,
(B.6)
where A and B are constants. Using eqs. (B.5), (B.6) in (B.l) and evaluating the integral, we find that
(
0-iw
+ ?)B
A
+-BeciwT\ cr+ioj
ea(7--T) AeiWT
=
0.
(B.7)
Since eq. (B.7) holds for arbitrary values of z (0 5 z 5 T ) , the two terms on the left-hand side of this equation must vanish separately, i.e., (B.8a) A eiWT(o-iw)-l+ BeciWT(o+iw)-l = 0,
A (o+iw)-l+ B(o-iw)-l
=
0.
CB.8b)
If we ignore the case w = 0, which leads to the trivial solution +(z) = 0, we find, on eliminating A and B from eqs. (B.8), that 1 -
0
tan w T
=
~
20
-
.
w2-02
The roots of (B.9) are real, and because of the form of eq. (B.6) we need consider only the non-negative roots.
436
THEORY O F PHOTOELECTRON
COUNTING
LVIII,
App. B
The roots CO, of the eq. (B.9) may be separated into two groups p i and v, which satisfy tan (+p,T)= o;
,LC,
Y,
cot (&v,T)=
-0.
(B.lO)
It may be shown from eqs. (B.6) and (B.8) that the corresponding normalized eigenfunctions of the integral equation (B.1 ) are given by the following expressions: (B.1l a ) and (B.1 l b ) Finally let us evaluate the generating function G ( s ) [eq. (4.101 given by ~ ( s=) (i+crs~k)-l, (B.12)
nk
where ilk are the eigenvalues [eq. (B.5)] (R.13) and wlCare the non-negative roots of eq. (B.9), i.e. of the equation (B.14) From eqs. (B.12) and (B.13), we may write (B.15) where
z2 = 02T2+2aT(n)s, ( n )
=
a(I)T.
(B.16)
The series on the right-hand side of (B.15) may be summed by the standard techniques of contour integration: Let 1 d
F(6) =
~
-
t2+z2 d t
where 8 is a complex variable. The only finite singularities of F ( 6 ) are simple poles at 6 = &iz and at 6 = f w , T . Consider now the
VIII]
437
REFERENCES
integral
If the closed contour C is properly chosen, it may readily be shown that this integral tends to zero as the number of poles in side C tends to infinity. Hence the sum cf the residue a t all the poles of F ( 6 ) vanishes and we have 1
coshz+
1 oT -
-
2 ( 2
1
+ : T ) ' sinhz -
= 0.
(B.18)
From (B.15), (B.16) and (B.18) we therefore obtain (B.19)
where the constant of integration is evaluated from the requirement G(O) = 1.
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AUTHOR INDEX A ABERL,M., 352, 371 ABRAMOWITZ, M., 414, 430, 437 ADAM, M., 167, 190, 197, 198 G. S., 282, 291, 429, 430, 438 AGARWAL, AHN, B. H., 194, 200 M. D., 437 ALDRIDGE, C. T. J., 136, 139, 140, 149, ALKEMADE, 197, 198 ALLEN,R., 298, 338 ALLEY, C. O., 242, 268, 270, 292 ALPERN,M., 128, 129 ALPERT,S. S., 137, 146, 192, 194, 197 AMESJR., A , , 127, 129 A. E., 365, 371 ANDERSON, ARECCHI,F. T., 140, 145, 156, 191, 197, 241, 242, 251, 254, 259, 260, 268, 269, 273, 283, 285, 286, 291, 292, 410, 411, 437 ARMITAGE, J . D., 302, 338 J . A., 241, 242, 244, 266, 268, ARMSTRONG, 292, 294, 375, 437 ARNULF,A., 87, 98, 100, 104, 105, 127, 129, 353, 355, 370 ARTZ,V., 247, 251, 292 .L\SELTINE, J. s., 175, 176, 197 ASPNES,D., 194, 199
B BACHL, A, , 46, 48 BALACHANDRAN, A . P . , 388, 439 BALL,It., 224, 237 BALLIK,E. A,, 136, 199 BALZARINI, D., 137, 192, 197 BARNES,C. W., 39, 48 BAUMEISTER, P., 224, 236
BEAVERS, W. I., 15, 48 B ~ D A RG., D , 140, 175, 176, 197, 265, 266, 268, 292, 375, 377, 395, 400, 401, 403, 407, 412, 414, 422, 428, 437 BELLISIO, J. A , , 375, 437 BENEDEK, G. B., 136-138, 145, 164. 175, 181, 185, 189-192, 197-200 BENNETT, W. R., 135, 199 BERAN,M., 310, 338 BERGE,P., 138, 167, 189-191, 194, 195, 197, I98 BERGER-L’HEUREUX-KOBARDEY, S., 85, 107, 129, 353-355, 370 BERNE,A,, 140, 197, 241, 291, 437 BERNY,F., 127, 129 BHARUCHA-REID, A. T., 250, 292 BILLARD, R., 138, 191, 198 G., 241, 294 BIRNBAUM, BLACKWELL, H. R., 115, 129, 359, 360, 365, 370 BLACKWELL, 0. M., 115, I29 BLANC-LAPIERRE, A , , 307, 338 BLAQUIERE,A , , 241, 258. 292 BLOCH, F., 245, 294 BOLWIJN, P. T., 140, 198 BOND, W., 241, 292 BOND, W. L., 136, 199 BORN,M., 6, 48, 165, 198, 256, 292, 299301, 303, 307, 311, 332, 338, 368, 370, 383, 392, 437 BOSCHLOO, G. A,, 140, 198 BOTCH,W. D., 195, 198 BOTHE,W., 375, 437 BOUMAN, A., 84, 131 BOUMAN, M. A ,, 355, 358, 362, 363, 366, 367, 371 R. M., 128, 129 BOYNTON,
442
AUTHOR I N D E X
BRACEWELL, R. N., 4, 48 BRANNEN, E., 411, 437 BRAULT, J. W., 298, 338 BRAUNSTEIN, R., 429, 437 BRBMMER, H., 312, 338 BRINDLEY, G. S., 115, 129 BROWN, R. HANBURY, 8, 19-21, 48, 198, 269, 292, 375, 377, 421, 437, 438 BRUCE,C. F., 337, 338 BRUNNER, W.. 242, 292 BRYNGDAHL, O., 100, 129, 321, 339, 355358, 366, 367, 369, 371 BUCK, G. J., 39, 45 BURLMACCHI, P., 241, 291, 437 BYRAM, G. M., 110, 129 C CALMETTES, P., 167, 195, 198 F. W., 77, 88, 98, 100-103, CAMPBELL, 110, 113, 114, 117, 124, 125, 129, 131, 351-356, 358, 371, 372 CANTRELL, C. D., 422, 423, 438 CARLSON, F. U., 159, 198 CATALAN, Id., 326, 327, 339 CHANDRASAKHAR, S., 248, 292 CHANG,J. C., 400-403, 437 CHANG,R . F., 242, 268, 270, 292 CHU, B., 138, 194, 195, 198 COBB,1’. W., 67, 106, 129 It. J., 241, 292 COLLINS, J . W., 365, 371 COLTMAN, CONNES,P., 298, 339 COTTON, A , , 316, 339 Cox, J. T., 222, 224, 226, 227, 236, 237 CRANE,It., 21, 48 CRAWFORD, B. H., 56, 131 CUBISCH, I<. W., 77, 113, 114, 124, 125, 129 H. Z., 136-138, 146, 154, 156, CUMMINS, 157, 159, 160, 163, 185, 189, 195, 198200, 376, 438 CURRIE,D. G., 11, 18, 48, 377, 439 L. J . , 4, 36, 48 CUTRONA,
