Progress in Optics, Volume 9

PROGRESS IN OPTICS VOLUME IX

E D I T O R I A L A D V I S O R Y BOARD

M. FRANCON,

Paris, France

E. INGELSTAM,

Stockholm, Sweden

K. KINOSITA,

Tokyo, Japan

A. LOHMANN,

San Diego, U.S.A.

W. MARTIENSSEN,

Frankfurt am Main, Germany

M. E. MOVSESYAN,

Erevan, U.S.S.R.

A. RUBINOWICZ,

Warsaw, Poland

G. SCHULZ,

Berlin, Germany (G.D.R.)

W. H. STEEL,

Sydney, Australia

G. TORALDO DI FRANCIA, Florence, Italy

W. T. WELFORD,

London, England

PROGRESS I N OPTICS VOLUME IX

EDITED

BY

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

Contributors A. L. BLOOM, A. J. D E M A R I A , J. W. S T R O H B E H N , E. 0. AMMANN,

L. A L L E N , D. G. C. J O N E S , V. M. A G R A N O V I C H , V. L. G I N Z B U R G ,

K. G N I A D E K , J. P E T Y K I E W I C Z , B. R. F R I E D E N

1971 N O R T H - H O L L A N D P U B L I S H I N G COMPANY

- AMSTERDAM *

LONDON

Q NORTH-HOLLAND PUBLISHING COMPANY

- 1971

AU Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner LIBRARY OF CONGRESS CATALOG CARD NUMBER: N O R T H - H O L L A N D ISBN:

61-19297

0 7204 1609 8

A M E R I C A N E L S E V I E R ISBN:

0444 10111 X

PUBLISHERS:

-

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - LONDON SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:

AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017

PRINTED I N THE NETHERLANDS

CONTENTS O F VOLUME I (1961) I.

THE MODERNDEVELOPMENT OF HAMILTONIAN OPTICS,R. J. PEGIS.. . . . . . . . . . . . . . . . . . . . . .

.

.

11.

1-29

WAVEOPTICSAND GEOMETRICAL OPTICS IN OPTICALDESIGN, K. MIYAMOTO. . . . . . . . . . . . . . . . . . . . 31-66 111. THE INTENSITY DISTRIBUTIONAND TOTAL ILLUMINATION OF ABERRATION-FREE DIFFRACTION IMAGES, R. BARAKAT. . . 67-108 IV. LIGHTAND INFORMATION, D. GABOR. . . . . . . . . . 109-153 ON BASIC ANALOGIES AND PRINCIPAL DIFFERENCESBETWEEN V. OPTICAL AND ELECTRONIC INFORMATION, H. WOLTER . . . . 155-210 v1. INTERFERENCE COLOR,H. KUBOTA. . . . . . . . . . . . 211-251 VI I DYNAMIC CHARACTERISTICS OF VISUALPROCESSES, A. FIOREN-

.

.

TINI.

.

.

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

VIII. MODERNALIGNMENT DEVICES,A. C. S. VAN HEEL. .

253-288 289-329

C O N T E N T S O F V O L U M E I1 ( 1 9 6 3 ) I.

11. 111. IV.

V. VI.

RULING,TESTING AND USE OF OPTICALGRATINGS FOR HIGHRESOLUTION SPECTROSCOPY, G. W. STROKE.. . . . . . . 1-72 THEMETROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS, J. M. BURCH. . . . . . . . . . . . . . . . . . . . . 73-108 DIFFUSIONTHROUGH NON-UNIFORM MEDIA,R. G. GIOVANELLI 109-129 CORRECTION OF OPTICALIMAGES BY COMPENSATION OF ABERRATIONS AND BY SPATIAL FREQUENCY FILTERING, J. TSUJIUCHI131-180 FLUCTUATIONS OF LIGHT BEAMS,L. MANDEL. . . . . . . 181-248 METHODSFOR DETERMINING OPTICALPARAMETERS OF THIN FILMS, F. ABELBS . . . . . . . . . . . . . . . . . . . 249-288

C O N T E N T S O F V O L U M E I11 ( 1 9 6 4 )

..

I. THEELEMENTS OF RADIATIVE TRANSFER,F. KOTTLER. 11. APODISATION, P. JACQUINOT AND B. ROIZEN-DOSSIER . . . 111. MATRIXTREATMENT OF PARTIAL COHERENCE.H. GAMO. .

. .

1-28 29-186 187-332

CONTENTS O F VOLUME I V (1965) I. 11. 111. IV. V. VI . VII.

HIGHERORDERABERRATION THEORY,J. FOCKE ... .. . 1-36 APPLICATIONS O F SHEARING INTERFEROMETRY, 0. BRYNGDAHL 3 7-83 SURFACE DETERIORATION O F OPTICAL GLASSES,K. KINOSITA 85-143 OPTICAL CONSTANTSOF THIN FILMS, P. ROUARDAND P. BOUSQUET. . . . . . . . . . . . . . . . . . . . . . . 145-197 THE MIYAMOTO-WOLF DIFFRACTION WAVE,A. RUBINOWICZ 199-240 ABERRATION THEORY O F GRATINGS AND GRATING MOUNTINGS, W. T. WELFORD. . . . . . . . . . . . . . . . . . . 241-280 DIFFRACTIONAT A BLACKSCREEN, P A R T I: KIRCHHOFF'S THEORY,F. KOTTLER. . . . . . . . . . . . . . . . . . 281-314

.

C O N T E N T S O F VOLUME V (1966) OPTICALPUMPING, C. COHEN-TANNOUDJI AND A. KASTLER. 1-81 NON-LINEAR OPTICS,P. S. PERSHAN. . . . . . . . . . . 83-144 TWO-BEAhl INTERFEROMETRY, w. H. STEEL. . . . . . . . 145-197 INSTRUMENTS FOR THE MEASURING OF OPTICALTRANSFER FUNCTIONS, K. MURATA.. . . . . . . . . . . . . . . . 199-245 LIGHTREFLECTION FROM FILMS OF CONTINUOUSLY VARYING V. REFRACTIVE INDEX,R. JACOBSSON. . . . . . . . . . . 247-286 AS A BRANCH VI. X-RAY CRYSTAL-STRUCTURE DETERMINATION OF PHYSICAL OPTICS,H. LIPSON AND C. A. TAYLOR .. . 287-350 ELECTRON, J. PICHT.. . 351-370 VII. THEWAVEOF A MOVINGCLASSICAL

I. 11. 111. IV.

.

..

C O N T E N T S O F VOLUME V I (1967) I.

RECENT ADVANCES IN HOLOGRAPHY, E. N. LEITHAND J. UPAT-

.

.

1-52 .. .... . . . ......... ..... OF LIGHTBY ROUGH SURFACES, P. BECKMANN 53-69 SCATTERING OF THE SECOND ORDERDEGREEOF COHERENCE, 111. MEASUREMENT 71-104 AND S. MALLICK. . . . . . . . . . .. M. FRANFON . . . . 105-170 IV. DESIGN OF ZOOMLENSES, K. YAMAJI . . . . . SOMEAPPLICATIONS OF LASERS TO INTERFEROMETRY, D. R. V. HERRIOTT . . . . . . . . . . . . . . . . . . . . . . . 171-209 STUDIES O F INTENSITY FLUCTUATIONS IN VI . EXPERIMENTAL 2 11-257 LASERS,J. A. ARMSTRONG AND A. W. SMITH. . . . . . VII. FOURIER SPECTROSCOPY, G. A. VANASSE,H. SAKAI.. . . 259-330 AT A BLACK SCREEN,PART 11: ELECTROMAGNETIC VIII DIFFRACTION THEORY,F. KOTTLER . . . . . . . . . . . . . . . . 331-377 NIEKS

11.

.

.

.

.

.

.

C O N T E N T S O F VOLUME V I I (1969) MULTIPLE-BEAM INTERFERENCE AND NATURAL MODESIN OPEN RESONATORS, G. KOPPELMAN. . . . . . . . . . . . . . 1-66 FOR DIELECTRIC MULTILAYER FILTERS, 11. METHODSOF SYNTHESIS E. DELANOAND R. J. PEGIS. . . . . . . . . . . . . . 67-137 111. ECHOES AT OPTICALFREQUENCIES, I. D. ABELLA. . . . . 139-168 FORMATION WITH PARTIALLY COHERENTLIGHT,B. J. IV. IMAGE THOMPSON. . . . . . . . . . . . . . . . , . . . . . . 169-230 QUASI-CLASSICAL THEORYOF LASERF~DIATION,A. L. V. MIKAELIANAND M. L. TER-MIKAELIAN . . . . , . . . . . 231-297 IMAGE,S. OOUE. . . . . . . . . . 299-358 VI . THE PHOTOGRAPHIC OF VERYINTENSE LIGHTWITH FREE ELECTRONS, VII. INTERACTION J. H. EBERLY.. . . . . . . . . . . . . . . . . . . . 3 5 9 4 1 5

I.

.

C O N T E N T S O F V O L U M E V I I I (1970) I. 11.

SYNTHETIC-APERTURE OPTICS, J. W. GOODMAN. . . . . . 1-50 THEOPTICALPERFORMANCE OF THE HUMAN EYE, G. A. FRY 51-131

111.

LIGHTBEATING SPECTROSCOPY. H . . CUMMINSAND H . L. SWINNEY. . . . . . . . . . . . . . . . . . . . . . . 133-200 IV . MULTILAYER ANTIREFLECTION COATINGS.A . MUSSETAND A . THELEN ........................ 201-237 STATISTICAL PROPERTIES OF LASER LIGHT.H . RISKEN. . . 239-294 V. VI . COHERENCE THEORY OF SOURCE-SIZE COMPENSATION IN INTERFERENCE MICROSCOPY. T . YAMAMOTO . . . . . . . . . . . 295-341 VII . VISION IN COMMUNICATION. L. LEVI . . . . . . . . . . . 343-372 VIII . THEORY OF PHOTOELECTRON COUNTING.C. L . MEHTA . . . 373-440

This Page Intentionally Left Blank

PREFACE

It is ten years since the first volume of PROGRESS IN OPTICS appeared, and it may be appropriate to reflect on this occasion upon the way in which this series of publications has developed and to ask whethei it has fulfilled the aim that the Editor and the Publishers had in mind at its conception. In the late nineteen fifties, it was our opinion that the rapidly increasing volume of scientific literature was making it impossible for workers in the various fields to keep abreast of the numerous developments. It seemed to us that the time had come for introducing a new kind of publication, devoted entirely to reviews of research in specific branches of the various disciplines. The new series that we envisioned was to be devoted to optics and related fields and its chief aim was to help optical scientists and optical engineers to be well informed about progress in these areas. As it turned out, the timing was fortunate indeed. For the year in which the first volume of PROGRESS IN OPTICS went to press was also the year in which the first lasers were made. The development of the laser triggered numerous activities in many branches of optics which we have attempted to cover in the ensuing volumes of PROGRESS IN OPTICS.We were fortunate in obtaining contributions from leading experts and, judging from the reviews of these volumes, this series is indeed fulfilling the aim originally set for it. We approach our second decade confident that the recent revival of interest in optics will continue and will lead to many more interesting developments. We will endeavour, as we did in the past, to provide useful review articles that reflect the progress in a field that has so greatly enriched modern science and technology.

EMILWOLF Department of Physics and Astronomy, University of Rochester, N.Y., 14627 Jane, 1971

This Page Intentionally Left Blank

CONTENTS I . GAS LASERS AND THEIR APPLICATION TO PRECISE LENGTH MEASUREMENTS by ARNOLDL . BLOOM(Mountain View and Stanford. California)

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

.

1 INTRODUCTION 2 RELATED TOPICS 2.1 Ancillary techniques 2.2 Multi-mode operation .

.

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

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

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

3. POWER OUTPUTVERSUSFREQUENCY 3.1 The gain profile 3.2 Travelling-wave lasers 3.3 Standing-wave lasers . . . . . . . . . . . . . . . . . . . . 3.4 Application of standing-wave techniques t o absorption lines

3 4 4 4 6 6 7 11 15

..... ............... ............. ............ .......................... 5. ACKNOWLEDGEMENTS .........................

29

REFERENCES

29

.

4 PRESENT STATUSOF EXPERIMENTAL WORK 4.1 Absorption cell experiments in the visible 4.2 Absorption cell experiments in the infrared 4.3 Emission lines

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

17

18 22 24

I1. PICOSECOND LASER PULSES by A. J . DEMARIA(East Hartford. Connecticut)

. .

............................ .................... 3. GENERATION OF PICOSECOND LASERPULSES .............. 3.1 Active mode-locking . . . . . . . . . . . . . . . . . . . . . . . 3.2 Passive mode-locking . . . . . . . . . . . . . . . . . . . . . . . 3.3 Chirping and pulse compression . . . . . . . . . . . . . . . . . . 3.4 Optical pumping with mode-locked pulses . . . . . . . . . . . . . 4. MEASUREMENT TECHNIQUES FOR PICOSECOND LASERPULSES . ...... 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Linear optkal pulse-width measurement techniques . . . . . . . . . 4.3 Nonlinear optical pulse-width measurement techniques . . . . . . . . REFERENCES ........................ ...... 1 INTRODUCTION

33

2 BASIC@SWITCHING PRINCIPLES

35 37 40

43 49

55

67 57 58 61 70

XI1

CONTENTS

I11. OPTICAL PROPAGATION THROUGH THE TURBULENT ATMOSPHERE by

JOHN

W . STROHBEHN (Hanover. N.H.)

............................ THE ATMOSPHERE. . . . . . . . . . . . . . . . . . 3. SPECTRA FOR AMPLITUDE AND PHASE FLUCTUATIONS .......... 3.1 Spectral expansions and transform relations . . . . . . . . . . . . 3.2 Rytov's method . . . . . . . . . . . . . . . . . . . . . . . . . 1. INTRODUCTION

75

IN 2. TURBULENCE

77

. .

............ ..................... OF THE PLANE WAVE RESULTS. . . . . . . . . . . . . . 6. EXTENSIONS 6.1 A randommediumwithsmoothlyvaryingcharacteristics . . . . . . . . 6.2 Spherical wave and finite beam sources . . . . . . . . . . . . . .

BETWEEN THEORY AND EXPERIMENT 4 COMPARISON

5 RENORMALIZATION METHODS

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

6.3 Finite aperture a t the receiver 6.4 Phase screen approach 6.5 Coherence functions . . . . .

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

.

7 APPLICATIONS

82 83 86 92 100 104 104 106 111 112 114 116 119

I V. SYNTHESIS O F OPTICAL BIREFRINGENT NETWORKS

. .

by E 0 AMMANN(Mountain View. California)

.

1 INTRODUCTION .................... 1.1 General 1.2 Lyot and Solc filters . . . . . . . . . . . . . . . 1.3 Understanding how birefringent networks function 1.4 Specifying the desired response

........ ............................. ........ ......... .................. 2. SYNTHESISPROCEDURES ....................... 2.1 Procedure 1 (single-pass) . . . . . . . . . . . . . . . . . . . . . 2.2 Procedure 2 (single-pass) . . . . . . . . . . . . . . . . . . . . . 2.3 Procedure 3 (single-pass) . . . . . . . . . . . . . . . . . . . . . 2.4 Procedures 4A and 4B (double-pass) . . . . . . . . . . . . . . .

.

2.6 Procedures 6A and 6B (double.pass) .

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

RESULTS. . . . . . . . . . 3 EXPERIMENTAL 3.1 Lyot and Solc filters . . . . . . . . . . 3.2 A technique for measuring transmittance 3.3 Synthesized birefringent networks

.

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

OF SYNTHESISPROCEDURES TO ELECTRO-OPTIC NETWORKS 4 APPLICATION

.

.......................... APPENDIXA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ACKNOWLEDGMENTS

.............................. APPENDIXC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RJWERENCES .............................. APPENDIXB

125 125 128 129 134 136 137 146 149 167 163 168 168 169 170 170 173 173 176 176 177

CONTENTS

XI11

V . MODE LOCKING I N GAS LASERS

. . . JONES (Brighton. England) 1. INTRODUCTION ............................ 2. FORCED LOCKING. . . . . . . . . . . . . . . . . . . . . . . . . . by L ALLEN AND D. G C

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

2.1 Introduction 2.2 Experimental 2.3 Theory . . .

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

181 183 183 184 196

............................ 212 212 . . . . . . . . . . . . . . . . .218

3. SELF.LOCKING 3.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Maximum emission principle theory 3.3 Self-consistent field theory . . . . . . . . . . . . . . . . . . . . 3.4 The pulse train approach . . . . . . . . . . . . . . . . . . . . .

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

221 229

4. CONCLUSIONS

232

REFERENCES ..

233

VI . CRYSTAL OPTICS W I T H SPATIAL DISPERSION

.

by V M. AGRANOVICH and V. L. GINZBURG (Moscow. USSR)

. ............................ 2. EQUATIONS OF THE ELECTROMAGNETIC FIELD.TENSORSeu(w.k) AND 1 INTRODUCTION

&. k).

E ~ .

DISPERSION LAWFOR NORMAL WAVESIN CRYSTALS ..

3. DEPENDENCE OF TENSOR Ei,(m. k) ON THE WAVE VECTOR k IN BAND . . . . . . . . . . . . . . . . . . . . . . . . .

THE

....

237 241

OPTICAL

......

252

4 SPATIAL DISPERSION EFFECTS IN CRYSTALS ............... 4.1 Introduction. optical anisotropy of cubic crystals . . . . . . . . . . . 4.2 New waves in the neighbourhood of absorption lines . . . . . . . . 4.3 Gyrotropic bi-refringence and new waves in antiferromagnetic non-gyrotropic crystals . . . . . . . . . . . . . . . . . . . . . . . . . .

269 269 260

CONCLUSION ..

278

.

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

271 279

V I I . APPLICATIONS OF OPTICAL METHODS I N THE DIFFRACTION THEORY O F ELASTIC WAVES by K . GNIADEKand J . PETYKIEWICZ (Poland) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 2. KIRCHHOFF'S DIFFRACTION THEORY 284 3. HWGENS' PRINCIPLE FOR ELASTIC MEDIA 286 290 4. PROPERTIES OF THE SCREENAND THE PROBLEM OF EDGE INTEGRALS 5. THE YOUNG-RUBINOWICZ INTERPRETATION IN KIRCHHOFF'S DIFFRACTION 292 THEORYOF ELASTICWAVES . . . . . . . . . . . . . . . . . . . . . 6 TENSORPOTENTIAL OF ELASTIC WAVES 293 1. FRESNEL DIFFRACTION OF ELASTIC WAVES 291 8 FRAUNHOFER DIFFRACTION OF ELASTIC WAVES 304 9. CONCLUSION 309

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

. .

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

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

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

REFERENCES

309

CONTENTS

XIV

.

VIII EVALUATION. DESIGN AND EXTRAPOLATION METHODS FOR OPTICAL SIGNALS. BASED ON USE OF THE PROLATE FUNCTIONS

.

by B ROY FRIEDEN (Tucson. Arizona) 1. INTRODUCTION ............................ 1.1 Linear prolate functions . . . . . . . . . . . . . . . . . . . . . 1.2 Circular prolate functions . . . . . . . . . . . . . . . . . . . . . 1.3 Normalization and notation . . . . . . . . . . . . . . . . . . . . 1.4 The space-bandwidth product and prolate functions . . . . . . . . 1.5 Past use in boundary-value problems . . . . . . . . . . . . . . 1.6 Optical applications. a synopsis . . . . . . . . . . . . . . . . . 1.6.1 Laser mode patterns . . . . . . . . . . . . . . . . . . . . 1.6.2 An apodization problem. maximizing the encircled energy . . . 1.6.3 Post extrapolation beyond the optical bandwidth . . . . . . 1.6.4 Post extrapolation of an image piece . . . . . . . . . . . . 1.6.5 Real-time extrapolation beyond the optical bandwidth . . . . 1.6.6 Degrees of freedom in an image . . . . . . . . . . . . . . 1.6.7 Analysis of image quality for generally aberrated optics . . . . 1.7 Discussion, and some questions of a fundamental nature . . . . .

2. MATHEMATICAL PROPERTIES OF THE

. .

. .

. . . . . . LINEARPROLATE FUNCTIONS ....

Relation to classical functions . . . . . . . . . . . . . . . . . . . Defining relation . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonality and completeness on the finite interval . . . . . . . . Invariance to the infinite Fourier transform operation . . . . . . . . The A, as eigenvalues of the sin ( x ) / x kernel . . . . . . . . . . . . Duality of orthogonality . . . . . . . . . . . . . . . . . . . . . Orthogonality and completeness on the infinite interval . . . . . . . An extrapolation formula for bandlimited functions . . . . . . . . . 2.8.1 Comparison with a familiar interpolation formula . . . . . . . 2.9 Behavior of the eigenvalues A,, (c) . . . . . . . . . . . . . . . . . 2.10 The maximum fractional “energy” within 1x1 5 x,, over the class of bandlimited functions . . . . . . . . . . . . . . . . . . . . . . 2.11 Expansion of some familiar functions in terms of the y,, ( x ) . . . . 2.12 A class of functions whose finite Fourier transform is asymptotic t o a Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

.

. . . . . . . . . . ......... ......... . . . . . . . . . . CIRCULAR PROLATE FUNCTIONS ...

3 NUMERICAL CALCULATION OF THE A,, AND y,, ( x ) 3.1 Values of c 5 10 . . . . . . . . . . . . . . 3.2 Values of c 2 10 . . . . . . . . . . . . . . 3.3 An alternative to closed formulae? . . . .

. . . .

. . . .

4 . MATHEMATICAL PROPERTIES OF THE 4.1 Relation to functions of Slepian and Heurtley . . . . . . . . . . . 4.2 Defining relation: invariance to the finite Hankel transform . . . . . 4.3 Orthogonality and completeness over a finite radial interval . . . . . 4.4 Invariance t o the infinite Hankel transform . . . . . . . . . . . . 4.5 The AN. II as eigenvalues of a “reflection kernel . . . . . . . . . . 4.6 Duality of orthogonality . . . . . . . . . . . . . . . . . . . . . 4.7 Orthogonality and completeness over the infinite radial interval . . . 4.8 An extrapolating formula for bandlimited functions of radius . . . . 4.9 Behavior of the eigenvalues . . . . . . . . . . . . . . . . . . . 4.10 The maximum fractional “energy” within 0 5 r 5 r0 over the class of bandlimited. radial functions . . . . . . . . . . . . . . . . . . . 4.11 Expansion of some familiar functions in terms of the @ N . “ ( Y ) . . . 4.12 A class of functions whose finite JN-transform is asymptotic t o a Dirac delta function of radius . . . . . . . . . . . . . . . . . . . . .

313 313 314 314 315 315 316 316 317 317 318 318 319 319 320 320 321 321 321 322 324 324 324 325 326 326 327 332 333 334 335 336 337 337 337 339 340 340 341 341 342 342 342 345 345 346

CONTENTS

.

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

AN. AND @N. (r) 5.1 Small values of c 5.2 Large values of c 6 APPLICATIONS 6.1 Fraunhofer-Fresnel image formation 6.1.1 Focussing device in the aperture . . . . . . . . . . . . . . . 6.1.2 No focussing device in the aperture 6.2 Analysis of laser modes 6.2.1 Confocal laser with identical endplates . . . . . . . . . . . . 6.2.2 Confocal laser with non-identical endplates . . . . . . . . . . 6.2.3 General laser geometry . . . . . . . . . . . . . . . . . . . 6.2.4 Discussion 6.3 Problems of optimal “concentration”: of encircled energy. point amplitude variability. and error of restoration or smoothing . . . . . . . . . . 6.3.1 Applications t o apodization theory: Maximizing the enclosed energy 6.3.2 Variability of the point amplitude response and other bandlimited functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Application t o filtering theory: maximum possible error of restoration or smoothing over a finite interval . . . . . . . . . . . 6.4 The extrapolation of image data . . . . . . . . . . . . . . . . . 6.4.1 Application to the incoherent image . . . . . . . . . . . . . 6.4.2 Experimental determination of the optical transfer function 6.4.3 Bandlimited restoration from extrapolated image data . . . . . 6.6 Extrapolation beyond the bandwidth 6.5.1 Effect of noise and truncation 6.5.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . 6.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Real-time extrapolation beyond the bandwidth . . . . . . . . . . . 6.6.1 Derivation (one-dimensional case) . . . . . . . . . . . . . . 6.6.2 Generally rectangular field . . . . . . . . . . . . . . . . . . 6.6.3 Generally circular field . . . . . . . . . . . . . . . . . . . 6.6.4 Ultimate response from a lensless aperture . . . . . . . . . . 6.6.5 Ultimate depth of focus . . . . . . . . . . . . . . . . . . . 6.6.6 Behavior of the solutions . . . . . . . . . . . . . . . . . . 6.6.7 Strehl flux ratio and sensitivity to pupil wave error . . . . . . 6.6.8 Real-time extrapolation of functions using pupil Pol(@) 6.6.9 A thought-experiment: picture extrapolation by use of a coherent processor . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.10 A laser superposition mode with an arbitrarily small spot size 6.7 Degrees of freedom in the image . . . . . . . . . . . . . . . . . 6.7.1 By counting sampling points within the interval . . . . . . . 6.7.2 By including contributions from sampling points outside the interval 6.7.3 An “effective number” . . . . . . . . . . . . . . . . . . . 6.7.4 Prolate degrees of freedom . . . . . . . . . . . . . . . . . 6.7.5 Effective degrees of freedom in the presence of noise . . . . . 6.7.6 Degrees of freedom intrinsic t o the “extrapolating pupil” . . . 6.8 Evaluation of wave-aberrations . . . . . . . . . . . . . . . . . . 6.8.1 Formal solution . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Past work by Zernike and Nijboer . . . . . . . . . . . . . . 6.8.3 A new approach to the problem . . . . . . . . . . . . . . . 6.8.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . 6.8.5 Rectangular pupil case . . . . . . . . . . . . . . . . . . . 7. ACKNOWLEDGEMENTS REFERENCES 5 NUMERICAL CALCULATION OF THE

.

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

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

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

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

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

....

......................... .............................. AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv 347 348 349 349 349 349 350 351 351 356 360 362 363 363 366 367 368 368 370 371 375 376 378 379 380 381 381 382 383 383 385 388 389 391 393 393 393 394 394 394 395 398 399 399 400 400 404 405 405 406 408 411

This Page Intentionally Left Blank

I GAS LASERS AND THEIR APPLICATION T O PRECISE LENGTH MEASUREMENTS BY

ARNOLD L. BLOOM Spectra-Physics, Inc., Mountain View, California. USA and Department of Electrical Engineering, Stanford University, Stanford, California, USA

CONTENTS

9 9 9 9 3

.

PAGE

2

.

. . . . . . . . . . . . . . . . . RELATEDTOPICS . . . . . . . . . . . . . . . .

3

.

POWER OUTPUT I’ERSUS FREQUENCY . . . . . .

6

.

PRESENT STATUS OF EXPERIMENTAL M-ORK . .

17

ACKNOWLEDGEMENTS . . . . . . . . . . . . . .

29

1

4 5

.

INTRODUCTION

REFERENCES

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

3

4

29

Q 1. Introduction Ever since the discovery of the laser, it has been realized that the coherence properties of laser radiation give the laser special significance in the field of precision length measurements. Lasers have found many other uses in which the precise wavelength of their output makes little difference, but application to length measurement, specifically by interferometric means, remains one of the most important applications in a fundamental sense. On the one hand, the long coherence lengths available from laser outputs has stimulated new developments in long-path interferometry; and on the other hand, the integrity of the output wavelength is intimately tied to the physics of laser operation. However, the ultimate benefit in the field of precision length measurements that can be provided by the laser is the fact that it may make possible a primary standard of length, or a set of primary standards, whose definition is several orders of magnitude more precise than has been heretofore available. In fact, it may be that the entire concept of the definition of standard of length may have to be ultimately revised; for example, by giving the speed of light an arbitrary definition and then defining wavelengths instead as frequencies to be compared with a primary frequency standard. This report considers the gas laser from two aspects. One of them is as a generator of a precisely known wavelength, and the other is as a “driver” for an absorbing medium whose resonance characteristics then determine the wavelength. It appears that the latter will provide the ultimate primary standard of length, although the former may be more suitable for secondary standards under conditions where a lesser degree of precision is needed. From either point of view, this paper considers only the gas laser, as opposed to solid, liquid or semiconductor lasers, because up to now only the gas laser has shown the necessary reproducibility. When the gas laser is used as a wavelength source in its own right, even without mode control or stabiliY

4

GAS LASERS A N D LENGTH MEASUREMENTS

[I, §

2

zation, it is at the very worst able to duplicate the best that has been done with incoherent spectroscopic sources that have been used up to the present time as primary or good secondary standards of length. When the absorption technique is used to generate the wavelength standard, then it is not clear that the gas laser is all that necessary. For example, it is conceivable that a solid-state laser of some sort could be tuned to approximately the same wavelength as the absorption resonance in order to drive it, with a stabilization technique used to tune the output of solid material. This sort of technique has not been used up to the present time, to the author’s knowledge, but has some attractive possibilities if the right laser materials and tuning means can be found. For purposes of this report, the characteristics that we are concerned about in the use of narrow absorption resonances are the same whether one uses a gas laser or some other type of laser. For that reason, and because the gas laser can also be used independently as a wavelength reference, we restrict the discussion to gas lasers.

Q 2. Related Topics 2.1. ANCILLARY TECHNIQUES

Several topics not discussed here are nevertheless closely related to that of precise interferometric measurements and we list appropriate review references for them. These topics are: refractive index variations in air (OWENS [1967]), and the use of lasers for long distance measurements (BENDER[1967]). By “long distance” we mean measurement techniques that do not depend upon interferometry of single wavelengths but rather of radar-like techniques employing coded or modulated light. In the latter application, the attributes of laser radiation that are useful are its spatial coherence and compatibility with narrow bandwidth filters, rather than its temporal coherence as such. The gas laser has proved useful in this application although other types of lasers can be used also (as well as some incoherent sources). 2.2. MULTI-MODE OPERATION

The usefullness of any line spectrum source lies in its ability to produce, in an interferometer, fringes that are countable for long interferometer path differences. Since this distance of fringe visibility (which we may refer to very loosely as “coherence length”) is inversely proportional to the wavelength spread in the source, the object is to

I. §

21

RELATED TOPICS

5

obtain a source that is as narrow as possible. Incoherent line spectrum sources are limited in width to the Doppler width, which is a function, and not a very sensitive one, of temperature and molecular mass only. In a laser, however, the corresponding width that one has to be concerned about is the width for which gain is greater than loss, which is always less than the Doppler width, sometimes very less so [see 3, eq. (3.2)]. Thus, for a given laser, one should be able to obtain greater coherence lengths for fringe visibility under the worst possible coherence conditions; for example, those of highly multi-mode operation (in both axial and transverse modes) than are obtainable from incoherent sources. Under better circumstances, for example single-mode operation, the uncertainty in wavelength may be limited by the bandwidth for possible laser operation, but the actual fringe visibility distance will be much greater. Finally, of course, there is the possibility of obtaining single-mode operation form a laser where the mode frequency is stabilized to the center of the Doppler curve or to some other marker, in which case much greater coherence lengths than are obtainable from an incoherent source are realized and the uncertainty in wavelength is drastically reduced. Most of this report deals with the stabilized single-mode types of operation; however, before going into the details, it may be well to consider the alternative of highly multimode operation under a truncated Doppler profile. If there are very many modes operating simultaneously in the laser and if the truncated gain profile is symmetrical, unperturbed by such factors as multiple isotopes in the constitution of the gain medium, then the source should indeed represent a good source for interferometric work. Experiments making use of this type of laser operation have been reported by, for example, BYERet al. [1965]. This particular laser was operable pulsed only, so that experiments of the type reported would be useful only for photographic recording of fringes. However, there are other lasers, the heliumneon visible laser, for example, that can be designed so as to operate simultaneously in a large number of axial and transverse modes. It is important to note that very highly multi-mode operation, not merely operation in a plurality of three or four modes, is necessary to avoid dispersion-pulling effects that shift the apparent center of the gain profile. The results of such pulling effects upon the measurements of isotope shifts in cadmium have been described in detail by HOPKINS and FOWLES [1968] and could be disastrous in metrological work if not thoroughly understood.

6

GAS LASERS A N D LENGTH MEASUREMENTS

[I,

§ 3

The special advantage of a multi-mode laser source for interferometric work, if it can be used, is that it requires no resonator length stabilization and is therefore a much simpler device to build and to operate. Of course, the center of the gain profile is subject to pressure shifts and related perturbations in exactly the same manner as the single-mode lasers to be discussed in the remainder of this report.

Q 3. Power Output Versus Frequency 3.1. THE GAIN PROFlLE

The emission profile of an incoherent source is defined to be the intensity function I (w) describing the intensity of the measurement made by a narrow band photodetector in the region of frequency w. More specifically, we shall limit the definition to refer to the intensity function around a single spectrum line, and it is assumed that I ( w ) = 0 except in the vicinity of each such line. Experience with many incoherent line emitters shows that the emission profile can be expressed in many cases as given by simple Doppler broadening, so that the emission function is of the form

where k.u ln2 is the “Doppler width” whose value can be obtained from standard formulas found in many textbooks on spectroscopy. When the emitting source is a laser, there is a function that corresponds to (3.1). More specifically, the function involved is given by

where NT will be referred to as the relative loss. The quantity g, given in (3.2) is given the name of “net unsaturated gain”, and has the following significance. It represents the net amplification of a small signal passed through the laser assuming that the losses represented by the laser structure are present but that otherwise it is non-resonant, i.e., we have not included in the equation the passband of the resonator that commonly surrounds the laser structure. The actual emission profile, or frequency-dependent profile, which is of interest to us in regard to length measurement by a laser, is not at all identical to the function given by ( 3 . 2 ) . The following factors, a t the least, contribute to and complicate the problem of determining the true output power profile of the laser:

1,

I

31

POWER OUTPUT VERSUS FREQUENCY

7

1. The fields inside the laser are strong enough to give rise to satu-

ration effects which modify the population distribution under the saturated gain profile. 2. The fields, and therefore the saturation effects, generally have a spatial variation within the laser structure. 3. The laser output frequency, being dependent upon optical path lengths, may be affected by dispersive effects caused by the laser resonance. Most lasers used for precise length measurement employ standing waves within the active medium. This is the configuration found in the typical small laser consisting of two highly reflective mirrors making up the resonator which contains the active medium. Travelingwave resonators, consisting of an active medium within a ring resonator, are less often used. However, in attempting to analyze the power output function of a laser, parts of the mathematical analysis are greatly simplified if they are taken initially in the traveling-wave approximation. For this reason, we shall analyze the traveling-wave case first, going on then to the case of standing waves. 3.2. TRAVELING-WAVE LASERS

LAMB[1964] considers a two-level quantum-mechanical system in which the off-diagonal matrix elements of the Hamiltonian are given by terms of the form

?iV(t)= -@E(Y,V , t ) ,

(3.3)

where p is the electric dipole matrix element for the transition, E is an electric field with frequency components near the optical resonance frequency, and Y, v , t are position, velocity and time coordinates of a radiating atom. E(r, V , t ) is a complicated and, in general, random function of these coordinates. In order to work with analytic expressions it is desirable to express the problem in terms of the actual optical frequency components of the field, E,(t), where each En is slowly varying compared to the optical frequency, rather than with the random local field E ( r , v , t). For the present discussion we will be concerned with one, or at most two, components En so that the field can be described by El cos &+E, cos o,t, if more than one E , is present. Lamb’s approach is to solve for the elements of the density matrix, p, in terms of a power expansion in En.For the off-diagonal elements,

8

G A S L A S E R S A N D LENGTH MEASUREMENTS

[L § 3

such as expansion consists of odd-power terms only, and Lamb restricts his analysis to the first- and third-order terms. The firstorder solution is a fairly trivial one, resulting in a linear superposition of stimulated transitions over and above the spontaneous ones. The interesting results begin to show up in the third-order terms, particularly when more than one component E , is present as is the case in standing-wave lasers. For a single-mode traveling-wave laser ( E = En) the third-order solution is

Here, yB and yb are the inverse lifetimes of the upper and lower states, respectively, and B is the polarization (proportional to the square of the transition matrix element). This equation predicts a power output that merely follows the Doppler curve when it is above the threshold level and is zero otherwise. It does not allow for saturation broadening effects and is certainly not valid when the Doppler width is as narrow as the natural line width. Attempts to carry the theory to higher orders than third-order have not been very fruitful. UEHARA and SHIMODA [1965] have done fifthorder calculations for the standing-wave laser and find only minor modifications from the results of third-order theory. On the other hand, in many lasers the expected value of E is large enough that one would have to go to many orders, perhaps as many as 30, before the expansion would show distinct signs of converging. We shall therefore take another approach which bypasses the convergence problem for high fields and which, for the traveling-wave laser at least, allows one to obtain solutions by numerical methods. It is based on basic equations that have been applied successfully in magnetic resonance applications where saturation phenomena are important. In this approximation, the response of a homogeneously broadened inverted population of M atoms having the identical resonance frequency wo is given by a dispersive component, u, and an emissive component, v, given by: -w1 (w-wo)ML (w -wo) H(0,

E) =

v(0,

E) =

‘2 Yab

O1ML(w -wo) I

Yab

1, §

31

P O W E R OUTPUT VERSUS FREQUENCY

9

where

and

PE 6

w1=-.

Here, yab and y:b are relaxation rates for effects, respectively, of spontaneous emission and hard collisions, and of the aforementioned effects plus soft collisions (SZOKEand JAVAN [1963]). More detailed formulations of the relaxation phenomena are possible and have been used by various authors, but the above will suffice for this discussion. These equations represent exact solutions for a quantum-mechanical system consisting of an isolated pair of energy levels, and are applicable to the present situation to the extent that the quantities involved, particularly the relaxation rates yaband ylb, are definable. The “approximation” that we make is that we ignore effects of the dispersive component, m, and base all our results on the v component. The effect of the m component is to make the refractive index of the laser medium a function of the excitation conditions as well as of frequency. This is important when one has to consider the exact operating frequency for a given laser structure or if one has to consider interactions between modes, as Lamb does. However, in the present case, we assume that the frequency is to be swept continuously through the gain profile and we only want to know the power output at any given frequency. Thus, all we are concerned with is the fact that the output power P is a single-valued function of the frequency o,a fact which we accept intuitively, and we ignore the concomitant variations in path length. The power output at a given frequency w is obtained by solving for the condition that, at steady state, the power input from the active medium equals the power lost to the resonator.* Thus, we have to solve an equation like the following:

* The quantities M , u and v are relative populations and the power input is thus proportional to v. Without loss of generality, we assume the proportionality constant to be unity.