D DAVENPOORT JR., W. B., 388, 438
DAVIDSON, F., 242, 270, 292, 410, 438 M., 356, 358, 365, 368,369, 371 DAVIDSON, DAVIS,S. P., 140, 200 DEBYE,P., 194, 198 V., 241, 242, 286, 291, 292 DEGIORGIO, DE GROOT,S. G., 359, 371 0 . E., 189, 198 DELANGE, DE LOTTO,I., 377, 412, 438 DE MOTT,D. W., 54, 129, 353, 371 J . J.. 86, 124, 129, 130, 355, DE PALMA, 357, 358, 366, 367, 369, 371 R. W., 242, 268, 270, 292 DETENBECK, DIALETIS,D., 422, 423, 425, 438 P., 429, 430, 440 DIAMENT, DUBIN,S. B., 138, 191, 198 DUFOUR,C., 210, 237 P., 307, 311, 338, 339 DUMONTET, DUPUY,O., 87, 98, 100, 104, 105, 127, 129, 353, 355, 370 DYSON,J . , 299, 328, 329, 339
E EINSTEIN, A., 160, 198, 375, 383, 434, 438 EKMAN, G., 348, 371 ELDERTON, W. P., 428, 438 ELLIOT,J. L., 15, 18, 48, 49 ELLIS, G. W., 305, 339 E M M A N U E L , C. B., 11, 49 ENGEL,G. K., 89, 117, 129 ENOCH, J. M., 56, 129 C., 62, 129 ENROTH-CUGELL, EPSTEIN, I,. I., 224, 237 EVANS,J . V., 33, 48
F FELLER, W., 412. 438 FERGUSON, H. I. S., 411, 437 FERRELL, A,, 187, 198 G. L., 397, 423, 438 FILLMORE, FIORENTINI, A,, 119, 128, 129 FISHER, I. Z., 160, 199 FISHER,M. E., 191, 198 FIXMAN, M., 195, 198 FLAMANT, F., 124, 127, 129, 351, 371 FLECK,J. A,, 242, 254, 292 FORD, N . C., 137, 145, 185, 192, 198
AUTHOR I N D E X
FORRESTER, A. T., 135-137, 143, 148, 164, 175, 198 F R A N ~ OM., N , 297,299, 306,314, 320, 321, 323, 325-330, 334, 335, 339, 340 FREED, C., 140, 198, 241, 292, 375, 437 FRIEDEN, B. R., 39, 42, 48 FRY,G. A., 53, 54, 56, 58, 60,71, 72, 78-83, 89-92, 94-97, 100, 108, 111-113, 116, 117, 120, 122, 123, 126-128, 129, 130, 355, 358, 366, 371 FURTH, R., 375, 383, 438
G GABOR,D., 305, 339, 378, 378, 438 GAMO,H., 21, 48, 422, 438 GAMPEL,L., 136, 198 GARRET, C. G. B., 241, 292 W., 45, 48 GARTNER, GATES,J . W., 337, 339 GATTI,E., 410, 411, 437 GEBHARD, J . W., 359, 371 U., 298, 339 GERHARDT, GHIELMETTI, F., 383, 438 GIGLIO,M., 242, 292 GIVENS,M. P., 75-77, 79, 130 K. J., 141, 150-153, 168, 199, GLAUBER, 242, 282, 284, 286, 292, 375, 377, 380, 381, 396, 400, 438 GOEPERT-MAYER, M., 429, 438 GOLAY,M. J. E., 139, 199 GONTIER,M. G., 322, 325, 339 GOODMAN, J . W., 35, 48 J . P., 242, 273, 287, 292 GORDON, Goss, W. P., 305, 339 GOTZE,W., 194, 199 I. S., 253, 292 GRADSHTEYN, GRAHAM, C. H., 369, 371 K., 243, 244, 251, 292 GRAHAM, GREEN,D. G., 88, 98, 100-103, 110, 115, 129, 130, 354, 371 GREEN,D. M., 347, 364, 371 T. J., 199 GREYTAK, GRIFFIN,D. R., 111, 131 GRIVET,P., 241, 258, 292 L., 407, 440 GROPPER, GROSSKOPF, H., 348, 371 GUBISCH, D. G., 126, 130
443
GUBISCH,R. W., 351-353, 356, 371 R. A , , 135, 164, 175, GUDMUNDSEN, 198 GUILD,F., 322, 325, 339 GUILLARD, M., 337, 339 J. J., 39, 48 GUSTINCIC,
H HAAKE,F., 242, 243, 273, 278, 294 HAGFORS, T., 33, 48 HAKEN,H . , 241-244, 247, 251, 254, 259, 273, 280, 287, 292-294 HALL,G. O., 4, 36, 48 HALPERIN,B. I., 195, 199 D., 194, 199 HAMBLEN, HAMELIN, A,, 138, 167, 190, 191, 197, 198 HANSEN,G., 298, 299, 305, 339 HARRIS,J . L., 39, 48 HARRIS,R. W., 39, 42, 43, 49 HARTLINE, H . K., 124, 131 HARTMAN, P. L., 429, 439 HARTRIDGE, H., 111, 113, 130 HASS,G., 222, 224, 226, 227, 236 HAUG,H., 251, 293 HAUS,H . A , , 140, 174, 198, 199, 241, 292, 375, 437 HECHT,K., 194, 199 HELLER,P., 192, 199 C. W., 394, 404, 438 HELSTROM, HEMPSTEAD, R. D., 242, 254-256, 293, 395, 438 HENRY,D. L., 194, 195, 199 HERBERT,T. J., 159, 198 HERPIN, A,, 210, 237 HERRIOTT, D. R., 135, 199 HILZ,I<., 97, 131, 358, 371 HIWATASHI, K., 356, 359, 365, 369, 371 HOCHULI,U., 242, 268, 270, 292 HODARA, H., 21, 48 HOHENBERG, P. C., 195, 199 HOPF,L., 375, 438 HOPKINS,H . H., 103, 130, 299, 300, 306, 309, 311, 321, 332, 334, 339 HUFNAGLE, R. E., 34, 48 HYDE,W. L., 305, 340
444
AUTHOR I N D E X
I INGELSTAM, E., 337, 339, 366, 371 INOUE, S., 305, 340 IVANOFF, A , , 113, 127, 130 IWATA, G., 312, 340
293, 375, 377, 380, 439 J., 124, 130, 351, 371 KRAUSKOPF, KRUG,W., 297, 321, 322, 340 KUBOTA,H., 224, 231, 237 E. S., 16, 49 KULAGIN,
L J JAISWAL,A. I<., 392, 394, 395, 397, 398, 405-407, 423, 429, 430, 438, 439 JAKEMAN, E., 140, 199,375,394,397, 398, 407, 438 JAMIN,J . , 328, 340 JANOSSY, L., 411. 438 J A V A N , A,, 135, 136, 199 JENNISON, K. C., 13, 48 JOHANSSON, L. P., 337, 339 J O H N S O N , F. A , , 377, 416, 417, 438 J O H N S O N , P. O., 135, 164, 175, 198 J O N E S , I<.,377, 416, 417, 438
K KAC,M., 376, 387, 438 L. P., 193, 194, 199 KADANOFF, KAISER,W., 241, 292 KANE,J., 194, 199 ICANO, Y., 426, 439 I
LACHS,G., 242, 293, 396, 397, 439 LACOMME, P., 46, 49 LAMB,W. E., 241, 242, 247, 254, 273, 286, 293, 294 LAMBJR., W. E., 395, 404 K. L., 75, 130 LAMBERTS, LASTOVKA, J. B., 136, 137, 189, 199 LAU,E., 46, 49 I<.W., 35, 48 LAWRENCE, L a x , M., 241, 242, 250, 254-256, 258, 273, 278, 293, 395, 438 LEBEDEFF, A., 297, 327, 328, 340 LE GRAND, Y., 84, 86, 98, 110, 130 13. I<., 375, 377, 380, 439 LEHMBERG, LEITH,E. N., 4, 36, 48 LENK,H., 298, 339 LEVI,L., 350, 361, 371, LEVY,J. D., 116, 117, 120, 122, 130 LEWIS,E. A. S., 194, I99 LINDBERG, O., 327, 328, 340 LINNIK, W., 297, 340 E., 137, 146, 194, 197 LIPWORTH, LIT, A , , 128, 130 LITTLE,A. G., 411, 440 LOEWENFELD, I. E., 128, 130 E. M., 429, 439 LOGOTHETICS, A,, 45, 48, 49, 302, 335, 338, LOHMANN, 340 LONDON, F., 407, 439 LORENTZ, H. S., 383, 439 LOUISELL,W. H., 258, 273, 287, 293 LOWENSTEIN, O., 128, 130 LOWRY,E. M., 86, 124, 129, 130, 355, 357, 358, 366, 367, 369, 371 LUKOSZ, W., 45, 46, 48, 49 LUNACEK, J. H., 138, 191, 198