10

GAS LASERS A N D LENGTH MEASUREMENTS

[I, §

3

Here, T is the transmissivity of the output mirror, A stands for the other resonator losses, and go (7 = 0) is the peak unsaturated gain. This is the most general representation of the problem, where we assume that the variation of the field intensity in different parts of the mode has an effect on the final solution. In practice, to simplify the process of obtaining a solution, one can assume an average value of field, EAV, to be the same throughout the active medium, and eliminate the integration over the radial coordinate 7. Equation (3.9) then takes the form

Equation (3.10) can be solved by numerical (i.e. computer) methods. When the ratio g,/(A+T) is just slightly greater than unity, the steady-state value of E is small at any frequency. The saturation terms in ZJ drop out and the solution reduces, with suitable choice of constants, to eq. (3.4).At the other extreme, g o / ( A + T )is very large, the field E is so large that the term ~ i / ~ dominates ~ ~ y i the ~ other terms in eq. (3.7). The “saturated line width” is then greater than the Doppler width and the power output, as a function of frequency, follows the saturated Lorentzian rather than the Doppler profile. However, when one is far from Doppler line center, the field E must be weak and the Lorentzian shape cannot be followed. Although, as has been pointed out, the weak-field case is the one that has been analyzed extensively, both weak and strong-field cases are known in practice. Many helium-neon lasers used in singlefrequency operation, particularly the short ones designed for this purpose whose output is of the order of one mW, are quite accurately described by weak-field approximations. On the other hand, the CO, laser at 10.6 p and other lasers farther in the infrared have Doppler widths comparable to the natural linewidth yhb even without saturation, so that saturation will definitely add to the observed width Another example of strong-field operation has been observed in the Cdf laser line at 4416 A. In this line, although ribis probably about 9 ku, the matrix element is quite large and saturation effects show up in lasers whose output power is of the order of 10 mW. A Cdf laser using Cd in the natural isotopic constitution has an unsaturated gain profile consisting of several peaks, corresponding to the various isotopes. When such a laser is operated single-mode, however, the power output function does not show the isotope structure, but only

1, §

31

POWER OUTPUT VERSUS FREQUENCY

11

a broad maximum. Therefore, in such a laser, if the laser is constructed with only one isotope, the power output function will be broader than the Doppler curve and must be analysed by solving the general eq. (3.10).Figure 3.1 shows such a solution, with parameters chosen to be typical for those of many lasers.

a:

5

0

a

w

2 I-

4

w

a

Fig. 3.1. Comparison of output power given by eq. (3.10) (solid line), with truncated Gaussian (dashed line) for the case NT = 0.1. The parameters used in calculating the solidline are: yib = yab = 0.1 ku. These parameters are typical ones in theoperation of many gas lasers.

3.3. STANDING-WAVE LASERS

The solution for the standing-wave problem is a much more difficult one to obtain, albeit more interesting than that for traveling waves, because ,standing-wave lasers are the more common type. The difficulty arises because the standing wave consists of two traveling waves, thus possibly of two different frequencies even in a singlemode laser. Atoms whose relative velocity is slow in the direction of the resonator axis see a field with essentially one frequency but large spatial variations. Fast-moving atoms, on the other hand, see the two traveling waves as having two distinct frequencies, one or both of which may be outside the resonance bandpass of response of the atoms. The problem, then, is to obtain a solution that satisfactorily covers the response of both the fast and the slow atoms.

12

GAS L A S E R S A N D L E N G T H M E A S U R E M E N T S

[I,

§ 3

The first useful solution to the problem is the third-order expansion given by LAMB[1964]. The result is well known but we repeat it here for completeness:

This equation predicts an output power, as a function of frequency, that is a truncated Gaussian except near line center where there may be a decrease in power (the so-called “Lamb dip”). The extent of this dip depends strongly on the degree of excitation, being small or non-existent for a laser near threshold but prominent in a high-power laser, particularly if ylb is small compared to kzt. If, for comparison purposes, the maximum output at the center of a Gaussian is taken to be P,, then in the limit of very high power the output at the center of the Lamb dip is (3.12)

Thus, for some lasers under some operating conditions, the output at the center of the dip may be as little as 4 the power to be expected if the Lamb dip were not present. Typically, however, the dip is much less prominent. For small helium-neon lasers operating at 6238 A, for example, the power at the center of the dip is 90 % or more of the power to be expected without the dip. The width of the dip, measured as a Lorentzian width from the center to the half-power point, is always determined by ribalthough the apparent width may be wider if the dip is masked by the curvature of the Gaussian itself. The fact that the width of the dip is determined by ylb and not by kzt is, of course, what makes the dip useful for laser frequency stabilization and wavelength determination. Equation (3.11) has been astonishingly successful in predicting the observed experimental results, considering the fact that the equation was derived from a third-order theory. Some insight into why this is so has been given by the results of attempts to extend the theory to strong-field conditions. The aforementioned fifth-order treatment of UEHARAand SHIMODA[1965] gives only minor corrections to the shape of the Lamb dip as given by third-order theory. Recently, there has been a flurry of activity in methods of extending the theory of the Lamb dip to strong fields, among these are the work

POWER O U T P U T V E R S U S F R E Q U E N C Y

13

of STENHOLM and LAMB[1969], RAUTIAN and SHALAGIN [1969] and FELDMAN and FELD[1970]. All of these treatments calculate the rate of energy input from the active medium using the field E and the frequency as independent variables; they do not consider the energy balance with the resonator as is done in eq. (3.10). The method of approach used in most of these papers, since a power expansion in E is not practical, is to use a Fourier expansion of the spatial variation of the field in the standing wave; then, using only the first few terms of the expansion, calculate the interaction of moving atoms with the field. The most important result of these strong-field theories is that the Lamb dip retains the Lorentz width yibr regardless of the strength of the field. This explains why third-order theory has been so successful: The deviations in high-field theory show up mainly in small changes in the depth of the Lamb hole; however, these are difficult to determine experimentally because the experimenter rarely knows

Fig. 3.2. (Top): Illustration, perhaps exaggerated, of the Lamb dip in the weak-field approximation. (Bottom): The same, in the high-field approximation showing the expected “bump” in the middle.

14

G A S LASERS A N D LENGTH MEASUREMENTS

[I. §

3

the saturation characteristics of his laser to the necessary accuracy. Exact values of the depth are not important in applications, as long as the center of the dip is well defined. There is, however, one new effect that has come out of these strongfield theories and that may be of more than theoretical interest. It concerns a small “bump” or “antidip” that should be observable at the very center of the Lamb dip under conditions of very strong fields (Fig. 3.2). This effect may be understood from the following rather elementary explanation. Consider in detail the interaction, with the standing-wave field, of the “slow” atoms that contribute the output at the bottom of the Lamb dip. As a rough approximation, an atom is “slow” if it crosses a distance $1 in a time longer than (yib)-I, so that it sees the standing wave as one frequency, rather than two frequencies. However, on further investigation, we can further distinguish between “slow” atoms and “stationary” atoms, i.e. those whose velocity along the mode axis is exactly zero. The “slow” atoms, although responding to the standing wave as a single frequency, nevertheless average the value of the standing-wave field. Let this field be described as E = E , cos cut cos kz (3.13) ( k = 2n/A). Then, the mean value of this field, averaged in space and time, is +Eo and the contribution of the “slow” atoms in a half-wavelength interval is proportional to (3.14)

The “stationary” atoms, on the other hand, can be thought of as being arrayed along the mode axis, each atom interacting with the local field at its particular value of z. The contribution of the “stationary” atoms in a half-wavelength interval is thus 3’

+

E, cos kz dz (E~ ‘OS kz 2/YabvLb

(3.15)

where E, = E,/z/i. Clearly, v, and vb are not identical. The slight difference between them is the origin of the “bump”; how useful this effect may be for defining line center has yet to be determined. A “complete” solution for the standing-wave laser in strong fields must take into account not only the velocity effects mentioned above, but also the detailed interaction with the resonator as is done for

POWER OUTPUT V E R S U S F R E Q U E N C Y

15

traveling waves in eq. (3.10). The mathematical difficulties of implementing this solution are formidable and, while it will modify the overall Doppler profile as in the case of the traveling-wave solution, it will probably not modify the Lamb dip. The reason for believing this is that, in third-order theory at least, the Lamb dip appears in energy balance equations as a change in saturation constant rather than as a change in gain. And since the saturation term only serves as a constant of proportionality connecting gain and power output, it does not, in itself, introduce terms in the power output equation that depend on the resonator parameters. 3.4. APPLICATION O F STANDING-WAVE TECHNIQUES TO ABSORP-

TION L I N E S

The use of absorption lines as wavelength-determining markers has been known for a long time - it is a standard practice in infrared spectroscopy - but the full potential of the technique has only recently been recognized when the absorption line is used as part of, or in conjunction with, a stabilized laser. In particular, if a standing wave can be set up in the absorption medium, then an analogue of the Lamb dip, actually a decrease in absorption or an increase in transmitted light, may be observed. The width of this marker is determined by natural line widths, rather than by a Doppler width, just as is true for the Lamb dip in an active medium. There are a number of advantages to the use of an absorption line, if one can be found to match a laser source in wavelength, but the chief one is the fact that absorption lines can be made to have tremendously narrow natural linewidths. This fact was recognized by LETOKHOV[1967], who has suggested using a transverse molecular beam of CH, in conjunction with a 3.39AI , helium-neon laser. However, the technique presently being developed in a number of laboratories is that of detecting the central tuning marker (Lamb dip analogue) in an absorbing gas at very low pressure. By operating at low pressures the two chief advantages as standards are realized a t once: a ) the observed linewidth is essentially the linewidth of the free atom or molecule, unaffected by collisions, and b) pressure shifts are reduced to a negligible value. The active medium of the laser itself, on the other hand, is subject to a primary constraint in that it must operate within a certain pressure range, and with a certain mixture of gases in order that it have optical gain, thus it is rarely free of pressure broadening and shift.

16

G A S LASERS A N D LENGTH MEASUREMENTS

[I,

§ 3

In order to observe the central tuning feature in an absorption line, one must set up standing waves in the absorbing medium. The first experiments to demonstrate the utility of the method placed the absorption cell within the laser resonator, in tandem with the active medium (Fig. 3.3a). This results in a resonator containing high losses from the accumulated effects of many Brewster angle windows and from the absorbing gas itself. Also, the resonator must be made longer in order to accommodate the absorption cell, thus making it more difficult for the laser to operate in a single mode. A better method is to set up the absorption cell and standing-wave field outside of the laser resonator. Fig. 3.3b shows a configuration which has been used in practice and which, though not providing a perfect standing wave, does provide complete isolation of the absorption cell from the laser resonator. The standing wave field produced in this way consists of a single pass of the output forward and backward, and thus is considerably weaker than the multi-pass standing wave field in the laser resonator itself. This is not a disadvantage; the fact that the natural linewidth of the absorber is much narrower than that of the laser active medium makes it saturable by a correspondingly weaker field. On the other hand, establishment of a standing wave in a long-lived absorber brings in a new problem if the full potentiality of the method is to be realized. The absorbing particle must experience a field of the form E cos ot,where w is the optical frequency and E is constant L A S E R

ABSORPTION

L A S E R

CELL

ABSORPTION

CELL

-

OUTPUT

Fig. 3.3. Showing two possible methods of obtaining saturated absorption in an absorption cell.

1,

5

41

P R E S E N T S T A T U S O F E X P E R I M E N T A L WORK

17

(time-independent) over the life-time of the particle. Any time- or frequency-dependence in the envelope E may be interpreted by the resonant absorber as modulation sidebands and cause spurious responses. Consider now that the particles that are “slow” or “stationary” along the optical axis actually have a mean speed (2kTlm)’ transverse to the axis. If the radiative lifetime of the particle is z, then the field must be spatially uniform, in all planes perpendicular to the propagation axis, over distances of approximately (2kTz2/m)* in order to avoid spurious responses. Some useable absorption lines, such as the CH, line at 3.39 ,u are thought to have natural lifetimes as long as 0.01 sec. In order to realize the correspondingly narrow linewidth, one would need a cell filled with uniform field to a diameter of 6 meters! This is obviously not practical, but it does suggest that the maximum practical diameter for a laboratory cell, perhaps 10 cm, should be used whenever possible. The corresponding pressure for the absorbing gas is several microns of Hg, and the cell length must be of the order of a meter. The optical problems involved in maintaining a uniform beam over a 10 cm diameter are severe, The beam must be blown up from the usual 1- or 2-mm diameter by beam-expanding optics. Such a beam is not really “uniform” in cross-section in the sense of having a constant value; instead, it should have a Gaussian profile if the laser is operating properly, in the lowest order transverse mode. The Gaussian profile broadens the observed linewidth in the absorbing medium, but only slightly. A more serious problem is the optical noise introduced into the beam by scattering and unwanted reflections within the beamexpanding optics. Reflections from air-to-glass interfaces, in particular, cause observable interference fringes even with the best available antireflection coatings on the surfaces, and these cause a modulation of the beam at high spatial frequencies. These effects seem to be more difficult to eradicate the larger the expanded beam. They may set an engineering limit to the definition obtainable for the resonant wavelength of the absorbing particle.

8 4.

Present Status of Experimental Work

In what follows, we present the results, as presently known to the author, of experiments in which a high degree of definition has been given to a measured wavelength in a laser-excited experiment, and in which the experiment appears to make possible the definition of a

18

GAS LASERS A N D LENGTH MEASUREMENTS

[I, § 4

highly reproducible wavelength standard. Results are given both for saturable absorption lines accessable to laser-excited wavelengths, and for sharply defined laser emission lines. The results are listed approximately in order of decreasing priority of experiments which, the author believes, should be candidates for further research. 4.1. ABSORPTION CELL E X P E R I M E N T S I N THE VISIBLE

In the past there has been a strong preference among metrologists toward the use of visible light sources, as opposed to UV or IR sources, whenever possible. This is understandable in the case of incoherent sources because of the difficulty of aligning interferometers with such sources, but it cannot be justified with laser sources. An interferometer system for the 3.39 p helium-neon laser, for example, can be aligned with the aid of a visible laser beam passed through the bore of the 3.39 p laser, even though the interferometer has no finesse in the visible. In some cases, the same laser which provides the 3.39 p radiation for measurements can provide 6328 A light for alignment purposes. Nevertheless, it must be admitted that, if all other factors were equal, it would be generally preferred if the primary standard of length were defined in the visible region of the spectrum. We therefore list first the known absorption cell experiments in the visible region, despite the fact that, up to the present time, a greater degree of definition has been obtained in certain experiments on infrared lines. For the same reason, we list other laser transitions in the visible which can be obtained fairly easily with commercially obtainable equipment, and which may be usable for the purpose if suitable absorption lines can be found to match them. The physical criteria for a good absorption line, usable as a candidate for a primary frequency standard, have been discussed by BARGER and HALL[1969]. We summarize their conclusions and ours together as follows: 1. The transition should have a good absorption coefficient in a gas

existing under reasonable and reproducible laboratory conditions. 2. The natural lifetime for the resonant absorption should be very long. 3. The transition should be free of fine structure or overlapping lines, and it should have negligible (or at least predictable) Zeeman and Stark shifts.

1,

I

41

P R E S E N T STATUS O F E X P E R I M E N T A L WORK

19

4. I t must, of course, lie within the gain profile of an existing gas

laser operating under some reasonable conditions. 5. Preferably, the laser used as an emission source should be one that is easily available or has been constructed in many different laboratories, and about which much is known regarding pressure and current shifts, as well as of gain and power output. 6. Finally, the transition should be in the visible, if possible.

A consideration of these factors leads immediately to the conclusion that the best candidate for a primary standard would be an absorbing gas, similar to methane as used by Barger and Hall at 3.39 y, that has a saturable absorption at the center of the helium-neon 6328 A line. It would be even better if this gas provided no other absorption lines within the helium-neon gain profile. This ideal absorber has not yet been found. We list, then, what is presently known in the entire visible region. 4.1.1. Neon absorption at 6328 A

The first demonstration of the central tuning marker in saturable [1967], who used a lowabsorption was that of LEE and SKOLNICK pressure neon discharge as an absorption cell within the laser resonator. Their results showed a width for the central tuning marker of approximately 60 MHz, which is probably the limit determined by the natural linewidth of the isolated excited neon atom. The pressure that they used in the absorption cell was 0.1 torr, which, while providing negligible pressure shifts compared to the helium-neon mixture in the laser active medium, does not guarantee as small a shift as might be obtained with absorption cells having much lower pressures. The neon tuning marker, as demonstrated in the photographs of Lee and Skolnick, occupies a relatively large proportion of the total gain curve of the helium-neon exciting laser, and therefore the peak of the center tuning marker cannot be used as a highly defined wavelength standard unless corrected for the tilted base line bias of the exciting laser output profile. For this reason - basically the relatively broad linewidth compared to other types of absorption lines - the neon absorption cell probably does not make a very good wavelength standard, except when one attempts to relate to a neon atomic beam or to other strictly atomic standards. It has the further disadvantage of requiring a discharge in the absorption cell and its attendent com-

20

GAS LASERS A N D L E S G T H MEASUREMENTS

[I,

P

4

plications. On the other hand, it may have some advantage in that it relates directly to the helium-neon laser itself and thereby it can be used to provide a direct calibration of pressure and current shifts in helium-neon lasers used as emission secondary standards. 4.1.2. Iodine absorptiolz at 6328

A

Experimental results on this saturated absorption used with a helium-neon laser have been published by HANESet al. [1969a and b]. The absorption of iodine vapor at 6328 contains at least 14 distinct hyperfine components within the gain curve of a helium-neon laser, and, while a given component could be defined as a primary standard, the existence of this structure will complicate the problem of designing automatic equipment to lock upon the selected line. The wavelength of one of these hyperfine components has already been compared to the present krypton standard to approximately 2 parts in log. This should be regarded as a preliminary measurement and does not indicate the full potentiality of the system after it is refined to obtain maximum signal-to-noise ratio from the saturated absorption. The use of iodine vapor as a standard material does not produce any great difficulties in the laboratory, inasmuch as the low vapor pressure desirable in an absorption cell is well below the vapor pressure over solid iodine at any conceivable ambient laboratory temperature. The natural linewidth of the iodine absorption is not as narrow as one would like, however, having been measured by these authors at about 2.3 MHz. From their results it does not appear likely that the natural width can be reduced significantly by going to lower pressures or larger absorption cells than the ones they used. The signal-to-noise ratio obtained by Hanes and his co-workers is also not particularly large, however, this might be improved by removing the absorption cell from the helium-neon laser resonator and using an external standing wave as suggested in Fig. 3.3b. It is clear from these remarks that, although the iodine absorption at 6328 A does not fulfill all the conditions of the ideal primary standard, it certainly deserves further investigation, particularly to the design of systems employing iodine vapor at lower pressures and producing greater signal-to-noise ratio in the output. 4.1.3. Iodine vapor at 5145

A

Experiments employing a molecular beam of iodine have shown the existence of laser-induced fluorescence from the emission of an argon

1,

5

41

PRESENT STATUS O F EXPERIMENTAL W O R K

21

laser at 5145 A (EZEKIAL and WEISS[1968]). These experiments also demonstrated the presence of saturation in this fluorescence. However, the experiments employed a traveling wave through the molecular beam and did not directly demonstrate the existence of a central tuning marker. Such experiments, which would directly demonstrate the natural linewidth that might be obtained from the iodine vapor (not necessarily the same as that demonstrated at 6328 A since the transitions are different and natural lifetimes might very radically even with small variations in J values) are not known to the author, if they have been performed. An argon laser is not as convenient a source, either in a laboratosy or in the field, as a small helium-neon laser. Therefore, unless the iodine absorption at 5145 A possesses special characteristics, such as narrow linewidth, or greater signal-to-noise ratio, it probably possesses no advantages over the absorption at 6328 A.The existing experiments show that iodine vapor possesses considerable structure under the 5145 A gain profile, perhaps more than can be tolerated in a usable standard. 4.1.4. Other prospects in the visible region

Table 4.1 lists wavelengths of CW lasers that are likely to be widely available now or in the future. This list is not exhaustive either with regard to wavelengths or to the known CW visible gas lasers (each of the lasers listed in Table 4.1 can generate other wavelengths besides the ones listed in the table). The wavelengths listed are the strongest and best-known from the particular type of laser, and the list disTABLE4.1 Visible laser transitions (between 0.4 and 0 . 7 p ) that may be useful in connection with absorption cell standards __~ Type of laser Wavelength, p ____

0.4416 0.4880 0.5145 0.5308 0.5681 0.6328 0.6471 0.6765

Helium-cadmium ion Argon ion Argon ion Krypton ion Krypton ion Helium-neon Krypton ion Krypton ion

22

GAS LASERS A N D L E N G T H M E A S U R E M E N T S

[I.

§ 4

tributes the wavelengths rather uniformly throughout the visible spectrum. The assumption that is made here is that, if a suitable absorber is found for a given wavelength A, then it will probably also be usable at one of the listed wavelengths in the vicinity of A. Of course, situations could arise where this is not so, for example, the natural lifetime of the absorbing transition might turn out to be strongly dependent upon small changes in wavelength as suggested by HANES and DAHLSTROM 11969bl. In this case, the reader should consult published tabulations of wavelengths, such as that of BENNETT[1965], for further information as to wavelengths that are available from wellknown visible gas lasers. On the other hand, it may turn out that several different types of lasers might be usable with the same absorbing material, in which case the convenience of using the laser may be the overriding factor. In that case, it is suggested that the following be the order of priority in terms of convenience: helium-neon, helium-cadmium ion, argon ion, and krypton ion. The helium-cadmium ion laser, although a relatively new type, should be given a higher priority than the other ion lasers because of its relatively narrow Doppler width for the gain profile (approximately 700 MHz versus 3000 to 5000 MHz for the other ion lasers).* * As for the standard absorber, one should consider any simple symmetric molecule that is known to produce a visibly colored vapor at room temperature or lower temperatures. Examples of candidates for the absorber would be all of the halogens plus NO,. There may, of course, be others. It is obvious that, unless something special is noted about a given molecule, transition lifetimes for example, the simpler materials will be explored first. 4.2. ABSORPTION CELL EXPERIMENTS I N THE INFRARED

4.2.1 Methane at 3.39 ,u

The work of BARGER and HALL[1969] has demonstrated a saturable absorption tuning marker that appears to have all of the characteristics that are desirable for a primary standard except that of being in the visible. Not only is the central tuning marker sharp and unique, ** A t the present time, the maximum finesse that is obtainable from interferometers of the Fabry-Pbrot type decreases rapidly when the wavelength is reduced below about 4500 A. This may be an important consideration in some types of measuring devices.

1,

I

41

PRESENT STATUS O F EXPERIMENTAL WORK

23

but it corresponds in wavelength to a very high gain laser line of the commonly available helium-neon laser. The high gain of the 3.39 p transition allows one to manipulate the helium pressure in the laser so that the center of the emission Doppler profile is shifted to correspond to the central tuning marker of the methane, thus eliminating possible biases due to a sloping base line. In addition, because of the high gain, it is a simple matter to make the laser operate single-mode, either by using a short laser or by inserting intracavity etalons. As pointed out above, the lifetime of the methane molecule in its excited state for this transition is so long that the observed linewidth almost certainly depends upon instrumental parameters, i.e., the length and diameter of the absorption cell itself. It would appear then that the accuracy to which the natural frequency of this transition can be defined will not be determined by the natural transition rate, but rather by such factors as the homogeneity of the laser beam and the maximum size to which one wishes to construct a standard apparatus. A methane-stabilized laser might have the following approximate parameters: laser Ne pressure, 0.1 torr; He pressure, 12 torr; laser length, 20 cm; laser gain, 3-10 dB (enough to allow for the losses of a beam-expander within the resonator) ; absorption cell diameter, 15 mm; absorbed beam diameter, 12 mm; cell length, 1 m; methane pressure, 0.01 torr. Under these conditions, the observed marker width is about 10 kHz and the signal-to-noise ratio for defining the marker is about 50/1. 4.2.2. Sulphur hexafluoride at 10.6 p

There are a number of saturable absorbers available in the region of the CO, laser transitions around 10.6 p , but the one that has been studied the most, both for Q-switching purposes and for wavelength definition, is that of sulphur hexafluoride (SF,). Observations of the central tuning markers of several SF, transitions that lie within the gain curve of a given CO, transition, have been reported by BASOV et al. [1969] and by RABINOWITZ et al. [1969]. Both groups of investigators report observed linewidths for the central tuning marker of the order of 1 to 1.5 MHz. The apparatus of Rabinowitz et al., at least, was such that a narrower natural linewidth should have been detectible. The definition of the wavelength given by this saturable absorber is, therefore, not as well defined as that given by methane at 3.39 p, for example. The usefullness of a saturable absorber at 10.6 ,u would lie in the

24

GAS LASERS A N D LENGTH MEASUREMENTS

[I*

§ 4

fact that it is quite simple to build high power, single-mode CO, lasers with an output of several watts or so, which would be sufficient power for any sort of detector that one wished to use in this wavelength region. It also has a possible future advantage of being accessible to frequency markers from frequency-multiplied millimeter wave sources, thus making possible a definition of the resonance absorption in terms of an absolute frequency. 4.3.EMISSION LINES

We list here the work that has been done on emission lines that are of principal interest for wavelength measurement, listing the work in order of increasing wavelength, without regard to other possible features of importance. All of the emission lines listed here have natural widths (and therefore widths of Lamb dips) that are broader than even the poorest of the absorption lines. This does not necessarily mean that lesser degrees of stabilization can be obtained from systems employing emission Lamb dips than those making use of saturated absorption: witness the relatively high degree of definition claimed for the krypton primary standard, which is an incoherent source whose width is determined entirely by the Doppler profile. The degree of definition that can be obtained in any wavelength determining system depends only on the signal-to-noise ratio, and the signal-to-noise ratio obtainable in these emission line Lamb dips is probably greater than that of most of the listed absorption lines (with the significant exception of the methane absorption at 3.39 p ) . On the other hand, as has been pointed out earlier, the centers of the Lamb dips in the emission lines are subject to pressure and current shifts that can be avoided in the saturated absorption standards, so that on an absolute basis these emission lines are probably not as desirable as the absorption lines. 4.3.1. 6328

A helium-neon

Since this laser is the most generally available, it is not surprising that the greatest amount of work has been done on this particular transition for length determination. The measurement of the center of the Lamb dip of this line, relative to the present krypton primary standards, has been performed in a number of laboratories. A definitive report of the results of this work has been given by MIELENZ et al. [1968] which also refers to earlier work on the topic. This paper reports on measurements, in three dif-

1.

I

41

P R E S E N T S T A T U S O F E X P E R I M E N T A L WORK

25

ferent laboratories, on a particular helium-neon laser of commercial type designed to stabilize automatically on the center of the Lamb dip. The results obtained by combining the work of these laboratories, il = 632.991 418&0.000 03 nm, is a compromise value which is in good agreement with earlier results obtained independently in the same laboratories on other lasers. It is also in good agreement with the value obtained by HANESand BAIRD[1969] for a particular iodine absorption line which occurs within the NeZ0 gain curve, 1 = 632.991 39&0.000 2 nm. The paper of Mielenz et al. points out, however, that the usefulness of 6328 A helium-neon lasers as absolute standards is severely limited by known pressure and current dependent shifts in the wavelength. A series of measurements of the pressure shift performed over the past 3 or 4 years has given conflicting results: Some investigators reported shifts to the red, others to the blue, suggesting that the measured direction of shifts by some of the investigators was either completely backward or subject to unknown experimental biases. All of the published results agree, however, on the magnitude of the shift which is approximately 10 MHz per torr, total pressure, with individual detailed differences depending upon whether the helium or the neon pressure is changed. The accumulated weight of the evidence, including some particularly careful work performed recently by SOSNOWSKI and JOHNSON [1969], is that the shift is to the blue. This also agrees with the relative position of the central tuning marker in LEE and SKOLNICK’S [1967] saturated absorption experiment in pure neon. A great deal is also known about the parameters that determine the signal-to-noise ratio of the Lamb dip, specifically the lifetimes. Measurements made by FORK and POLLACK [1965] can be reduced to the following simple approximation, where p is the pressure in torr,

(lo+ 10p)MHz

(4.1)

yLb/2?t.= (lO+GOP)MHz.

(4-2)

yab/2?t.=

From eq. (3.11) it is clear that the relative depth of the Lamb dip depends upon the ratio yab/yAb.Visible helium-neon lasers typically operate at total pressures of 2 to 3 torr, and certainly not less than 1.5 torr if lifetime of the laser is of any importance at all. On this basis, it appears that the depth of the Lamb dip is approximately 10% that of the total amplitude of the Gaussian profile and this agrees quite well with experiment. A calculation of the second derivative of the power curve at the center of the dip shows that it is not appreciably

26

GAS LASERS A N D LENGTH MEASUREMENTS

[I, §

4

greater in absolute magnitude than that of the Doppler gain curve itself without the dip. Discussions of the mechanical methods that have been employed to lock the laser output to the center of the Lamb dip, or an equivalent marker, are given by several papers listed in the references (WHITE [1965], WHITE[1967], BIRNBAUM 119671). 4.3.2. 1.15 ,u helium-neon

This wavelength of the helium-neon laser has been the subject for almost as much investigation as the red line although the red line is obviously preferred for its visibility. Other than this fact the 1.1523 micron line is probably better suited for use as a wavelength standard for a number of reasons, some of which are the following: 1. The line has a higher unsaturated gain per unit length, and operates

at lower currents than the visible line. This, particularly the higher gain, makes it possible t o design a 1.15 ,u helium-neon laser for wavelength measurement use which operates at lower total pressures than are permissible with the red line, thus greatly decreasing the pressure- and current-dependent shifts. 2 . Under these operating conditions, the 1.15 p line has been observed to have a more sharply defined Lamb dip than is obtainable in the visible. In fact, the earliest published experimental observations of the Lamb dip were performed on this line (SZOKEand JAVAN [1963]). Their results depict a much clearer Lamb dip than is ever obtained in the visible. 3. For the same pressure change, the 1.15 ,u line is expected to have

a much smaller pressure shift than the visible lines because the infrared transition starts from an excited state that is more closely bound to the nucleus of the neon atom than the upper state of the visible transitions, and thus is less likely to be perturbed by collisions.

A detailed study of the absolute pressure shift in the 1.15 ,u line similar to those performed at 6328 A, has not been done, to the author’s knowledge; however, a detailed examination of the shape of the Lamb dip at 1.15 ,u has been performed by SZOKEand JAVAN [1966] with the intent of studying effects of asymmetry in the Gaussian that occur concurrently with the pressure shift. The exact location of the center of the Lamb dip, relative to the center of the Doppler profile, is a

1,

§ 41

PRESENT STATUS O F EXPERIMENTAL W O R K

27

sensitive function of this type of asymmetry. The measurements of SZOKEand JAVAN give a relative shift, of the center of the Lamb dip to the center of the Doppler profile, of 200 kHz per torr. The shift is a blue shift, in accord with the absolute pressure shift observed at 6328 A. The absolute shift might be expected to be approximately a factor of 10 greater. The width of the Lamb dip at 1.15 p, based on the published curves in the original report, is approximately 50 MHz at the pressure that was used by these investigators. This is of the same order of magnitude as the width of the power dip at 6328 A at this pressure. 4.3.3. 1.52 p helium-neon laser

This transition arises from the same family of excited states of the neon atom as the 1.15 ,u line, and therefore its Lamb dip is expected to have similar properties. However, the 1.52 p transition has a special characteristic in that it is a transition of the form J = 1 --f J = 0, which gives it a simpler Zeeman structure than can be obtained from the other laser lines of this family. Advantage has been taken of this property by TOMLINSON and FORK [1969] to devise a stabilized laser system that has greater sensitivity than that of most other systems making use of the Lamb dip. The system makes use of the strong coupling and competition effects between opposite-sense circularly polarized modes in a single-mode laser when a magnetic field of approximately 1 to 2 gauss is supplied in a direction parallel to the axis of the mode. These effects are strongest when the transition is of the J = 1 -+ J = 0 type, and, as explained by Tomlinson and Fork, the competition provides a built-in amplification of the discriminant that defines the center of the Lamb dip. It is calculated that a signal-tonoise ratio of 100 in detecting the polarization of the laser output is sufficient to provide for a stabilization of one part in lolo. Although this particular stabilization technique shows high promise for use with emission lines it is restricted to the transitions having the requisite simple Zeeman structure. Among the visible lines of the helium-neon laser, the only ones that might be suitable for this type of stabilization are the lines at 6351, 7305 and 6401 A. The first two are weak lines, while the last is very close to a strong neon absorption line and therefore unsuitable for standards work because of possible biases caused by sloping base lines and anomalous dispersion from the absorption line.

28

GAS LASERS A N D LENGTH MEASUREMENTS

:I>

§ 4

4.3.4. 3.39 p heliwn-neon laser

The Lamb dip in this line has been studied experimentally and theoretically by DIETEL and JAHN [1969], who carry through an analysis of the shape of the dip on the assumption that ribis no longer small compared to ku. The results for Yab and rib agree closely with those obtained on the 6328 A line, which is to be expected because both lines share the same upper excited level. It can be expected, for the same reason, that the pressure shifts in the two lines should be similar. The pressure shift in the 3.39 ,u line is of particular interest because it is desirable to shift the center of the gain curve so that it corresponds to the location of the sharp absorption tuning marker in methane. Experimentally, it is found that this shift can be accomplished by the use of approximately 12 torr of helium in the laser. If the laser is operated at low pressure, then it is found that the methane transition lies on the blue side of the helium-neon line center by approximately 100 MHz, which thus corresponds to a pressure shift in the heliumneon mixture of approximately 8 MHz per torr. At the high pressures used in connection with saturated absorption in methane, a Lamb dip is not expected to be seen in the power output, nor would it be desirable for this purpose. 4.3.5. 10.6 ,u CO, laser

Lamb dips are not normally seen in the lines of this laser when it is operated at the high pressures that are desirable for high output. However, at much lower pressures the Lamb dips become quite prominent, and these have been studied by BORDBand HENRY [l968]. The published work shows quite clear photographs of the Lamb dip at pressures of the order of 300 p, on a number of lines. Of these the P22 line is perhaps the most suitable because of the sharpness of the dip and isolation of the line from other lines that might cause perturbing biases. These authors also discuss the use of a stabilized CO, laser for examining the absorption properties of other gases. The advantages of a CO, laser for this purpose have already been mentioned; namely, ease of construction and relatively high power output when operating under single-mode conditions. 4.3.6. Far infrared

The use of far-infrared transitions as interim wavelength standards

I1

REFERENCES

29

is attractive because their frequencies can be compared with microwave sources and thus, eventually, with the cesium beam frequency standard. Frequency determinations have been made on a number of laser transitions in the vicinity of 100 p, but determinations of frequency at the center of a Lamb dip has been reported only for the water vapor discharge line a t 118.6 p (FRENKEL et al. [1967] and references to earlier work contained therein). The accuracy claimed for this work was about 1 part in 108.

5 5.

Acknowledgements

The author wishes to acknowledge particularly helpful discussions with J. L. Hall, M. S. Feld and V. P. Chebotaev.

References BARGER, R. I-. and J . L. HALL,1969, Phys. Rev. Letters 22, 4. BASOV, N. G., I. N. KOMPANETS, 0. N . KOMPANETS, V. S. LETOKHOV and V. V . NIKITIN,1969, Zh. Eksperim. i Teor. Fiz. Pis’ma Red. 9, 568. BASOV, N. G. and V. S. LETOKHCV, 1969, Zh. Eksperim. i Teor. Fiz. Pis’ma Red. 9, 660 (transl. JETP Letters 9, 409). P. L., 1967, Proc. IEEE 55, 1039. BENDER, BENNETT Jr., W. R., 1965, Appl. Opt. Suppl. 2, p. 3-33. RIRNBAUM, G., 1967, Proc. IEEE 55, 1015. BORDE, C. and L. HENRY,1968, IEEE J. of Quant. Elect. QE-4, 874. BYER,R. L., W. E. BELL,E. HODGES and A. L. BLOOM, 1965, J. Opt. SOC. Am. 5 5 , 1598. DIETEL,W. and G. JAHN, 1969, Z. Angew. Phys. 26, 206. EZEKIAL, S. and R. WEISS,1968, Phys. Rev. Letters 20, 91. FELD, M. S., A. JAVAN and P. H. LEE, 1968, Appl. Phys. Letters 13, 373. FELDMAN, B. J . and M. S. FELD, 1970, Phys. Rev. A l . 1375. FORK, R. L. and M. A. POLLACK, 1965, Phys. Rev. 139, A1408. FRENKEL, L., T. SULLIVAN, M. A . POLLACK and T. J. BRIDGES,1967, Appl. Phys. Letters 11, 344. HANES,G. R. and K. M. BAIRD,1969a, Metrologia 5, 32. HANES, G. R. and C. E. DAHLSTROM, 1969b, Appl. Phys. Letters 14, 362. HOPKINS, B. D. and G. R. FOWLES, 1968, IEEE J. of Quant. Elect. QE-4, 1013. KAN,T., H. T. POWELL and G. H. WOLGA,1969, IEEE J . of Quant. Elect. QE-5, 299. LAMBJr., W. E., 1964, Phys. Rev. 134, A1420. LEE,P. H. and M. L. SKOLNICK, 1967, Appl. Phys. Letters 10, 303. LEE, P. H., P. B. SCHOEFER and W. P,. BARKER, 1968, Appl. Phys. Letters 13, 373.