M MCCLEAN,T. P., 400, 439 F. J., 429, 440 MCCLUNG,
AUTHOR I N D E X
MCCUMBER,D. E., 242, 293, 395, 439 MCKENNA,J., 377, 439 MCLEAN,T. P., 377, 416, 417, 438 MACPHIE,R. H., 21, 49 MAGILL,P. J., 305, 340, 396-398, 439 MAIMAN,T. H., 241, 293 D., 306, 340 MALACARA-HERNANDEZ, MANDEL,L., 4, 20, 49, 138-143, 165, 199, 200, 242, 243, 256, 260, 265, 266, 270272, 292, 293, 375-378, 382, 384, 387, 392, 393, 400, 401, 403, 404, 407, 409411, 421, 424, 425, 437, 438, 440 MANFREDI,P. F., 377, 412, 438 MARATHAY, A. S., 8, 49, 422, 440 E. W., 73, 130 MARCHAND, MARCHAND, M., 46, 49 M A R ~ C H AA L,,, 335, 340 K. B., 124, 130, 366, 371 MARIMONT, MARKLE,D. A,, 21-25, 49 MARTIENSSEN, W., 139, 200, 410, 418, 439 MEHTA,C. L., 8, 49, 139, 200, 266, 294, 375, 377, 381, 387, 388, 390, 393, 397, 398, 405-407, 409, 422, 423, 425, 431, 433, 438-440 MENZEL,E., 99, 130, 332, 335, 340, 355, 357, 371 MERTZ,L., 298, 340 MEYER-ARENDT,J. R., 11. 49 MICHELSON,A,, 7, 49 D., 376, 385, 391, 439 MIDDLETON, MILLER, K. S., 431, 439 MILLER,N. D., 111, 130 MILLER,R. H., 10, 18, 49 MILLER, U., 352, 371 MILLER,W. H., 124, 131 G., 302, 340 MINICWITZ, MOLLOW,B. R., 377, 429, 440 MORGAN,B. L., 140, 200, 410, 411, 440 MORGAN,R. H., 362, 363, 365, 371 MORGAN,S. P., 3, 49 MORI, T., 356, 359, 365, 369, 371 MOUNTAIN, K. D., 160, 161, 193, 200 MOYAL,J . E., 282, 293 R. B., 212, 237 MUCHMORE, MURRAU,A. E., 231, 237 MURRAY,B. C., 32, 33, 49 MURTY,M. V. R. I<., 306, 340
445
N NACHMIAS, J., 97, 130 NAGATA,S., 356, 359, 365, 369, 371 NARAY,Z., 411, 438 NELSON,D. F., 241, 292 G., 322, 324, 328, 340 NOMARSKI, NOVICK,R., 137, 192, 197 K., 244, 293 NIMMEDAL, NUSSENZVEIG, H. M., 422, 440
0 O’BRIEN,B., 111, 130 OCKMAN, N., 429, 437 OGLE, I<. N., 69, 130 OLIVER,C. J., 140, 199 O’NEILL,E. L., 347, 371 O’NEILL,G. K., 28, 49 ONLEY,J . D., 128, 131 J. S., 194, 200 OSMUNDSON, OSTERBERG,H., 222, 224, 237, 305, 341 MSTERBERG,H., 59, 60, 62, 131
P PALCIAUSKAS, V. V., 194, 199 PARIS, D. P., 45, 49 PARRENT, G. B., 310, 338 I’ATEL, A. S., 97, 99, 131, 355, 357, 358, 367, 371 PAUWELS,H. J., 241, 293 PEASE, F., 7, 49 PEASE, F. G., 7, 49 PECORA, K., 156, 159, 160, 191, 200 PERRIN, F. H., 74, 75, 131 PHILLIPS, D. T., 140, 200 PHILPOT, J . ST.,330, 340 Prcus, G. S., 429, 440 PIKE,E. R., 140, 199, 375, 377, 394, 397, 398, 400, 407, 416, 417, 438, 439 POLLAR, H. 0.. 42, 49 POLYAK, 54, 56, 131 POLZE,S., 306, 340 PORCELLO, I,. J., 36, 48 POUND, K. V., 411, 440 PRAT, K., 323, 325, 339 PRINCIPI, P., 377, 412, 438
446
AUTHOR I N D E X
PRINGSHEIM, 297 PROCTOR, C. A,, 1 2 7 , 129 PROTHEROE, N. M., 11, 49 PURCELL, E. M., 377, 440 PUTNER,T., 224. 237
Q QUBRZOLA, R., 241, 242, 292
R KANTSCH, I<., 305, 340 KATCLIFF,F., 93, 97, 119, 124, 131 RAWCLIFFE,13. D., 36, 49 RAYL,M., 194, 199 RAYLEIGH, 247, 293 REBKA,G. A,, 411, 440 REED, 1. S., 143, 200, 433, 440 RICE, S. C., 376, 385-387, 440 KIENETZ,J., 297, 321, 322, 340 KIGGS,L. A., 128, 131 RISKEN,H., 241, 242, 244, 247, 350, 251, 254-256, 258-260, 273, 278, 280, 287, 291, 292-294, 376, 395, 407, 409, 410, 440 ~IOBSO J.NG., , 62, 117, 129 f
S SAGNAC, 297 SAUERMANN, H., 241, 247, 251, 273, 292294
SAUNDERS, J. E., 348, 371 SAWAKI,T., 224, 237 SAWYER, r<. A,, 84, 131 A. C., 192, 200 SAXMAN, SCHADE,0. H., 74, 89, 97, 131 SR., 0. H., 355, 358, 362, 366, SCHADE 367, 371 SCHARF, P. T., 231, 237 SCHAWLOW, A. I)., 241, 292 SCHAWLOW, A. L., 241, 247, 293 SCHERB,F., 15, 18, 48, 49 SCHMID, C., 241, 247, 251, 278, 281, 292294 SCHOBER, H. A. W., 97, 131, 356, 371 F. J . , 138, 194, 195, 198 SCHOENES, SCHOLNIK, R . , 127, 130 SCHULZ, G., 297, 299, 300, 302, 321, 322, 340 SCULLY, M., 242, 254, 273, 286, 294, 395, 440 SEIGEL, L., 137, 192, 193, 197, 200 SENITZICY, J. K., 273, 294 SENGERS,J . V., 192, 200 SERGENT,B., 314, 339 SHAPIRO, S. hf., 190, 200 SHERWIN, C. W., 36, 49 SHLAER, S., 67, 84, 108, 131 SIEGERT,A. J . F., 376, 387, 438 S ~ E G M AA.XE., , 19, 49 SIRKS,297 SLANSKY, S., 299, 332, 339, 340 SLEPIAN,D., 42, 49, 376, 385, 387, 389, 440 SLEVOGT, IT, 299, 340 A ,, 205, 237 SMAKULA, SMEKAL, A , , 375, 440 SMITH, A. W., 241, 242, 244, 266, 268, 292, 294, 375, 437 SMITH,F. H., 321, 326-329, 339 SMITH,S . D., 210, 237 S N O W , I<., 305, 340 SONA,A,, 140, 197, 242, 251, 254,259, 260, 268, 269, 273, 283, 285, 292 SONI,R. P.,396-398, 439 SOUTHALL, J . P., 352, 357, 371 SPILLER,E., 139, 200, 410, 418, 439 STEEL,W. H., 297-302, 306, 310, 319, 321, 322, 325, 335, 340
AUTHOR I N D E X