30

GAS LASERS A N D LENGTH MEASUREMENTS

[I

LETOKHOV, V. S., 1967, Frequency Stabilization of an Absorption Line in a Molecular Beam, U.S.S.R. Acad. Sci., P. N. Lebedev Inst., Preprint No. 36. MIELENZ,K. D., D. F. NEFFLEN,W. R. C. ROWLEY,D. C. WILSONand E. ENGELHARD, 1968, Appl. Opt. 7, 289. OWENS,J., 1967, Appl. Opt. 6, 51. RABINOWITZ, P., R. KELLERand T. T. LATOURETTE, 1969, Appl. Phys. Letters 14, 376. RAUTIAN, S.G. and A. M. SHALAGIN, 1969, Zh. Eksperim. i Teor. Fiz. Pis’ma Red. 9, 686 (transl. J E T P Letters 9, 427). SHIMIZU, F., 1969, Appl. Phys. Letters 14, 378. SOSNOWSKI, T. P. and W. B. JOHNSON, 1969, IEEE J. of Quant. Elect. QE-5, 151.

STENHOLM, S.and W. E. LAMB Jr., 1969, Phys. Rev. 191, 618. SZOKE,A. and A. JAVAN, 1963, Phys. Rev. Letters 10, 512. SZOKE,A. and A. JAVAN,1966, Phys. Rev. 145, 137. TOMLINSON, W. J. and R. L. FORK,1969, Appl. Opt. 8, 121. UEHARA, K. and K. SHIMODA, 1965, Japan J. Appl. Phys. 4, 921. WHITE,A. D., 1965, IEEE J. of Quant. Elect. QE-1, 349. WHITE,A. D., 1967, Micro-waves, Jan. 67, pp. 51-61.

I1

PICOSECOND LASER PULSES BY

A. J. DEMARIA United Aircraft Research Laboratories, East Hartford, Connecticut, U S A

CONTENTS

PAGE

0 3 5 3

. . . . . . . . . . . . . . . . . 2 . BASIC Q-SWITCHING PRINCIPLES . . . . . . . .

33

3 . GENERATION OF PICOSECOND LASER PULSES

37

1 . INTRODUCTION

.

35

4 . MEASUREMENT TECHNIQUES FOR PICOSECOND

LASER PULSES . REFERENCES

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

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

57 70

Q 1. Introduction Soon after the operation of the first laser, HELLWARTH [1961] proposed an experimental technique for generating large output bursts of radiation from laser devices. The experimental technique for obtaining these large output bursts of radiation is now referred to as “Q-switching” a laser and was first experimentally accomplished by MCCLUNG and HELLWARTH 119621. The availability of these Qswitched optical pulses has made possible the experimental investigation of previously unobserved phenomena such as optically generated plasmas; optical harmonic generation; stimulated Raman, Brillouin, and Rayleigh-wing scattering; photon echoes; self-induced optical transparency; optical self-trapping; and optical parametric amplification. These short-duration, high-peak-power Q-switched laser pulses have also found practical applications in the field of semiactive guidance, ranging, illumination, high-speed photography, holography, and material working and removal. The minimum pulse widths obtainable with existing Q-switching techniques are limited to approximately 10-8 sec because of the required pulse build-up time. Peak powers of approximately 5 x lo8 W have been obtained with various straight-forward, Q-switching, experimental arrangements of oscillators without any additional stages of amplification. For the sake of completeness this paper will briefly review the basic fundamental principles and concepts envolved in Qswitching lasers. The major content of this paper will be concerned with reviewing the methods of generation and measurement of the second generation of short time-duration, high-peak power laser pulses (i.e., laser pulses having picosecond time duration and peak power in excess of lo9W (DEMARIA et al. [1969] and DEMARIA et a1 [1967]). There are numerous reasons why researchers have become interested in the generation of picosecond light pulses having peak power in the gigawatt range. Picosecond light pulses can be conveniently generated in laser media having wide spectral bandwidths and long fluorescent 33

34

PICOSECOND LASER PULSES

[II.

§ 1

lifetimes, such as: YAG: Nd3+, ruby or Nd3+: glass. The availability of optical pulses of such high power and short time duration has aroused considerable interest among military, academic, and industrial researchers. For example, a pulse of sec at a wavelength of 1 micron has a length in free space of 0.03 cm and, therefore, offers the possibility of measuring long distances to fractions of millimeters. An event would have to move an appreciable fraction of the velocity of light in order to realize the full potentials of such pulses in highspeed photographic applications. Electrical pulses having a rise time of less than sec, amplitudes of 60-100 V, and repetition periods as short as 1.5 nsec have been generated by detecting these ultrashort laser pulses with fast photodiode detectors. These electrical pulses were previously unattainable and should find application for determining the location and severity of internal reflections in wide-bandwidth transmission systems, studying propagation delays, and transient response of wide-bandwidth electronic systems, etc. The application of high-energy, picosecond laser pulses in high temperature plasma experiments, optical radar, optical information processing, spectroscopy, nonlinear optical properties of materials, transient response of quantum systems, radiative lifetime measurements, stimulated scattering, and ultrashort acoustic shock research appears very promising. As a result of these many promising potential instrumental applications of ultrashort laser pulses in applied and basic research, great strides have been made in generating them over the last three to four years. For example, it took approximately two years (i.e., from 1966 to 1968) for researchers to accomplish a five order of magnitude jump in their ability to generate short duration laser pulses (i.e., from low8 sec to sec) having peak powers greater than lo6 W. During this same two-year time period, researchers were also able to accomplish a three order of magnitude jump in their ability to generate highpeak powers (i.e., from 109 W to 1012 W). To keep up with these breakthrough, researchers had to devise techniques for measuring the time duration of these ultrashort pulses. These new measurement techniques represent a three order of magnitude improvement (i.e., from sec to sec) over past, direct, optical pulse measurement technique. In addition to these accomplishments, the detailed behavior of these pulses on a picosecond time scale have also been investigated. For approximately two years, it was not understood why the pulses generated by a simultaneously, passively, Q-switched and mode-locked Nd3+:

11, §

21

BASIC

Q-SWITCHING PRINCIPLES

35

glass laser were 10 or more times longer than the theoretical limit. I n 1968, Treacy found that this discrepancy was partly caused by an approximately one percent positive frequency sweep of the optical carrier within the picosecond duration of the pulses. The theoretical sec was obtained by TREACY [1968a] limit of approximation 4 x by optical pulse compression techniques. The availability of a 4 x l O - l 3 sec pulse enabled TREACY [1969] to learn some details of the pulse shape of the sec pulses by the use of optical sampling techniques. GLENNand BRIENZA [1967] found a gradual increase of the pulse widths throughout the pulse train output of their Nd3+: glass laser. The phenomenon responsible for most of these interesting behavior are not presently known. As a result, the study of the generation and measurements of pico;econd laser pulses will continue for quite a few more years.

Q 2. Basic Q-Switching Principles If a cylindrically shaped laser medium is continuously excited or “pumped”, the excess population difference of the upper quantum laser level over the lower quantum laser level will reach some steady state value denoted by AN,, if no feedback reflectors are placed on the medium. The value of AN,, is determined by the pumping intensity and various radiative time constants of the laser medium. The excess population difference required for self-oscillation to occur between these two levels when parallel feedback reflectors are placed on both ends of the laser medium of length L is given by AN

-

1

--hZLo

1

(R,R,))

where o is the radiation cross-section for the transition and R, and R, are the reflectivities of the mirrors at the oscillating frequency. This excess population is called the threshold population or a = AN,o is the threshold gain coefficient per unit length. If one assumes that the radiative lifetime of the upper laser level is long, one can obtain a large excess population of this level if the threshold required for self oscillation is maintained so high as t o quench any stimulated emission or laser action. This can be performed with the insertion of a closed optical shutter in the laser cavity. This high inversion serves as an energy reservoir. Assume that at some time t o , oneinstantaneously opens the shutter. The threshold population

36

P I C O S E C O N D L A S E R PULSES

[II,

§ 2

will decrease from infinity down to some finite value determined by eq. (2.1), (see Fig. 1).The excess population initially begins to slowly decrease because the radiation level within the optical cavity is small at the moment of opening the shutter (i.e., stimulated emission is small). The time required for this background radiation to build up sufficiently for a rapid exponential build up to occur is denoted by zB in Fig. lb. Typically, zB is 50 to 100 nsec for ruby and Nd3+: AN

TIME

Fig. 1. A qualitative illustration of the dynamics of Q-switching in the high inversion, fast switching case; a) the variation of the inversion AN as a function of time and b) the pulse output as a function of time. Various characteristic times referred t o in the text are indicated.

glass lasers. Presently available electro-optical shutters have switching times well within this time limit. The performance of the Q-switching operation in a time much less than zBis called the fast-switch mode of operation. Switching times within this time limit are not as easily obtained with mechanically rotating devices and, therefore, represent the slow-switch mode of operation. After the time period zB,there is

11,

§ 31

GENERATION OF PICOSECOND LASER PULSES

37

sufficient radiation within the cavity so that a rapid exponential build up occurs. The build up of this second process is essentially the rise time (tR) of the pulse. It is experimentally found that tR > 1, eq. (3.1) can be assumed to have arithmetric rather than geometric symmetry and we can assume that 2 ( w o - w ) m A o L where AmL is the bandwith of the medium of length L , where as: Aw is the bandwith of the medium of unit length. The gain of an optical amplifier operated far from its center frequency tends to unity rather than to zero as for electronic amplifiers.

38

P I C O S E C O N D LASER PULSES

[II,

§ 3

The reduction of the over-all bandwidth Am, of an optical amplifier when compared with the broadened atomic-line bandwidth can be found from

EOL --l]*. In ['{exp (a,L)+lj]

If exp(a,L)

(3.3)

>> 1, then eq. (3.3) can be approximated by AmL/Am M [In 2/(aoL)]*

(3.4)

An optical amplifier with positive feedback can be considered in terms of a transmission line analog, where L --f co (Fox and LI [196l]). For the case L + co, eq. (3.4) reveals that AmL --f 0. This result is to be expected, since for sufficient optical feedback, the effective length of the amplying medium appears infinite; which is another way of stating coherent oscillation at the peak of the spectral line is obtained or that Am, + 0. The proceeding simple analysis indicates that the natural tendency for an optical oscillator is to oscillate in a continuous sinusoidal manner and not in a pulsating manner. The fastest rise time pulse ( A t L A )that can be propagated in an amplifier without distortion is given by 2n(0.7 to 0.9) AtL.4 M

>

(3.5)

AWL

and the fastest rise time pulse (AtLo) that can be generated in an oscillator is (3.6) AtLo w AyLA/2&. The factor 2-4 appears in AtLo because the pulse makes two passes through the optical medium for one round trip between the feedback reflectors (DEMARIA and MILLER 119681). The value of AtLo can be very short for laser type devices. The relatively narrow spectral line widths of gas lasers limit pulse widths to the order of 10-lo sec while the broader line widths available in solid-state lasers can yield much sec for Nd3+: YAG, narrower pulses (i.e., approximately sec for ruby, and 10W3sec for Nd3+: glass lasers). Since the natural tendency for an oscillator is to operate in a continuous sinusoidal manner, one has to perform an additional experi-

11,

§ 31

GENERATION O F PICOSECOND LASER PULSES

39

mental manipulation beside providing feedback in order to obtain a pulsating output. A laser oscillator essentially consists of an active medium, having a complex refractive index as given by n = n,-ik, which is inserted between two reflectors separated by a distance L. The parameter a = wk/c is the gain per unit length (or Lorentz’s index of absorption if the sign of the complex refractive index portion is positive) where cc) is the optical angular frequency and c is the velocity of light in free space. The resonances of such a system are determined by the spectral line width of the laser transition and by the number of half-wavelengths existing between the two reflectors. The gain of a laser line profile superimposed on the resonances of a Fabry-Perot interferometer is illustrated by Fig. 2. An interferometer resonant frequency v, is given by v, = mc/2Ln (3.7) where m is an integer representing the number of half wavelengths stored between the two reflectors. These resonances are the axial mode resonances of the laser’s interferometer. The frequency separation

Fig. 2. Visual representation of the gain of the laser line profile with the resonances of the Fabry-Perot interferometer superimposed.

40

PICOSECOND LASER PULSES

[II, §

3

f between two adjacent axial resonances is given by

f = c/2Ln.

(3.8)

It is important to note that the round-trip transit time (t)for a pulse of light to bounce back and forth between the two mirrors is given by t = 2Llz/c = l/ f. (3.9) The field E ( t ) , at a fixed point inside the optical Fabry-Perot resonator represents an aggregate of the fields of the various oscillation axial modes or

E ( t ) = Re 2 A , ( t ) exp [io,,,t+i@,]

(3.10)

m

where A,,,, @, and omrepresent the amplitude, phase and frequency of the modes, respectively. It is well known from the Fourier theorem, that any repetitive pulse train having a fixed period t can be represented by eq. (3.10) if the frequencies are all multiples of l / z and have a fixed phase relationship between the many discrete frequencies. It is also well known that the narrower the pulses in the train become, the more sinusoidal terms (i.e., axial modes) are required to fatefully represent the pulses. Equation (3.9) informs us that the frequencies are multiples of l/f if dispersion is neglected. The only additional requirement for obtaining a pulsating output from a laser is a means of phase-locking the discrete = 0, CP,,,+~-@,,,= 0, oscillating frequencies of the laser (i.e., @,,,+l-@m etc.). 3.1. ACTIVE MODE-LOCKING

In the preceding sections of this paper we have seen that laser oscillators consist of a resonant system with dimensions that are large compared to the oscillating wavelengths. Consequently, mode density is high and there are m = 2Lnv,lc (3.11) axial interferometer resonances within the line width of a laser transmission. In normal operation of a laser, these modes are to a great extent, uncoupled; and therefore, have no fixed phase relationships. As these frequencies drift in and out of phase, their interference causes the output of the laser to have random and systematic amplitude fluctuations. Such random and systematic amplitude fluctuations can

11,

§ 31

41

GENERATION O F PICOSECOND LASER PULSES

be eliminated by coupling the modes together. The interference of the phase-locked modes then produces a pulsating laser output having a repetition rate equal to f . HARGROVE et al. [1964] were the first to couple the modes of a laser for the generation of optical pulse trains of ultrashort laser pulses. They added the normal missing phase-locking element within the laser as follows: It is resonable to assume that one reasonance v, will be nearest the peak of the laser gain profile and will, therefore, be the first to oscillate. If an amplitude modulator operating at a frequency f is inserted into the laser’s feedback interferometer, the mode of frequency v, will be amplitude-modulated at a frequency f such that its time-dependent electronic field amplitude E,(t) will be of the form E,(t) = A,(l+M

cos 2nft) cos w,t.

(3.12)

Since the modulating frequency was chosen to be commensurate with the axial mode frequency separation, superposition of the upper (v,+f) and lower (v,-f) side bands of the amplitude-modulated radiation with the adjacent resonances v,+~ and v,-~, respectively, couple the three modes with a well defined amplitude and phase (see Fig. 2). As the v,+f and vm-f side bands pass through the resonator, they also become amplitude modulated. Their side bands in turn couple the v,+~ and v,-~ axial modes to the previous three modes. This process continues until all axial modes falling within the oscillating line width, Av, are coupled. Let us assume that at time t = 0 , m sinusoidal components of amplitude A and circular frequencies wl, w 2 , . . ., w, all have identical phases such that they are all aligned along the x-axis of the phaser diagram of Fig. 3a. Let us assume also that all the frequencies are integral multiples of the same angular frequency w . The resultant ( R ) of all amplitudes is R = mA. At some later time t’; the frequency w, has rotated through an angle 8 = w,t’. The angle 8 between any two adjacent modes is given by (see Fig. 3b)): (3.14)

At some later time t = A t , the angle 8 = 2nIm and R lustrated by Fig. 3c. Solving eq. (3.14) for t = A t gives

AT = (m-l)/mAv

=0

as il(3.15)

42

PICOSECOND LASER PULSES

[II,

s3

Fig. 3. Phasor diagram of m phase-locked laser axial modes. The vector sum of the vectors represent the amplitude variation of the optical electric field within a modelocked laser cavity.

where Av

=

For large m,

( W ~ - W ~ ) / ~ Z .

At

M

l / A v = ;ii/c A;l

(3.16)

where A,, = c/vm and A1 = Av A i / c . At a still later time, t = t, the components again have identical phases and the angle 8 equals 2n and R = mA (see Fig. 3d). Solving eq. (3.14) fort = t yields t=

(m-l)/Av.

(3.17)

For large m, t = m/Av = l// as previously stated by eq. (3.9). A schematic representation of the constructive and destructive interference of the electric field of m axial modes of a “mode-locked’’ or “phase-locked’’ laser is illustrated by Fig. 4. One then has a pulse of

Fig. 4. Visual representation of the amplitude variation of the optical electric field within a mode-locked laser cavity.

11,

§ 31

GENERATION O F PICOSECOND LASER PULSES

43

light bouncing back and forth between the two mirrors of the laser. The physical length of such a pulse can be much shorter than the separation distance L of the mirrors. The average power Pa is proportional to mA2 and the peak power P, is proportional to m2A2.As a result

P,

= mP,

(3.18)

for “phase-locked” or “mode-locked’’ lasers. A detailed theory of the operation of actively mode-locked lasers has been given by MCDUFFand HARRIS[1967], LETOKHOV and MOROZOV[1967] and by CUBEDDUand SVELTO[1969]. 3.2. PASSIVE MODE-LOCKING

It is well known that a feedback loop, as illustrated by Fig. 5a, encompassing an amplifier, a filter, a delay line, and a nonlinear element that provides less attenuation for a high-level signal than for a low-level signal, behaves as a regenerative pulse generator. When the loop gain exceeds unity, a pulse recirculates indefinitely around the S&)

1

I -

REFLECTOR

+ DELAY=^I - - , (b) Fig. 5. Equivalance of a) an electronic regenerative pulse generator and b) a laser with a saturable absorber inserted within its feedback interferometer.

44

PICOSECOND LASER PULSES

[II,

§ 3

loop and each traversal gives rise to an output pulse at the output terminal. It is evident that such a pulse would soon be degraded unless the effects of noise and distortion can be counteracted. The nonlinear element (called an “expandor” by CUTLER [1955]) has the effect of: 1) emphasizing the peak region of the recirculating pulse while reducing the lower amplitude regions, 2) discriminating against noise and reflections, and 3) acting to shorten the pulse until the pulse width is limited by the frequency response of the circuit. The output of the regenerative oscillator has a pulse rate equal to the reciprocal of the loop delay, pulse widths equal to the reciprocal of the overall system bandwidth, and center frequency determined by the filter frequency. An ordinary laser posses all the basic elements of the regenerative pulse generator operated in the microwave region with the exception of the expandor elements. The laser medium serves as the amplifier, the combination of the Fabry-Perot resonances and the linewidth of the laser transition serve as the filter, and the time required for an optical pulse to traverse twice the distance between the reflectors serves as the loop-time-delay (see Fig. 5b). The optical analog of the electronic expandor circuit element is a saturable absorber, such as the reversible bleachable dye solutions commonly used as laser Qswitches. The fundamental requirements of the saturable absorber are 1 ) that it have an absorption line at the laser wavelength, 2) that it have a linewidth equal to or greater than the laser linewidth and 3) that the dye recovery time be shorter than the loop-time-delay of the laser. A simplified explanation of the operation of the regenerative pulse laser oscillator illustrated by Fig. 5b can be given with the aid of Fig. 6. If an optical carrier frequency vo along with two sidebands at vo&f are superimposed, an amplitude modulation of the light results at a frequency f with some peak-to-peak variation M I and a peak intensity I, (see Figs. 6b and 6c). When this beam is passed through a saturable absorber having the typical characteristics illustrated by Fig. 6a, the initial sinusoidal amplitude fluctuation of the input beam will be found to be distorted, the peak-to-peak excursions of the fluctuation will be increased (i.e. M , > M , > M I ) , and the time duration of the fluctuation will be shorter as a result of the nonlinear transmission characteristics of the saturable absorber. With the sharpening of the amplitude variation, additional sidebands are added to the spectrum. This process is repeated over and over again by reflecting the light

11,

§ 31

GENERATION O F PICOSECOND LASER PULSES

45

INTENSITY

(a)

SATURABLE ABSORBER

SATURABLE ABSORBER

(b)

, I ,v,-f

v,

v,+f

:.i"i;,-v;

vrn,-[kI

Vm-2f vrnt2f Vm-2f

UfiL

Vmt2f

Fig. 6. Operation of a ) the transmission characteristics of a saturable absorber in generating a periodic pulse train in b) the time and in c) frequency domain.

beam back and forth between two mirrors placed on both sides of the saturable absorber cell. The fluctuation will continue to sharpen until a discrete pulse is circulating in the cavity. The laser media provides the gain to compensate for the residual saturation loss of the absorber and the mirrors. The pulse will eventually acquire a steady-state width At determined predominately by the bandwidth of the laser media. The repetitive output pulse train emitted from the laser will have discrete spectral components defined by the Fabry-Perot resonance of the cavity, as illustrated by Fig. 6c. This result is to be expected since, as it was pointed out earlier, it is well known from Fourier's theorem, that any repetitive pulse train can be represented by a series of discrete sinusoidal functions having integrally related frequencies and fixed phase relationships. The frequencies are all

46

PICOSECOND LASER PULSES

[II,

§ 3

multiples off, and the narrower the pulse width At, the larger the bandwidth required to reproduce the repetitive pulse. A quantitative feeling of the magnitude of the sharpening experienced by an optical pulse in passing through a saturable absorber can be obtained by a simple calculation given below. Assume a pulse whose time duration is long compared to a saturable absorber relaxation time, the absorber will act as an intensity dependent absorber. Such a case has been treated by HERCHER [1967]. The intensity transport equation is given by (3.19)

where the propagation time has been neglected, a. is the small-signal absorption coefficient, and I, is a saturation intensity, i.e., the intensity at which the absorption coefficient of the dye is reduced to onehalf of its small-signal value. The solution to this equation is

I: exp I;%, = TI;,,, exp

(3.20)

where T is the small-signal transmission of the finite length dye cell, and Ii0,and I&,are the normalized intensities at the entrance and exit as a function of distance in the dye cell, respectively. A computer experiment can be performed to illustrate the operation of Fig. 6 using eq. (3.20). Fig. 7 illustrates the pulse shape normalized to unity intensity after every tenth pass of a trial Gaussian-shaped 1.O

5

0.8

Z

2

Z 0.6 D

w 0.4 -=l

2

0 0.2

z

0

t

Fig. 7. Optical pulse sharpening as a function of passes through a saturable absorber having a short relaxation time with respect to the pulse duration. The assumed parameters in the calculation were intensity of the initial Gaussian pulse 0.1 I,, gain between successive passes G = 1.4, and dye transmission T = 0.7 (see DEMARIAet al. [1969])..

11,

§ 31

GENERATION O F PICOSECOND LASER PULSES

47

pulse initially having a peak intensity of 0.1 I , through a saturable absorber cell having a small-signal transmission of T = 0.7. The absorber is most effective in sharpening when the intensity is in the neighborhood of I,. In the actual operation of a laser, the absorber concentration is much weaker and the transmission is correspondingly higher, of the order of 0.95. It is also of interest to obtain a quantitative feeling of the magnitude of the sharpening when an optical pulse is propagated through a twolevel system in which the duration of the optical pulse is less than the relaxation time of the absorber. A rate equation treatment of this case and NODVIK[1963] and BELLMAN et al. has been given by FRANTZ [1963]. Their solution is applicable to the saturable gain in the laser medium as well as the saturable absorption in the absorber. When the relaxation time is assumed infinite, an exact solution of the rate equations is possible. For this case, it is more appropriate to deal with the integrated pulse intensity

U ( t )=

SI,

I ( t ) dt

(3.21)

rather than the intensity. The solution for the saturable two-level medium is W,,(t) = ln[l+G(exp Win(t)-l)l, (3.22) when the normalization W ( t )= U(t)wo/hvis used, cr is the cross section per active atom, and G = exp(cr,l) is the small-signal gain or attenuation at the peak of the line of the medium. Figure 8 illustrated the computer simulation of a laser-medium saturable-absorber combination utilizing eq. (3.22) for both the active medium and the absorber. The cross section uL of the laser atoms was taken to be much smaller than for the absorber molecules cra (i.e., hv/20L = 16 J/cm while hv/2cra= 0.016 J/cm). The loss due to the reflectivity of the laser mirrors was assumed to be 0.2, the initial laser gain 2.0, the absorber transmission 0.8, and the starting Gaussian pulse energy 1 . 6 10-3 ~ J/cm. Figure 8a illustrates the amplitude increase of the pulse and the gain decrease of the laser medium as a function of time. The absorber was assumed to relax to its initial state between the successive passes of the pulse but the laser gain decreases as a result of all the pulses that have passed through it. The amplitudes of the pulse train for the first 30 passes are small, in the range of 0.1 to 1.0 hv/2cra. When the energy finally grows to approximately the

48

PICOSECOND LASER PULSES

[II,

§ 3

(a) >.

2z

$200

4

W

z W

[L

w

?I00

lu,

4

3

a

0

20

30

40 50 60 PULSE NUMBER

0

70

Z

D

0.4 -

W

L

t

Fig. 8. Computer simulation of an optical pulse regenerative oscillator. The assumed parameters in the calculation were mirror reflectivity 0.8, starting pulse energy = laser saturation energy, laser saturation energv E L = 16 J/cm, dye saturation energy = 10-3 EL, and the dye relaxation time was assumed much greater than the pulse duration (see DEMARIA e t al. [1969]).

assumed absorber saturation energy (0.016 J/cm), a rapid increase in pulse energy takes place. The growth in amplitude is finally limited by the onset of saturation in the laser. The laser gain then begins to decrease. In this region, the energy of the pulses reaches a maximum and then begins to decrease. Figure 8b illustrates the sharpening of the pulses as a function of the number of passes. The peak intensity of each pass has been normalized to unite energy. The most pronounced sharpening is found to occur in this model between the twentieth and fourtieth passes. Referring to Fig. 8a, it is seen that this is just the region where the pulse energy reaches and exceeds the absorber saturation energy. I t can be concluded that much can be learned about the operation of the laser saturable-absorber combination by means of such computer simulation. Neither of the two models described here is capable of explaining the formation of the initial pulse. In addition, the absorber concentration is much weaker in actual operation of such systems than

11,

§ 31

GENERATION O F PICOSECOND LASER PULSES

49

the values assumed in these simulations. As a result, the sharpening action is much weaker. It is apparent that a more elaborate model of the dye is needed. 3.3. CHIRPING AND PULSE COMPRESSION

The system schematically shown in Fig. 5a lends itself to simplified analysis by repeated application of Fourier transforms, applying the frequency or amplitude characteristics of each element, and finally equating the characteristics of the returning signal to the characteristics of the assumed initial signal. Consider a signal entering the expandor given by S,(t)exp (iyt). The signal leaving the expandor is

S,(t)

= K[S,(t)]qe'vt

(3.23)

where K is a constant indicating an amplitude change of the signal and the superscript can be taken to indicate a nonlinear operation, not necessarily a power law. Notice that the nonlinear operation was performed only on the envelope portion of the signal and not on the phase portions. Such nonlinear operation on the phase portion would yield higher frequency terms which are not experimentally found. If this signal is now passed through an amplifier of gain G, the output signal S3(t)is S3(t)= GS,(t). (3.24) We now Fourier transform S,(t) into the frequency domain (3.25)

so that we can have the signal be operated on by the filter having a transfer function F ( w ) . As a result, F 4 ( w )= F ( w ) F , ( w ) and the signal leaving the filter is given by 1

S4(t)= 2,/-,

"

F 4 ( w )e-i"t dw.

(3.26)

After passing through the delay line, the signal has the form

S,(t) = S,(t-t).

(3.27)

We now require that S,(t) be equal to S,(t), except for the phases so that

(3.28)

50

PICOSECOND LASER PULSES

[II,

§ 3

where 8 is the phase shift of the optical wave relative to the pulse time. Given a filter characteristic F ( w ) , an expandor nonlinear law K , and time delay z, then eq. (3.28) specifies the time function Sl(t)ect. In the event a function S,(t) is found to be a replica of S,(t), a solution to eq. (3.28) has been found. Gaussian functions are particularly well suited for such types of solutions for eq. (3.28) if a power law for K is assumed. So lets assume a pulse having the form

S , (t)eW = Eoedt2ei+

(3.29)

and a filter having a phase (8)variation with frequency given by

8 = a+~(o--Loo)+y(o--oo)2+.

..

(3.30)

so that its Fourier transform function has the form

F(,-,,)

= Bed(~-~o)’,-i@

(3.31)

where d = (2/Ac0)~,B is the normalized peak amplitude of F ( o ) , a = AT)^ and E, is the peak amplitude of S,(t). Upon carrying on the calculations, with the assumed forms of S,(t) and F ( o ) ,one finds it impossible to obtain a function for S,(t) having the same form as S,(t). The reason being that the ~ ( w - c o , ) term ~ in F ( o ) transforms into the time domain as exp(-t2). As long as we assume a dispersive filter (i.e., a filter having a phase term varying as the square of the frequency) we have to assume an input signal having the form given by S,(t) = E,e d f Z e i(wotftbt’) (3.32) in order to obtain a closed form solution for eq. (3.28). The pulse described by eq. (3.32) is called a linear “chirped” pulse. A chirped pulse is one in which the carrier frequency changes smoothly over the duration of the pulse, i.e., the time variation of the carrier wave phase can be expressed in the form ot++bt2+et3+ . . .. A linear chirp has e and other higher order coefficients equal to zero. For a linear chirp, b is the angular frequency sweep rate. Chirped pulses are familiar in the field of microwave radar (COOKand BERNFELD [1967] and bat sonar (GRIFFIN [1958]) where greatly increased information content of the echo over that from a constant carrier frequency pulse of the same pulse width results from the phase modulation. It is well known that the equations for the amplitude and phase in the theory of the laser are coupled to each other (LAMB[1964]). It is thus expected that optical pulses generated by lasers will have some

11,

§ 31

GENERATION OF PICOSECOND LASER PULSES

51

phase modulation. The dispersive properties of the host medium of the laser will also influence the phase structure (ARMSTRONG and COURTENS[1969]). Further phase structure can be added by the saturable dye and its solvent and by nonlinear optical properties of the mediums in the laser cavity (FISHER et al. [1969]). Upon substituting eq. (3.32) into eq. (3.28) and equating the constants of similar parameters (i.e., o,t and t2, respectively) on both sides of the resulting equation, the following expression for the angular frequency sweep rate, b -_-- ~ ( 7 - 1 )

x

q+l

YAW‘ 16+y2Aw4’

(3.33)

the pulse width

and the time interval between pulses T =

p/2n

(3.35)

are obtained. Equations (3.33) and (3.34) are obtained by equating the imaginary and real parts of the coefficients of t2 in eq. (3.28) respectively. The pulse width equation is then obtained from the relationship AT = 2 / ( ~ ) & . (3.36) For the condition where y2A04 + 0, eq. (3.34) can be approximated by AT M l/Av. As a result of its large bandwidth, Am, the y2A04 term apparently plays a dominant role in the Nd: glass lasers. Due to the relatively small bandwidths of the ruby, YAG, etc. lasers, the y2A04 term is negligible. As a result, the output of these lasers are not naturally chirped, and their time-bandwidth product (AT Av) is essentially equal to unity. The bandwidth of a laser can be increased by staggered tuning (DEMARIA and MILLER [1968]). The output of small bandwidth lasers can also be chirped by increasing the y term in eq. (3.33) or (3.34). An increase in the y term can be obtained by [1968b] twoinserting high dispersive elements such as TREACY’S [19641 interferometer, into parallel gratings or the GIRES-TOURNOIS the laser’s feedback cavity. To date, only passively mode-locked Nd3+: glass lasers have been found to automatically emit chirped pulses. Ruby and Nd3+:YAG

52

PICOSECOND LASER P U L S E S

[II,

§ 3

laser have not been found to do so to any practical extent. Chirped laser pulses can also be obtained by a ) internal modulation of the optical resonator (TREACY [ 1968c]),b) electro-optical modulation outside the laser (GIORDMAINE et al. [1968]), c) passive modulation associated with self-steepening effects in a Kerr active liquid (FISHER et al. [1969]), and d) propagation of a short pulse through a dispersive medium or optical system. Method (a) produces a frequency sweep range approximately equal to the bandwidth of the laser medium above oscillation threshold. Methods (b) and (c) involve nonlinear effects which increase the bandwidth of the pulse. Method (d) on the other hand, involves a linear operation and leaves the bandwidth of the pulse unchanged. For approximately two years, it was not understood why the pulses generated by a simultaneously passively Q-switched and mode-locked Nd3+: glass laser were 10 to 20 times longer than the theoretical uncertainty principle limit given by eq. (3.6). Some researchers suspected that a phase modulation in the pulses existed which utilized the bandwidth not appearing in the envelope of the pulses. The question arose as to how to detect such optical phase variations in time periods of 10-l2 sec. An experimental technique which has been utilized to detect optical phase variations in picosecond laser pulses (i.e., chirped picosecond pulses) is the pulse-compression technique common to the compression of frequency-sweep wavetrains in microwave chirp radar (KLAUDER [1960], or PRICEet al. [1969]). (b)

(C )

AMPLITUDE

FREQUENCY

t

INPUT SIGNAL+

1

so

OUTPUT SIGNAL

"'[->! v2 -L--FREQUEN{i[:g TIME

TIME

Fig. 9. Visual representation of pulse compression of a ) an input negative chirped pulse by b) a dispersive filter, and c) is a representation of the compressed pulse.

11,

§ 31

53

G E N E R A T I O N O F P I C O S E C O N D L A S E R PULSES

In pulse-compression systems the signal pulse is of relatively long duration AT >>. l / ( v 1 - v 2 ) and the instantaneous frequency is swept over a given frequency range. In Fig. 9a, we have assumed a negative chirp (i.e., the instantaneous frequency decreases with time). In order to compress the pulse duration, the signal pulse is passed through a dispersion filter having the opposite dispersion of the pulse (see Fig. 9b). As a result, the energy at the beginning of the pulse is delayed by the proper amount so as to reach the end of the filter at the same time as the energy at the end of the pulse (see Fig. 9c). The duration AT of the compressed pulse produced in this manner is given by eq. ( 3 . 6 ) .The ratio A T / A t by which the pulse is compressed is called the dispersion factor, D.The ratio of the amplitude of the signal output to signal input is given by SOIS,= A T / A t

=

AT Av.

(3.37)

GIRESand TOURNOIS [1964] were the first to discuss the compression of optical pulses. They proposed the use of a pair of dielectric-coated flat mirrors spaced some millimeters apart and aligned parallel to one another as in a Fabry-Perot. The mirrors have unequal reflectivities; one is highly reflective and the other has a low reflectivity (i.e., for example 30 percent) (see Fig. 10). A third mirror is placed some distance from, and parallel to the low reflectivity mirror. The third mirror is used to reflect the pulse many times through the two-mirror structure. It is also useful to have the surface of the third mirror slightly curved to reduce diffraction losses. As one tunes the spacing of the Fabry-Perot, the transmission, reflectivity, and dispersion (or phased delay) of the device vary as the device goes through ANTI - REFLECTIVE SURFACE

----HIGH REFLECTIVE SURFACE

REFLECTIVE SURFACE

--

REFLECTIVE SURFACE

Fig. 10. Schematic diagram of a Gires-Tournois interferometer suitable for compressing or chirping short optical pulses.

54

PICOSECOND LASER PULSES

[II.