STEGUN, I. A , , 414, 430, 437 STEVENS,S. S., 348, 349, 368, 371 STILES, W. S., 56, 115, 131 STRATONOVICH, K. L., 254, 294 STROKE,G. W., 28, 49 A , , 428, 439 STUART, SUDARSHAN, E. C. G., 138, 142, 152, 154, 199, 200, 242, 282, 293, 294, 375-378, 381, 382, 439, 440 SWETS,J . A , , 347, 364, 371 SWIFT, J.. 193, 194, 199, 200 SWIFT,W. D., 15, 48 SWINNEY,H. L., 185, 192-195, 198-200, 376, 438
T TEICH, M. C., 429, 430, 440 THELEN, A,, 206, 207, 210, 212, 214, 222, 224, 226, 227, 236, 237 THETFORD, A,, 210, 222, 237 THORNTON, B. S., 337 340 TOLMAN, F. R., 337, 340 DI FRANCIA, G., 39, 45, 49 TORALDO TOUSEY, R . , 127, 130 TOWNES, c. Er., 137. 200, 241, 247, 294 TSURUTA, T., 325, 340 TWISS,K. Q., 19, 21, 48, 49, 198, 269, 292, 375, 377, 411, 421, 438, 440
U UHLENBECK, G. E., 250, 294, 407, 440 ~ J H L I G ,M., 322. 325, 340 V VAN DE HULST,H. c., 158, 200 VANDER POL, B., 247, 294 K., 305, 340 VAN DEWARDER, VAN HOVE,L., 160, 200 R. F., 305, 341 V-4N LIGTEN, VANNES, F. L., 84, 97, 131, 355, 357, 358, 366, 367, 371 VARGA, P., 411, 438 VIVIAN,W. E., 4, 36, 48 VOLLMER, H. D., 242, 251, 255, 256, 259, 294 B., 138, 167, 194, 195, 198 VOLOCHINE,
447
VON BAHR,G., 115, 131 VONHELMHOLTZ, H., 110, 130 Vos, J . J., 55, 131
W WAETZMAN, E., 328, 341 WAGNER, W . G., 241, 294, 429, 440 WALD,G., 111, 131 WALRAVEN, P. L., 131 WALTHER,A,, 8, 49, 422, 440 WANG,M. C., 250, 294 WANGNESS, R. I<., 245, 294 WATANABE, A,, 356, 359, 365, 369, 371 WATSON,G. N., 347, 372 WEHLAU,W., 411, 437 WEIDLICH,W., 241-243, 247, 251, 254, 273, 278, 280, 287, 291, 292-294 WEIL, A. R., 322, 324, 328, 340 G., 87, 98, 110, 124, 127, WESTHEIMER, 128, 131, 351, 352, 354, 357, 372 WHITE,J . A , , 194, 200 WIGNER,E., 282, 294 WILCOX,L. R . , 193, 200 J . S., 29, 31, 49, 50 WILCZYNSKI, WILD, B. W., 119, 131 WILLIS,C. K., 242, 273, 294 WILSON,A. D., 305, 340 WOLF,E., 4, 6, 8, 20, 48-50, 139-142, 165, 198, 200, 256, 260, 265, 266, 282, 291294, 299-301, 303, 307, 311, 332, 338, 341, 368, 370, 375-378, 382, 384, 387, 388, 392, 393, 405, 407, 409, 421-423, 437, 439, 440 WOLTER, H., 39, 50 WOOD,J . G., 337, 340 TVOODS, G., 159, 198
Y YAMAMOTO, T., 299, 312, 314, 323, 325331, 334, 335, 339, 340 YATISV,S., 429, 440 YEH, Y., 136, 137, 146, 157, 190, 192, 194, 195, 197, 198, 200 YOUNG, L., 212, 224, 237
z ZERNIKE, F., 307, 341
SUBJECT INDEX A aberrations, 38, 126 c t seq., 297, 299, 305, 352 abrasion resistance, 225 abrupt contrast border, 73 absolute phase, 32 absorbencc, 231 absorption-free materials, 204 accomodation, 54 accuracy, 346 active figure controle, 22 et scq. - illumination, 21, 32 - interferometer, 32, 33 acuity, 66 et seq. adaption, 348 brightness, 350 - level, 362, 366 additive mixing, 63 adhesion, 224 - test, 225 admittance matching, 222 afferent nerves, 351 air, 217 aligning, 36 amacrine cells, 61 ambiguity sensors, 24 amplitude contrast function, 335 - fluctuation, 169 - spread function, 314 analytical signal, 306 angular diameter, 8, 15 - resolution, 39 separation, 69 annihilation operator, 275, 381 anterior focal point, 91 antireflection coatings, homogeneous mul~
~
tilayer, 204
_ _ , inhomogeneous, 204 -~, multilayer, 204 et seq., 210 - _ , single layer, 205, 206 ~- , three layer, 214 .-, two layer, 206 et seq., 212 ~t seq. antiresonant terms, 277, 279 aperture, 297 - , completely filled, 29 , one dimensional, 27 - , hexagonal, 30 , partially filled, 26 , super-gain, 45 hrnulf-Dupuy method, 104 artificial pupil, 84, 89, 98, 106 astigmatic aberration, 314 astigmatism, 80 - , retinal 62 astronomical observation in space, 29 atmospheric effects, 12 - phase translation, 14 - seeing, 10 et seq. - turbulence, 20, 33 autocorrelation function, 8, 143 et seq., 156, 185, 397 automatic gain control, 350 axial chromatic aberration, 7 1 ~
~
~
~
B background noise, 42 bandwidth, 138 -, electrical, 20 -, optical, 5, 9 et seq., 19, 20 barred grid device, 15 beam, multilobed, 33 beam-spli tters, 3 12
SUBJECT INDEX
Bessel functions, 75 et seq. binary critical mixtures, 194 - diffusion coefficient, 194 bipartite pattern, 70 birefringent elements, 320 - systems, 326 blackbody radiation, 375, 383 bleaching, 57 et seq. Bloch equations, 245 Bloch’s law, 349 blue sensitive cones, 59 blur, 77 et scq., 93 -, index of, 80 et sey., 120 borders, 63, 92 - contrast, 65, 71, 118 - _ sensitivity, 65 - _ threshold, 65 - perception, 65 Bose-Einstein distribution, 140, 242, 268, 418, 426 - _ statistics, 140 boson, 140, 407 - fluctuation, 400 brain, 86, 351 Brewster’s fringes, 303 brightness, 347 - difference, 348 _ _ threshold, 65 - function, 346 et seq., 348, 365 - _ , instantaneous, 348 et seq. - _ , steady state, 349 et seq. - response, 59 bril, 349 Brillouin components, 136 Brownian motion, 248 bunching effect, 407
C c-number equation, 286 et seq.
- function, 377 Campbell-Green method, 100 cavity, 244 center of curvature, 23 central acuity, 66 - limit theorem, 175, 386, 393, 430 - lobe, 27 cerium fluoride, 222 characteristic functions, 390 et seq., 402
449