§ 3

Fig. 11. Schematic diagram of the Treacy tandem grating pair for compressing or chirping short optical pulses.

resonances. Near a resonance, the structure has high dispersion and low loss. Depending on which side of resonance the carrier is on, the dispersion can be either positive or negative. DUGUAY and HANSEN [1969] utilized such a pulse compression interferometer to reduce the width of 0.5 nsec visible He-Ne laser pulses down to 0.27 nsec. GIORDMAINEet al. [1968] considered the propagation of chirped optical pulses through dispersive mediums such as bromobenzene and nitrobenzene. In general, one finds that the dispersion is relatively small in optical media and, therefore, one requires long tubes of commonly available dispersive media in order to obtain appreciable compression or chirping. TREACY[1968] utilized a pair of diffraction grating with their faces and ruling parallel as illustrated by Fig. 11 to detect a positive frequency sweep (i.e., from the red toward the blue) automatically occurring in picosecond pulses emitted by Nd3+:glass lasers. One disadvantage of the grating pair is that the time delay is always an increasing function of the wavelength, whereas the Gires-Tournois interferometer can be made with this function either increasing or decreasing in the wavelength region of interest. A fundamental understanding of the operation of the tandem grating pair as either a pulse expander or as a pulse compressor can be obtained with the aid of Fig. 11 (TREACY [1969]). A light beam having a spectral width extending from 2, down to 2, is diffracted by the first grating. The path of the light ray of 2, travels a distance A longer than the light ray 2,. The diffracted light beam is subsequently also diffracted from the second grating. Light ray A, travels an ad-

11,

§ 31

GENERATION O F PICOSECOND LASER PULSES

55

ditional distance B greater than light ray l2 for a total additional distance of A f B . In this manner, the tandem gratings have the property of producing a time delay that is an increasing function of the wavelength. An alternative statement of this property is to say that the phase shift through the system as a function of frequency contains a quadratic frequency term. The grating pair can thus be used to compress optical chirped pulses or to generate chirped pulses out of short unchirped pulses. The Treacy tandem grating pair can also be inserted into the interferometer of a mode-locked laser to compensate for the dispersion generating the chirp or to provide sufficient dispersion to generate chirped pulses. 3.4. OPTICAL PUMPING WITH MODE-LOCKED PULSES

There are many organic dye solutions which exhibit laser action when excited with a short-duration, high-intensity pump pulse. Pump pulses from Q-switched lasers and from specially constructed flash tubes have been employed to serve as sources of short intense pumping radiation. The output spectra from such dye lasers are quite broad, extending in some cases over a few hundred Bngstroms. This suggests the possibility of mode-locking an organic dye laser to produce ultrashort pulses over a wide range of the spectrum due to the wide range of oscillating wavelengths available from dye laser sources. It is well known that periodic modulation of the gain of a laser medium will lead to the generation of upper and lower sidebands in an analogous manner as one obtains in active loss modulation of a light beam. If the frequency of the gain modulation is equal to or a multiple of the difference frequency between longitudinal modes of the laser, coupling of the modes will result as described in Section 3.1 of this chapter. If the pumping signal for a dye laser consists of a mode-locked train of pulses, the gain of the laser will have a periodic variation with a period equal to the spacing between the pumping pulses. This occurs because the fluorescent decay time of the dye is very rapid. These decay times vary between 1 to 10 nsec. If the length of the dye laser cavity is equal to or some multiple of the length of the cavity of the laser producing the pump pulses, then the mode-locking condition will be satisfied and the output of the dye laser will consist of a series of pulses. GLENNet al. [1968] succeeded in mode-locking rhodamine 6G and

56

PICOSECOND L A S E R P U L S E S

[TI,

§ 3

rhodamine B by pumping these dyes with the second harmonic of a passively mode-locked Nd3+ : glass laser. This arrangement thus provides a general method of producing short light pulses over a wide spectral range. If a diffraction grating is used in a Littrow arrangement as one of the reflectors, the output of a dye laser can be tuned over a broad spectrum. If the grating is utilized in the cavity of a dye laser to be mode-locked by pumping with another mode-locked laser, a tunable mode-locked laser can be obtained. The spectral narrowing by the grating in such an experiment would limit the ultimate shortness of a pulse that could be obtained. A bandwidth of 1 A, however, at 5 000 A is sufficient to produce a pulse duration less than sec. The output from such an experiment was found to be tunable over slightly more than 300 A from the green through the orange with a molar solution of rhodamine 6G. One striking characteristic of many of these dyes is the very high gain which can easily be obtained. Laser action in a laser pumped rhodamine 6G solution can be obtained in lengths of approximately a millimeter with mirror reflectivity of only a few percent. Such high gain suggests that solutions having thickness of a few centimeters can be superradiant if one can pump with sufficient intensity. Of particular interest is the case of pumping such high gain dye lasers with picosecond duration pulses having power in the range of lo9 W. The dye inversion is then a wave traveling with a velocity approximately equal to that for the pump pulse. Since the length of the pumping pulse is very short, the emitted dye radiation is amplified only in the direction of propagation of the pumping pulse. This technique offers greater simplicity and ease than the method previously described. MACK [ 19691 observed superradiant traveling wave emission as described above for three dyes: DDI (1, 1'-diethyl 1-2, 2'dicarbocyanine iodide), cryptocyanine, and DTTC (3, 3'-diethylthiatricarbocyanine iodide), each dissolved in methanol. In the frame of reference moving with the emitted pulses, the growth of the energy density, W, per unit frequency interval per unit solid angle follows the relation

dW(z' t , -hvS(v)B AN(z, t ) dt

= hvS(v)AN,(z, t ) ,

(3.38)

where S ( Y )is the lineshape function, A and B are the Einstein spontaneous and stimulated emission coefficients per unit solid angle,

11,

§ 41

57

MEASUREMENT TECHNIQUES

respectively, for the given polarization and N , and AN are the excited state population and inversion in the dye, respectively. Saturation effects are ignored in eq. (3.38). In general N , and AN are functions both of position ( 2 ) in the moving frame and of time ( t ) . However, if the difference in group velocities of the emitted dye laser pulse and the incident pump is ignored, N , and AN are not explicitly dependent on time and equation (3.38) can be integrated. The results of pulse narrowing as a function of gain G are shown in Fig. 12 for an excited state population of the form N2(tO) = NCexp(--to/z,)

-exP(--to/~r)h

(3.39)

The position ( 2 ) was replaced in eq. (3.39) by to = Z/D, where ZI is the common group velocity, tfis the fluorescence decay time, and t, is

to ( p sec)

Fig. 12. Pulse narrowing of the fluorescence of a dye laser as a function of gain (see MACK[1969]).

the population risetime. For Fig. 12, tf = 50 psec and tr = 2 psec were assumed. These values are representative for the polymethine cyanine dyes. Figure 12 indicates that a fairly modest gain will initially give a substantial pulse sharpening. However, once the dye laser pulse duration becomes comparable to the pumping pulse duration, a considerable increase in gain is required to achieve even a slight decrease in pulse duration.

Q 4. Measurement Techniques for Picosecond Laser Pulses 4 . 1 . INTRODUCTION

Direct measurement of the duration of relatively long optical pulses is most often made by displaying, by means of an oscilloscope, the

58

PICOSECOND LASER PULSES

[II,

5

4

output of a suitable photodetector illuminated by the optical radiation. Since it is not expected that direct electronic techniques will be capable of measuring time durations down to l O - l 3 sec, new measuring techniques had to be found for the measurement of the time duration of picosecond laser pulses. It will be shown in this section that any linear interferometer measures the autocorrelation function of the pulse amplitude. Since the power density spectrum and the amplitude autocorrelation function are a Fourier transform pair, knowledge of one uniquely specifies the other. The power density spectrum is usually measured with a spectrometer. The linear interferometer can also be used to measure the coherence length AD and the coherence time At,; these two parameters are related to the spectral bandwidth Am and to each other by the following relationships:

Since the relation &OAT, >= 2n provides only a lower limit to a pulse duration for a given Aw, it can be concluded that amplitude autocorrelation measurement taken with a linear optical system can provide only information establishing a lower limit to the time duration of a pulse. It will also be shown that measurements taken with nonlinear optical instruments can provide information for determining the actual time duration of a pulse. Nonlinear optical instruments measure the autocorrelation of the pulse intensity. 4.2. LINEAR OPTICAL PULSE-WIDTH MEASUREMENT TECHNIQUES

For purposes of illustration, let us consider a linear optical instrument such as the Michelson interferometer diagrammed in Fig. 13. An incident pulse having an amplitude E ( t ) is split into two pulses, each with an amplitude E (t)/@?. Each of these pulses is made to traverse a separate orthogonal arm of the interferometer. After traversing their respective paths 2 0 , and 2D,, the pulses are recombined on a square law detector such as a photographic plate or photodetector. If D, # D,,the pulses can be represented by (4.2)

11,

P

41

59

MEASUREMENT TECHNIQUES

SQUARE LAW DETECTOR

Fig. 13. A typical linear optical instrument using a Michelson interferometer.

and

where t = 2(D,-D,)/c and Eo(t)is the slowly varying envelope of the pulse with respect to o.The intensity incident on the detector is given by I ( t , z) = *lE0(t)+Eo(t--t)12. (4.4) The response of the detector is assumed to be slow compared to the pulse duration or the delay time z, so the output signal S ( t ) of the detector is given by

where W is the pulse energy and A (7)is the autocorrelation function of the pulse amplitude, i.e.,

j-,E;(t) dt 00

W=

(4.6)

60

PICOSECOND LASER PULSES

and [m

A ( t ) = J -m

E,(t) E,(t-t)dt rm

(4.7)

J -m

When t = 0, then S ( t ) / W= 2, and when t is large enough so that no overlap between E,(t) and E,(t) exists, then S ( z ) / W = 1 . From Fourier analysis, we find that

where ] P ( u )is/ ~the power density spectrum of the original laser pulse (DAVENPORT and ROOT[1958]). Since the power density spectrum and the amplitude autocorrelation function are a Fourier transform pair, knowledge of one uniquely specifies the other. The power density spectrum is the quantity measured with a spectrometer and the amplitude autocorrelation is the quantity measured with an interferometer. The two results are essentially equivalent and the equivalence arises from the fact that both instruments are linear optical systems. Similar considerations hold for any linear optical instrument. It is clear from Fig. 1 3 that an interference pattern will be observed in the plane of the detector, if the interfering pulses have a relative retardation in time equal to z = 0, 2L/c, 4L/c, . . ., i.e., if the difference in length of the interferometer arms is D,-D, = 0, L , 2L, . . ., where L is the separation between the laser reflectors. The visibility of the interference pattern changes with the relative delay time of the pulses; it is a maximum at D,--D, = 0, L , 2L, . . ., and gradually falls to zero in moving away from these points. The difference in path lengths, AD, over which the interference pattern is visible is called the coherence length. The coherence length and the spectral bandwidth of the pulse are related by 2ADlc = 2nlAw = At,, where Atc is the coherence time of the pulse. The coherence time At, will equal the duration of the pulse only in the special case where the entire spectral content of the pulse is due to the short duration of its envelope. In other words, the relation Am A t 2 2n provides only a lower limit to the pulse duration. Figure 14 schematically summarizes the results of measurements with a linear optical instrument for pulses having a pulse-width (At) bandwidth (Av) relationship given by A t = Arc = l / A v and by A t >> l/Av.

MEASUREMENT TECHNIQUES

61

Fig. 14. For a pulse (a) having a bandwi th Av determined by the width of the pu se envelope At, = I/Av, a linear optical instrument gives a signal-to-background ratio (b) of 2 for t = 0 and unity for t >> Arc. For a pulse (c) having the characteristic AT z l/Av, the identical result is obtained for the contrast ratio and pulsewidth measurement. The instrument cannot distinguish between the two cases.

The spectral width of the pulses emitted by a passively switched and mode-locked Nd3+: glass laser has been measured and found to be typically 120 A per pulse. The coherence length of the pulses was measured to be typically 6 x cm per pulse. 4.3. NONLINEAR OPTICAL PULSE-WIDTH MEASUREMENT TECHNIQUES

4.3.1. Simple technique

In the preceding part of this section, it was noted that linear optical systems can yield only a lower limit on the pulse width of optical pulses, whereas measurements performed with nonlinear optical instruments can yield the actual pulse width. The manner in which a nonlinear optical system can perform a true measurement of a pulse can be explained as follows. Suppose one passes the two output pulses E,(t) and E,(t) from the linear interferometer of Fig. 13 through a

62

PICOSECOND LASER PULSES

[II, §

4

nonlinear optical crystal. The second harmonic output from the nonlinear optical crystal will be given by

2"E(t) = [E1(t)+E2(t-z)]2

(4.9)

if one neglects the constant representing the second harmonic generation efficiency of the crystal. The output signal S(z) from a detector having a slow response time with respect to w and z is given by 00

S(z)

I2"E(t)l2dt= ""w(l+2G(z))

(4.10)

where 3"W is the second harmonic pulse energy and G(z) is the autocorrelation function of the pulse intensity, i.e.,

J-, E4(t)dt 00

20W =

(4.11)

and cn

E 2 ( t )E 2 ( t - z ) dt G(t)=

I-, 00

(4.12)

E4(4dt

When t = 0, S(z)/2"W = 3, and when z is large enough so that no overlap between El(t) and E,(t) exists, S ( Z ) / ~ " = W 1. Measurement of the variation of S(z)/2"W as a function of t gives the time duration over which the energy of the pulse is distributed. Figure 15 schematically summarizes the results of measurements with a nonlinear optical instrument for the two basic types of pulses Az = At, = l / A v and A t > > IIAv. 4.3.2. Zero background technique

A nonlinear interferometer can also be constructed in which S(z)/"W = 0 when t is large enough so that no overlap between E,(t) and E,(t) exists. Suppose one passes an optical pulse through a birefringent crystal of length L with the optical polarization 45" with respect to the optic axis (see Fig. 16). The crystal will resolve the single pulse into two pulses of equal amplitude and orthogonal polarization. One polarization component of the pulse propagates as an ordinary ray and the other as an extraordinary ray. For a crystal of thickness L, the delay t so introduced is given by z = L(n,-n,)/c, where ne and no are the extraordinary and ordinary indices of refraction. For

11,

§ 41

63

MEASUREMENT TECHNIQUES

Fig. 15. For a pulse (a) having a bandwidth Av determined by the width of the pulse At, = l/Av, a nonlinear optical instrument gives a signal-to-background ratio (b) of 3 for t = 0 and unity for t >> At, and a true pulse-width measurement. For a pulse (c) A t > l/Av, a measurement of the true pulse width A t can be obtained. The shaded area of (b) and (d) represent interference fringes. It is important t o remember that t h e contrast ratio measurement is also important in such measurements as explained in the text.

calcite, the delay is 0.59 psec/mm at il = 1.06. The two orthogonal polarized pulses are then caused to interact in a nonlinear optical crystal, in this case ADP, to produce second harmonic radiation a t 5300 A. The orientation of the nonlinear crystal is chosen such that no second harmonic is produced by either component of the pulse

I

I

I

I

SHG

OPTld AXIS

Fig. 16. Schematic experimental arrangement for one method for measuring picosecond laser pulses with zero background.

64

PICOSECOND LASER PULSES

[II,

§ 4

acting alone, but only when both are present. The method thus, measures the "overlap" of the pulse with a delayed replica of the pulse. When a delay is introduced such that no overlap of the two pulses occurs, the second harmonic drops to zero. For ADP, the angle at which the second harmonic is proportional to the product of two orthogonal polarizations was reported by WEBER [1967] to be 61' 12'. This angle produces phase matching for this combination of vertical and horizontal polarization and produces second harmonic radiation with horizontal polarization. If we let input signal Eito the ADP crystal be represented by

E (t) = EH(t)ei"'+E,(t--t)ei"('-")

(4.13)

the second harmonic signal 2mEH(t) is given by

2WEH(t) = E H ( t E,(t-t) ) eiW(2'-")

(4.14)

where EH(t) and E,(t-t) are the slow-varying envelope of the horizontal and vertical polarized electric fields with respect to w . The constant representing the second harmonic generating efficiency has also been neglected. If t >> At, then 2WEH = 0, and if z = 0, then 2WEH= max. The output signal from a detector having a slow response time with respect to 2w is given by

=I-, m

S(t)

E k ( t )E $ ( ~ - - zdt. )

(4.15)

Since a separate experimental run is required for each data point and since each experimental run is independent of previous runs, normalization is required in the data processing. A convenient means of normalizing is to take a pulse of any polarization and let it alone produce second harmonic energy. This reference signal can be represented by eq. (4.11).The experimentally measured quantity is S ( T ) / ~ " W = I ( z ) as a function of z by selecting different lengths of delay crystals. = 1, and when z is large enough so that no When t = 0, .S(t)/2mW overlap between E H ( t )and E,(t--t) exists, S ( T ) / ~ " W = 0. Figure 17 schematically summarizes the results of measurements with the orthogonally polarized nonlinear optical instrument for the two basic types of pulses A t = Atc = l/Av and A T > l/Av. The nonlinear optical systems utilized in the measurement of picosecond pulses have been reported by KRASYUKet al. [1968], ARMSTRONG [1967], GLENNand BRIENZA [1967], and MAIER et al. [ 19661.

11,

§ 41

66

MEASUREMENT TECHNIQUES

I

c

t

(b)

Fig. 17. For a pulse (a) A t = l/Av, an orthogonally polarized nonlinear optical instrument gives an infinite signal-to-background ratio (b) for t = 0, zero for t >> At,-, and a true pulse-width measurement. For a pulse (c) A t > l/Av, a measurement of the true pulse width AT can be obtained with no background signal. The shaded areas in (b) and (d) represent interference fringes.

In the analysis of the nonlinear optical systems for the measurement of picosecond pulses, it has been assumed that phase matching in the nonlinear crystal is maintained over the entire bandwidth of the laser pulse. This sets an upper limit to the thickness of the crystal or a lower limit to the time resolution. ADP and KDP crystals having a thickness of 1 mm provide resolutions of approximately 1 psec. It is important to note that the group velocities of the fundamental and second harmonic frequencies must be approximately equal in order to have the envelope of the second harmonic pulse equal to the square of the fundamental pulse (GLENN [1969]). If the group velocities of the two harmonically related frequencies are not equal, the second harmonic pulse will have a flat top with a time duration L,/vg(2w)L,/v,(w), where L, is the crystal length, and v,(w) and v g ( 2 w )are the group velocities of the fundamental and second harmonic frequencies, respectively. The phase matching angles as a function of wavelength for KDP is relatively flat around A,, = 1.06 microns, therefore making it possible to phase match easily over several hundred Qngstroms.

66

P I C O S E C O N D LASER PULSES

[II,

s4

4.3.3. Two-photon absorption-fluorescence technique

GIORDMAINEet al. [ 19671 reported the two-photon absorptionfluorescence technique for the measurement of picosecond laser pulses. This technique is also an intensity-correlation system. Its simplicity is its major advantage over the second harmonic systems described previously. A schematic diagram of the two-photon absorption-fluorescence measurement experimental arrangement is illustrated by Fig. 18. The correlation of one pulse with succeeding pulses (crosscorrelation) illustrated by Fig. 18a, or the correlation of a pulse with itself (autocorrelation) illustrated by Fig. 18b, can be obtained. Both cross-correlation and autocorrelation experiments have been performed by the author and identical results were obtained in the two measurements. E2

I

(a) CAMERA

Fig. 18. Experimental arrangement for performing intensity cross-correlations and intensity auto-correlation measurements of picosecond pulses by the two-photon absorption-fluorescence technique.

If we consider the center of the cell to be the origin ( z = 0) and denote pulses approaching the cell from the left and right as sin(Kz-wt)

and

E,

sin(-kz-wt),

(4.16)

respectively, the intensity I,,,in the dye cell will then be given by

11.

s 41

MEASUREMENT TECHNIQUES

67

If we assume that the fluorescent intensity I, is proportional to the intensitysquared, I, cc I:, then the fluorescent intensity is given by

The photographic film essentially records the value of I, but it is time averaged through the photographic process and spatially averaged over several optical wavelengths due to the limited resolution of the film. The value recorded by the film is

(4.19)

The normalization of eq. (4.19) gives (4.20)

which is identical to eq. (4.10). If E,(t) = E,(t) at z = 0 is assumed, a bright vertical line will be recorded with a contrast ratio of 3 to 1 with respect to the background. This line will have a width AL = Arc/%, where AT is the pulse duration, c is the velocity of light, and n is the refractive index of the two-photon absorption-fluorescence dye. I t is important to realize that if a noise pulse having a time duration A T and a spectral bandwidth of Am, in which A T >> 2n/Aw, is passed through a two-photon absorption-fluorescencecell, a bright line having a length corresponding to the coherence time AT w 2n/Aw will be recorded. The evaluation of the pulse duration by this method requires caution because a very similar fluorescence structure is obtained from the radiation of an ideally mode-locked laser and from a freerunning laser with the same oscillating bandwidth (WEBER[l968]). The proper interpretation of the data depends strongly on the contrast ratio in the photographic record. For a bandwidth-limited short pulse (AT l/Av), the maximum ratio of eq. (4.20) is 3. For a freerunning laser, the maximum ratio of eq. (4.20) was calculated by Weber to be 1.5. Figure 19 presents typical data obtained by the two-photon ab-

-

68

PICOSECOND LASER PULSES

111, §

4

Fig. 19. Photographs of picosecond laser pulses taken with the two-photon absorptionfluorescence measurement technique.

sorption-fluorescence measurement technique. Figure 19a shows the bright fluorescent line due to the overlap of a pulse with itself or with its immediate predecessor. The fact that the bright line is the overlap point of two pulses was confirmed by the observation that a movement of the reflecting mirror in (a) or the dye cell in (b), through a given distance, caused the bright line to move a corresponding distance in the dye cell. Rhodamine 6G was utilized as the dye medium. In an ethanol solution this dye has its primary absorption peak at a wavelength that is quite close to the 5300 A second harmonic of the Nd: glass laser. In addition, it has another strong absorption line which peaks very close to the second harmonic of the ruby laser line. The two-photon absorption and subsequent fluorescence a t 5500 A of this dye is relatively much larger than for the 1, 2, 5 , 6-dibenzenthracene (DBA) dye previously used in such experiments. An additional advantage of rhodamine 6G is that the experiment can be performed by direct irradiation a t either 1.06 micron or 6943 A without requiring the conversion of these wavelengths into their second harmonic before irradiating the dye. The use of rhodamine

11,

5 41

MEASUREMENT TECHNIQUES

69

6G dye in this experimental technique has made possible the measurement of a single ultrashort pulse with only one firing of the laser. Under certain conditions multiple bright lines were obtained as shown in Figs. 19c and 19d. The optical path length between the bright lines corresponds to one-half of the actual pulse separation in the laser output. The time scales in Fig. 19 have taken this factor into account, and therefore represent the actual pulse separation in time. These multiple pulses are caused by mode selection within the laser bandwidth. The presence of a reflecting external mirror can act as a secondary cavity external to the main laser cavity, and gives rise to a channeled spectrum in the frequency domain or multiple pulses in the time domain (see Fig. 19c). Experiments were also performed with the insertion of optical flats into the laser cavity normal to the laser axis. Multiple bright lines were again obtained with the separation of the lines in time equal to twice the optical thickness of the flat in the laser (see Fig. 19d). Pulse widths as short as 2 psec and as long as 25 psec were recorded by this technique. It should be noted, however, that the photograph of Fig. 19 is an average over the entire pulse train and not the width characteristic of a single pulse in the train. WERER[1968] has noted the danger in drawing the conclusion that normally free-running or Q-switched wide-bandwidth lasers (such as ruby, Nd: glass, or YAG: Nd) consist of picosecond pulses on the basis of data obtained solely by the two-photon absorption-fluorescence measurement technique (DUGUAY et al. [1967]; BRASSand WOODWARD [1968]). He has pointed out that a unique assignment can be made only if the contrast ratio is known with great accuracy, and has shown that the fluorescence record from N modes of equal-amplitude, equal-frequency separation and random phase relationships yields a maximum contrast ratio of 1.5. For the same case, but with fixed phase relationships between the modes, a maximum contrast ratio of 3 is obtained with this experimental technique. This second case is the mode-locked case. Similar results have also been obtained by KLAUDER et al. [1968] and their measurement of the contrast ratio of the modelocked pulses from glass: Nd3+yielded 2 instead of 3. The simplicity of the two-photon absorption-fluorescence technique is one of its major advantages if a measurement of the contrast ratio does not have to be taken. If the pulse-width measurement obtained by this technique is greater than the inverse of the spectra bandwidth of the pulse, then the measured width is the actual pulse width and a

70

PICOSECOND LASER PULSES

[XI

contrast ratio measurement does not have to be performed. If a periodic pulse train consisting of subnanosecond laser pulses having zero energy between the pulses are displayed on an oscilloscope, then the measurement of the pulse width obtained from the two-photon absorptionfluorescence can be considered to be the true pulse width with a high degree of certainty without the contrast ratio measurement.

References ARMSTRONG, J. A., 1967, Appl. Phys. Letters 10, 16. ARMSTRONG, J. A. and E. COURTENS, 1969, IEEE J . of Quant. Elect. QE-5, 249. BASS,M. and D. WOODWARD, 1968, Appl. Phys. Letters 12, 275. BELLMAN, R., G. BIRNBAUM and W. G. WAGNER, 1963, J. Appl. Phys. 34, 780. COOK,G. E. and M. BERNFELD, 1967, Radar Signals (Academic Press). CUBEDDU,R. R. and 0. SVELTO, 1969, IEEE J. of Quant. Elect. QE-5, 495. CUTLER,C. C., 1955, Proc. IRE 43, 140. DAVENPORT Jr., W. B. and W. L. ROOT,1958, An Induction to the Theory of Random Signals and Noise (McGraw-Hill,New York) p. 87. DEMARIA, A. J., W. H. GLENNJr., M. J. BRIENZA and M. E. MACK,1969, Proc. IEEE 57, 1. A . J., D. A. STETSER and W. H. GLENNJr., 1967, Science 156, 1557. DEMARIA, DEMARIA, A. J. and C. MILLER,1968, J. Opt. SOC.Am. 58, 467. DUGAY, M. A. and J. W. HANSEN, 1969, Appl. Phys. Letters 14, 14. DUGUAY, M. A,, S. L. SHAPIRO and P. M. RENTZEPIS, 1967, Phys. Rev. Letters 19, 1014.

FISHER, R. A,, P. L. KELLEY and T. K. GUSTAFSON, 1969, Appl. Phys. Letters 14, 140.

Fox, A. G. and T. LI, 1961, Bell System. Tech. J. 40, 453. FRANTZ, L. M. and J. S. NODNIK, 1963, J. Appl. Phys. 34, 2346. GIORDMAINE, J. A,, M. A. DUGUAY and J. W. HANSEN, 1968, IEEE J . of Quant. Elect. QE-4, 252. J. A,, P. M. RENTZEPIS, S. L. SHAPIRO and K. W. WECHT,1967, GIORDMAINE, Appl. Phys. Letters 11, 216. GIRES,F. and P. TOURNOIS, 1964, C. R. Acad. Sc. Paris 258, 6112. GLENN,W. H., 1969, IEEE J. of Quant. Elect. QE-5, 284. GLENN, W. H. andM. J. BRIENZA, 1967, Appl. Phys. Letters 10, 221. GLENN, W. H., M. J. BRIENZA and A. J. DEMARIA, 1968, Appl. Phys. Letters 12, 54.

GRIFFIN,D. R., 1958, Listening in the Dark (Yale University Press). HARGROVE, L. W., R. L. FORK and M. A. POLLACK, 1964, Appl. Phys. Letters 5, 4.

HELLWARTH, R. W., 1961, Advances in Quantum Electronics, ed. J. R. Singer (Columbia University Press) p. 334. HELLWARTH, R. W., 1966, Lasers, ed. A. K. Levine (Marcel Dekker, Inc.) p. 253. HERCHER, M., 1967, Appl. Opt. 6 , 947. KLAUDER, J. R., 1960, Bell. Syst. Tech. J. 3 9 , 809.

111

REFERENCES

71

KLAUDER, J . R., M. A. DUGUAY, J. A. GIORDMAINE and S. L. SHAPIRO, 1968, Appl. Phys. Letters 13, 174. KLAUDER, J. R., A. C. PRICE, S. DARLINGTON and W. J. ALBERSHEIM, 1960, Bell Syst. Tech. J . 39, 745. KRASYUK, I. K., P. P. PASHKIN and A. M. PROKHOROV, 1968, Soviet Phys. JETP Letters 7, 89. LAMB Jr., W. E., Phys. Rev. 134A, 1429. LETOKHOV, V. S. and V. N. MOROZOV, 1967, Soviet Phys. JETP 25, 862. MACK,M., 1969, Appl. Phys. Letters 15, 166. MAIER,M., W. KAISER and J. A. GIORDMAINE, 1966, Phys. Rev. Letters 17, 1275. MCCLUNG,F. H. and R. W. HELLSWARTH, 1962, J. Appl. Phys. 33, 828. MCDUFFand S. E. HARRIS, 1967, IEEE J. of Quant. Elect. QE-3, 101. TREACY, E. B., 1968a, Appl. Phys. Letters 14, 112. TREACY, E. B., 1968b, Phys. Letters 2 8 A , 34. TREACY, E. B., 1968c, Proc. IEEE 56, 2053. TREACY, E. B., 1969, IEEE J. of Quant. Elect. QE-5, 454. WEBER,H. P., 1967, J. Appl. Phys. 38, 2231. WEBER,H. P., 1968, Phys. Letters 27A, 321.

This Page Intentionally Left Blank

OPTICAL PROPAGATION THROUGH THE TURBULENT ATMOSPHERE BY

J O H N W. S T R O H B E H N Radiophysics Laboratory, Thayer School of Engineering, Dartmouth College, Hanover, N.H., U S A

CONTENTS

PAGE

4

1.

4 4

2.

3.

. . . . . . . . . . . . . . . . . TURBULENCE I N T H E ATMOSPHERE . . . . . .

INTRODUCTION

SPECTRA F O R AMPLITUDE AND PHASE FLUCTUATIONS.. . . . . . . . . . . . . . . . . . . . .

75 77

82

4 4. COMPARISON BETWEEN T H E O R Y AND E X P E R I -

. . . . . . . . . . . . . . . . . . 4 5. RENORMALIZATION METHODS . . . . . . . . . 5 6. EXTENSIONS O F THE PLANE WAVE RESULTS . . 4 7 . APPLICATIONS . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . MENT . . . .

92 100 104 116 119

Q 1. Introduction With the recent invention of the laser and the rapid advances in optical technology, there has been renewed interest in optical propagation over line-of-sight paths through the atmosphere. We wish to distinguish at the outset between two problems with similar names but very different approaches and results. There is a distinction between “propagation through turbid media” and “propagation through turbulent media”. The distinction between the two words “turbid” and “turbulent” is related to the size and character of the scattering mechanism relative to a wavelength. In a turbid medium the scatterers are roughly the size of a wavelength or smaller or have sharp discontinuities in refractive index compared to a wavelength. For example, the propagation of a light wave through clouds or the scattering of light by aerosols falls into this class. In contrast, propagation through a turbulent medium refers to those cases where the wavelength of the optical wave is much smaller than a typical distance over which the refractive index varies. This condition is met by the refractive index variations in the atmosphere caused by temperature fluctuations. These fluctuations cause the familiar “twinkling” of stars and “shimmering of heat” over hot pavement. It is this latter problem with which this article is concerned. We are interested, therefore, in the effects of temperature fluctuations in the atmosphere on optical waves. Since these temperature fluctuations are only significant in the troposphere (the lower 10 km of the atmosphere), our interest is confined to this region. Furthermore, we will only discuss random scattering and not the problem of refraction, e.g. the bending of a wave caused by the decrease of the mean value of the refractive index with height. The random fluctuations in the refractive index cause variations in the amplitude, phase, angleof-arrival, and polarization of the optical wave. The problem in which we are interested is to calculate statistical descriptions of the parameters of the optical wave in terms of statistical descriptions of the random medium through which the optical wave passes. Though this 75

76

ATMOSPHERIC PROPAGATION

;Ill*

1

article is concerned with optical propagation through the atmosphere, many of the results can be directly applied to similar problems such as acoustical propagation in the atmosphere or ocean. Astronomers were among the first to be interested in the effects of the atmosphere on optical waves, since this is one of the controlling factors in determining the limitations of optical telescopes. CHANDRASEKHAR [19521 calculated the phase and angle-of-arrival fluctuations of starlight resulting from atmospheric turbulence. He used a geometrical optics approximation and assumed the major effects were from a turbulent layer at about four km. The contributions of other workers, particularly in other fields, has been given recently (STROHBEHN [1968]) and will not be repeated here. KELLER[1962, 19641 and HOFFMAN [19641 give some excellent discussions of the mathematical approaches to this problem. The translations of the two monographs by CHERNOV[1960] and [1961] made available two complete treatments of the TATARSKI problem of wave propagation in random media. Chernov calculated the angular distribution function of an optical ray and its displacement from its original direction using a method based on applying the Einstein-Fokker-Kolmogorov equation. In this method the propagation of a ray is regarded as a continuous Markov process. In this monograph amplitude and phase correlations are also calculated using the method of small perturbations applied to the wave equation and including diffraction effects. In this work an approach is followed that was originally applied by OBUKHOV [1953] to the problem of wave propagation in random media. This method, commonly called Rytov’s method, consists of applying the method of small perturbations to the Riccati equation, which arises from making a logarithmic transformation of the scalar wave equation. RYTOV[1937] had used this method when concerned with the problem of diffraction of light by an ultrasonic grating. The most important difference in the work of Tatarski was the application of spectral expansions (Fourier transform techniques) to the differential equations. As a result he was able to find algebraic expressions relating the phase and amplitude spectra (i.e. the Fourier transform of the phase and amplitude covariance functions) to the spectrum describing the random medium. In a recent book TATARSKI [1967] has presented much of his earlier work with a great deal more attention to detail, particularly to the approximations and limitations involved when applying the method of geometrical optics or Rytov’s method.

111,

§ 21

TURBULENCE IN THE ATMOSPHERE

77

The major theoretical problem is that the early approaches using geometrical optics or Rytov’s method, based on a perturbation theory, are not valid for strong fluctuations in the wave. However, over reasonably short horizontal paths (sometimes as short as 500 meters) during conditions of strong fluctuations in refractive index, the amplitude fluctuations of the optical wave are no longer small. Under these conditions the perturbation theory is not adequate, and efforts at including higher order terms have not been successful. I n an attempt to circumvent some of these limitations, Tatarski’s more recent book describes several new approaches, including an extension of geometrical optics and application of the renormalization techniques of quantum field theory. These attempts have not been entirely satisfactory, and as this article is being prepared a series of papers by Tatarski and Klyatskina are appearing in the Soviet Journal of Theoretical and Experimental Physics (JETP). Good experimental work in the area of optical propagation through random media has been extremely meager. Though there have been numerous measurements of various optical parameters, both by astronomers and others, these optical measurements have rarely been made in conjunction with thorough meteorological measurements. As a result the parameters of the random medium have not been known well enough to permit careful quantitative comparison between theory and experiment. Though such measurements are difficult or impossible in certain types of experiments, it was through such measurements that the limitations in earlier theories were first revealed (GRACHEVA and GURVICH[1965]; GRACHEVA[1967]). Further work in this direction is still needed.

Q 2. Turbulence in the Atmosphere Fluctuations in an optical wave are caused by random variations in the refractive index of the atmosphere. The refractive index is a random function of both space and time and may be written as

n ( r , t ) = no+%(r, t )

no = ( 4 r , t ) ) ,

where the brackets denote an ensemble average. If we make the reasonable assumption that the time variations in the refractive index are slow compared to one period of an optical wave, then it may be shown that the time variations in the refractive index can be suppressed, and only the spatial variations are important.

78

ATMOSPHERIC PROPAGATION

[IIL

§ 2

One of the unavoidable complications in considering the atmosphere is that it is not statistically homogeneous, i.e. whatever conditions exist at one point in the atmosphere do not generally exist at another, even in a statistical or average sense. However, as will be discussed below, the only situation in which much progress has been made is when the atmosphere is considered homogeneous. Under this assumption the autocovariance function of the refractive index, B n ( r ) ,and , be defined as its spectrum, Q n ( ~ ) may

where K is a spatial wavenumber. We will make the additional assumption that the medium is statistically isotropic, though this assumption is really not needed in the development. In this case

B*(r) = -

Q,(K)K

GK J, B,(r)r sin (Kr)dr. 1

Qnk)

=

sin ( ~ rdtc )

nca

In actual fact, the condition of statistical homogeneity can be relaxed somewhat and some authors have used the concepts of structure functions and locally homogeneous fields. This approach establishes weaker conditions on the form of the spectrum but, because of space limitations, will not be followed here. Detailed descriptions of this approach are given by TATARSKI [1961, 19671 and a summary of the [1968]. pertinent formulas is given by STROHBEHN At optical wavelengths the refractive index can be related to meteorological parameters through the relation (STRAITON [ 19641)

N = (a-1) x lo6 = 77.6 P,,/T

(2)

where n is the refractive index, P,, is the partial pressure of dry air in millibar, and T is the temperature in degree Kelvin. Since pressure

111,

§ 21

TURBULENCE IN THE ATMOSPHERE

79

changes are unimportant, and temperature variations about the mean value are small, the fluctuations in refractive index may be written as

The temperature fluctuations normally originate from large scale phenomena such as heating of the earth’s surface, but are broken and mixed by the wind until temperature fluctuations exist at all scales. Since the velocity fluctuations control the temperature variations, a description of the velocity fluctuations will be given first. Despite the fact that atmospheric conditions vary widely, the only case for which there is a well developed model for the atmosphere is for the case of homogeneous isotropic turbulence, i.e. the case of well-mixed random [1961]) that fluctuations in velocity. It has been argued (TATARSKI when the conditions for homogeneous turbulence exist, the form of the refractive index spectrum is the same as for a correctly defined velocity spectrum. However, strong mechanical turbulence may result in a well mixed atmosphere having weak temperature fluctuations, while a stable atmosphere with little wind may give rise to strong temperature fluctuations (BEAN[1969]), therefore it is only the form of the two spectra that are related. Further work on these relations is still needed. Detailed descriptions of homogeneous turbulence are presented [1959] and LUMLEY and PANOFSKY in the monographs by BATCHELOR [1964]. A more concise summary, but with particular attention to the aspects of interest in propagation, is given in the two books by TATARSKI [1961, 19671. Since our primary concern is in relating the parameters of the optical wave to the parameters describing the refractive index, we will only summarize the most pertinent results of turbulence studies. The covariance tensor of the velocity fluctuations is defined as

where i and j represent unit vectors in the three orthogonal directions, and the primes denote fluctuations about the mean. The spectral tensor of the velocity field is defined by

80

ATMOSPHERIC PROPAGATION

[III,

p

2

It can be shown that the form of the spectral tensor is given by

where E ( K ) represents the spectral density in three-dimensional wave vector space of the distribution of the velocity fluctuations. The following is an intuitive description for the form of the spectral density, E ( K )(see Fig. 1). With a wavenumber, K , is associated a scale ~ . major sources of energy input in the size, I, given by 1 = 2 ~ 1 The atmosphere are from large scale phenomena such as wind shear or convective heating from the ground. Obviously this energy input is determined by local conditions such as meteorological factors, the surface geography, and the height from the earth's surface. In many circumstances the energy producing eddies will be at scales sizes larger than some minimum value, Lo, called the outer scale of turbulence, which corresponds to a wavenumber, K~ = 2n/L0. For wavenumbers smaller than K~ the form of the energy spectrum cannot be determined since it depends on local conditions. It is very likely that the turbulence is neither homogeneous nor isotropic in this range. Near the ground the value of the outer scale is thought to be on the order of the height above the ground. In the upper atmosphere Lo may be on the order of 100 meters or more, but if there is strong stratification, e.g. in an inversion layer, it may be on the order of a few meters. I

Fig. 1. Three-dimensional spectrum of the refractive index fluctuations (from STROHBEHN [1968]).