chromatic aberration, 71 chromatically restricted objects, 47 classical particles, 382 - wave field, 382 clock telescope, 29 closed-loop servo-control system, 21 clusters, 271 coating, quarter-half-quarter, 220 coherence, 4, 8, 9, 17, 32, 84, 163 et seq., 299, 307, 310, 312, 320 - areas, 164, 174, 187 - diffraction theory, 306 - functions, 338, 375, 409 - -, higher order, 419 et seq. - length, 38 -, longitudinal, 9 - matrix, 404 - spatial, 142 et seq., 149, 154, 163, 305 spread function, 314 - temporal, 305 - tensor, second order, 409, 421 - theory, 142 et seq. - time, 378, 385 et seq., 393, 412, 422 ~
et seq. transmission, 312, 318 - transverse, 9 coherent detection, 19 - fields, 397 - illumination, 32 - light, 76, 84 - local oscillator, 146 - optical data processing, 28 - receiver, 36 - reference wave, 31 - source, 146, 375 coincident counts, 411 color blindness, 114 - index, 231 - mixture, 59, 60 - photography, 231 common-path interferometer, 298 communication system, 345 compensation systems, 299 completely filled aperture synthesis, 29 complex analytical signal, 142 - coherence factor, 6, 7, 14, 27 - degree of coherence, 4, 6, 20, 299, 301 et sey., 305 -
450
SUBJECT INDEX
- visibility function, 6, 14 compound interferometer, 2 1 computor, 185 concentration fluctuations, 180 condensation effect, 271 process, 272 condensor-objective system, 318, 324 conditional probability, 409 cones, 53 e t seq., 351 contrast, 70 et seq., 359, 370 - sensitivity, 114 control of thickness, 224 controlled evaporation, 224 convolution, 39, 71 cooperative retroreflector, 38 cornea, 351 correlation function, 185, %42,256 et seq., ~
384, 389
-, second order, 143 correlator, 20 cortex, 351 counting distribution, 398 et seq. - timc, 425 - _ intervals, 415 creation operator, 275, 381 critical point, 191 - - singularity, 193 cross-coherence, 31 1, 316, 318, 332 - -correlation function, 409 - spectral density, 393 CW laser, 170 - system, 37 D
delay compensation, 305 demodulation, 97 density matrix, 244, 2'76 operator, 245, 377, 381 - -, reduced, 276 depolarization, 305, 31 4 detecting time, 376 detection, 345 -, fringe, 15 - probability, 361, 364 detector, 15, 411 - fluctuations, 16 - noise, 361 -, quantum efficiency of, 21 detuned coatings, 222 detuiiing, 222, 251 difference frequency, 138 diffraction, 299 - limit, 34, 39, 43 - limited eye, 76, 89, 106 et s q . - _ optics, 3, 22, 37 diffusion broadening, 146 - coefficient, 250 digital computor, 35 dilute solution of particles, 154 rt srq.,lYO directed vision, 84 disk-annulus pattern, 70 distribution function, 262, 281, 287 - -, joint, 265 - _ , stationary, 252 -, photoelectron counting, 242, 264 -, transient, 26% Doppler broadening, 157 - shift, 34, 136, 157 -spread imaging, 33 double focus interfcrometer. 298, 304 - -objective tclescopc, 28 et seq. - reflection, 233 - refracting systems, 321 - slit interference pattern, 98 rt seq., 110 drift coefficient, 250 -
-
damped harmomc oscillator, 273 data processing, 31 dark light, 361 dead timc correction, 412 _ _ effects, 411 c t seq , 425 Debyc-Sears effect, 136 decay constant, 257 decimal acuity, 68 defocusing, 310, 322 deformation of frlnges, 323 degree of coherence, 4, 6, 20, 299, 301 rt seq , 305 - - polarization, 392, 404, 407
E earth-based interferometer, 10 echo, pulse, 37 effect sub-linearity, 368 effective interfaces, 210 ~t seq., 222
SUBJECT INDEX
eigenmodes, 255 eigenvalues, 258 et seq. electric field, 246 et seq. electrical bandwidth, 20 electronic correlator, 20 - detection noise, 41 - fringe detection, 11, 12, 15 emmetropic eye, 82 energy fluctuations, 383 entrance pupil, 76 et seq., 332 entropy fluctuations, 161 enucleation, 54 error, position, 38 -, profile, 25 estimation, 345, 348 evanescent wave, 39 excess noise, 139 excitation, 60, 61, 71, 94 exit pupil, 82, 317, 320 expectation values, 271 eyeglass lens, 233
F Fabry-Perot formula, 210 Fechner's law, 349 feedback-controlled optics, 21, 36 field operators, 381 field-widening, 298 film grain, 41 figure analyser, 22 - error, 23 filter admittance, 176 - bandwidth, 176 -, low pass, 42 Fixman's modification, 195 Fizeau fringes, 302 fluctuating intensity, 414 fluctuations, 241, 269, 272, 375, 380 et seq. -, intrinsic, 380 - of light beams, 375 focus, 27 -, common, 27 Fokker-Planck equation, 242, 249 et seq., 272 et seq., 283 Foucault target, 67 Fourier analysis, 31, 185, 299 - spectra, 31
451
transformation, 8, 31, 75, 149, 257, 307, 332 fovea, 53, 59, 60 foveal acuity, 60 Fraunhofer diffraction, 69, 75, 77 - image, 84 frequency, difference, 135 - plane, 31 - _ transparency, 31 frequency pulling, 241 - pushing, 241 - shift, 135 - shifter, 2 $ - stability, 10 Fresnel image, 77, 83, 84 - _ , monochromatic, 81 - point spread function, 77 fringe, 5, 6, 15 et seq., 298 - amplitude, 15 - detection, 15 et seq. - frequency, 18 - localization, 299, 302 phase, detection of, 17 et seq. - sensitivity, 15 et seq. - visibility, 13, 17, 301, 316 fringed field, 304, 319, 323 fringes of equal inclination, 302 - _ - thickness, 302 Fry-Cobb index of blur, 80 et seq., 112 -
-
G gain, 362 factor, 362 ganglion cells, 60 et seq., 93 Gaussian distribution, 252 _ _ , complex, 431 - functions, 47, 366 - noise, 41, 407 - probability distribution, 422 - process, 260, 384 - random process, 143, 139, 375 - spread function, 72, 77, 117 - statistics, 21, 175 glass, 217 glow discharge cleaning, 225 granularity, 346 grating, 67 -