111,

5 21

T U R B U L E N C E IN T H E A T M O S P H E R E

81

At spatial wavenumbers above K, the spectrum, E ( K ) ,is better understood. If the conditions for homogeneous turbulence are met, then both theory and experiments have led to a K-* wavenumber dependence. This form, originally predicted by KOLMOGOROV [ 1941a, b] and OBUKHOV [1941], is based on a number of assumptions: the energy is input at much smaller wavenumbers and dissipated at much larger, the Reynolds number must be much greater than 1, and buoyancy effecrs are negligible. In this wavenumber range, called the inertial subrange, the Reynolds number of the large eddies is sufficiently high that the eddies are unstable, break up, and transfer energy to smaller scales. This process continues to smaller and smaller scale sizes (higher wavenumbers) until a scale size is reached where the Reynolds number is no longer greater than 1. For scale sizes smaller than this critical size, I,, called the inner scale of turbulence, the energy is dissipated into heat. In the viscous dissipation region, i.e. for K > K, = 2n/l,, the form of the spectrum is not well understood. It is known that the energy in the spectrum falls off much more rapidly than in the inertial range, and an exp(--aK2) form has been suggested. The inner scale of turbulence is on the order of 1 to 10 mm. Assuming that the form of the temperature spectrum, and hence the refractive index spectrum, is the same as for the velocity spectrum, E ( K ) ;TATARSKI [1967] has used the following form for the refractive index spectrum Qn(K)

= 0.033

c:

K-*

eXp(-K'/K:),

(3)

with the following conditions: (1) .,lo = 5.92, (2) the equation is a poor approximation for K 5 K,,, and (3) the equation is a reasonable approximation for K 2 K,. The constant C:, called the structure constant, is directly related to the intensity of the refractive index fluctuations, and typically ranges between 1O-l' to 10-14 m a . Because of the singularity at K = 0, the above spectrum does not have an autocovariance function, and is normally used in conjunction with structure functions and locally homogeneous fields, which were mentioned earlier. In order to avoid difficulties with the singularity at the origin, the following form for the spectrum has also been used (STROHBEHN [1968]) Q n ( ~ )=

0.063 (n:) Lz

eXp (- K 2 / K : ) (1 ~ 2 L ; ) l l ' ~

+

(4)

where (n:) is the variance of the refractive index fluctuations. The

82

ATMOSPHERIC PROPAGATION

-111.

3

two forms of the spectrum agree at large wavenumbers provided that C i w 1.9 (n4) Lot.

Before leaving this discussion a few remarks need to be made. First, the above spectra are reasonable approximations or models for the atmosphere under reasonably special conditions. At the present time there does not seem to be enough data concerning the range of variability of the spectral models nor their frequency of occurrence. It has been predicted (BOLGIANO [1960]) that when buoyancy effects are important, the form of the spectrum may change from K-11’3 to K-17’5, particularly at the lower wavenumbers. There have been some studies of the effects of different atmospheric models on different optical quantities (STROHBEHN [1966, 1968, 19701; GURVICH[1968]). The second point is that there is one atmospheric model which, though used extensively, should be discouraged. This is the gaussian model, with covariance and spectral density functions given below

B,(r)

= (n:)

exp (-r2/r:)

There are two problemswith these relations in modeling the atmosphere. One is that there is only one scale size while there are needed at least two. For example, as will be shown later, phase fluctuations tend to be dominated by large scale fluctuations, i.e. by the region of the spectrum near K ~ In . contrast, amplitude fluctuations on very short paths are dominated by small scale fluctuations on the order of I,, or the spectral region near K,. If the gaussian model is used, however, both amplitude and phase fluctuations seem to be influenced by the same scale size, ro, and it is not clear whether ro should be on the order of a millimeter or greater than a meter. The second problem with the gaussian form is that the spectrum is relatively flat out to a wavenumber on the order of K = 2n/r, and then falls off extremely rapidly. This very fast decrease with wavenumber tends to mask other effects which may be important when more realistic models are considered. These problems can give rise to misleading or erroneous predictions when the gaussian model is used.

Q 3. Spectra for Amplitude and Phase Fluctuations Because the refractive index of the atmosphere is a random function of both space and time, when an optical wave is propagated through

111,

S 31

83

SPECTRA FOR AMPLITUDE

the atmosphere random fluctuations are induced in its phase, amplitude and other parameters. Because of the random nature of the problem, statistical averages of different optical parameters are calculated in terms of averages of the refractive index field. In this section we will concentrate on the variance and spatial covariance function of the amplitude and phase fluctuations. However, the spectrum of these two quantities will be calculated, and transform relations used to find the covariance functions. 3.1. SPECTRAL EXPANSIONS AND TRANSFORM RELATIONS

Before the electromagnetic problem is presented, it is necessary to discuss in some detail representations of random fields. A homogeneous random field can be represented in spectral form by a Fourier-Stieltjes integral (YAGLOM [1962]) "1

(r) =

/sty 1exp ( i - ~r ) (drc), rl

(6)

--oJ

where T(dic) is the random amplitude associated with nl(r). The random amplitudes satisfy the following relation when averaged

where cDn (rc) is the three-dimensional spectral density function which has already been introduced in eq. (1). The basic relations between three-dimensional covariance functions and spectral density functions have already been given in 5 2 for the refractive index under the assumption that the refractive index field is statistically homogeneous and isotropic. For some applications it is desirable to make a spectral expansion of a random field in only two dimensions, even if it is homogeneous in all three. In this case n,(x,

-

Y,2) = /-l/exp

(irc p ) v(drc, x),

drc = dKydKZ.

(8)

The random amplitudes v(drc, x) satisfy the relation (Y(dK1, X) v*(dK2, x'))

= S ( K ~ - K ~ ) F , (xK- ~ %, ' ) d ~ l d ~ 2 ,

(9)

where Fn(rc,x) is an even function of x. F,(K~, K,, x) is related to @, ( K ~K ,~ K,) , through the Fourier-transform relations m ~ n ( K y tK z ,

x) = /-m cos

(KEX) @n(Kzj Ky,

Kz)dKn:,

POa)

84

[III. §

ATMOSPHERIC PROPAGATION

3

m

Qn(K,,

Ky, K z )

j

= -

2n

cos ("2) F,(Ky, K z , x)dx.

-m

(lob)

It can be shown (TATARSKI [196l]) that F,(K,x) falls off to zero very rapidly when I K I X > 1. This fact permits numerous approximations when considering the electromagnetic problem. When an optical wave is propagated through a random medium it is apparent that the statistics of the received field will not be (in general) statistically homogeneous. Furthermore, assuming the random medium is homogeneous, the direction of propagation of the wave is a unique or preferred direction. For example, assume an infinite plane wave propagating in the x-direction is incident on a homogeneous random medium in the half-space x > 0. Then the statistics of some optical parameter, such as the amplitude fluctuations, is probably different at a point r = ( L ,y, x) than it is a point r = (2L, y , 2). However, because of the symmetry of the problem, the statistics of the fluctuations should be the same at any point in the plane x = L , i.e. the fluctuations are homogeneous in any plane x = const. Since the optical fluctuations are homogeneous in the plane x = const. but not in all three coordinates, covariance functions and spectra can only be defined in two dimensions. If f ( r )is a quantity that is homogeneous in the plane x = const. but not in all three dimensions, then the form of the spectral expansion given in (8) is valid, but that in (6) is not. Eq. (9) is correct provided x' = x, but obviously eq. (10) no longer holds. The two-dimensional covariance function is defined as B,(O>P )

=

(f(% PI) f ( x , P I + P ) ) >

P =

(Y,x)

(11)

and is related to the two-dimensional spectrum F ( K ,0) by

-I:/

1

F,(K,

0) =

B,(O, p ) =

B,(O, p)e-i"'P dp,

(2nd)

K

=

(Ky, K Z )

(12a)

Im / F,(K,

0)e'"'P drc.

-m

If the fluctuations are isotropic in the plane x relations reduce to 1

nm

rm

= const. the above

111,

§ 31

SPECTRA FOR AMPLITUDE

85

3.2. RYTOV’S METHOD

Detailed derivations of the covariance functions for amplitude and phase fluctuations have been given in the monographs of CHERNOV [1960] and TATARSKI [1961, 19671. In Chernov’s work the covariance functions are solved for directly, but Tatarski has first solved for the spectra of the amplitude and phase fluctuations. The covariance functions may then be found from the relations given in 5 3.1. The following derivation is a composite of several different approaches, but is most closely associated with the work of TATARSKI [196l, 19671. However, his derivation is considerably shortened by using a variation of a technique suggested by CLIFFORD [1969] (CLIFFORD and STROHBEHN 119701). If we assume we are concerned with an atmosphere in which there are no free charges, the magnetic permeability is constant and that the time dependence of the electric field is given by exp (-id), then the vector wave equation for the electric field becomes

V2E+k2fi2E+2V(E- V log f i ) = 0 ,

(14)

where k = 2n/il, and il is the wavelength of the radiated wave. Since the last term contains the interaction between orthogonal components of the field, it gives rise to depolarization effects. Since the wavelength of light, A, is much smaller than the scale size associated with the smallest inhomogeneities, Z, it can be argued that the last term may be neglected. The effect of this term has been calculated theoretically (STROHBEHN and CLIFFORD [1967]) and it has been predicted that the depolarized component is typically 160 db weaker than the incident wave (STROHBEHN [1968]). SALEH [1967] attempted to measure the depolarized signal experimentally and was unsuccessful. Dropping the last term in (14), assuming the wave is polarized in the z-direction, propagating in the x-direction, and letting E = E,, we have V2E+k2n2E = 0. (15) In Rytov’s method E = exp ( y ) and the scalar wave equation transforms to the Riccati equation

-

V2y(r)+Vy(r) Vy(r)+k2n2(r= ) 0.

(16)

If E = A exp (is) where A and S are the amplitude and phase of the wave, then y = XfiS, where x = In A . The imaginary part of y is the phase and the real part is X, which we will call the log-amplitude.

86

ATMOSPHERIC PROPAGATION

[III,

s:

3

Rigorously, a perturbation expansion should be performed on eq. (16) (BROWN[1966], TATARSKI [1967], STROHBEHN [1968]). The results we desire come from the first order terms, and their range of validity can be estimated by calculating the second order terms. Define n ( r ) = no+nl(r), no = < n ( r ) > , and Y ( r ) = YO(')+Yl(')> where we are only keeping first order terms. By equating terms of the same order (16) becomes

-

V2y,+Vy, Vy,+k2ni = 0,

V 2 y , + 2 ( V y , * Vy,)

=

(174

-2k2n,n,.

(17b)

Since no FV 1, it will be set equal to 1 from here on. The first equation corresponds to the free space wave equation and is easily solved. Solving the second equation is our primary concern. Tatarski goes through numerous approximations and transformations to reduce (17b) to a single first-order differential equation. We will work with equation (17b) directly and make the approximations later, an approach that seems more straight-forward. Since y l ( r ) is a random variable which, by virtue of the assumptions of an incident plane wave and a homogeneous medium, should be homogeneous in the plane x = const., it may be written as a two-dimensional spectral expansion

[ i *PI ~ e(dx, x),

r

=

(x,P) = (x,Y , 2).

(18)

A similar expansion may be written for n l ( r ) n l ( r )=

Im

1 e x p [ix * p ] v(dK, x).

(19)

-W

Assuming that the incident plane wave was of the form A , exp ikx, then yo = In A,+ikx. Substituting this relation, along with eqs. (18) and (19) into (17b), we find

111,

5 31

where K~

87

SPECTRA FOR AMPLITUDE

The solution to this equation is

= I

B(dK, X ) =

(1-k2/K2)9

+ /:u(dx,

LJ

Y (dx, x ’ ) exp { -ik

x ’ ) exp {-ik[l+

[1- (1- 2 / k 2 ) 9 ] (x-x’)}dx’

1

(l-~~/k~)9](x--~’))dx’

(21)

+

+C, exp { -ik[ 1- (1- K 2 / k 2 ) * ] X } + C ,

exp { -ik[ 1 (1--2/k2)+]x).

Since we are only interested in the forward-scattered wave, we set B(dq 0) = 0, which implies C, = C, = 0. (A more careful consideration of the back-scattered wave shows that this term is of the order of (TATARSKI [1967]).) Also we desire to eliminate the second integral in (21) based on the argument that it is small compared to the first. The following discussion is not rigorous, but can be made so. As already discussed, for optical propagation in the atmosphere A = exp ( - 4 n Z k 2 ~ ~ ~ ~ d ~ [ 1 - J O ( ~ l ~ 1 - ~ 2 1 ) ~ Q ~ ( l o ) *

(41)

This result can also be calculated directly from the results in 9 3.2 (see 5 6.5), and has also been found by HUFNAGEL and STANLEY 119641 using some rather questionable mathematics (CHASE [1965]). DEWOLF[1967, 19681 has also used a renormalization type procedure to attack the multiple scattering problem. He calculated the expected value of the field, ( E ) , and the mutual coherence function, and obtains results that are equivalent to (39) and (41) respectively. In addition he considers higher order moments of the intensity ( I N ) . In his second paper DEWOLF[1968] illustrates how the calculation of the second moment leads to a saturation effect. However, there are some differences in Dewolf's notation and that of other workers. Dewolf defines & as ln((12)/(1)2) instead of the variance of lnl, as is used by

111,

$ 51

103

RENORMALIZATION METHODS

most workers. If In1 is normally distributed then (I2)as

0 :

agrees with the

cr2 used by most workers He calculates

(12) = I: [2--exp(-o;)] 0; =

4 4 = 1.23 C337/6L11/6

(42)

where 0; is the variance of the log-intensity fluctuations as calculated using the first order perturbation theory, i.e. by eq. (31). Dewolf also claims the moments of I imply a Rice-Nakagami distribution for the amplitude in contrast to the log-normal distribution predicted by Rytov's method. Using his definition for o2 Dewolf calculates an expression for & given by

cr;

= ln[2-exp(-&)].

(43)

This expression saturates a t a value of clnl= 0.833, a value which appears to agree with the data of GRACHEVA [1967] and DEITZand WRIGHT [1969]. However, if Dewolf actually predicts a RiceNakagami distribution for the amplitude fluctuations, then eq. (43) is not correct, and a new expression should be found. For example, if the Rice-Nakagami distribution is approximated by the m-distribution (see NAKAGAMI [1960]), then it can be shown that

m=

(44)

I

1-exp

(-0;) '

It is clear that as 0; goes from 0 to co,o$ will go from 0 to some finite value equal to l/n2 = i n 2 .That is, a Rice-Nakagami distribution for the amplitude fluctuations leads to a saturation value, old = 1.28. However, as already mentioned, there appears to be little experimental evidence pointing to a Rice-Nakagami probability distribution (see 9 4). In general, it appears that the renormalization technique has led to very few new results, despite the tremendous increase in difficulty in the mathematics. None of the workers have yet calculated expressions for the covariance functions of a log-amplitude and phase which can be compared with experimental results. Except for Dewolf's prediction of the saturation effect, there has been no important experimental result determined for the first time. The major problem

xzl

104

ATMOSPHERIC PROPAGATION

[III,

§ 6

in this approach is that to keep the problem mathematically tractable, it is necessary to delete or approximate a very large number of terms. Furthermore, it is extremely difficult in most cases to estimate the sizes of the terms that are ignored. For this reason, there is a fair amount of disagreement in the field about the validity of the approximations that any particular worker has made.

Q 6. Extensions of the Plane Wave Results The theoretical results given in the previous sections have all been based on the assumption of an infinite plane wave propagated through a medium that is statistically homogeneous, i.e. its statistical properties are not a function of space. In this section the results are given when certain of these assumptions are removed or changed. For example, how strongly are the results altered when the source is changed from a plane wave to a spherical or a beam wave, or the receiving aperture is of finite size, or the statistics of the medium are not constant in space. In addition, we will discuss a unique and elegant approach of LEE and HARP [1969] which enables many of these questions to be answered with relative ease, as well as the work of Beran and others on the propagation of the mutual coherence function through a random medium. 6.1. A RANDOM MEDIUM W I T H SMOOTHLY VARYING CHARACTER-

ISTICS

In the preceding discussion it has been assumed that the random medium through which the wave propagates is statistically homogeneous, but for many cases of practical interest this is clearly not the case, e.g. earth-to-space propagation. As a consequence several authors have developed a model of a turbulent medium whose characteristics are slowly changing in space (TATARSKI [1961], SILVERMAN [1957, 1958,19591). Essentially these models assume that the statistical quantities of the medium can be written as the product of two functions, one which has the same functional form as would occur if the medium were homogeneous, and the second a function that depends on the spatial coordinates. As an example, consider the refractive index spectrums given in eqs. (3) and (4) of 2. These can be generalized to the form Gn(K,r) = C 3 - 1 @ : ( K ) >

G:(K)= 0.033 K

exp(-K2/K;);

- ~ ~ / ~

(45)

111.

§ 61

E X T E N S I O N S O F THE P L A N E WAVE RESULTS

105

The above equations imply that the spectral shape of the fluctuations remains constant throughout space, but that the intensity of the fluctuations depends on where in space the fluctuations are measured. The use of this model infers that the primary wavenumbers of interest r ) large compared to l/L,, where L, is a distance that in @ ( ~ , are characterizes the changes in C,(r). This assumption is often met in problems of interest in optical propagation. As an example, in 3.2 it was pointed out that the log-amplitude fluctuations are only influenced by wavenumbers greater than 1/2/(Ax),where x was the path length; or, L, must be much greater than the width of the first Fresnel zone. For a 10 km path and red light, L, must be much larger than 8 cm, a condition that should be relatively easy to meet. In contrast, phase fluctuations are strongly influenced by low wavenumbers, and the suggested model may not be as suitable. To introduce the added complexity of this model into the calculations of the phase and log-amplitude spectra, it is necessary to repeat the calculation given in 0 3.2, though the derivation is unchanged [1961,1967]has carried out this calculation through eq. (28). TATARSKI and finds that eq. (29) should be replaced by

and

Using the spectral model in eq. (45), it can be shown that

4 = 0.56 k7’6

LZ

J O

Ca(y) ( x - v ) ~ ’dy, ~

and

It can be seen from the above formulas that the phase fluctuations are equally sensitive to the strength of the refractive index fluctuations

106

ATMOSPHERIC PROPAGATION

[III,

5

6

no matter where they occur along the path. In contrast, the logamplitude variance is most strongly influenced by refractive index fluctuations near the transmitter and very weakly by fluctuations near the receiver. These two conclusions may be explained by noting that phase fluctuations are essentially proportional to the integral of the refractive index fluctuations over the entire path, while the amplitude fluctuations are primarily the result of the interference of waves scattered at some distance from the receiver. FRIEDand CLOUD [1966] have used the above equations to calculate the logamplitude covariance function and the phase structure function for an infinite plane wave through the atmosphere -a calculation applicable to the propagation of starlight or a signal from a space satellite. 6.2. S P H E R I C A L WAVE AND FINITE BEAM SOURCES

Except for stars or other sources far from the earth, the assumption of an infinite plane wave is only a crude approximation to the true source distribution. When introducing more realistic source distributions, most workers have followed approaches similar to those employed in the plane wave case. If we start with eq. (17) in § 3, then the first step is to solve the free space wave equation (17a) for y o , or equivalently, E,, for the source distribution desired. This result is substituted in (17b) and y 1 is then calculated. Two difficulties arise when increasing the complexity of the source. First, there is the obvious increase in mathematical complexity just because yo is no longer a simple function. Second, the method of spectral expansions that was used in the plane wave case must be reexamined. The form of this expansion is based on the assumption that the wave was statistically homogeneous in the y-z plane. Obviously this assumption is not true for a spherical or beam wave, but some authors still use the expansion with little discussion of the justification. Other authors solve directly for the covariance functions of log-amplitude and phase, which avoids this problem. In this last approach it is usual to start with the Green’s function solution of eq. (17b), i.e.

For a spherical wave, the free space wave E o ( r ) ,is given by

111,

§ 61

EXTENSIONS O F THE PLANE WAVE RESULTS

107

where A is a normalization constant a n d t the distance from the source to the point of observation. For a beam wave, most of the workers have assumed that the beam has a gaussian amplitude distribution whose width is given by m’, and a radius of curvature R,, i.e.

Eo(O,y,z )

=A

exp

Following ISHIMARU [1969b], define

then the free space solution for the beam wave may be written as

a result obtained earlier by SCHMELTZER [1967]. Schmeltzer, in a paper which is reasonably careful from the mathematical viewpoint, derives the mean value of y ( r ) = lnE(r) to second order, and the spatial covariance functions of the log-amplitude and phase fluctuations, also to second order, for the source distribution given in (52). Though having completed the problem from a mathematician’s point of view, his results are left as reasonably complex double or triple integrals, which will not be given here. His results are given in a general form which permits the assumption of any spectral form for the refractive index fluctuations, including one whose intensity is slowly varying in space. In his calculation of the mean value of y ( x , y, z ) only the zeroth order term depends on y and z. In other words, the mean value of y ( r ) is equal to the free space value of ?y plus a correction term that depends only on how far the beam has traveled in the medium. GEBHARD and COLLINS[1969], based on Schmeltzers’ general expressions, have calculated the log-amplitude mean for a laser beam propagating over a homogeneous path when on(^) = 0.033 cz K-11’3, and present their results graphically. Their plots indicate that if the normalized source-beam size 52 = kWt/2x is greater than 10, the mean value is relatively independent of Q, and reaches a value which can be predicted by conservation of energy arguments. These authors also point out that Schmeltzer’s result for the mean value is not valid as Q approaches zero, i.e. in the spherical wave limit. These authors

108

ATMOSPHERIC PROPAGATION

[III.

§ 6

also calculate a forward antenna gain factor which they define as

G 3 10 log,,

(T) ( I > (db).

(55)

This quantity is found to depend on both Q and the variance of the log-amplitude fluctuations. In Schmeltzer’s calculations for the covariance functions of log-amplitude and phase he also makes the assumption that the observation point is in the far field of the beam, i.e. x >> I/.. If R, = 00, i.e. a collimated beam, then this condition is equivalent to the normalized source-beam size Q being much greater than 1; if R, > IR,l. Since one case of particular interest is when x = R,, i.e. a beam that has its focal point located at the receiver, it should be noted that Schmeltzer’s results do not apply to that situation unless Q >> 1. Because of the assumption that the observation point is in the far field, he concludes that the covariance functions do not depend on the location of the observation point in the beam. A t first inspection it appears that the above conclusion is reasonable near the center of the beam, but would be questionable near the edge. However, because the quantities calculated are phase and log-amplitude covariances, and neither of these quantities are strongly affected by the relative amplitude of the signal at the measuring point, the results may be a reasonable approximation near the edge of the beam also. It should also be pointed out, that though Schmeltzer does not discuss the crosscovariance between phase and log-amplitude explicitly, it is contained in his calculations -it is equal to the imaginary part of his quantity C,, and may be found from his expression (6.22). ISHIMARU [1969a, 1969bl has also studied the problem of a beam wave in some detail. In particular he has an interesting interpretation in terms of spectral expansions. He indicates that the fact that y1 is non-homogeneous can be represented by complex wavenumbers which are analytic continuations of the wavenumbers occurring in the homogeneous case. He indicates that eq. (54) for E , ( r ) is valid for x

n3~313,

which would indicate this expression is not valid for spherical waves, i.e. in the limit as W,+ 0. However, even for a 1 mm aperture, nSW$l3 lo8 m, which shows it is not a strong restriction. Ishimaru also calculates the log-amplitude and phase covariance functions, though his expressions appear to be somewhat different than

-

111,

§ 61

E X T E N S I O N S O F THE P L A N E WAVE RESULTS

109

Schmeltzer's. I n particular, his results do depend on the location of the observation point relative to the center of the beam. However, because the results are again expressed as double integrals, it is not obvious whether the expressions strongly depend on this quantity. Using eq. (3) for the refractive index spectrum Ishimaru calculates expressions for the log-amplitude covariance function and the logamplitude variance. In his expression for (x2) and B,(+) it seems apparent that in the limit assumed by Schmeltzer, i.e. ax >> 1, his expressions are only weakly dependent on the location of the observation point in the beam, in agreement with Schmeltzer's results. It appears that one advantage of Ishimaru's formulation of the problem is that he has never made the assumption that ax >> 1, and therefore his expressions should be valid in the near field as well as the far field. His result for ( x 2 ) behaves as a plane wave in the limit ilx/nWi > 1. It is interesting that for all three types of waves (plane, spherical, or beam), (x2) eventually varies as C ~ G ~ " ? XThis ? ~ ~conclusion . allows us to conclude that the saturation effect observed experimentally and discussed in 3 4 does not occur just because the experimental results were compared against [1969b] has a plane wave theory. I n his second paper ISHIMARU plotted (x2) as a function of the quantity x/R,. When x = R, the beam is focused at the receiver, and he finds a big (but very narrow) drop in the variance. This result points out one very definite limitation in both the work of Schmeltzer and Ishimaru. Their work is based on the first order perturbation solution, and hence is not valid in the region of strong fluctuations. On physical grounds, it does not seem likely that such a minimum would exist when there are strong fluctuations, since the entire concept of focusing a beam at some point in the path undoubtedly breaks down. FRIEDand SEIDMAN[1967] have, using Schmeltzer's results as their starting point, calculated the log-amplitude variance as a function of the normalized transmitter beam size for a collimated beam focused a t the receiver. It is possible to make a comparison of a point calculated by Ishimaru [1969b] with one calculated by Fried and Seidman, i.e. they both calculated ( x 2 ) for a beam focused at the receiver for 52 = 5. Their values appear to be within a factor of two of one another, despite the fact that they used slightly different models. Ishimaru included the effect of the inner scale of turbulence, I,, ;while Fried and Seidman did not. Ishimaru finds a fair dependence on this quantity.

110

ATMOSPHERIC PROPAGATION

[III,

$ 6

FRIED[1967d] has extended the paper by himself and Seidman to the case where the transmitter is on the ground and the receiver in space. He assumes a collimated beam, and calculates the reduction in the log-amplitude variance, (x2), as the normalized beam diameter is increased. Values of l2 on the order of 1 to 10 lead to reductions in the variance of 1 to 2 orders of magnitude. KINOSHITA et al. [1968a, b, c] have also studied the propagation of a focused beam, but unfortunately in all their work they employ a gaussian covariance function to describe the refractive index fluctuations (see eq. (5)).As a result most of their graphs are normalized by the parameter, y o , which makes the interpretation extremely difficult when referring to the atmosphere. For example, many of their graphs are normalized by the parameter, f = 4x/kr;, where Y, is the correlation length for a gaussian covariance function. However, as we have already discussed, the refractive index fluctuations are normally described by two scale lengths, I , 1-10 mm and Lo 1-100 m. For optical waves, setting yo = I , gives i 1 . 6 1OP2x ~ while yo = Lo gives f 4 x lO-’x. In one of their graphs KINOSHITA et al. [1968a] show that if 2 > 1 then there is a significant reduction in (x2). Does their graph imply x must be on the order of 100 meters or lo7 meters to find this reduction? Similarly, in calculations of the spatial covariance functions of log-amplitude and phase the determining scale size is y o . When using a Kolmogorov type model for the atmosphere, these two quantities normally depend on 2/(Ax) and Lo respectively. It is for these reasons that it appears as if this work will be of little value when attempting to predict the important effects of a focused beam in optical propagation through the atmosphere. Several workers have studied the problem of spherical wave propagation in some detail. TATARSKI [1961, 19671 calculated the logamplitude variance and found that in the geometrical optics limit, 2/(Ax) CK I,, then (x:) = 0.1 (x:) where the subscripts s and p refer to the spherical and plane wave cases respectively. When Zo

4

=

{Vl, t+z)

V"(r2,t)}

(60)

where V ( r ,t ) is an analytical signal associated with the real electromagnetic field. He then considers an ensemble average of the mutual coherence function, < W l > r 2 >z)>

=

({Wl,t t z ) V*(rz,

Q)>.

Under the normal assumption that V ( r ,t) is an ergodic process the double averaging is not necessary, provided that the time average is given by dt.

(61)

However, since Beran's V ( r ,t ) also includes the rapid fluctuations due to the mean frequency of the generating wave (the discussions

111,

§ 61

EXTENSIONS O F THE PLANE WAVE RESULTS

115

are almost always restricted to quasi-monochromatic waves), the time average can conceptually be considered as an average long compared to the period of the original light wave but short compared to the time variations induced by the atmosphere (or the slow fluctuations of a quasi-monochromatic source). In this framework, for a monochromatic wave {V(rl, t + t ) V * ( r 2 t, ) } would be equal to A(rl, t+t ) A ( r 2 ,t ) X exp [i[S(rl, t+t)-S(r,, t)]] whereA(r, t ) a n d S ( r , t ) are the amplitude and phase of the wave as used throughout this article. For a quasimonochromatic wave, it is necessary to first consider a Fourier decomposition of the wave, and discuss a single frequency component, an approach also used by Beran. Therefore, in the notation of this article

BERAN[l966, 19671 calculates the above quantity for the case of a plane wave and spherical wave source by deriving a wave equation for the coherence function, which he then solves - essentially using a perturbation approach. Normally, the mutual coherence function is only considered in a plane perpendicular to the direction of propagation, say in the plane x = L. For the monochromatic wave, Beran finds

inB. 0

sinO,

-m8,

0

11

n

o

0

0

(1

I1

0

. .

. .

n

0

...

...

II

11

? ~ p (tb,)

case,

I1

I)

0

11

11

I)

I)

cxp (-!b,) cosf3,

11

11

II

0

i!

0

cxp(-ib,)cosB

(2.14) where b, is the retardation (in radians) of the compensator. In procedure 1, eq. (2.9d) led to the expression for the rotation angle of the ith crystal given by eq. (2.13a); in procedure 2 the corresponding expression for the jth stage is

which gives (2.16a) and (2.16b)

IV.

5 21

SYNTHESIS PROCEDURES

149

Thus the rotation angle of the stage is determined by the magnitude of (-F&JS$) and the compensator retardation by the phase angle. An example has been carried out by AMMANN and YARBOROUGH [1966] using this procedure. 2.3. PROCEDURE 3 (Single-pass)

2.3.1. General

Synthesis procedure 3 (AMMANNand CHANG [1965]) provides a third method for synthesizing a birefringent network having an arbitrary amplitude transmittance. For this procedure, the Ci of eq. (1.4) can be complex. The form of a three-stage network obtained from procedure 3 is shown in Fig. 13. The network consists of an input polarizer followed by a series of stages, each stage containing a birefringent crystal, optical compensator, and polarizer. The quantities determined for each stage by use of the synthesis procedure are the angle to which the crystal (and compensator) are rotated, the retardation of the optical compensator, and the rotation angle of the polarizer. Note that the network contains internal polarizers, and hence cannot be termed a lossless network. Procedure 3 is the only procedure which does not yield a lossless network. The resulting amplitude transmittance of the synthesized network is not C(w),but rather K C ( w ) , where K is a complex constant whose magnitude is unity or less. Thus unlike procedures 1 and 2 which

Fig. 13. Basic form of birefringent network obtained from synthesis procedure 3.

150

0P T I C A L B IK E F R I N G E S T N E T W 0K K S

IIV,

s

2

always produce networks yielding exactly the desired C ( w ) , an attenuated C(w) sometimes results from the networks obtained via procedure 3. Because of this, and because of the greater number of network components required (procedure 3 required three components per stage versus two for procedure 2), procedure 3 is in general less useful than the other procedures. We have considered it here, however, because it can occasionally offer an important advantage. With all three procedures, it is sometimes possible to realize a C ( o ) using fewer than n network stages, but with longer crystals in those stages. With procedure 3, it is a simple matter to predict from a given C(w) whether this situation will occur. With procedures 1 and 2, however, it has not been possible to predict this occurrence. Thus with procedure 3, we know how to tailor the desired C ( w ) to cause this situation, while with procedures 1 and 2 this is not presently possible. Recalling that the network of procedure 1 resembles the Solc birefringent filter, we note that the network obtained from procedure 3 resembles the Lyot filter. I n fact, AMXANNand CHANG[1966] have shown that the Lyot filter can be obtained by the use of synthesis procedure 3. 2.3.2. Procedure The formalism required by procedure 3 is quite different from that of procedures 1 and 2. This is not surprising since the forms of the networks are also considerably different. Computationally, procedure 3 is easier to carry out than procedure 1 or 2. We begin by assuming that a C ( o ) in the form of eq. (1.4) has been chosen. Equation (1.4) can be considered to be a polynomial in exp(-iiao) and rewritten as (2.17a)

- - - (-z,+e-ia"),

= Cn(-z~+e-iuw)(--Z2+e-iao)

(2.17b)

where the z's are the zeros of the polynomial. These zeros are, in general, complex. If we express each zero in terms of its magnitude IzI and phase angle eia, eq. (2.17) becomes

C(o)= Cn(-Izlleial+e-ia")(-~z21eia~+e-ia")*

-

(-Iz,Ieia~+e-""'). (2.18)

IV, §

21

SYNTHESIS PROCEDURES

151

Fig. 14. Components which make up a single stage of the network: birefringent crystal, optical compensator, and polarizer.

We now show that each factor in eq. (2.18) can be associated with one stage of the network of Fig. 13. Consider one stage (say the jth stage) of the network of Fig. 13. This stage contains a crystal, an optical compensator, and a polarizer as shown in Fig. 14. Using the Jones-calculus formulation (JONES [1941]), the input and output of the jth stage are found to be related by

(2.19)

where E denotes the complex amplitude of the electric-field vector of the light. The 21 axis is parallel to the transmission axis of the polarizer of the preceding stage. The birefringent crystal introduces a phase difference of am radians between the S and F components, while the compensator introduces a phase difference of bj radians. The retardation b, is always assumed to be between 0 and 2n radians. The assumption that the compensators are achromatic is the only approximation used in procedure 3.

162

[IV, § 2

OPTICAL BIREFRINGENT NETWORK5

Perhaps a more familiar form than that used in eq. (2.19) for the matrix of a birefringent crystal is (2.20)

If we factor exp(4iao) out of (2.20), the matrix used in eq. (2.19) results. The factored exp(4iao) term can be dropped without affecting the results. From eq. (2.19) we see that it is again more convenient to deal with relative angles than absolute angles. Consequently, we define (2.21)

and (2.22)

Performing the matrix multiplication indicated in eq. (2.19) and noting that E , is zero, we obtain E,, = eibj cos Oj cos p,(-eibi tan 8, tan pr+e-iaio)E,.

(2.23)

The amplitude-transmittance function for the entire network (all stages) is the product of the amplitude transmittances of the individual stages. Thus we have for the entire network

el cos p1 cos e2 cos pz - . cos 8, tan el tan /Il+eAam)

C l ( w ) = e-ibT cos

x (-eibl x (-elb, tan 0, tan 82+e-iatu)

-

*

cos B,

(-eibm tan

(2.24)

en tan

pn+eiaio),

where (e-ibl) (e”bz) (e-ibs)

. . (e-ibn) = e-ibT.

(2.25)

C ’ ( w ) refers to the actual amplitude-transmittance function of the

birefringent network, while C ( w ) refers to the desired function. As mentioned earlier, the two differ only by a constant multiplier K.

IV. §

21

SYNTHESIS PROCEDURES

153

We see that eqs. (2.18) and (2.24) have the same form. Equating similar terms, we find tan 8, tan pl = ~ z , ~ , 6, = a,,

(2.26a)

tan 8, tan pa = Iz21, b, = a,,

(2.26b)

tan 8, tan /?, = Iz,~,

(2.26~)

b,

= a,.

I n eqs. (2.18) and (2.24), the order of the factor is, of course, unimportant. It is only necessary to equate each factor in eq. (2.24) with one of the factors in eq. (2.18). Note that if a root is real and positive, a = b = 0, and an optical compensator is not required for that stage. Terms in eqs. (2.18) and (2.24) may also be equated in a slightly different manner, namely, tan 8, tan 8, = -(zll, b, = al+z,

(2.27a)

tan 8, tan /3, = -Iz21,

(2.27b)

b, = a2+7c,

tan 8, tan j?, = - - ~ z , ~ , b, = a,+n.

(2.27~)

Here if a root is real and negative, an optical compensator is not required for that stage. Either eqs. (2.26) or eqs. (2.27) may be used for obtaining 8, and b for a particular stage. We see from eqs. (2.26) that 8 and /3 are not yet uniquely determined for a particular stage. There are an infinite number of combinations of 8, and /I, which satisfy the equation tan Oj tan /Ij = Izjl. Thus an additional criterion may be imposed when choosing the particular 8, and p, to be used. One important possibility is to choose 8, and p, so that the magnitude of the output is maximized. When we satisfy eq. (2.26~)(or (2.27c)), we ensure that the C ; have the proper relative values. However the magnitude of the entire response (i.e., the magnitude of K ) depends upon the particular choice for the 8, and p,. We therefore wish to maximize the magnitude of one of the Ci subject to the constraints imposed by eqs. (2.26). (Since the relative values of the C’s are fixed, maximizing any one of them maximizes all of them.) Let us maximize the magnitude of C i which is found by comparing eqs. (2.18) and (2.24) to be

154

OPTICAL BIREFRINGEST S E T W O R K S

[IV,

5

2

This problem can be solved by the method of Lagrange multipliers. We obtain for the result tan 0 , tan 0 ,

tan p1 = f l z , l h , = tan p, = f,z,13, =

(2.29)

tan Oj = tan pj = &Izj;*.

If we maximize eq. (2.28) using eqs. (2.27) instead of eqs. (2.26) as the constraint, we obtain tan 0, = -tan p1 = 6lzll*, tan 0, = -tan p, = fIz,lt, (2.30) tan O j

=

-tanpj

=

f:zjl*.

2.3.3. Networks containing crystals of unequal lengths In general, the synthesis procedure just described requires an nstage network to realize a C ( w ) containing (n+ 1) terms. Occasionally, however, it is possible to realize a C ( w ) using fewer stages with longer crystals in those stages. This comes about in the following manner. We have shown that each of the factors of eq. (2.17b) can be associated with one stage of the network. Each stage contains a birefringent crystal of length L , an optical compensator, and a polarizer. Suppose now that two of the factors of eq. (2.17b), when multiplied together, give a term of the form (-z+

e-i2cuu).

(2.31)

It is apparent that (2.31) can be realized by a single stage containing a crystal of length 2L, an optical compensator, and a polarizer. In general, if r factors of eq. (2.17b) multiply together t o produce a term of the form ( -z+e-iram), (2.32)

this term can be realized by a single stage containing a crystal of length rL, an optical compensator, and a polarizer. As before, the crystal angle, polarizer angle, and the optical compensation are calculated from the magnitude and phase angle of z. Once the zeros of C ( w ) have been found, it is a n easy task to determine whether factors can be combined to produce a term similar to (2.32). The following rule is useful. If r of the zeros have the same magnitude and have a’s which differ by 2 ~ 1 7 , the corresponding

Iv,

8

21

SYNTHESIS PROCEDURES

166

factors can be multiplied together to produce a term of the form of (2.32). In some cases, it may even be feasible to adjust some of the zeros so that they satisfy the above condition. I t is necessary, of course, to make certain that this does not cause an unacceptable change in

c

(0).

2.3.4. Magnitude of amplitude transmittance of network

It has been stated in Section 2.3.2 that the actual transmittance of the synthesized network will differ from C ( w ) by a multiplicative constant K . This occurs because we are trying to use n stages to produce the ( n f l ) different Cj. Thus one less degree of freedom is present than is necessary. If C(o) is chosen to have a maximum magnitude of unity at some portion of its period, the magnitude of K will always be equal to or less than unity. In this section, we show how the magnitude of K can be determined. In order for the network of Fig. 13 to have unity transmittance at some frequency w l , each stage must have unity transmittance at that frequency. Consider one stage of the network. From eq. (2.23), its transmittance can be rewritten as (sin2 el+e-ible-ia"

cos2

el),

(2.33)

where we have set = 8 as prescribed by eq. (2.29) for maximum transmittance. In ordex for (2.33) to have unity magnitude at o = w l , it must be true that e-ible-ia~l

=

1

8

which gives b1 = q n - m , ,

(2.34)

where q is an even integer such that b, is positive, but less than 2n. Now consider a second stage of the network with transmittance given by (2.35) (sin2 8, +e-ibse-"~ cos2 e,). In order for (2.35) to also have unity transmittance, it will similarly require that b, = qn-awl. (2.36) Subtracting (2.36) from (2.35), we obtain the result bl-b,

= 0,

156

OPTICAL BIREFRINGENT NETWORKS

[IV,

§ 2

which is equivalent to the restriction that all compensators of the network must have the same retardation. Since from eqs. (2.26), the bi are determined by the air the result may be restated as follows. I n order for the network of Fig. 13 to have unity transmittance at some frequency ol,the phase angles (the a t ) of all roots of C ( w ) must be equal. If the roots of C ( w ) do not satisfy this condition, the actual transmittance C ’ ( w ) of the network will be smaller than the desired transmittance. In order to determine how much smaller, we will compare the magnitudes of C i and C, . From eqs. (2.28) and (2.29), we have that

IC;~ = cos2

el cos2 e2 - - - cos2

8,.

(2.37)

We can solve for cos2 0, in terms of lzjl by noting from eq. (2.29) that tan2 8, = ]z,], which leads to cos2

ej = i/(i+iz,l).

(2.38)

Equation (2.37) can now be rewritten as (2.39) which is the desired result. Thus by calculating ICkl from eq. (2.39) and comparing this to lC,J, a comparison between actual and desired transmittances is obtained. 2.3.5. Summary of synthesis procedure 3 (1) Choose the desired spectral amplitude transmittance C(o)and write it in the form of eq. (1.4).I t is usually desirable to choose the Cj so that C(o)has a maximum magnitude of unity. The Cjmay be real or complex. (2) Rewrite eq. (1.4)in the form of eq. (2.17~~) and solve for the roots (the zj) of the polynomial. Each root should be written in polar form, i.e., in terms of a magnitude lzjl and a phase angle exp(ia,). Each factor of eq. (2.17b) can be associated with one stage of the network of Fig. 13. (3) If r of the zeros have the same magnitude and have a’s which differ successively by 2 4 1 , the factors containing these zeros can be multiplied together to produce a term of the form of (2.32). This makes possible the replacement of r stages by a single stage containing

N.

8 21

SYNTHESIS PROCEDURES

157

a crystal of length rL, where 7L = 7uc/(qs-qF). Otherwise, the required crystal length L for a stage will be L = uc/(qs-qp). (4) If maximum transmittance through the network is desired, the crystal angle 8,, polarizer angle p,, and optical compensator retardation bj of the jth stage should be calculated from 8, = pr = tan-l(j-lz,l*),

b, = ajr

6, = -pj = tan-'(&lz,/*),

bj = aj+n.

or

If the condition of maximum transmittance is not necessary, eqs. (2.26) or (2.27) should be used. Some other criterion is then necessary to determine 6, and p, uniquely. (5) If the 8, and pj are calculated to give maximum transmittance, the amplitude of the network's actual spectral transmittance can be compared to the amplitude of the ideal spectral transmittance by calculating ICil from eq. (2.39) and comparing it to ICJ. The actual transmittance will be equal to or smaller than the desired transmittance with IRI = IC)/lC,l. 2.4. PROCEDURES

4A ASD 4B (Double-pass)

2.4.1. General

For a certain class of amplitude transmittances C ( w ) , the birefringent networks which result from using synthesis procedure 1 have a particular symmetry. Because of this symmetry, the last half of the network can be replaced by a mirror which reflects the light back through the first half. Since the light makes two passes through the network, fewer stages are required to realize a given C(o). The double-pass procedures of this section can be used only when all Ci are real; each stage of the resulting double-pass network (theoretically) consists only of a birefringent crystal. The doublepass procedures which will be described in the following Section 2.5 are applicable when the C i are real or complex; each stage of the double-pass network contains a crystal and compensator. In both of these cases, C(o) must also satisfy certain additional symmetry requirements before double-pass procedures can be used. In this section, we present two different techniques for synthesizing lossless double-pass birefringent networks. These techniques were described by AMMANN [1966a]. The first of these, synthesis procedure 4A, resuIts in the type of device shown in Fig. 15a. Here the output

158

0 P T I C A L B I R E F R IN G E K T N E T W 0R K S

;nil

e

BlREFRlNGENl NETWORK

INPUT

rl BIREFRINGENT

INPUT

QUTWT

6) Fig. 15. General form of network and corresponding spatial relationship of output t o input which results from (a) procedure 4A, and (b) procedure 4B.

coincides with the input, i.e., the input and output are spatially degenerate. The second technique, synthesis procedure 4B, results in the type of device shown in Fig. 15b where the output and input are spatially separated. These two techniques complement each other and are useful in different situations. 2.4.2. Procedure 4A Consider the single-pass and double-pass networks of Figs. 16a and 16b, respectively. We assert that the transmittances of these two networks are the same. This can be proved by noting that (provided we use the same coordinate axes) the Jones matrix of a crystal or polarizer is the same for light propagating forward or backward through the components. Hence if we write transmittance expressions for the networks of Figs. 16a and 16b, we find they are identical except for a minus sign which is introduced by reflection at the mirror. This minus sign is of no practical importance, and is neglected in what follows. The restrictions which are imposed on the absolute crystal angles

SYNTHESIS PROCEDURES

INPUT F'OLARIZER

169

MIRROR

b) Fig. 16. Two equivalent birefringent networks: (a) single-pass network, and (b) corresponding double-pass network.

of Fig. 16a are 9, = 0", +5 = dl,#, = +2, or in the general case of n crystals, = 0", bn = +1, rjnA1 = +2, +n-2 = c p 3 , . . . etc. If we rewrite these restrictions in terms of the relative crystal angles (the O i ) , wehave 0P = -01, 8, = - 0 ,, 8n-l - - 0 3 , . . . etc. We now use a result which was proven in AMMAXS [1966a]. Theorem 5b of that paper states that if the C i of the desired transmittance are chosen so that C, = -Cn, C, = -Cn--l, C, = -Cn-,, . . . etc, then each of the resulting lossless birefringent networks has e, = -epf900, e, = - e n , 0, = -en+ 8, = . . . etc. We see that the ei restrictions upon the network of Fig. 16a are almost satisfied by the networks described in theorem 5b. The only discrepancy is that in Fig. 16a, we have 0, = - el, while in theorem 5b we have 0, = - O1f90". Therefore, if we take the network of theorem 5b and rotate the output polarizer 90" (thereby obtaining D ( w ) as an output instead of C ( w ) ) , the Bi satisfy the restrictions of Fig. 16a. Armed with this information, we can now outline a synthesis procedure for realizing a given transmittance via a double-pass network of the type shown in Fig. 16b.

+,

160

OPTICAL BIREFRINGENT NETWORKS

[IV,

8 2

(1) Suppose an ideal amplitude transmittance G ( w ) is desired. Find the complementary ideal function H ( o ) , where G ( w )G* (a)+ H ( w ) H * ( o )= 1. (2) Approximate H ( w ) by a C(w) of the form of eq. (1.4). In addition to the usual requirement that each Ci must be real, the following restrictions must be satisfied. For n odd, it must be true that C, = -C,,, C, = -Cn-,,. . . Ct(n-l) = -Ct(n+l). For n even, the Ci must satisfy C, = -C, , c, = -Cn-,, . . . = -c+(,+a), c,,=o. These restrictions are automatically satisfied if a sine series is used to approximate H ( w ) , i.e. if H ( o ) has odd symmetry with respect to w . (3) Determine the lossless birefringent networks which produce this C(w) by using synthesis procedure 1. From theorem 5b, we know that the Oi of each network have the symmetry shown in Fig. 17a. In the expression 8, = -88,&90" in Fig. 17a, the upper sign applies when 0" < 8, < 180°, and the lower sign when -180" < OP < 0". (4) Form the new network shown in Fig. 17b by rotating the output polarizer of the network of Fig. 17a by T90". Each of the newly formed networks has D ( w ) as its transmission characteristic.

INPUT POURIZER

(a)

OUT?Ur MLAREER

Fig. 17a. Form of single-pass birefringent network which results when the desired amplitude transmittance C(o) satisfies the restrictions C, = -Cn, C , = -Cn-,, C , = -Cn-*, . . etc.

.

( 5 ) The network of Fig. 17b is symmetric about a vertical line which bisects the center crystal. Form a new network which consists of the left half of the network of Fig. 17b followed by a mirror. This new network is shown in Fig. 17c. If n is odd, the center crystal of the network of Fig. 17b is in general of length L , and hence the last crystal of Fig. 17c is of length 4L. If n is even, the center crystal of Fig. 17b is in general of length 2L, and the corresponding length of the last crystal of Fig. 17c is L. The double-pass transmission of the network of Fig. 17c is identical to the single-pass transmission of the network

1v.

0 21

161

SYNTHESIS PROCEDURES

(b)

INPUT K X M m

01

62

end-

03

ewl-e3

en-

ep-

-

-el -e2 81 Fig. 17b. Symmetrical network obtained from that in Fig. 17a by rotating the output polarizer by 90".

of Fig. 17b and is therefore equal to D ( o ) . Hence the synthesis is complete. The restrictions imposed upon the Ci in step (2) are not as severe as they may seem. If the Fourier approximation is used for determining the C,,the restrictions are equivalent to specifying that only sine terms may be used in approximating H ( o ) . The requirement of a Fourier sine series means that the symmetry of H ( w ) has been restricted, but not its detailed shape. Since we are ordinarily interested in using only a portion of one period of H ( o ) (most often one-half), it is not important what value H ( o ) has over the remainder of the period and hence whether it is even, odd, or neither. It should therefore be possible to suitably approximate most functions while satisfying the Fourier sine series requirement. Finally we should note that the output from the network of Fig. 17c is coincident with the input, a situation which is undesirable for many applications but desirable for others. If, for example, the network were to be placed inside a laser cavity, the coincidence of input and output would be required. In many other instances, however, it is necessary for the input and output beams to be separated. A method of accomplishing this is given in procedure 4B which is a modification of procedure 4A. (C) INPUT KKARIZER

Fig. 17c. Double-passnetwork which is equivalent t o the network of Fig. 17b.

162

[IV,

OPTICAL HIREFRINGEXT XETWORKS

s

2

2.4.3. Procedzcre 4B

Let us return to the network shown in Fig. 17c. This network results from procedure 4A and, as noted earlier, has as its output D ( w ) . Stated slightly differently, the x component (the component along the transmission axis of the polarizer) of the reflected light leaving the first crystal has been operated upon by D ( w ) . Similarly, the y component (the component perpendicular to the polarizer’s transmission axis) of the reflectedlight leaving the first crystal has been operated upon by C ( w ) . In procedure 4A, the C(w) component is stopped by the input polarizer. Suppose that we now replace the polarizer of Fig. 17c by a polarizing beam splitter, i.e., a component which separates the x and y polarization components of the beam but transmits them both. As an example, let us substitute a calcite slab followed by an iris for the input polarizer as shown in Fig. 18. The calcite has flat and parallel end faces, and its optic axis lies in the y, z plane at some angle a with respect to the face normal. Light which enters the crystal normal to the face is divided by double refraction into an ordinary ray, which propagates within the crystal undeviated, and an extraordinary ray which is deviated within the crystal by some angle /I. Now let us consider what happens to light which is incident upon the network. IRIS

CAlClTE

I

MIRROR

Y

Fig. 18. Double-pass birefringent network which results from procedure 4B.The optic axis of the calcite polarizing beam-splitter is shown dotted. The x direction is out of the page.

As in procedure 4A, light polarized in the x direction is passed by the calcite and is incident upon the first crystal of the network. However, light polarized in the y direction is deviated upward by the calcite and is stopped by the iris. Thus for the incoming light, the calcite and iris act as a polarizer. The x component of the input light passes through the crystals, is reflected by the mirror, returns through the crystals, and emerges from the first crystal with an x component D ( w )

IV,

8 21

SYNTHESIS PROCEDURES

163

and a y component C ( w ) . The x component again passes through the calcite undeviated and is therefore coincident with the input; this component of the output is not used. The y component C(o)is displaced downward by its passage through the calcite and emerges parallel to, but displaced from the input; this constitutes the output of the device. We can summarize this procedure as follows. (1) Suppose that an ideal amplitude transmittance G ( w ) is desired. Approximate G ( w ) by a C ( w ) of the form of eq. (1.4). In addition to the usual requirements that each C, be real, the following restrictions must be satisfied. For n odd, it must be true that C, = -C,, C, = -Cn-,, . . ., Ct(n-l)= -Ct(n+l). For n even, the Ci must satisfy C, = -C,, C, = -Cn--l, . . ., C+[n--2)- -C+(n+z), C+n = 0. These restrictions are automatically satisfied if a Fourier sine series is used to approximate

G(w). (2) Determine the lossless birefringent networks which produce this C ( w ) by using synthesis procedure 1. From theorem 5b of AMMANN [1966a], we know that the Oi of each network have the symmetry shown in Fig. 17a. (3) Form the new network shown in Fig. 17b by rotating the output polarizer of the network of Fig. 17a by 90". (4) The network of Fig. 17b is seen to be symmetric about a vertical line which bisects the center crystal. Form a new network which consists of the left half of the network of Fig. 17b followed by a mirror. This new network is shown in Fig. 17c. (5) Replace the input polarizer in Fig. 17c by a calcite slab. This change puts the birefringent network into the final form of Fig. 18. The output beam is displaced from the input beam and has been operated upon by C ( w ) . Procedure 4B is probably the more useful of the two schemes, since in most applications separated input and output beams are required. In addition, the advantage of imposing the symmetry restrictions directly upon the quantity which represents the network's transmittance is an important one. 2.6. PROCEDURES 5A AND BR( Double-pass)

2.6.1. Gelzeral

Double-pass synthesis procedures 4A and 4B result in birefringent networks containing only half as many stages as normally required.

Fig. 19. Network symmetry which is required in order for procedures 4A and 4B t o be applicable. (a) Lossless network without compensators, and (b) lossless network with compensators.

It was seen in Section 2.4 that these procedures can be used when the Ci of eq. (1.4) satisfy certain rather broad requirements. AMMANN and YARBOROUGH [1967] showed that double-pass networks could also result from other more general classes of amplitude transmittances. Their results are described in this section. Fig. 19a shows the network symmetry required in procedures 4A and 4B in terms of both the q$ and Oi. Although compensators were not present in the networks of procedures 4A and 4B, equivalent symmetry requirements can be stated for a network which contains them. These requirements are shown Fig. 19b, where the bi are the compensator retardations. This section shows two methods whereby networks obtained for still other classes of C(o) can be made to have this symmetry. Having done this, it is then possible to use the procedures 4A or 4B directly. 2.5.2. Procedure 6 A

AMMANN and YARBOROUGH [1967] derived a property of birefringent networks which is central to the procedures of this section. Their result states that the transmittance of a birefringent network remains unchanged if the network is altered by (a) for the jth stage changing 8 to - 8 and adding n radians of optical compensation, and

IV, § 21

186

S Y N T H E S I S P R O C E D U R BS

(b) for the preceding stage, adding n radians of optical compensation. Procedure 6A is applicable when the Ci of C(o)are real and satisfy C, = C,, C, = Cn-,, C, = Cn--2, . . . etc. As proved by AMMANN [1966b], the resulting birefringent networks have = 8,F90°, 8, = On, 8, = 8,,, . . . etc. This symmetry is not appropriate for double-pass procedures 4A or 4B, but we now show how it can be made appropriate when n is odd. We illustrate by considering the case of n = 9. If C ( w ) is chosen so that C, = C,, C, = C,, C, = C,, C, = C6and C, = C,, the networks obtained using synthesis procedure 1 will have the symmetry tabulated in Table 1. We now make use of the result discussed above. If we change 8, to -02, b, to n, and b, to n, the transmittance of the network is unchanged. Similarly we can change O7 to - Ba , 6, to n, b6 to n; 8, to -& b, to Z, b, t o n; 8, to -3,, b, to n , b , to n;and 8, to - 8 , , 6, to 3~ without affecting the transmittance of the network. The network now has the symmetry shown in Table 2, which is the symmetry required. Hence methods 4A and 4B are directly applicable. Similar techniques apply for other odd values of n. If n is even, however, the above procedure does not result in a symmetrical network, and hence these techniques are not applicable. In methods 4A and 4B, the symmetric network is halved by cutting it through the middle stage. For this example, the 6th stage is the TABLEI Network symmetry which results for n = 9 when the (real) C1 are chosen to satisfy C, = C,, C , = C , , C , = C,, C , = C, and C, = C,.

Rotation Stage 1 2 3 4 5

6 7 8 9

output polarizer

angle

el = el e2 = er

e, = e, e, = e, es = es e, = 3, e, = e, es = ea e, = er

ep = e,*goo

Compensator retardation (radians)

bl = 0 bs = 0 ba = 0 b4 = 0 bl = 0 b, = 0 b, = 0 ba = 0 be = 0

b, = 0

166

OPTICAL BIREFRINGENT

NETWORKS

TABLE2 Network which is equivalent t o that listed in Table 1. ..~.

Stage

el +-el er = e,

1 2 3 4 5 6 7 8

output polarizer ..

b, b, b, b, b, b, b, b, b, b,

e, -+ -e3 e, = e, er +--8, es = e, e, -+ -er es = es eB+-e2 e, = e,+goO

9

__

Compensator retardation (radians)

Rotation angle

~

. .. .

-

+n +n

+n +n +z

+n +n +n +n =0

..

middle stage and consists of a crystal of length I. and compensator whose retardation is n rad. When this stage is halved, the components are a crystal of length i L and a compensator whose retardation is in rad. 2.5.3. Procedure 5B

We now show a double-pass procedure that can be used with complex

Ci.We begin by stating a theorem from AMMANNand YARBOROUGH [ 19671. If the C iof the desired transmittance are chosen so that C, = Cn*, C, = CnF1*, C, = Cn-,*, C, = Cn--3*, . . . etc. then the resulting lossless birefringent networks have 8, = -08,+900, 8, = On, 8, = On_,, O4 = O n - , . . . . etc., and b, = bn-n, b, = bn-,, b, = bn-,, b, = bn-,, . . etc. We again use a network with n = 9 for illustration and note that the technique is applicable only when rt is odd. If the Ci satisfy the requirements listed above, the resulting networks will have the symmetry shown in Table 3. We next apply the result stated earlier by noting that C ( w ) for the network remains unaltered if we change Os to -03, b, to b , f n , b, to b,+n; 0, to - 0 5 , b, to b,+n, b5 to b5+n; O4 to - 0 4 , b, to bp+n, b, to b,+n; and 0, to -O,, b, to b,+n, b, to b,+n. In addition, b, is changed to 0. This new network, whose Oi and bi are listed in Table 4, has the symmetry necessary for application

IV,

5

2;

167

SYNTHESIS PROCEDURPS

TABLE3 Network symmetry which results for n = 9 when the (complex) C, are chosen to satisfy C, = C,*, C , = C,*, C, = C,*,C , = C,* and C , = C,*. -.-

Stage

Rotation angle

el = el er = e,

1 2 3 4

5 6 7

8 9 output polarizer

es = es e, = e, e6 = e, es = e, e, = e, es = e, e, = ea ep = - e,+goo

Compensator retardation (radians) b, = b, b, = b, b, = b, b, = b, b6 = b, b, = b, b, = b , b, = b2 b, = b,+n b, = bp

__

of procedures 4A and 4B,and hence the desired result has been obtained. For other odd values of n, the same technique can be applied. Several items should be briefly mentioned at this point with respect to procedure 6B.The first is that with procedure 5B,the manipulations are begun at the (n-1)th stage while with procedure 5A, they TABLE4

---

~

Stage

Network equivalent to that listed in Table 3. ~-~ Compensator Rotation retardation angle (radians)

168

OPTICAL B I R E F R I N G E N T N E T W O R K S

[IV,

8 3

begin at the nth stage. The second point concerns setting b, to zero. This step is necessary if the input and output polarizers are to be the “mirror-images” of each other. The result of setting b, to zero is that the transmittance of the double-pass network will be exp(ib,) C(w) instead of the desired C(w). The difference is only a constant phase factor, which in most instances is unimportant. Furthermore, if C, is originally chosen to be real, b, = 0 and the need for the final compensator is automatically eliminated. Finally, if the Ci satisfy C, = -Cn*, C, = -C,-,*, C, = -Cn--2*, C, = -Cnd3*, . . . etc., the technique of this section is also applicable. This can be seen by noting that if we multiply such a C(w) by the factor i, the new C(w) satisfies the constraints of the theorem given in this section.

-

2.6.4. Disczlssion

Let us consider how severely the restrictions which are placed on the Ci in this section limit the choice of C(w). In discussing this, we find it convenient to deal with K ( w ) as well as with C(o).K ( w ) ,the noncausal transmittance mentioned in Section 1.4, is formed by multiplying C(w) by exp(4inao) and therefore has the form K ( a ) = C,etinaw+ Clei(tn-l)am . . .+Cn-le-i(fn-l)aa’+Cne-+jnaw. (2.40)

+

The usefulness of K ( w ) stems from the fact that it is often real, whereas C(w)is complex. In choosing a desired transmittance, it is often written first in the form of eq. (2.40) and then converted to C(w). In terms of K ( o ) , the Ci restrictions of this section have the following effects: The restrictions C, = C,, C, = Cn-,, C, = C,-,, C, = Cn--3, . . . etc. of procedure 5A are equivalent to requiring K ( o ) to be real and to have even symmetry. The restrictions C, = Cn*, C, = Cn-,*,C, = Cn-,*,C, = Cn-3*, . . . etc. which were necessary with procedure 6B are equivalent to requiring K (w) to be real. The symmetry of K (w ) is not restricted in any way. Thus these restrictions impose relatively little constraint upon the choice of desired amplitude transmittance. Q 3. Experimental Results 3.1. LYOT

AND

SOLC FILTERS

Numerous Lyot and Solc filters have been built and successfully operated by various workers. OHMAN[19381 independently conceived

IV.

8 31

EXPERIMENTAL RESULTS

169

the Lyot form of filter and using quartz crystals and Polaroid film, constructed a four stage monochromator. EVANS [1940] employed the same materials to build a six-stage Lyot filter with a 5 A halfwidth. Ammonium dihydrogen phosphate (ADP) was utilized by BILLINGS et al. [1951] to produce a Lyot filter having a 1 A bandwidth. More recently a Q A Lyot filter was constructed by STEELet al. [196l] using calcite and quartz for the birefringent materials. The first Solc filter was built by SOLC [1953], who apparently developed his filter experimentally. EVANS [19631 designed and built a quartz Solc filter having a bandwidth adjustable from 40 to 550 A. Although these and other birefringent filters have operated satisfactorily, precise measurements of transmittance were difficult to obtain due to the narrow bandwidths of the filters. EVANS [1963] measured the transmittance of his Solc filter photoelectrically, but such data has not been given for the vast majority of birefringent filters. SOLC[1959] and YARBOROUGH and AMMANN [1968b] have independently suggested a technique which is useful for accomplishing such measurements. 3.2. A TECHNIQUE FOR MEASURING TRANSMITTANCE

The difficulty in measuring birefringent network transmittance is, of course, in the requirement of a very narrowband tunable optical source of adequate intensity. A spectrometer can be employed as this source, but high-resolution spectrometers are not always accessible. Furthermore, for networks with very narrow bandwidths, resolutions are needed which are difficult to obtain with even the best spectrometers. The technique described here does not use a tunable source, but employs a fixed-frequency source (such as a laser) and “tunes” the transmittance of the birefringent network. The technique makes use of the fact that the birefringence of crystals is, in general, a function of temperature. This is well known, of course, and accounts for the rigid temperature control which must be applied to birefringent filters. The basis for this measurement can be easily understood by considering a single stage (i.e., a single crystal) of a birefringent network. The phase difference in radians (Av) introduced between fast and slow axis components of light propagating through the crystal is given by

AT

=z=

~ L ( ~ j l ~ - q= s ) ~/ L ~A~jllc.

(3.1)

We note from (3.1) that Ay has a linear dependence upon both co

170

OPTICAL B I R E F R I N G E N T NETWORKS

[IV, §

4

and Ay. Thus the transmittance of a birefringent network varies with A7 in exactly the same way that it varies with m. But for many crystals (including calcite) Aq is in turn linearly dependent upon temperature, and hence Ap also depends upon temperature in the same way that it depends upon LO. Thus the transmittance of the network vs. temperature for a given frequency will be the same as the transmittance versus frequency at a fixed temperature. 3.3. SYNTHESIZED BIREFRINGENT NETWORKS

Several birefringent networks have been built which were designed using the procedures of Section 2. Many of these were tested using the procedure outlined in Section 3.2. YARBOROUGH and AMMANN [ 1968bl constructed and tested singleand double-pass birefringent networks. A three-stage network (designed using procedure 1) was tested in the single-pass experiments, while three-, five-, and seven-stage networks (designedusing procedure 4B) were used in the double-pass experiments. Each stage of the networks consisted of a calcite crystal 2 cm in length followed by a quartz compensator. Good agreement between theory and experiment was obtained, serving both as a verification of the synthesis procedures and as a demonstration of the usefulness of the measurement technique. AMMANNet al. [1968] built and tested a nine-stage double-pass filter having a 4 A bandwidth; the filter was designed using synthesis procedure 4B. MCINTYRE and HARRIS[1968] have constructed and tested six- and ten-stage achromatic quarter-wave plates which were designed using a variation of procedure 1. These wave plates, which employed sapphire crystals, had a retardation of 90°& 1"over the entire range 4000 A to 8000 A.

Q 4. Application of Synthesis Procedures to Electro-Optic Networks Until now, we have assumed that naturally-birefringent crystals are employed in the stages of the synthesized birefringent networks. AMMANN [1966c] has shown that the procedures of Section 2 can also be used to synthesize networks composed of electro-optic rather than naturally-birefringent crystals. With naturally-birefringent crystals, one can synthesize a network whose transmission versus frequency is arbitrary; with electro-optic crystals, one can synthesize a network

IV, §

41

A P P L I C A T I O N OF S Y N T H E S I S PROCEDURES

171

whose transmission (at a given optical frequency) versus applied voltage is arbitrary. To understand how this is accomplished, consider the naturally birefringent crystal of Fig. 3. This crystal is the basic building block of a naturally birefringent network. Light entering the crystal with its electric field polarized in the F direction is operated upon by the frequency transfer function exp (him), while light polarized along S is operated upon by exp( -4iao). The basic building block of an electro-optic network is the cell shown in Fig. 20. The cell consists of a pair of electrodes immersed in a fluid (or crystal) which becomes birefringent upon the application of an electric field. In what follows, we assume for convenience that a material exhibiting the Kerr effect is used in each cell. The derivation could be carried out equally well assuming each cell exhibits the Pockels effect; one would simply use ZJ in place of v2. The position of the F and S axes shown in Fig. 20 assumes that the electro-optic material has a positive Kerr constant, as does nitrobenzene and the majority of other commonly used materials. If a material having a negative Kerr constant were used, the F and S axes would be interchanged.

I

VOLTAGE

F

EL

I

ELECTRODE

Fig. 20. Cell used as basic ‘‘building block” of an electro-optic network.

172

0 P TI C A L B I R E F R I N G E K T N E T W 0 R K S

[IV, § 4

Light entering the cell with its electric field polarized along F is operated upon by the transfer function exp(ai@o), while the light polarized along S is operated upon by exp(-)i@o) .The quantity @ is the difference between the transit times in the cell of the F and S light components. This "time delay" @ is due to the electrically-induced birefringence of the cell, while in the crystal of Fig. 3, the time delay a is due to the natural birefringence of the crystal. Generally, a is much larger than @. The quantity $@o can be written in terms of the cell length L , the Kerr constant K of the medium, and the applied electric field E as

3@0

= nLKE2.

(4.1)

Furthermore, if we express the applied electric field E in terms of the electrode spacing d and the applied voltage v, eq. (4.1)becomes $@o = nLKv2/d2.

Finally, if we let

vz,= d 2 / 2 L K ,

then eq. (4.2) becomes +@o = +n(v/V0)2.

The quantity V ois simply the half-wave voltage for the electro-optic medium. Eq. (4.4)states that light entering the cell of Fig. 3 polarized along F is operated upon by the transfer function e~p[$in(v/V,)~], while light polarized along S is operated upon by e~p[-$in(v/V~)~]. We note that the crystal of Fig. 3 and cell of Fig. 20 behave similarly except that the quantity o for the crystal is replaced by v2 for the cell. That is, the transmittance of a cell or series of cells depends upon (or varies with) v 2 in exactly the same fashion that the transmittance of a crystal or series of crystals depends upon o.Hence since we are able to synthesize a network composed of suitably oriented identical birefringent crystals which has an arbitrary transmittance vs. frequency characteristic, we are also able to synthesize a shutter composed of suitably oriented identical cells which has an arbitrary transmittance vs. voltage-squared characteristic (at a single optical frequency). The form of an electro-optic network synthesized using procedure 1 is shown in Fig. 21.Such a network has as its amplitude-transmittance vs. voltage characteristic, the desired function C (v2),where C ( 2v ) - c0 +Cle-ix(o/Y,)a+CZe-z~x(u/V,)"+

. . , +cne

--nix(v/V0)*.

(4.5)

APPENDIX A

173

Fig. 21. Basic configuration of electro-optic network.

The network consists of a series of n cells suitably oriented at angles calculated by the synthesis procedure, with the output polarizer angle c&, also calculated from the synthesis procedure. Any of the synthesis procedures of Section 2 may of course be used to synthesize an electrooptic network. AMMANN [1966c] discusses the use of procedure 1 to synthesize electro-optic shutters. AMMANN and YARBOROUGH [1968] have considered the selection of C (v) for multistage electro-optic amplitude modulators, while YARBOROUGH and AMMANN [1968al report on the synthesis, construction, and testing of a three-stage modulator. Acknowledgments The author wishes to acknowledge several colleagues who were important contributors to the synthesis procedures described here. Special thanks go to Prof. S. E. Harris who pioneered and participated in much of the early work. The contributions of Drs. I. C. Chang and J. M. Yarborough to various phases of this work are also gratefully acknowledged. Financial support for much of this work was provided by the U.S. Air Force, the National Aeronautics and Space Administration, and the Sylvania Independent Research Program.

Appendix A In this appendix, we give a method for calculating D ( o ) from I D ( W ) In ~ ~addition, . we show that at least one real set of Di exists,

174

OPTICAL BIREPRINGENT NETWORKS

provided ID(w)l2 never becomes negative. Suppose we are given the positive semidefinite polynominal )D(w)12= A 0 + 2 A , cos aw+ . . . +2An cos nao. Rewriting ( A l ) we have

+ . . . +A,e'""+A,

ID (0)12 = Aneinao+ A n-lei(n-l) +~,~-iaw+

IV

(All

~ U J

. . . + ~ , - ~ ~ - i ( n - l ) a " +Ane-i"a".

(A2)

Xotice that the zeros of (A2) appear in reciprocal pairs. Equation (A2) can, therefore, be factored as lD(w)12 = (DO+D,eia"+D2ei2""+

x (D,+D,e-'""+

. . . +Dneinocu)

. . . +Dne-inao).

(-43)

The D iare not unique, but rather there are 2"+l possible sets of them. ~ and always positive, we may write it as Since I D ( W )is~ even

ID (w)12 = D (w)D*(w).

(A4)

Comparing (A3) and (A4), we see that (A4) can be satisfied only if the D iof (A3) are real. Therefore, at least one real set of the coefficients must exist. The following method of obtaining the D iis due to PEGIS[1961]. For more details and explanation of the procedure, the reader should refer to his paper. We begin with )D(w)I2as given by eq. ( A l ) . Form the equation A,x"+A,-,xn-1+ . . . +A,xfA,+A,x-'+ ... (-451 +An-,x-(n-l)+Anx-n = 0. Put eq. (A5) into the form B , ( x + ~ - ' ) ~ + B , _ , ( ~ + x - ' ) ~ - .~.+. +Bo = 0 (-46) and obtain the Bi from the A i by equating similar coefficients. Make the substitution x+x-l = y (A7 1 into eq. (A6), which gives

Bny"+Bn-,yn-'+

. .. +Bo = 0.

(4

Solve for the n roots of (A8) and call them yl, y 2 , . . ., y n . Using eq. (A7), solve for the reciprocal pairs of roots: x1,

1/%,

IVI

175

APPENDIX B

Next, construct all possible equations having real coefficients di using one root from each row of (A9); e.g., one possibility would be ( x - x l ) ( x - l/xz)(x-xJ

. . . (x-x,) = P+d,-lX"-l+

. . . +dflc2+d1x+do.

(A10)

The Di are proportional to the di

U,

= qd,.

The quantity q is found from

q 2 ( 4 + d ? + . * +d:) = A , ,

(A12)

and upon substituting this value into ( A l l ) , we obtain the Di. The number of real sets of Djwill depend upon the number of yi which are complex. If m of the yi are complex, there will be 2[n-*m+11 real sets of Di.Half of these, however, will just be the negative of the other half, for q can be negative or positive. Appendix B We derive here the conditions which the F iand Si satisfy because of conservation of energy. Assume, for convenience, that we are dealing with a four-crystal network. Since the birefringent crystals are assumed to be lossless, the energy in the fast-axis output plus the energy in the slow-axis output of the fourth crystal must equal the input energy. Mathematically, this is expressed by F4(w)F4*(o)+S4(w)S4*((w) = (1;)'.

(B1)

Writing out (Bl) and equating similar coefficients, we obtain

(F,4I2+(W+ ( F p + (W+(Sf)2+( S p

+ (SQ4)2+cS:lc2

=

(qZp (B2)

F:F:+ F:F;+

F;F;+s:s;+s;s;+s;s~ = 0,

F: F;+ FfF t

+SiS; +SiS: = 0,

(B3) (B4)

h G E N T li E T W 0 R K S 0 P T I C A L B I R E F R I'

176

[IV

Similarly, we can derive for the ith stage

(F;)'+ (F:)'+

. . . + (FfJ'+

(S:)'+ (S;)'+

. . .+(Sf)2= (I:)', (Be)

F'Fi1+ F < FI+ ' . . . +F:-,Fi-,+S:S~+.SiS~+ F;Fi+

. . . +F:-,Ff-,+S~S~+ . . .

F;F:-,+s;s~

. . . +Sf-,Sf +S:-,S:

=

0,

= 0, (B7)

(B8)

= 0.

Appendix C This appendix gives a systematic and rapid method of calculating the input to a crystal once the output is known. This is simply a formalized procedure of solving for the F'-l and Si-l of eq. (2.9d) once the Fi and S' are known. In matrix form, the expressions are

REFERENCES

I-

177

Si-1

Sf-'

Sf-1 a-1

One convenient check is that the calculated values should always be zero. References

and S:-'

AMMANN, E. 0.. 1966a, J. Opt. SOC.Am. 56,943. AMMANN, E. O., 1966b, J. Opt. SOC.Am. 56, 962. AMMANN, E. O., 1966c, J. Opt. SOC.Am. 56, 1081. AMMANN, E. 0. and I. C. CHANG,1966, J. Opt. SOC.Am. 55, 836. AMMANN, E. 0. and J. M. YARBOROUGH, 1966, J. Opt. Soc. Am. 56, 1746. AMMANN, E. 0. and J. M. YARBOROUGH, 1967, J. Opt. SOC.Am. 57,346. AMMANN, E. 0.and J. M.YARBOROUGH, 1968, IEEE J. of Quant. Elect. QE-4,209. AMMANN, E. 0.. J. M. YARBOROUGH, E. J. SLEEPand M. L. SHOEMAKER, 1968, Rirefringent Devices (finalreport), NASS-20670, Sylvania Electronic Systems Mt. View, California. J. A., 1968, Transform Method in Linear System Analysis (McGrawASELTINE, Hill, New York) p. 134 BILLINGS, B. H., S. SAGEand W. DRAISIN,1951, Rev. Sci. Instr. 22, 1009. EVANS, J W., 1940, h b l . Astron. Soc. Pacific 52, 305. EVANS, J. W.. 1949, J. Opt. SOC.Am. 39, 229. EVANS, J, W., 1958, J. Opt. Soc. Am. 48, 142. EVANS, J. W., 1963, Appl. Opt. 2, 193. GUILLEMIN, E. A., 1963, Theory of Linear Physical Systems (JohnWiley and Sons, New York) p. 430. S. E., E. 0. AMMANN and I. C. CHANG,1964, J. Opt. SOC.Am. 54, 1267. HARRIS, LYOT,B., 1933, Compt. Rend. 197, 1693. LYOT,B., 1944, Ann. Astrophys. 7, 31. MCINTYRE, C. M. and S. E. HARRIS,1968, J. Opt. SOC.Am. 58, 1576. MERTZ,L., June 1960, J. Opt. SOC.Am. 50, advertisement facing p. xii. MURDOCH, D. C., 1947, Linear Algebra for Undergraduates (John Wiley and Sons, New York) p. 50. OHMAN,Y., 1938, Xature 141, 167. PEGIS, R. J., 1961, J. Opt. SOC.Am. 52, 1266. SOLC,I., 1953, Casopis pro Fysiku 3, 366. SOLC,I., 1954, Casopis pro Fysiku 4, 607, 669. SOLC,I., 1965, Casopis pro Fysiku 5, 114. STEEL,W. H., R. N. SMARTT and R. G. GIOVANELLI, 1961, Australian J. Phys. 14, 201.