452
SUBJECT INDEX
gravitational stress, 21 green sensitive cones, 59 Green’s eye, 101
H Haidinger rings, 303 Hamilton operator, 245, 274 Hanbury Brown-Twiss effect, 164 harmonic signal, 397, 407, 409 He-Ne laser, 135 Heisenberg picture, 273 heterodyne detection, 19, 137, 146 et seq., 173, 191 - detector, 34 - efficiency, 167 spectroscopy, 188 et scq. high index substrate, 212, 214 numerical aperture, 297 hole burning, 241 holographic arrays, 34 et seq. - interference microscope, 305 holography, 31, 34 homodyne detection, 137, 143 r t seq., 191 - spectroscopy, 185 et seq. - spectrum, 172, 174, 182 et seq. homogeneity, 368 human eye, 51 et seq. - observer, 345 - sensory detection system, 345 - vision, 345 Huygens-Fresnel principle, 307 hyperbolic fringes, 314 -
-
I illumination-to-noise ratio, 362 image blurring, 11 - contrast, 236 - dancing, 11 - formation, 299, 306 - function, 8, 20 -, improcessed, 44 - intensity distribution, 39 et scq. - modulation, 347 -, retinal, 71 et seq. - space, 317 imaging, Doppler spread, 33
incoherent detection, 19 illumination, 39 index of blur, 120 - - performance, 66 - - refraction, 204 inhibition, 60, 61, 71, 94 effect, 368 input modulation, 347 - signal, 345 instabilities, 32 integration time, 20 intensity distribution, 8, 14 - fluctuation, 241, 243, 249, 256, 375, 380, 384 et seq. - interferometry, 19 et seq. - stabilized laser light, 415 interference fringes, 36, 353 - microscope, 297 - spectroscopy, 422 interferometer, 8, 10, 14 -, active, 32 -, optical, 13 -, radio frequency, 19 - spacing, 12 -, three element, I 2 et seq. interferometry, 4 et seq. intermode heats, 171, 174 interocular comparison, 348 intra-lens reflections, 229, 233 _ _ surface reflections, 228 inversion problem in photoelectron couiiting statistics, 423 ionization potential, 429 irrelevant images, 233 isobaric entropy fluctuations, 161 isoplanatic condition, 309 - p a t h , 15 isoplanatism, 368 Ives pattern, 67 -
-
J Jamin interferometer, 328
K Kohler illumination, 307 Koenig bar target, 66
SUBJECT INDEX
L Landau-Placzek equation, 195 Landolt C target, 66 Langevin force, 272 et seq. - methode, 248 - operator, 273 large-aperture optical systems, 21 laser, 9, 10, 32, 34, 101, 135, 305, 375, 383 - action, 241 - beam, 429 - cavity, 264 - equation, 244 et seq., 248 et seq. - fields, 375 - fluctuations, 241, 376 - Fokker-Planck equation, 243 e t seq., 252 light, 24'2, 395 et seq., 406 et seq. _ _ , intensity stabilized, 415 - line, helium neon, 225 - master equation, 273, 286 et seq. - monitoring methods, 12, 15 - oscillation, 139, 241, 260 - radiation, 140 - resonator, 243 -, single mode, 21 - technology, 32 - threshold, 242, 244 lateral geniculate body, 351, 361 layer thickness, 224 Le Grand's condition, 85 lens, 351 - of the eye, 56 light beating, 135 -~ spectroscopy, 135 et seq., 376 _ _ spectrum, 145 - bucket receiver, 32 - gathering power, 11 - scattering, 137 _ _ theory, 154 et seq. line, 62 - shape, 135 - spread function, 60 et seq., 71 et seq 347, 352, 355 - width, 192, 251 linear theory, 241 Linnik system, 298 lobe, 32 ~
453
local oscillator, 34, 146, 167, 173, 189 longitudinal coherence, 9 Lorentzian profile, 144, 149, 159, 162, 174, 257, 389, 391, 400 et seq. low frequency signal, 19 - -level detection, 137 - -spatial-frequency con-,orient, 29 lumiriance-equivalent modulation transfer function, 369 - level, 108 luminosity curve, 57
M Mach bands, 369 -2ehnder interferometer, 304 Macphie, compound interferometer, 21 macro-molecule, 159, 190 et seq. magnesium fluoride, 203, 222 magnitude estimation, 348 Malus-Dupin theorem, 301 Mandel's formula, 399, 41 1 Markov process, 250 maser principle, 241 master equation, 242 Maxwell-Boltzmann statistics, 157 medium index, 212 Michelson interferometer, 304 - stellar interferometer, 7, 10, 18, 32 micro-interferometer, 297 - -Jamin, 297 et seq. - -Mach-Zehnder, 297 et seq. - -Michelson, 297 et seq. micronystagmoid movements, 95, 128 microscope imaging, 44 microwave technique, 37 mirror, 21 et seq. - segments, 22 mixing, 136 - additive, 63 mode, 139 - competition, 241 -, laser, 21 - structure of lasers, 136 modulation, degree of, 32 - depth, 170 - index, 33 -, perceived, 99 -
454
SUBJECT INDEX
sensitivity, 70 e t scq. threshold, 89 et seq., 95, 98, 115 r t spy. - transfer characteristic, 346 - - factor, 75 e t seq., 86 ... - function, 27, 75 c t srq., 80, 89 e t seq., 95 e t sep., 100 e t seq., 347, 351 et seq., 365, 368 _ _ _ , visual, 365 modules of coherence factor, 8, 9, 1 7 _ _ cross, coherence, 320 Moiri. effect, 45 monochromatic Frcsnel image, 81 - light, 113 et srq. monochromator, 84 moving-grating technique, 46 motor adjustments, 127 multi-level system, 244 multimode laser, 167, 170 multimoding, 32 multiple correlations, 418 e t seq. mutual coherence function, 307, 310 intensity, 316, 320, 331 - -. function, 310 - spectral density, 307, 309 et seq.
-
-
N narrow band Gaussian noise, 169 fields, 152 near-infrarcd. sighting, 227 nerve fiber, optic, 61 neural intcrconnections. 351 night-driving conditions, 233 nodal point, 69, 72, 75 noise, 39, 41, 242, 249, 346, 359, 370 -, background, 42 -, bandlimited measurement, 43 -, detector, 361 - level, 361 -, neural, 361 -, quantum, 362 -, radiation, 361 - sensation, 361 et seq. - sources, 188, 361 ~. spcctrum, 136, 363 -, white measurement, 43 noiseless imaging system, 39 et seg. lion spherical scatterer, 158 - - .- random
nonlinear equation, 243 non-linearities of area effects, 366 normalized line spread function, 8 0 - point spread function, 80
0 objcct modulation, 347 restoration, 39 r t seq. space, 317 thickness, 336 objective compensation, 330 r t seq. obliquity effect, 337 one-dimensional aperture, 27 - - ring laser, 246 optic nerve, 351 . - fiber, 81 optical bandwidth, 5, 19, 20 - blur, 57, 75, 97, 99, 109 - d a t a processing, 28, 31 -. difference frequcncy, 136 - interferometry, 13 path compensation, 30.5 - systems, large aperturc, 21 - thickness, 337 - lransfer function, 42, 316, 331 optics, feedback controlled, 21 orbiting instruments, 22 o u t of focus imagery, 76 et s r q . output signal, 344
-
-
-
-
P P-representation, 152 partially coherent light, 315 filled apcrture, 26 et seq., 36 polarized light, 391, 409 path length, 9, 30 perceived modulation, 99 peripheral retina, 59 perturbation expansion, 429 phase, absolute, 32 -~ autocorrelation function, 157 - changes, 211 -- contrast function, 335 distribution, 8 error, 13 - - fluctuation, 168 . - modulation, 33 -
-
-
-
SUBJECT INDEX
-
quadrature, 18 reference, 38 shift, 14 space, 400 - transition, 243 - translation, 10, 13, 14, 33 photocurrent, sinusoidal, 33 - spectrum, I74 photodetector, 33, 34, I63 et seq., 383 - surface, 163 photoelectric counts, 378 - detector, 375 - effect, 378 - emission, 380 - mixing, 135 - surface, 138 photoelectron counting distribution, 138, 242, 264 _ _ formula, 377 -, continuous, 267 - -, discrete, 267 photoelectrons, 265, 277 photoionization, 429 photomultiplier tubes, 18 photon, 20, 244, 381 - arrivals, 41 - counting distribution, 286, 399 detector, 264 - generator, 151 - statistics, 244 photons per unit bandwidth, 21 photopic brightness, level, 367 - luminosity curve, 57, 59 - trolant, 57 photopigment, 56 et seq. photoreceptor, 53 et seq., 93, 351 physiological blur, 97 picket fence device, 15 pigment, 57 et seq. - epithelium, 54 pillbox distribution, 78, 83 plexiform layers, 361 point, 63 - -spread function, 27, 39, 41 et seq., 71 et seq., 91, 347, 351 pointing skill, 60 Poisson distribution, 139, 242, 268, 426 - transformation, 266, 377 -
455
polarization, 45, 245
et seq., 404, 409, 418 interferometer, 310 polarized light, 384 - thermal light, 399 polystyrene diffusion broadening, 146 190 position error, 38 post-detection bandwidth, 19 - _ - object restoration, 39 - _ - processing, 44 power pattern, 32 probability density, 272, 376 et seq., 380, 385, 391, 425 - distribution, 376 protective glass, 232 psychophysical investigations, 344 et scq. - techniques, 84 pulse echo, 37 pump parameter, 257 et seq., 268 et seq.,286 pumping, 250 - power, 244 pupil, 309, 312, 352 - function, 310, 333 - size, 106 - surface, 324 pupillary compensation, 330 et seq. -, degree of, 392
-
Q quantum efficiency, 21, 151, 378, 419 - noise, 362 - theory of optical coherencc, 150 et sey. quarter-half-quarter coating, 220 - -wave layer, 224 quasielastic light scattering, 137 quasi-monochromatic optical wave, 5 - optical signal, 138 - source, 300
R radar, 36 et seq. - astronomy, 33 -, optical synthetic-aperture, 36 et scq. radial object motion, 36 radiation field, 381 radio astronomy, 4, 13 - frequency, 13 - - interferometer, 19 - interferometry, 9
456
SUBJECT INDEX
- tclescope, 13 random effects, 346 - fluctuations, 364 - Gaussian complex process, 138 - phasc modulation, 172 - process, 376 ratio scaling, 348 IZavenau, rule of, 316 Raylcigh criterion, 43, 69, 91 - equation, 247 - -Garis criterion, 158 - limit, 39, 44, 46, 158 - linewidth, 162, 184, 190 receptive field, 62 receptor, 115 ~t seq. red sensitive cones, 57, 59 redundancy, 29, 30 -, undesired low-frequency, 81 refcrcnce fringes, 304 - star, 30 reflcctance, 203. 205, 211 - minimum, 207 -, residual, 203 -, zero, 214 reflected energy, 227 reflections, unwanted, 227 reflectometry, 124 refractive index 205 et seq., 337 refractometry, objective, 54 relaxation rate, 169 reliability, 346 reservoir, light ficld, 274 residual reflcctance, 203 resolution, 27, 34, 37, 135, -, angular, 39 -, azimuthal, 37 - -degrading effects, 33 - element, 42 et seq. -, spatial, 32 - threshold, 69 resolving power, 27, 39, 68 et seq., 85 et seq., 106 et seq. resonator losses, 261 response of the eye, 57 restoration, object, 39 et seq. restoring, 28 retina, 53 et seq., 351 et seq., 361, 366 -, anatomy of the, 55
-brain portion, 354 subsystem, 351 -, physiology of the, 55 retinal illumination, 358, 362 - image, 71 et seq. - mosaic, 109 optics, 56 - reflectometry, 124 rctinoscopy, 54 rigid rod, 158 rods, 53, 351 Roscoe-Bunscn principle, 57 running mode, 244 - wave, 244 -
_ . _