STROKE, G. W., 1964, Roc. IEEE 52, 865. YARBOROUGH, J. M. and E. 0. AMMANN, 1968a, IEEE, J. of Quant. ElWtr. QE-4, 230. J. M. and E. 0. AMMANN, 1968b, J. Opt. Soc. Am. 58, 776. YARBOROUGH,

This Page Intentionally Left Blank

V

MODE LOCKING I N G A S LASERS BY

L. A L L E N and D. G. C. J O N E S School of Mathematical and Physical Sciences, University of Sussex, Rvaghton, England

CONTENTS

PAGE

5 3 0 §

1. INTRODUCTION

. . 2 . FORCED LOCKING . 3. SELF.LOCK1NG . . . 4 . CONCLUSIONS . . .

R E F E R E N C E S.

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

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

. . . .

. . . .

181 183 212 232 233

Q 1. Introduction The phenomenon of forced oscillation has been a subject of study for a very long time. For example, RAYLEIGH[1877] discussed the important difference between “forced” and “free” vibrations of a resonant system, pointing out that “the period of the former is determined solely by the force which is supposed to act on the system from without; while that of thelatter depends only on the constitution of the system itself”. Long before any oscillating systems other than mechanical or acoustic could be studied, the principle of the coupled pendulum with one oscillating system driving another was analysed and the relative phases and amplitudes of the driving and driven oscillations discussed. The first lengthy analysis of such coupled oscillations in electronic circuits was made by VAN DER POL [1922, 19341, who considered the coupling between two LC circuits in one of which oscillation was maintained. Van der Pol observed that if the resonant frequency of the passive circuit was brought close to that of the other, the frequency of oscillation of the system jumped to that of the passive circuit. To explain this phenomenon, he found it necessary to extend the classical treatment of forced oscillation by including nonlinear saturation and coupling terms. He produced equations for the forced oscillation of a resonant system of the form,

E ( - (cr-,6E2)l?+w:E

= coTE‘sin w,t,

where u is the small signal gain coefficient of the system, ,6 describes saturation, and a signal of amplitude E’ and frequency w1 is impressed from outside on the circuit. This approach enabled him to make a successful analysis of the interactions in simple systems. Further work on coupled circuits was carried out by ADLER[1947] who dealt specifically with the problem of the locking of one signal on to another and whose results have a direct applicability to laser work. 181

182

M O D E LOCKING IN GAS LASERS

[v, 8 1

Adler analysed in detail the conditions under which oscillation in a resonant circuit could be locked to an external signal, and the consequences of such locking. A high Q system resonant with amplitude E at frequency mo, excited by a signal of amplitude &El and frequency m1 (near coo) is considered. Adler demonstrates that the rate of change of the phase of the signal in the resonant system relative to that of the impressed signal is governed under certain conditions by a differential equation of the form

El du - - -sin dt

2QE

u+Amo

where Am, = ml--oo and u is the phase angle between E and E l . This equation possesses a solution describing a signal in the resonant circuit stationary in phase with respect to the imposed signal and hence phase locked on to it. The analogy between the laser and the electronic oscillator was noted very early on in the development of the laser. Lasers operate at optical frequencies and usually produce an output composed of a number of closely spaced modes each with a very high Q. The spacing of the modes, c/ZL (referred to in this paper by the symbol d in units of hertz) is usually of the order of tens to hundreds of megahertz. Otherwise the analogy is exact, and many of the results of electronic oscillator theory can be carried over with minor modifications to the discussion of lasers. This is especially true of the concepts outlined above. The modes of oscillation can be shown to obey (at least to a first order of approximation) equations identical to Van der Pol’s, and Adler’s analysis can be taken over completely. Complications are caused by the multimode nature of the laser output and, experimentally, by the optical frequency of operation, but the electronic analogue provides possibly the most fruitful method of analysis of the locking of laser oscillators. The most obviously analogous experimental situation would be the locking of a laser signal to one injected into the laser from another laser, and this has been achieved by STOVER and STEIER[1966] (see 5 2.2.4). More common is the indirect injection of a locking signal which causes the various modes to lock to each other and produce a regular output. This occurs via various modulating elements (see 3 2). Finally, self-locking, a phenomenon directly analagous to Adler’s work has been observed (see 9 3). The present review attempts to describe a theoretical and experi-

v, 5 2:

FORCED LOCKIKG

183

mental basis for the study of locked laser systems. To produce a coherent picture, attention has been concentrated on those sections where experiment has gone more or less hand-in-hand with theory. Much of the present experimental work on high power, solid state systems has been omitted since not only are the processes occurring poorly understood, but the technology of such systems is advancing at such a rate as to make a review not worth while at this time. Clearly, though, in the long run this aspect of locking is going to be most important; the locking process has been used to produce chirped high-energy pulses of only picosecond duration. Similarly discussion of ring lasers, which have to a large extent a theoretical analysis and technology of their own, has been omitted.

52. Forced Locking 2.1. INTRODUCTION

Locking between the modes of both gas and solid state lasers has been induced in a variety of different ways. Internal time-varying perturbation, variation of cavity length, feedback via the R.F. excitation and light injection have all been successfully exploited. Internal timevarying perturbation implies an element whose optical path length, or whose loss, may be varied by an externally applied signal. A change of the first type is called a phase perturbation, while one of the second type is a loss perturbation. The effect of a phase perturbation is to associate a set of sidebands with each of the previously free-running modes. This will occur provided that the strength of the perturbation and the difference between its frequency and the axial mode interval are of the right order. In the limit of very small detunings and weaker perturbations the laser behaves in a very similar manner to the locking process obtained by a loss perturbation. That is, the frequency interval between adjacent oscillating modes is identical from mode interval to mode interval when locking occurs and the inter-mode phase relationship is well defined and constant in time. The result is a laser output in the form of a series of regularly spaced sharp pulses. The general development of the technique of induced, or forced locking is historically speaking a short one. The first example was et al. [1964]). The wide range of possible reported in 1964 (HARGROVE techniques and the piece-meal development of the associated theory,

184

M O D E L O C K I N G IN G A S L A S E R S

P,

§ 2

however, makes it difficult to separate the experimental and theoretical aspects of the problem in a completely meaningful way. It seems worthwhile, though, to give a description of the experimental work if only to make clear what is observed experimentally when locking occurs. Then in the section devoted to the theory of forced locking, reference back will be made to specific observed phenomena. 2.2. EXPERIMENTAL

2.2. I. Amplitde modulation

Hargrove, Fork and Pollack made the first experimental observation of forced locking in a 0.63 pm He-Ne laser. The laser was internally modulated at the same frequency as the beat between adjacent axial modes, d . A fused quartz block was placed at the Brewster angle, to minimise reflection losses, as near as possible to one laser mirror. The block was excited at a longitudinal acoustic resonance and the resultant standing acoustic wave in the block acted as a diffraction grating. The amplitudes of the diffraction orders produced fluctuated at twice the driving frequency; all except the zeroth order go out of the cavity. The light intensity in the zeroth order is modulated with fundamental frequency vmOd= 2v,,0,,tic. When locking occurred, the previously erratic laser output changed to a constant amplitude with, typically, a five-fold increase over the average unlocked amplitude. A scanning interferometer display showed that the randomly varying mode intensities changed to a stationary spectrum. Within the range of modulation frequency over which locking occurred, A -+ (1+10-6)d, different output pulse shapes were observed. At the upper range of locking frequencies essentially symmetric pulses with a 2.5 nsec half width and repetition period of 17.8 nsec were observed. As the modulation frequency was decreased, the pulses broadened and reduced in peak intensity. A t yet lower frequency a secondary pulse became resolved from the trailing edge of the primary pulse. Further reduction in modulation frequency produced a sinusoid of twice the modulation frequency. The observed locking range varied with laser gain as did the modulation threshold. The threshold, which was sharply defined, appeared to correspond to the peak round-trip loss required to produce nearly a 100 yo modulation of the internal energy density. Hargrove et al. suggest that the absence of significant changes in pulse shape with increase of peak round-trip loss shows that nonlinear effects play a role in pulse

v, § 21

FORCED LOCKING

185

sharpening. FOSTERet al. [1965] successfully achieved locking with the locking element external to the laser cavity. This had the great advantage, compared to the internal acoustic diffraction cell of Hargrove, that the optical losses of the modulator were of considerably less importance when not inside the cavity. A He-Ne laser was used together with an ultrasonic Bragg diffraction cell followed by a mirror. This mirror enabled the Doppler shifted light to be fed back into the laser. When the sound cell was driven at a frequency which was an even submultiple of the laser axial mode beat frequency ( A ) , power could re-enter the laser cavity after passing through the sound cell an even number of times. The frequency-shifted radiation then interacted with that within the cavity to produce locking. In the experiment actually performed the cell was driven at 57 MHz since A was 114 MHz. After one pass each laser mode was shifted upward by 57 MHz and was then reflected by the mirror back through the cell. In this way the mode was again shifted upward by 57 MHz and was therefore able to enter the laser and interact with the next higher frequency mode in the cavity. The evidence for locking was that when the cell was on, the audiofrequency second-order beats due to the slight differences between the beat frequencies from adjacent modes disappeared. These slight differences are, of course, the results of the cavity modes being displaced from the passive cavity resonances due to the result of pushing and pulling. Such beat notes are time dependent because of the drift, in time, of the cavity length. I t was found that when the total external path length was approximately equal to the optical length of the cavity optimum locking occurred. This presumably was because both the phase and frequency of the light was then suitable for locking. In 1965 Crowell attempted to put the theory of resonant intracavity modulation, and of self-locking, on to a good theoretical footing and tested the quality of his theory by his own direct experiments. The most important results to be experimentally verified he listed as 1. The width of the pulse is approximately equal to the (oscillating linewidth)-l or (Av)-l. Where Av is the range of frequencies over which oscillation is possible. 2. The ratio of peak power to the average power is approximately equal to the number of coupled modes or A Y / A . 3. The average output of the laser is a function of modulator drive and cavity tuning as given by eq. (2.6) in Section 2.3.1 of this chapter.

186

M O D E L O C K I N G IN G A S L A S E R S

[V?

J

2

4. The phase of the light pulse relative to the R.F. is linearly dependent upon the cavity tuning parameter @, as given by eqs. (2.4) and (2.8) in Section 2.3.1.

Experimentally Crowell found four distinct features. First the pulse repetition rate was twice the frequency of the R.F. applied to his quartz transducer. The pulse widths in He-Ne and A+-lasers were 0.5 and 0.25 nsec respectively, which are compatible with Av of 2000 and 4000 MHz. Second, the measured ratio of peak power to average power for the He-Ne laser was 17, which was approximately equal to the number of oscillating modes. In A+ he expected a ratio of 40 but measured between 20 and 30. The discrepancy was attributed to a nonuniform distribution of mode amplitudes. Third, the experimental result predicted by eq. (2.6) was essentially verified but Crowell considered that more refined experiments were necessary to establish its absolute accuracy. Fourth, the measured position of the light pulse relative to a constant frequency R.F. drive indicated that the position was essentially linearly dependent upon cavity length as predicted by eq. (2.8). Crowell concluded that the experimental results verified the ad hoc assumptions used in his analysis. Consequently the analysis based upon these assumptions ought to provide some physical insight into the characteristics of mode-coupled lasers. His theoretical treatment is discussed in Section 2.3.1. A related approach was that due to Di Domenico and Czarniewski. They achieved locking in a 0.63 pm He-Ne laser by means of synchronous intracavity acoustic modulation in coupled multi-mode confocal interferometers. The internal loss modulation was performed in a separate resonator coupled to the laser resonator. The coupled interferometers were in the confocal configuration with the coupling mirror, M,, located at the common focal plane of mirrors MI and M,. The acoustic modulator was a block of fused quartz placed at the Brewster angle and driven at 50 MHz. The two sub-resonators had a length of 150 cm and so had an axial mode frequency separation of 100 MHz. As the acoustic exciting frequency was tuned near 50 MHz mode locking occurred, as was shown by the fact that the 100 MHz axial beat frequency had a smooth envelope when displayed on the spectrum analyser and that the time-resolved output signal was in the form of a train of sharp periodic pulses with a 100 MHz ( A ) repetition frequency. It was not possible to check accurately whether or not the pulse width was equal to the reciprocal of the oscillating linewidth of the 0.63 pm transition or whether the peak power equalled

v, § 21

FORCED LOCKING

187

the average laser power times the number of locked modes, as theory predicted. It was found that the two resonators had to be well aligned for locking to occur and just as for the external Doppler cell experiment the position of the modulator was found to be important. The strongest locking occurred when the modulator was close to either mirror M, or M,. When the modulator was in the centre of the interferometer the mode locking ceased. This is explainable by considering the modulator to be a shutter which opens once during each modulation period. If alight pulse passes through the shutter when it is open in the centre of the resonator, it will return from the mirror a time (24)-l sec later and find the shutter closed. But when the modulator is near the mirror the pulse returns d-l sec later and finds the shutter open again. Thus the modulator must be close to a mirror, although the exact position is not critical. 2.2.2. Frequency modulation Very soon after the work of Hargrove et al. HARRISand TARG [1964] reported the operation of a He-Ne laser such that all of the laser modes oscillated with well defined phases and nearly Bessel function amplitude, so that the modes effectively played the role of sidebands of a frequency-modulated signal. A frequency modulated light signal has the form

E

= exp i(Znvt+B sin 2nv,t) 05

= exp(i2nvt)

2

Jn(6) exp (i2nvmt).

na-05

The resulting laser oscillation was, in effect, swept over the entire Doppler linewidth with a frequency which was approximately that of the axial mode spacing. This form of locking was induced by an intra-cavity phase perturbation. This was created by placing a KDP crystal oriented with its optic axis parallel to the axis of the laser, and with one of its electrically induced principal axes parallel to the direction of polarization of the laser light. In this way the KDP crystal introduces a pure phase perturbation and should, ideally, introduce no amplitude perturbation into the cavity. When the phase was perturbed at a frequency not quite that of the axial mode spacing the beat notes between the axial modes were suppressed by at least 25dB but the laser output power became increased by 0.2dB. A scanning interferometer showed that, under these

188

M O D E LOCKING I N GAS LASERS

[v. 8 2

conditions, the laser modes possessed approximately Bessel function amplitudes just as they should if they were spectral components of a pure F M signal of low modulation index. It is interesting to note that when the phase was perturbed at exactly the axial mode spacing frequency, the F M laser oscillation was not obtained. In fact under these circumstances the beat notes were found to be stabilized and enhanced and appeared similar to the AM locking observed by Hargrove et al. The F M laser output was passed through a Michelson interferometer with a path difference between its arms of 30 cm. With both arms of the interferometer unobstructed a strong signal was observed at the modulation frequency but this signal disappeared if either arm were blocked. The quality of the F M signal was also tested by converting the light to AM light using birefringent crystals (HARRIS;1963]). By use of a scanning interferometer the laser mode amplitudes were observed directly as a function of the depth of modulation. As the modulation depth was increased the central mode amplitude began to fall and the first pair of sidebands increased. At greater depths the second and third sidebands achieved appreciable amplitudes and hence there was a diffusion of power toward the wings of the Doppler broadened gain-line. Harris and Targ concluded that the process appeared to be “a regenerative parametric oscillation, with a resultant quenching of the original laser modes; as opposed to the phase-locking process considered by Hargrove et al.”. MASSEY et al. [1965] soon put the knowledge gained in this work to good use by producing a “super-mode laser”. This is a technique for producing essentially single-frequency light from the entire output of a high-power multi-mode laser without losing power. Hitherto a single mode laser implied the suppression of modes and, for an inhomogeneously broadened gain line at least, a consequent drastic drop in the output power. MASSEY et al. [1965] took the output of an F M laser and passed it through an external phase modulator driven 180” out of phase, but with the same peak phase deviation as the output of the FM laser. The light output of the F M laser may be described as

E = E, cos(w,t+Fsin w m t ) where E , is the peak amplitude of the optical field, we the central frequency of the optical output spectrum, w, the modulation frequency and r the peak phase deviation of the F M oscillation. If this signal is

vs § 21

FORCED LOCKING

189

passed through a second phase modulator driven at the same frequency as the internal modulator then the resultant signal is E‘ = E , cos (o,+Tsin o,t+r‘ sin(o,t+8)), where r‘ and 8 are the maximum phase deviation and relative phase respectively of the second modulator. When T = r’and 8 = ( 2 n f 1)n then E’ = E , cos o,t and the signal is monochromatic. For an improperly set relative phase, the signal is still F M with deviations up to twice that of the F M laser. Again a scanning Fabry-Perot interferometer was used to analyse the resultant laser output. When both modulators were on, and the relative phases and deviations adjusted correctly, it was found that within the limits of the 100 MHz resolution of the interferometer “all the energy of the original modes had been converted to a single frequency or ‘super-mode”’. The condition was found to be stable with time; only slight changes occurred in the output amplitude. In conclusion the authors pointed out that the super-mode technique was by no means limited to gas lasers and ought to be useful with high gain solid-state lasers. HARRIS and MCDUFF[ 19651 published a detailed non-linear analysis of the F M laser which included the effects of atomic saturation, powerdependent and power-independent pulling and finite width of gain profile. Simultaneously AMMANN et al. [1966] published an account of some detailed experiments on He-Ne F M lasers to check the applicability of the theory. They found that, in general, the agreement between theory and experiment was very good. The essential experimental observations they made are summarized below. The laser was operated in a single F M oscillation and a particular value of r the modulation index measured, where

r =6-L-l

nud

6 1

=_

_

3ta

(see Section 2.3.2 for meaning of symbols). F was calculated from the above formula and both the calculated and observed values plotted as a function of l/a. Calculated and observed values agreed within 10 yo,Ammann et al. noted that in regions of high r the observed values exceeded calculated values for positive values of a but were less than the calculated values for negative de-

190

[v. § 2

MODE LOCKING IN GAS LASERS

tunings. If the measurements are based upon a centre frequency of about 20kHz instead of zero the calculated and observed values agree more closely. This asymmetry is discussed in Section 2.3.2. For small values of F there was little competition between the various F M oscillations and the laser oscillated substantially like a freerunning laser. As predicted by Harris and McDuff substantial coupling between the various F M oscillations began to occur as was increased. The regions where multiple F M oscillations occur simultaneously are referred to by them as “unquenched”. As r was still further increased, the laser output became a single F M oscillation with steady Besselfunction amplitude and F M phase. As I‘ was further increased multiple F M oscillation occurred again and this second unquenched region marked the transition from F M to the final phase-locked region which occurred for very large values of P. The spectral characteristics of this region are very similar to those of the AM phase-locked laser observed by Hargrove et al. The F M laser power output was measured as a function of modulator detuning, ad, for several values of the parameter 6. For a given excess gain, the size of the unquenched region between the phase-locked and truly F M region became smaller and smaller as 6 was increased. This is again in agreement with the predictions of Harris and McDuff. For sufficiently large values of 6 there was a region over which the output power of the laser was zero. For a given value of laser gain the range over which the laser was extinguished became larger as 6 increased. In the region of real F M oscillation it was found that the laser power increased as r g o t smaller. In fact when r 5 2 the laser output effectively consisted of a carrier and a sideband on each side of it; all other sidebands had negligible amplitudes. Thus the laser modes were concentrated in the middle of the Doppler gain curve where the gain is greatest. When an F M laser is operated at higher values of r more energy is shifted into the sidebands which are displaced from the line centre and hence the F M power is limited by the absence of gain. Theory predicts that for sufficiently high 6 the F M power depends only upon r and is independent of the combination of 6 and ad which produced r. This was confirmed experimentally. The theory also states that in practice perfect F M oscillation is virtually unobtainable because the actual mode amplitudes and phases will deviate somewhat from the ideal. Experiment confirmed the prediction that for a given excess gain and 6, the distortion decreases as r gets smaller. Similarly for a given gain and r,the distortion decreases with larger 6.

r

v. 5 21

FORCED LOCKING

191

A very interesting effect was observed in the phase-locked regime. If the modulation frequency Y, was shifted slightly from d (the axial mode spacing) the mode envelope, but not the modes themselves, was shifted down in frequency some hundreds of megahertz and the laser power decreased. As v, was changed still further, the laser was finally extinguished. But if the detuning was stopped just prior to extinguishing the laser the coupled mode envelope intermittently jumped from its shifted position below the line centre say, to a mirror image position above the line centre. I t was found, further, that the even beats were steady but that the odd beats were unsteady. Thus the even beats behaved as for normal phase-locking but the odd beats had typical F M region behavior. In Harris and McDuff’s theory the phaselocked region is really a region of highly distorted FM. They predicted that the region should disappear for large 6 and this apparently was the case here. A prediction that all modes will be displaced from their original free-running position by an amount 4 = (6/n)d was also verified. Finally the effect of thermal drift on the F M and phase-locked regime is interesting. In the FM regime when the spectrum drifts so that the carrier is no longer the mode nearest the gain centre, the entire spectrum suddenly shifts and centres upon a new mode nearest the line centre. However, in the phase-locked case as the modes drift they continuously change their amplitudes such that their envelope is unchanged. The intermittent jump of the coupled mode envelope to a mirror position about the gain curve centre has been further investigated by [1969]. They demonstrated the existence of two HONGand WHINNERY phase-locked states; one with p-even and the other with p-odd, where dn+l-dn

= Pn

dn = ( - l ) P + ‘

24.

The two states were observed by the different time positions of their respective pulse trains measured with respect to the reference modulation signal. Either state can exist by itself or both simultaneously. The time position evidence was supported by observing the beat spectrum. Since the beats for the odd-set are out of phase with the even-set the beat note disappears when both sets oscillate simultaneously. When a pure neon isotope He-Ne laser was used the “intermittent jumps” occurred in an unpredictable fashion, but with natural neon predictable behaviour occurred. It was found that as v, was increased

192

M O D E LOCKING IN GAS LASERS

[v. 8 2

the mode envelope corresponding to the even state was shifted up in frequency by several intermode spacings. The envelope was no longer symmetric. The output pulses became distorted in shape. The odd state however, shifted toward the low frequency side of the Doppler gain curve in an exactly similar way. It was found that for positive detunings one state could switch to the other, hence the “intermittent jumps”. The effect was amplitude-dependent and a switching mechanism based on supermode competition was proposed. MASSEY [1966] reported F M and phase-locked operation of a small He-Ne laser with full output power at 0.63 pm with an internal phase modulator which was a crystal of synthetic quartz. The great advantage of synthetic quartz is that it has a small electro-optic effect but can be obtained with excellent optical properties and finished with negligible surface imperfections. The crystal was oriented such that the light entered the crystal at the Brewster angle and was refracted along the Y-axis. The optical polarization and applied electric field were parallel to the X-axis of the crystal. The exit face of the crystal was polished parallel to the entrance face and in this way reflections at both faces could be minimized by careful adjustment of the modulator orientation without affecting the angular alignment of the laser mirrors. When the modulation frequency was that of the axial mode beat frequency the laser was easily phase-locked. With a higher power input to the modulator and/or a slight detuning, the laser output was a highly distorted F M oscillation. With further detuning out to 100-200 kHz came full F M oscillation with low distortion. In this region the free-running modes were replaced by the spectral components of a single F M oscillation. For detunings greater than 300 kHz the modulator did not couple the modes. The spectrum analyzer revealed the characteristic spectral narrowing and amplitude enhancement of the intermode beat frequencies when the modulator was working in the phase-locked region, and the suppression of the R.F. beats by more than 20 dB was again seen when the modulator was detuned 200 kHz giving the F M region. Massey found that the low-frequency noise was 30 dB down in both cases compared with that of the free-running laser. The CO, laser normally oscillates in only one axial mode. Consequently mode-locked operation can be realized not by locking the phases of several oscillating modes, but rather by transferring energy from a central oscillating mode to nearby, but below-threshold, modes. CADDESet al. [1968] used intracavity loss modulation to obtain a mode-

v, § 23

FORCED LOCKING

193

locked pulse train with 5 nsec pulse widths. They were able to account for their observations by an extension of the work of MCDUFFand HARRIS[1967] which included the effects of nonlinear saturation. The experimental technique used by them to obtain locking was that of Hargrove et al. and consisted of a GaAs crystal used as a standing acoustic wave loss modulator. With no modulator drive no axial mode beats were detected. As the drive was increased, beats appeared at SA out to S = 5 at 183 MHz. As the depth of modulation increased, the average output power decreased to about 60 yo of the free running power which was rather higher than the theoretical analysis predicted. This was probably because there was a finite amount of gain in the modes adjacent to the central driving mode whereas theory assumed the presence of gain in only one mode. 2.2.3. Locking by use of a moving mirror

The convenient device of a moving cavity mirror forcing the modes to lock was devised by HENNEBERGER and SCHULTE[1966]. A 13layer high reflectance coating was deposited on to the end of a quartz crystal transducer and used to operate a 0.63 pm He-Ne laser. The crystal was operated at its ninth harmonic at 59.4 MHz to achieve a resonance with the axial mode spacing of a 252 cm cavity. Depending upon cavity adjustments, pulses at d or 24 were obtained for either the TEM,,, or TEM,,, modes. Surprisingly there was no suggestion of F M operation when the drive frequency differed from A , possibly because the amplitude of the variation in cavity length was insufficient. The authors of this work neatly explain that when vR.F.= d, light emitted at any one instant returns to that spot %/A sec later but has suffered in the meantime n reflections from a moving mirror and is Doppler-shifted by an amount [2n-v/c] v. For all large values of v ( t ) this results after very few reflections, in an optical frequency shift larger than the cavity linewidth. The only light which remains in the cavity is that which finds the mirror at rest. Thus two running waves would be expected in the cavity, resulting in an output pulse at frequency v, = 2vR,F.= 24. This is observed only well above threshold; below that only v = 4 is seen. This discrepancy seems to be not well explained. Phase-locking of the axial modes of a He-Ne laser operating at 0.63 pm was obtained by SMITH[1967] by translating one of the cavity mirrors at a constant velocity. When the laser was operating sufficiently above threshold so that self-locking was absent it was found that mov-

194

M O D E LOCKING IN GAS LASERS

[v, § 2

ing a mirror above a critical velocity changed the laser state from freerunning to mode-locked. For certain values of excitation this locking took place with output pulses with a repetition rate of 24 but at higher mirror velocities repetition rates of d observed. The laser output did not depend on mirror velocity until the velocity was as high as 30 cm/sec when the output began to decrease. Smith explained that at this velocity a laser mode is swept over the Doppler gain curve in the time it takes energy to bould up in the laser cavity. Hence for larger velocity the energy stored will not be at its highest possible level. At that time Smith was not able to explain the observed phenomena. AUSTON[1968a] predicted that if the transverse mode set TEM,,, where n = 0, 1, 2, . . . could be be phase locked with a Poisson intensity distribution and equal phases, a scanning laser beam would result. In a subsequent paper (AUSTON[1968b])he demonstrated the accuracy of his prediction. Forced locking was obtained in He-Ne using a modification of the vibrating mirror technique of Henneberger and Schulte. A rectangular slit was inserted into the laser resonator to limit the transverse modes to a single plane and to prevent very high order transverse modes from quenching the lower order modes. When the detector slit was placed on axis a pulse train of repetition rate equal to twice the transverse mode spacing was observed. This was because the beam swept past the slit twice per period. However, when the slit was located at the outer edge of the scan, a pulse train was obtained with a repetition rate equal to the mode spacing. Also the pulse width was a minimum on axis since the velocity of scan is greatest at this point. Sporadic self-locking was also made to occur when the modulator mirror was replaced by a good quality flat mirror. When the mirror separation was increased and the excitation set at its highest level a second type of spontaneous locking occurred which involved two sets of transverse modes having different axial indices, i.e. TEM,,,, n = 0 to 5 and q = qo, qo+l. The locking here was less sporadic but did not involve the complete locking of all modes. The beam did not appear to be sweeping back and forth in the transverse plane in a continuous manner, but appeared to be scanning in discrete steps, each step spaced in time by the reciprocal of the axial mode spacing, 2 nsec. This is because the mode-locked wave packet has been localized in the axial as well as transverse direction. Low-frequency cavity length modulation has been used by BAMBINI and BURLAMACCHI [1968] to obtain forced locking. One of their mirrors was made to oscillate at 50 Hz by means of a loud-speaker coil.

"3

§ 21

FORCED LOCKING

195

For a mirror velocity above a certain critical value locking proved to be possible for all sets of transverse modes with a pulse repetition rate of A . The critical velocity was found to depend on a number of parameters such as excitation level, cavity length, position of the laser tube in the cavity and on mode configuration. It was found that if the low-frequency noise spectrum was measured with the mirror at rest there resulted a half-width of about 20 kHz. This corresponds to a characteristic fluctuation time of about 50 psec. The amplitude fluctuation of any mode of the system, observed with a spherical mirror Fabry-Perot, was found to have the same spectral width. In this experiment the transit time of each mode across the Doppler curve for the critical mirror velocity was about 60 psec. Consequently since the mode was then being swept across the Doppler curve in a time comparable with the characteristic fluctuation time, each mode cannot exchange appreciable energy with other modes, hence each mode will maintain its phase relationship with the others. Thus if self-locking occurs at some instant it will continue for the lifetime of the modes. Moreover the new modes which begin to grow will grow with the correct phase because they are driven by the combination tones due to the interaction of the preceding modes. 2.2.4. Miscellaneous locking techniques

HUGGETT [1968] has shown how to maintain the mode-locked condition independent of cavity perturbations. The output of the laser was detected by a photo-multiplier to produce an R.F. beat signal at approximately A . This signal was amplified and fed back to the laser by the modulator. The modulator, in turn, perturbed the laser and the feedback loop is completed. Provided the phase of the feedback signal was correct the signal grew. A KD*P crystal was placed in the cavity to produce the phase perturbation. When the feedback loop was completed the laser immediately changed from free-running to mode-locked. As the cavity length changed the frequency of the fed-back signal automatically changed with it and locking persisted. As Huggett points out, if one incorporated an external modulator for demodulation, as demonstrated by Massey et al. and discussed previously, it would be possible to make a single frequency source which stayed single frequency indefinitely. Finally, some mention should be made of an experiment which is perhaps closest in spirit to the locking process discussed by Adler (see Section 1). STOVER and STEIER[1966] succeeded in phase-locking one

196

M O D E L O C K I N G IN G A S L A S E R S

rv. § 2

laser to another by direct injection of the first laser beam into the cavity of the other laser. One mirror of each laser cavity was mounted on a piezoelectric transducer so that the cavity lengths could be fine-tuned. An optical isolator ensured that laser 1 injected its signal into laser 2 but that the reverse possibility could not occur. The frequency of laser 2 was swept linearly with time by a sawtooth applied to the transducer. With the locking path blocked, the laser outputs mixed in a photomultiplier gave rise to beats. When the locking path was unblocked the beat notes vanished over the portion of the trace corresponding to the frequency range for which the lasers were locked. Adler has shown that the locking range 24, should be given by

E 2 4 , = wo --

Q

Ao

where (0, and A , are the frequency and voltage amplitude respectively of the free-running oscillator, E is the voltage amplitude of the effective locking signal, and Q is the loaded Q of the oscillator cavity. Adler assumes that E < A , and 24, +Az,

where the thickness of the acoustic diffraction grating or modulator is Az. Consider ooM nc/L. The appropriate Maxwell equations may be used and if the fields are written as normal modes (see SLATER [1969]) then, a

then the representative mode equation proves to be,

198

M O D E LOCKING IN G A S L A S E R S

[V’ § 2

where w, is the frequency of the nth mode, Q, the frequency of the lzth mode in the passive cavity and Q, the Q of the nth mode.

where R is the reflection coefficient of the mirrors. The mode equation is equivalent to Lamb’s equation with the additional assumption that the time-independent loss of the cavity results from imperfectly reflecting mirrors and with the addition of a time varying loss represented by the volume integral JCT E - En dV. When the two volume integrals are expanded in terms of the normal modes, coupled mode equations result. Crowell was able to evaluate the integral of P * E in terms of the real and imaginary parts of the laser medium susceptibility. Expressing the oscillation frequencies in terms of the resonant frequencies of the passive cavity yields, LO,

= nQ,-6,.

For a solution in terms of time invariant mode amplitudes, the frequency separation between oscillating modes must be equal to the frequency of the time varying loss. So, when the modes are coupled 6,,,-6,

= constant = 6,.

The coupling term which represents the projection of E onto the nth mode must also be evaluated. Substituting yields a complicated equation which may be simplified by equating real and imaginary parts separately giving,

and where (x,) is the real part of the laser susceptibility and Qns represents the gain of the medium in the lzth mode. It is extremely interesting to note that Lamb’s assumption that the loss due to the mirrors be replaced by a volume conductivity leads to

and

v, § 21

199

FORCED LOCKING

The results obtained by these different approaches will only agree when Qn is very large. The optical frequency of the nth mode differs from the passive cavity frequency by an amount proportional to (x,}. As Q is decreased the frequency pulling of the medium will be affected by Q;:. In order to solve the coupled-mode equations the following assumptions are made: 1. That the frequency pulling by the medium on the centre mode is unchanged by the time-varying loss, or 8; oc So where 8; = 6,,-8,. 2. All oscillating modes have the same amplitude,

E,.

3. That Q, is sufficiently high that the imaginary part of the right-hand side of eq. (2.1) is equal to Q;;E%. That, Q;; = Q;~-~m/?,,m~cmIz where p,, represents the saturating effect from the optical power :; represents in the mth mode on the gain of the nth mode and Q the small signal gain parameter.

4. That the phase relationships between modes may be expressed as, +n=

and

-nw,t,

+a,

+,= - n(w,t,+n)+a,

for cos (nz,/L) > 0 for cos (nz,/L) < 0

(2.3)

where a,, 0. If the modulator were moved to the opposite end of the laser without changing the phase of the R.F. the light pulse would be delayed by approximately one-half of the period between pulses. Crowell found that if a,, = n2@/2p where, @ = -4pQ,d0[Q~ COS (nZ,,/L)]-' (2.4) and Q,, describes the loss due to the modulator, then the relative phase between modes may be written as, @&m

M

+zI+m(@--o,t,)

(2.5)

where the 9 t h mode is the mode oscillating at the centre of the gain curve, and

200

M O D E LOCKING IN GAS LASERS

P o

§ 2

where ~m&,mlem12is replaced by #?W,since all modes are assumed to have the same amplitude. Equation (2.6) represents the fractional reduction in optical power in a typical mode as the R.F. drive is increased from zero. The term (Q2-Q;l) represents the degree that the laser is operating above threshold. If the loss of the cavity as represented by Q;' is just equal to the is small and the reduction in power small signal gain, then (Q$-Q;') is critically dependent upon cavity dimensions. But if the laser is operating well above threshold the term is much larger and the fractional reduction in power is much smaller. Locking the phase of each mode to its adjacent modes as in (2.4) causes the laser to operate as a pulse regenerative oscillator. When there are 2 N + l identical modes and E,, is the central mode field amplitude the total field may be summed to yield

where

y = wo(t-t,)+di.

Thus a light pulse occurs whenever y = 2nn where n = 0, 1, 2 . . .. From (2.8) the phase of the light pulse relative to the R.F. is linearly dependent upon the difference in frequency of the time varying loss and the fundamental cavity frequency. If di = 0 the pulse occurs when the standing acoustic wave is zero, independent of the position of the modulator in the cavity. The ratio of peak power to average power follows from (2.7), peak power average power

M

number of coupled modes

and the pulse width is given by [(N++)y]-l which is

-

[ ( N + + ) xnc/L]-' which is

-

AvD-l.

2.3.2. Freqzlency modzllatioiz

HARRISand MCDUFF[1964, 19651 outlined a linear theory of the F M oscillator. They demonstrated that from a physical point of view an intracavity phase perturbation associates a set of F M sidebands with each free-running laser mode. In this way each free-running mode becomes the centre frequency, and so carrier, of an F M signal. The

v,

§ 21

FORCED LOCKING

201

multiple F M oscillations then compete for the atomic gain as did the previously free-running modes. For the case where the perturbing frequency is approximately d the first upper sideband of an F M oscillation centred on the gain curve is within the same homogeneous linewidth as the centre frequency of an F M oscillation positioned one mode above the centre of the gain curve. The competing oscillations are thus much more tightly coupled than were the free-running modes. The strongest F M oscillation, usually that at the centre of the gain profile, is able to quench the weaker oscillations, but the sidebands deplete the inverted population distributed across the gain profile. The resulting laser consists of a set of modes which have nearly Bessel function amplitudes and F M phases and which is, in effect, swept over a large portion of the gain profile at a sweep frequency equal to that of the phase perturbation. The calculation begins with LAMB’S[1964] self-consistency equations identical to those discussed in Section 3.3:

and

If an integer no describes the ratio of the central frequency to A , i.e. Q, = 2znJ

and hence 52, = Q0+2znA is the frequency of the nth passive cavityresonance, and the phase perturbation is assumed to have a time varying susceptibility Ax’ (2, t ) of the form A x f (2, t ) = Axf ( 2 ) cos v,t where v , is the driving frequency of the perturbation. Then the contribution to the polarization due to the intracavity modulation is, P ( z , t ) = E ~ A x ’ (tz),E(z,t ) =

E ~ A X( 2 I)

cos ~,t E (2, t )

202

M O D E LOCKING I N GAS LASERS

and the contribution to the P ( t ) driving the nth mode is,

(2.9)

where U,(z) = sin (n,+n)nz/L. If we consider that only adjacent modes make large contributions then we need keep only the terms for which q = n + l . Defining,

:I

6 =-

(2.10)

A x ’ ( z ) U , ; ~ ( ZU,(Z) ) dz

as the coupling coefficient between modes, and expanding this assuming that the spatial variation of Ax‘@) is slow compared with U,(z) yields,

Ax’ (z)cos

ZZ

I,

dz.