-
S satellite-to-satellite imagery, 9 saturation, 63 - effects, 264 Savart platc, 305, 312 et seq., 321 et seq. scanning itnagc dissector, 24 scattering, 366 - amplitudc, 168 - theory, 154 Schawlow-Towncs formula, 247 scintillation, 10 11, 12 - frequency spcctrum, 11 scotopic luminosity curve, 57, 59 second nodal point, 72 - -order correlation, 143, 184 seeing, 11 scgmented mirror, 22 self-beat detection, 143 et seq. - compensating elements, 305 - -sustained oscillator, 247, 251 semiclassical theory, 243, 244 et seq. semiconductor lascr, 251 sensitivity, 9 sensor, 24 servo-control system, 21 et seq. - -controlled segmented mirror, 22 - mechanism, 10, 15 shear, 319, 331 shearing, 312 et seq. - interference microscope, 321 - interferometer, 298 shift factor, 212 short intervals, 265
SUBJECT INDEX
shot noise, 139, 144 et seq., 164, 172 sideIobes, 27 signal averager, 180 - plus noise model, 242 - -to-noise ratio, 19, 44, 163 et seq., 174, 187, 363 sine wave, 73 single layer antireflection coating, 205
et seq. single mode, 273 - - laser, 21, 139, 145, 168, 305, 415 - _ ring laser, 244 - surface transmittance, 229 Smith's method, 326 Snellen fraction, 68 letters, 66 source fringes, 321 - -size compensation, 297, 303, 317 et -
seq. space telescopes, 22 - -time correlation, 160 spacer layer, 211 spatial coherence, 142 et sep., 149, 154, 163, 305 frequency, 13, 31, 69 et seq., 331, 370 - _ response, 351 - resolution, 32 - rigidity of largc optical systems, 25 - structure, 43 spectral bandwidth, 305 - profile, 257, 375, 389 - reflectance, 231 - response, 346 - transmittance, 231 spectrum analyser, 34, 180 et seq. spherical aberration, 71, 106 et seq. - scatterers, 156 spheriodal wave functions, 42 spin resonance theory, 245 split-field photometry, 355, 365 spontaneous emission, 243, 248 - _ rate, 264 - photons, 264 spread functions, 71 et seq. spurious neural pulses, 359 square-wave gratings, 116 stabilization effect, 264 force, 244 -
-
467
stability, environmental, 225 standing wave type resonator, 244 star tracker, 30 states of polarization, 397 static scatterers, 156 stationarity, 368 statistical probability distribution, 138 stellar diameter, 8 - interferometry, 376 et seq., 421 step function, 73 - _ response, 355 Stiles-Crawford effect, 56, 59 stimulus, 345, 348, 365 stochastic noise force, 248 stray light, 231 et seq. subjective sensation, 345 sub-threshold response, 116 super-gain aperture, 45 superheterodyne, 189 - detection, 137 supra-threshold, 347, 363 levels, 99 response, 116 symmetric interferometer, 304 synthesized layers, 222 synthetic-aperture optics, 25 et seq.,'36 radar, 36
T telescope, 11 -, aplanatic, 30 -, clock, 29 -, double-objective, 28 et seq. -, large aperture, 15 -, space, 22 temporal coherence, 305 synthesis, 28 temporally restricted objects, 46 test fringes, 321 - patterns, 358 thalamus portion, 351 thermal conductivity, 193 - diffusion, 161 - equilibrium, 140 - evaporation, 224 fields, 397 - light, 242, 305, 384, 417 - noise, 396 et seq. -
-
458
SUBJECT INDEX
origin of light, 20 quanta, 249 - sources, 19 - stress, 21 three element interferometry, 12 et seq. - level system, 251, 278 threshold, 343, 248 et seq., 86 r t seq., 251, 263, 270, 279, 347 et seq., 354, 360, 363 - contrast, 360, 370 - of visibility, 99 -- techniques, 370 - measurements, 356, 359 t h u m b prints, 225 tilting, 312 ct seq. time domain, 46 - -invariant specimens, 44 transfer coefficients, 333 - function, 70 et seq., 98, 346 transient distribution, 262 - mean intensity, 263 variance, 263 transition rate, 431 translation diffusion, 157 transmission, 311 - of coherence, 312, 318 transmittance, 212 - improvement, 228 transmitting optics, 32 transversal aberrations, 309 transverse coherence, 9 triple interferometer, 12 et seq. troland, 72, 359 truncation, 261 tungsten lamp, 416 turbulence, 11, 20 two-beam interference, 306 - . _ interferometer. 297, 300 et seq., 331 two-level laser system, 244, 251 two photon absorbtion, 429 Twyman-Green interferometer, 33, 298 -
V V-coating, 206 et seq., 225 Van Cittert-Zernike theorem, 6, 165 Van der Pol equation, 247 - ..~ oscillator, 396, 409 variance, 269, 382 vibrating barred grid, 18 vibration, 38 visibility, 6, 10 et seq., 30, 116, 232, 300, 302 et seq., 331 - function, 14 - of a bar, 120 - of borders, 123 -- of lines, points and borders, 64 - of square-wave gratings, 116 - phase, 12 visual acuity, 66 et scq. - efficiency, 68 - estimation of fringe amplitudes, 12, 15 vitreous humor, 54
~
U ultrasonic waves, 136 uncoated optical surface, reflectance of, 203, 205 uniform field, 304, 325 unwanted reflections, 227
w W-coating, 206 r t seq. wave interference term, 383 wavefront tilt, 10, 11 - warping, 10, 11 Weber’s law, 349 wedge-shaped object, 336 Westheimer’s arrangement, 87 white light, 416 - noise, 158 wide-band coating, 227 Wiener-lihintchine theorem, 142 et seq., 147, 150, 156, 176 window, 312 Wollaston prism, 45, 305, 312 et seq., 324 working distance, 299
Y Young’s interference experiment, 5 - _ iringes, 98, 100 et seq.
z Zeeman splitting, 135 Zernike-Gabor object, 335 zero reflectance points, 217 zirconium oxide, 222 zoom lens, 331