Combining ( 2 . 9 ) and ( 2 . 1 0 ) gives,

The frequency of the nth mode is arbitrarily chosen to be in the vicinity of Q,. Putting, vn = Qo+nv,

then v,+l-v,

= Qo+ (n+ l)v,-vm

Y,_l+Vrn

= Q,+

= v,

and (n- l ) v m + v , = v,.

The effect of the atomic inversion on the polarization is accounted for by in-phase and quadrature contributions to the susceptibility, x; and xn respectively. Thus, I I

v, 9: 2;

I, 6, ( t ) = E O X ~ En

203

FORCED LOCKING

+

E08C

[-En+, sin

(+n+l-+n)

+En-,

sin

(+n-+n-1)1.

AV =z A-V,

Let

represent the detuning of the driving frequency from the inter-axial mode beat frequency, then

Qn-vn = nAv.

Substituting for Sn(t),Cn(t) and Qn-vn sistency relationships gives,

into the original self-con-

6C [ d n - n A v + i ~ ~ 2 E n= - 2L [En+lcos ( + n t l - + n )

+En-lcos

(+n-+n-l)I

(2.11)

For small signals,

where an is the round trip phase retardation seen by the nth mode as a result of the gain pulling and znm represents the effects of power dependent pulling and pushing. Letting 6, be the peak single pass phase retardation of the perturbing element of length a, then from (2.10)

A 1 6, = - X L V 2c

taking Ax1 to be independent of z over the length of the intracavity modulator. If the element is centred at distance zo from cavity mirror then

=

(sin

i)

(cos

F)

6,.

6, cos (nz,/L). If a / L > 60 rf 0, three steady state equations were obtained. These were solved for E,, El and C#I for various values of P, and 60, taking

If for a pair of IPS] and 6w, lEol is zero, then the oscillation at w, is quenched by the injected signal a t w,+Sw and locking occurs. This is found to be the case for

where E: corresponds to E , for the free running laser. In terms of laser power W , and injected power W , these imply, 6w(locking) = 2

ZQ

[3J '. W

This is identical to the result obtained by Adler for valve oscillator circuits. A parallel development of the theory of internal time-varying perturbation was carried out by Yariv and various co-workers over the time-span considered in this section. (See YARIV[1965, 19661, PETERSON and YARIV[1964] for example.) The work is not reviewed here firstly because the work has had only a minimal interaction with the development of the theory and experiment associated with internal time-varying perturbations that are outlined in this account. Similarly the theory of HAKENand PAUTHIER [1968] is ignored since it is especially suited for the case of homogeneously broadened lines. § 3. Self-Locking 3.1. EXPERIMENTAL

The phenomenon of self-locking was first observed by Javan who communicated it to W. Lamb (LAMB[1964]). Javan's experiment apparently comprised a gas laser operating on three modes at frequencies

v, J 31

SELF-LOCKING

213

vl, v 2 and v3; the beat notes v1-v2 and v2-v3 were observed. As the cavity tuning was changed so that the values of (v1-v2) and (v2-v3) approached each other, a frequency jump occurred. A similar phenomenon was noted at around the same time in the construction of ring lasers, when the (Doppler shifted) frequency separation between the waves travelling round the ring in opposite directions was reduced below a few kHz. Observation of beat notes between modes is not however, the most important way in which self-locking manifests itself. As discussed earlier when a number of modes are locked to each other, there is a constant phase relationship between the oscillating electromagnetic fields comprising the modes. In a free-running state, the phase relationship is time dependent and so the modes add incoherently to produce a laser output which has no marked amplitude modulation. If, however, locking occurs, the modes add coherently and a strongly modulated output occurs. For example, if we consider an idealised laser with n modes in oscillation which are locked together so that there is a zero phase difference between one mode and the next, the total output amplitude of the laser will be n

where the amplitude of one mode is described by

E ( t ) = E , exp {i(n+n,)dt} (if the modes are assumed to have equal amplitude). Thus the total amplitude is given by: n [exp (indt)- 1: -. E = E, exp (in&) zexp(ilzdt) = E , exp (in&) 5 [exp (idt)- 1: Simple optical theory tells us that this formula describes a train of pulses with repetition rate d and width llnd sec. Similarly, if a pattern of modes is chosen such that alternate modes are JZ radians out of phase, a train of pulses results with repetition rate 24 and width l/nd sec. In general the occurrence of self-lockingcan be observed by a sudden modulation of the laser output, usually into trains of sharply defined pulses. I t should be noted that the two cases written down above are the most commonly studied, because the most easily observed, although they are not by any means the only ones possible. I t can also

214

,MODE L O C K I N G I N G A S L A S E R S

[“.§ 3

be seen that the pulse length produced by self-locked lasers may well be very short. In the first case described above, if there were ten modes oscillating in a laser with cavity length 1 metre, the pulse length would be of the order of 0.5 nsec. Pulses this short are observed only with difficulty using conventional photomultiplier systems and thus quantitative analysis of self-locked systems usually requires observation of the beat notes between the various modes in the locked and unlocked condition. The first systematic analysis of the phenomenon occurred when experimentalists, notably CROWELLj19651 and UCHIDAand UEKI [1967], working on the locking of gas laser modes by the use of time varying perturbations discovered that quite often the lasers would lock themselves spontaneously. CROWELL[1965], while analysing forced locking in the He-Ne 0.63 pm laser, found that spontaneous locking could be obtained by “optimising” the R.F. beats observed between the modes and decreasing the Q of the cavity. Using a high speed photomultiplier, he was able to observe the pulse form of the output of the laser. He demonstrated that, as would be expected, the period between pulses corresponded to the addition of inphase modes separated by a frequency of A for a wide range of values of A . His experiment was obviously not designed to make a more thorough analysis of the conditions under which such spontaneous locking occurred. Crowell also attempted unsuccessfully to obtain a similar effect in a A+ laser. Such an effect was obtained by GADDYand SCHAEFER [1966]. As in Crowell’s experiment, the output of the laser was observed on a high speed photomultiplier and locking was obtained by decreasing the Q of the cavity, in this case by inserting a thin glass plate. Pulses with a repetition rate of A , were obtained but also, in some circumstances, trains of pulses with a repetition rate of 24. The analysis of the interactions in the A+-laser is complicated and little work has been done on it. It should be noted that the “disappear[1965], and the ing mode” phenomena noted by BRIDGESand RIGROD supermode” effect produced by passive etalons may well owe something to self locking effects. Most of the experimental work on self-locking has been carried out using the 0.63 pm He-Ne laser, and this laser lends itself best to an analysis of the results obtained. Unfortunately, the specifically experimental investigation of MCCLURE [1965] and NASH j1967j, although wide ranging, do not include enough information about the parameters of the laser systems used, to enable comparison with theory. Their I‘

V,

I

31

SELF-LOCKING

215

work is, however, worth describing in some detail as they give qualitative results which are useful in deciding which parameters of the system are of importance. XcClure studied a 0.63 pm He-Ne laser oscillating in TEM,, modes which had a plasma tube 92 cm long. The self-locking was produced using a diaphragm stopped down to a diameter slightly smaller than that necessary for the elimination of higher order transverse modes. The dependence of self-locked operation on cavity length was investigated for two positions of the plasma tube. With the plasma tube centred in the cavity, pulses with a repetition rate 4 could be observed for cavity spacing of 1.6 to 2.2 m. For spacings of 4.3-5.3 m, pulses with a repetition rate of 34 were observed. With the plasma tube asymmetric in the cavity, pulses with a repetition rate of 4 were observed for cavities up to 3.6 m long. For lengths of 3.6 m-5.2 m pulses with a 2 4 repetition rate were observed. NASH [1967] also studied self-locking for TEM,, modes. He used two similar He-Ne plasma tubes 1.4 m long, one filled with a mixture of natural Ne and 4He and the other with pure 20Neand SHe,both in a 7 : 1 ratio. Like the other workers, he produced self-locking by means of an adjustable iris, but unlike the others, his cavity length (1.5 m) was only very slightly longer than the length of his plasma tube. He observed both the beats between the modes and the modes themselves. As noted by other workers, more than one type of self-locking occurred. Most easily observed, and appearing using both tubes was “type I locking” giving pulses with a repetition rate of 4 as observed by Crowell and McClure. With the mixed isotope tube only, he also obtained what he referred to as “type I1 self-locking” where the repetition rate was 24. I n these experiments, beats at frequencies of 3 4 were not observed. Type I1 locking was obtained by increasing the aperture of the iris, but not so much that modes other than TEM,,,,,, occurred. Nash also noted that the output power of the laser did not vary more than 9 % between the unlocked and the self-locked regimes. Nash was able to explain the amplitude variation with frequency of the various modes on the basis of the isotopic mixture of Ne in the two tubes. All the preceding work was concerned with TEM,, type modes. A number of workers including AUSTOX [1968], KOHIYAMA, FUJIOKA and KOBAYASHI [1968] and SMITH[1968] have observed self-locking between modes with different radial and azimuthal field distributions. AUSTON[1968] demonstrated self-locking between transverse modes

216

M O D E L O C K I N G IN G A S L A S E R S

in He-Ne lasers. Using a short (16 cm) plasma tube in a 24 cm cavity, he produced self-locking by carefully aligning his optical system. In this case the beam scans across the cavity, so that the identification of the modes oscillating is not quite so direct. However two cases appeared. I n one case locking occurred between TEM,,, modes where q was constant and n took a value in the range 1-4. I n the other case TEM,,, modes were involved with q taking values in the range &5. In this case the locking was sporadic and did not apparently involve all the oscillating modes. SXITH [l968] obtained simultaneous locking of both longitudinaland transverse modes in a 0.63 pm He-Ne laser. Two situations occurred in his experiments. In one case all the longitudinal resonances corresponding to each transverse mode were locked and two or more separate pulse trains were observed in the output of the laser. The second type of self-locking was similar to that described by Auston in that the different transverse modes were locked together and the laser beam 'scanned' across the resonator. Smith used a two-metre laser cavity with a 130 cm plasma tube and self-locking was induced by a neon absorption cell (see below). FUJIOKA and UCHIYAMA [1969] obtained self-locking between TEM,,, and TEM,,, modes in He-Ne with a 1 m plasma tube and a 1.25 m cavity. They detected the output using a fast photo-diode and t u o different states of locking were obtained. In the first the set of TEM,,, modes were locked together as were the set of "EMd,,modes. The output of the laser consisted of two superposed trains of type I locked pulses. The relative amplitudes of the two sets were dependent on where the detector was placed with respect to the output pattern of the laser. The relative position of the two trains was determined by the position of the plasma tube in the cavity and the alignment of the mirrors. In the second type of self-locking, a single train of pulses of repetition rate 24 was obtained, which did not change with position of detector. Fujioka and Uchiyama suggest that this second type of locking was due to transverse modes being quenched alternately. It seems that this might not be the only possible explanation and that a 'scanning' type mode as described by Auston and Smith might be occurring. The first type of locking was the most common and the second achieved by chance. HIRANO and KIMURA[1969] investigated the possibility of producing pulses with repetition rates of nd where 'iz is relatively large. They used a He-Ne laser with fixed cavity length and fixed position of tube

V.

s

31

SELF-LOCKING

217

in the cavity. Various techniques were used for selective suppression of unwanted modes. The remaining modes then locked spontaneously. Suppression of alternate modes by either a Michaelson or Fabry-Perot interferometer enabled pulses with repetition rate of 24 to be obtained. Suppression of all modes between the first and the (n+l)st produced pulses with repetition rates of nd. Hirano and Kimura found that stable pulses were produced for 1z. = 2 or 3 but that for n 2 4 the pulse trains became unstable. In most of the experiments described above self-locking occurred spontaneously in the system by a judicious adjustment of either the cavity losses or the excitation of the plasma tube. Fox et al. [1968] demonstrated a more reliable method of producing self-locking effects using a non-linear absorber. A tube of pure ,OX'e, excited by a discharge was placed in the cavity of a He-Ne laser and allowed stable self-locking to occur under both high and low excitation conditions. The process by which such self-locking is produced is not clear. A simple suggestion is that the combination of the frequency dependences of the amplifying tube and the loss tube produces a loss minimum at the centre of the Doppler-broadened gain curve. A mode configuration centred about this frequency would be favoured and this is a highly favourable configuration for self-locking (see 9 3.3). A full explanation is likely to be considerably more complicated. CHEBOTAEVet al. [1968] have carried out an experiment similar to Smith's and shown that for certain combinations of gain and loss, self-locking occurs, while for others single mode operation is favoured. WOODand SCHWARZ [1968] observed self-locking in a passively Q switched CO, laser. The output frequency of the laser was selected by an intra-cavity prism and mode locked behaviour could be obtained on lines from P(20) to P(24) by rotating the prism. I t was not obvious whether the self-locking was produced by non-linear polarization in the SF, Q switch or whether, as seems more likely, the CO, is responsible, with the SF, merely controlling the Q switch behaviour. It should finally be noted in this survey of purely experimental observations the majority of observations of self-locking have been made on the He-Ne 0.63 pm laser purely for reasons of availability. There is no particular reason why self-locking should not occur on other laser lines and as far as is known, whenever this has been looked for it has been observed. From the point of view of analyses of the processes involved other lines, such as the 1.15 pm He-Ne line used in the selflocking experiments of JONES et al. [1969], may well be preferable.

218

M O D E L O C K I N G Ih’ G A S L A S E R S

3.2. MAXIMUM EMISSION P R I N C I P L E THEORY

3.2.1. Introdzlction

The first attempt at an extended theoretical analysis of self-locking was made by Statz, Tang and others in a series of papers starting in 1965. The approach used in these papers, although not theoretically foolproof enables a number of useful predictions to be made about self-locking phenomena. The main problem involved in the calculation of the output of the laser is the dialectical relationship between the electric field in the resonator and the amplifying medium which maintains that field. The system can be treated in classical terms, and the contribution of the medium can be expressed by a macroscopic electric polarization term which has, however, to be calculated from the atomic properties of the medium. This polarization drives the field but is itself produced by the field. One solution to the problem is the “self consistent field” approach discussed in the next section, but Statz et al. proposed, as a first approximation, a more simple method which could be used for relating field and polarization, the so-called “maximum emission principle”. In this case the polarization is calculated as a function of the field intensity, but the field intensities used are defined according to a rule that the relative phases between the various modes of oscillation of the electric field must be such as to maximise the total rate of stimulated emission of the medium. In the first paper of the series, STATZ and TANG[ 19651 considered a homogeneously broadened laser oscillating simultaneously in three longitudinal modes, TEMd,,, TEM(,+,,,, TEM(,+,,,. The calculation is one dimensional and deals only with plane polarized fields, but takes into account the fact that the excited medium does not completely fill the cavity. The macroscopic polarization produced by these fields and the non-linear terms which produce mode interaction effects, are calculated to a third order of perturbation. The rate of stimulated emission is then calculated. It is found that the rate of emission is determined by the relative phase between the modes. If the expressions for the electric fields of the three modes are given by ~ , ( t= ) E , cos ( ( w - ~ w ) t +el)

~ , ( t= ) E , cos (wt+ e,) and

E 3 ( t ) = E 3 cos ((w+Aw)t+O,).

v, 5 31

SELF-LOCKING

219

The relative phase between the modes Y is defined as Y = (e1+0,-2e,).

The maximum emission principle can now be used to find the value of Y which maximises the output. I t is found that in this case, or depending on the sign of a parameter describing the length and position of the excited medium in the cavity. x‘ and x” are the real and imaginary parts of the effective ‘susceptibility’ of the medium derived from the polarization calculated earlier. In the case of the ruby laser, Statz and Tang showed that x‘’ w 0, so that for certain positions of the ruby in the cavity, self-locking described by the first solution for Y should occur and for other positions, locking described by the second solution for Y should occur. It can be seen that these two situations are equivalent to type I1 locking and type I locking described earlier. This work was extended by STATZet al. [1967] to cover both homogeneously and inhomogeneously broadened systems and to deal with four or more modes. Their extension to inhomogeneous broadening is only qualitative. In the four and five mode systems, the non-linear polarization contains terms involving the summation of a number of three mode relative phases. Solutions equivalent to those given above are found using the maximum emission principle, and these are used to calculate in special cases the output waveform of the locked four or five mode system. The predicted output agrees qualitatively with results obtained using a ruby laser. 3.2.2. T h e Probability of occzlrrence of self-locking In the previous papers, the only cases investigated were those with the locked values of relative phase. STATZ[1967] attempted to calculate the magnitude that must be obtained by the non-linear parts of the macroscopic polarization if locking were to occur at all. Considering a number of systems, both homogeneously and inhomogeneously broadened, and using the same formalization as before, Statz used a and STATZ[1967a] (and discussed in 5 2.3.4) result derived by TANG to calculate the ‘locking range’ 60 of a three mode system.

220

As in

M O D E L O C K I N G IN G A S L A S E R S

3 2.3.4

this is defined by

6~ = +w0 IP,/EoI where coo is the frequency of the unlocked central mode, E , is the amplitude of that mode and P , is a non-linear polarization term. Given this definition, Statz considered the magnitude of the nonlinear polarization terms which might give rise to self-locking. Considering three modes as before, the interaction term at frequency wo is given by P ( z ) = &(z) E,*(z), where x is calculated as in the earlier work, for either homogeneous or inhomogeneous broadening. The above calculation gives a maximum value for 8co if locking is to occur. Longitudinal modes in a passive cavity are, of course, equidistant, but mode pushing and pulling effects ensure that for an operating laser the modes are displaced. Statz attempted to calculate the divergence from equidistant spacing for some special cases, and compare them with the calculated values of 6w, to decide whether selflocking was possible. Pulling is caused solely by the power-independent dispersion of the medium and the expected value of 6w can be fairly easily calculated. Statz showed that for ruby lasers, oscillations should be self-locked, so long as the frequency range of oscillation is less than 10’OHz. For inhomogeneously broadened systems, power-dependen t pushing effects must also be taken into account. Statz concluded that the achievement of self-locking in He-Ye, A.+ and Kd lasers would be an uncommon event. This, of course, is not in agreement with experiment. 3.2.3. Successes and failures of simple theory

Although this theory gives acceptable results for lasers with homogeneous broadening it is not so successful for inhomogeneously broadened systems. From the analysis given above it can be seen that the failure might either be due to underestimation of the locking range 80 or overestimation of the non-linear pushing and pulling effects. However, even if this were so, Statz’s predictions disagree with those of Lamb (see Q 3.3) who shows that self-locking will always take place if the modes are symmetrically disposed about the centre of an inhomogeneously broadened line. This diagreement does not stem from the use of the “maximum emission principle” even though this is essen-

V,

I 31

SELF-LOCKING

221

tially based on physical intuition. TANGand STATZ[1967b] have compared the results of calculations based on this principle with those obtained by the self consistent field approach. These results are in agreement, but it should be noted that the maximum emission principle results apply to the case when self-locking has already taken place, and do not have any bearing on the question of whether self-locking will occur. The answer to this question rests, in any particular set of circumstances, on the solution of the coupled equations connecting the electric field to the source parameters. Statz et al. effect an arbitrary separation of these equations by considering locking between a mode and an external polarization term, then calculating separately the pulling and pushing of the free-running mode by another polarization term Po.The two problems can only be separatedin thisway when Poand P, are not intimately related as is apparently the case in homogeneously broadened systems. For inhomogeneously broadened systems, such a separation cannot be made. The maximum-emission principle is thus most useful in considering the equilibrium self-locked state. For lasers with more than three modes in oscillation it enables calculations of the output waveforms to be made. Such calculations would be prohibitively complicated using a more sophisticated theory. 3.3 SELF-CONSISTENT FIELD THEORY

3.3.1. Outline of theory Mode self-locking in a laser system was predicted by LAMBj19641 as a consequence of his theoretical analysis of laser action in inhomogeneously broadened lines. Lamb’s analysis, although restricted in scope, leads to some important predictions and is worth considering in some detail. Lamb consider the use of a laser medium of arbitrary length in an optical resonant cavity, dealing only with the one-dimensional case of fields varying along the axis of the cavity. The standing-wave optical field set up is considered semiclassically and Maxwell’s equations relating the modes of oscillation of this field to macroscopic polarization source terms are applied. The equations can be reduced to a pair for each resonant mode of the cavity: V

E,+&-E,=

Q

V

-i-S,, 80

222

M O D E L O C K I P F G IN G A S L A S E R S

+

(vn

[v, 8 3

V

d n -Qn

1E n = -4 - C,.

(3.2)

EO

Q, is the frequency of the nth mode of the passive cavity, Q is the quality factor of the cavity at this frequency v M v,. The electric field of the nth mode in the cavity is described by

E ( z , t ) = E,(t) cos (v,t++,(t))

- sin(nnz/L)

(3.3)

and C, and S , are the amplitudes of the real and imaginary parts of the spatial Fourier component P , ( t ) of the macroscopic polarization driving the field.

:Io’

P,(t) = -

P ( z , t )sin

nm

-dz.

L

Assuming P , can be calculated, eqs. (3.1) and (3.2) can be solved to give the amplitude En and frequency v, of the nth mode. The macroscopic polarization P is calculated to a third order of perturbation in terms of the electric field existing in the cavity, and eqs. (3.1) and (3.2) are then used as self-consistency relations which can be solved to produce values for En and v,. As might be expected, the polarization contains components near the various resonant frequencies. However, the third order terms (concerned with saturation of the output) also contain so-called “combination tones” oscillating at frequencies given by combinations of the various cavity frequencies. These combination-tones are an important element in the production of the self-locking effects. For instance, in the case of a laser oscillating in three modes vl, v2 and vQ, the third order perturbation terms in P have components , 2v2-v3 near the original with frequencies vl, v2, v3, V ~ + Y ~ - - V ~ 2v2--vl, passive cavity resonant frequencies (see Fig. 1 ) . These polarization components provide what are essentially external oscillators to which the various modes can lock. As shown in the figures the mode a t v 2 has an adjacent source term a t (v1+v3-v2) to which it can be locked under certain circumstances. I t should be noted that the discussion so far parallels that of Statz et al. (9 3.3). The difference lies in the fact that in this case the locking or otherwise of the modes stems directly from the solution of the self-consistent field equations (3.1) and (3.2). To talk about modes oscillating a t frequencies vl, v2 etc. and then to

B

V,

31

223

SELF-LOCKING

time Centre !

J

Fig. 1. Cavity resonances R,,s), and 9, together with mode structure of an inhomogeneously broadened laser oscillating a t frequencies v l , v p and v 3 .

consider whether they will lock, is incorrect. However, for a qualitative picture of the processes involved, the discussion of the previous paragraph is useful. The Lamb theory produces very complicated equations when applied to more than one modes of oscillation. Full analytic solutions have only been worked out for the case of two and three mode oscillation in the so-called Doppler limit case when the Doppler broadened frequency with Av,, of the laser medium is much greater than the natural broadening. For two modes, the qualitative discussion shows that there are no combination tones t o produce locking. For three modes, however the situation is different. In the three mode case, the self-consistency equations resolve themselves into a set of six coupled differential equations, three concerned essentially with mode amplitudes and three with frequencies. 2 1

= a1E1--B1E~--1(E1,E2,E3)-E~E3(r123 cos

y+tZ3 sin u/)

2,= Q ~ E , - ~ , E ~ - ~ ~ ( E , , E , , E ~ ) - E ~ Ecos ,EY , (-4~13~ ~sin Y ) (3.3) 8,= a3E3--B,E~-F3(El,E2,E3)-E~El(~21 cos Y+tZ1sin Y ) v1+61=

vZ+&

S,+a,+G1(E1,E,,E3)-E~E3E1-1(r],3

= 92+a2+G,(E1,E2,E3)+E1E3(r113

sin

sin

~ - t , 3 cos

Y/+t13

cos

u)

u/) (3.4)

v 3 + d 3 = Q3+a3+G3(E1,E2,E3)- E ~ E 1 E 3 - 1 ( ~sin 2 1Y--tZ1cos y)

224

M O D E L O C K I N G IN G A S L A S E R S

!P =

(42-41)

-( 4 3 - 4 2 )

= 242-41-43.

[v, § 3

(3.5)

Although the equations are coupled, only the three frequency determining equations (3.4)need be discussed for the present. The important terms are as follows. (T, describes mode pulling due to the gaussian line shape. G , describes pushing effects due to “hole burning” and “hole repulsion”. The final terms on the right-hand side describe interaction effects between modes which depend both on the mode amplitudes and, more importantly, on the relative phases between modes which are defined, as in Statz’s work by the ‘relative phase angle’ between the three modes. The various components depend on the natural line width of the transition, the Doppler broadening of the line and the positions of the passive cavity resonances SZ, with respect to the laser line centre. More importantly they also depend on terms describing the distribution in the cavity of the excited population. These terms are of the form; L N, = dz “(z) p = 0, 2, 4. (3.6)

:Io

L

N ( z ) is the excitation density of the laser medium at point z. It can be seen that N o gives the mean excitation density per unit length of the cavity. I n general, to find out what will be the nature of the laser output, the excitation and cavity mode structure must be specified in equations (3.3)and (3.4).These can be solved numerically (assuming that the various atomic parameters are known). This will, however, be a tedious procedure and approximations can be made which enable quantitative results to be obtained with relative ease. First, the amplitude and frequency equations can be ‘decoupled’ and dealt with separately. This may be justified by the fact that in most applicable experimental cases, the variation in field amplitudes between locked and unlocked states is small (see for example KASH [1967]). In other words in most experimental situations, even in the self-locked state, the mode interaction terms in the amplitude determining equations are relatively small. The case of He-Ne is discussed extensively by SAYERSand ALLEN [1970]. Given this approximation, it is possible to consider, as Lamb did, the phase relationship implicit in eq. (3.4).By subtraction, this set of equations can be recast in the form

P = a + G ( E , , E , , E , ) f A sin Y + B cos Y,

(3.7)

V,

8 31

SELF-LOCKING

226

where a is a term dependent on pulling effects; G describes the effects of power-dependent “pushing” and the final term once again describes the interaction specifically dependent on the relative phases of the modes. This equation is of similar form to the general equation describing locking phenomena derived by ADLER[1947] and mentioned in 9 1. I t defines the time variation of the relative phase between the modes. Lamb pointed out that the equation had two very different solutions, depending if the term (A2+B2)*is greater or less than la+GJ. This can be seen by writing an implicit solution of the equation v dx ( Y = Yowhen t = 0). t(‘l) (a+G+A sin x + B cos x ) ~

=I,,

(3.8)

If la+Gl > (A2+B2)*,the integral has no singularities and an a s y m p totic solution results of the form Y

M

(a+G)t.

In this case the phases of the three modes are independent and the A and B terms in eqs. (3.3) and (3.4) may be considered to‘ time-average to zero. For such a case, the equations can be solved to obtain amplitudes and frequencies of the free-running oscillations. If however, la+GI < (A2+B2)),the integral diverges and Y(t)moves asymptotically to a value

Since in this case Y is not linearly dependent on time it follows that 2v2-v1-v3

=0

and

Thus the mode frequencies arrange themselves in such a way that their spacing is equal, and the relative phase of the modes is constant. 3.3.2 Modes symmetric about the line centre

In the special case of modes symmetrically disposed about the line centre, la+GI is always zero and A and B are non zero. Locking always

226

M O D E L O C K I N G IN G A S L A S E R S

iv, 5 3

occurs, and the three modes have a relative phase defined by Iy =

2#2-41-43

= - sin-' (0).

I n this case it can be seen that the self-locked signal will consist of the sum of three modes with their phases related by either 42-41

(3-9)

43-42

or 42-41

= d3-42+n.

(3.10)

In the case represented by (3.9) it was shown earlier that the output signal will be in the form of a train of pulses with repetition rate A , equivalent to the type I locking discussed in Section 3.1. In the case represented by (3.10) a train of pulses with repetition rate 24 will occur, equivalent to type I1 locking. It is not obvious at first sight whether type I or type I1 locking will occur in any particular situation. BAMBINI and BURLAMACCHI [1968] investigated the stability of the two solutions, using a number of approximations. Their method yields useful results, although the conclusions of their original paper are incorrect due to a misprint in the original Lamb paper. The stability of the solutions depends critically on the relative amplitudes of the three modes and in the relative values of No and N2. If it is assumed that E , = E3and that the natural line width of the transition is small compared with the mode spacing, criteria for stability can be written for the two cases. For stability with Y = 0: N, (2E:-E;) - > -. N o 2(2E?+E;)

(if 2Ef > E;)

or

(3.11) (if 2Ei

< Ei).

For stability with fv = ?I: (if 2E:

> Ei) (3.12)

(if 2E:

< Ei).

5

31

227

SELF-LOCKING

.b

I

0 .

Type I

.

locking

.6

4

Type -.I

Stable

1.0

II

U

1:b

Locking Stable

E:IE:

i

Fig. 2. Stability criterion for type I or type I1 locking in a laser with an inhomogeneously broadened medium. The line parallel t o the abscissa demonstrates the effect of increasing the laser excitation and hence its output, while keeping the geometry of thesystem constant.

These criteria for stability are shown in Fig. 2. For given value of N,/N, either type I or type I1 locking may be stable depending upon the relative values of El and E,. For any particular laser system, a diagram such as Fig. 2 enables a prediction as to what type of locking will occur. It should be noted that the ratio N , / N , depends only on the relative lengths of the discharge tube and the cavity so that changing the level of excitation in a laser will be equivalent in Fig. 2 to travelling along a straight line parallel to the E:/Ei axis. I t can thus be seen that different levels of excitation will produce either stable type I or stable type I1 locking. This is in qualitative agreement with the experimental work discussed in Section 3.1. Since the value of N , / N , is extremely sensitive to the position of the discharge tube in the cavity, and the stability condition is very sensitive to the value of N,/I?,, especially in the region E: O.SEi, a quantitative comparison with published experimental results is not possible. However, it would appear that, in default of any more thorough analysis of the stability criteria, the results of the approximate calculations shown in eqs. (3.11) and (3.12)enable some predictions to be made when the stability of a particular type of self-locked state is in question.

-

3.3.3. Asymmetric modes

Lamb originally envisaged mode self-locking in terms of an experiment in which the beats between three modes were observed as the central mode was tuned towards the line centre. At first, the pushed

228

M O D E L O C K I X G IN G A S L A S E R S

[v, J 3

and pulled modes would display two beat notes at frequencies near A , separated perhaps by a fe& kHz. Lamb postulated that as the central mode approached the line centre, the value of a+G would decrease to the point where transition to a self-locked state would occur with only a single beat note near A . This, in fact, is observed as will be noted later. Thus if eq. (3.7) is evaluated, using the values of the parameters calculated for some arbitrary position of the modes with respect to the line centre, the region over which self-locking will occur can be discovered. Calculation of the extent of this region depends once again on the solution of eq. (3.3). Such a calculation has been carried out by SAYERS AND ALLEN;1970] solving the equation numerically to simulate the time-development of the laser signal from a low level. The results bear out the qualitative analysis of the time variation of the relative phase in the free-running and self-locked situations and detailed locking characteristics are computed for a He-Ne laser. The results are used to study the quality of some of the approximate approaches previously employed. 3.3.4. Application of Lamb theory to exfieriment

The theory is complicated and requires the knowledge of a number of parameters of the atomic system. However, it provides a useful tool for the analysis of the important He-Ne 1.15 pm and 0.63 pm lasers, and can be used, with more care for the A+ laser system. In these cases the linewidth parameters ya, yb,defined in Lamb, are known approximately and for typical experimental conditions the approximation is valid:

In this case, the coefficients in the Lamb equations take on a relatively simple form and approximate equations equivalent to (3.7) can be produced. JONES et al. [1969] obtained approximate forms for the locking conditions and considered the appearance of locking in the He-Ne 0.63 pm and 1.15 pm systems. I t was found that type I locking was a common phenomenon. The approximate theory gives a minimum value for the frequency separation between the two unlocked beat notes v3-v2 and vz--vl before locking occurs, and the observed results agreed with theory. JONES [1969] extended this work to calculate, in particular cases, the values of the locking range (the maximum value of frequency separation between the central mode and the centre of the gain profile). Type I locking was normally observed, in

v,

§ 31

SELF-LOCKING

229

agreement with the discussion on stability of Section 3.3.2. Type I1 locking could be sporadically observed. The locking range was measured by modulating the cavity length and result were obtained for a range of values of Na/Noand L which agreed with theory to within the level of approximation involved. 3.3.5. Possible extension of Lamb theory

Lamb theory gives a self-consistent field solution to the problem of three oscillating laser modes, where the interaction between the radiation field and the atomic system is calculated to a third order of perturbation. Two major criticisms can be levelled at it. First, direct calculation of the locking parameters of three modes is difficult; for more than three modes the calculation would be prohibitively tedious. Second, the atomic system is treated in a highly simplified manner and at high power levels the perturbation approach may lead to serious errors. The question of the model for natural line broadening in the atomic system has been discussed by GYORFFYet al. [1968] and a modified theory which takes account of collision broadening has been proposed. This theory can be taken over without too much difficulty to the discussion of locking effects. The importance or otherwise of higher order perturbation terms can be dealt with to some extent by considering the “saturation parameter” defined by Lamb which gives a measure of the importance of higher order terms. In general, it is found that the third order terms are adequate for most low power gas laser systems. In any case, in any present experimental situation, there are so many unknown parameters that higher order terms would not be expected to play an important part in the analysis. For single mode oscillation, the accuracy of the perturbation approach has been discussed by STENHOLM and LAMB[1969]. Their conclusions also bear out the suggestion that for typical low power He-Ne laser systems third order theory would be a reasonable approximation. 3.4. THE PULSE TRAIK APPROACH

3.4.1. “n-pzllse” analysis

The theoretical analyses discussed so far have all considered the frequency spectrum of the modes of oscillation and attempted to calculate the conditions under which this spectrum is such that self-locking occurs. There is, however, another approach in which the fact that

230

[v. 5 3

M O D E L O C K I N G IN G A S L A S E R S

self-locking produces an output train of pulses is taken as the starting point. I t can be considered that the laser medium is being excited by a train of optical pulses travelling through it with a well-defined repetition rate. The question of the response of the medium to such a train of pulses, and the conditions under which its response to such a train is maximized, can then be considered. This problem is closely connected to that of propagation of a pulse of radiation through an amplifying medium which has been treated by a number of workers, notably WITTKEand WARTER[1964], ARECCHI and BONIFACIO [1956], ARMSTRONG and COURTENS [1968] and MCCALLand HAHN[1969]. The difference between the two cases lies in the fact that the pulse traverses the same excited medium many times, and thus the processes which go to produce a pulse stable in shape are intimately linked with the resonant nature of the laser cavity. The first attempt at such an approach was made by Fox and SMITH [1967]. They considered an analogy with the behaviour of a gyromagnetic spin system with spins antiparallel to a magnetic field. In the presence of a stimulating optical field, the spins are induced to precess about the magneticlines of force, and if the optical field consists of a pulse of exactly the right duration, the spins are flipped exactly 180” and the maximum amount of energy is given up to the pulse. Such a pulse is known as 180”or ‘n’pulse and this type of system has been extensively analysed (see HAHN[1950]). There is an obvious analogy with the situation of a population of excited atoms which can be stimulated to omit radiation. Fox and Smith postulated that the radiation field in a laser cavity might well adjust itself so that n pulses are produced. This physical postulate is exactly analagous to the “maximum emission principle” discussed in 3.2. Using results derived by LAMB[l960] for the interaction of a pulse of radiation with a population of two level atoms, Fox and Smith derive an optimum pulse length, t,for maximum emission where t =

x

[(w--oo)2+

(;)3

-t

;

o is the frequency of incident radiation, wo is the transition frequency of the atomic system, I/ is the perturbing potential between the radiation and an atom and is proportional to the amplitude of the incident radiation field. For a homogeneously broadened line, the equation predicts a linear relationship between t and laser output intensity. For an inhomoge-

v. g 31

SELF-LOCKING

231

neously broadened line, a value of t has to be produced which averages over the Doppler broadened line width. Fox and Smith compared their theory with experiments carried out on a He-Ne 0.63 pm laser. Pulses corresponding to type I1 self-locking were observed and a graph of pulse width as a function of average intensity was drawn. Reasonable agreement between theory and experiment was found, but only on the assumption of a value of (o--w0) that was not checked by any other method. The theory was extended by SMITH[1967] and compared more extensively with experiment. Using a rate-equation approach and assuming homogeneous broadening, Smith analysed the interaction of a train of rectangular z pulses with an excited medium occupying part of a laser cavity. Two criteria for self-locking to occur are deduced. First the maximum allowable small signal gain for C.W. oscillation must be less than the maximum gain achieved during pulsing. The second criterion is deduced from the assumption that the laser will oscillate in such a way as to maximize its internal energy. Then, using Smith’s assumptions, the output intensity in C.W. operation must be less than the average output intensity in pulse operation. These two criteria enable two formulae to be written down relating the pulse width z and pulse spacing T , to the known parameters of the laser system. The first criterion gives

where AvD is the Doppler width of the laser line. This gives a minimum value for t. The second criterion is:

where 7 is the ratio of the maximum unsaturated gain to the cavity loss (in C.W. conditions), and y is a parameter describing spontaneous relaxation from upper and lower levels. This criterion gives a minimum value for T , the pulse spacing. Smith tested his theoretical analysis using a He-Ne 0.63 pm laser consisting of a 35cm laser tube in a cavity which was variable in length up to 8 metres. He was able to obtain self-locked pulses with higher repetition rates than those of any other workers (up to 6d) and his results agreed reasonably with theory.

232

MODE LOCKING IN GAS LASERS

[V> § 4

3.4.2. Fzcrther work .sing

the a~ p d s e approach The 'n pulse' approach to self-locking was also considered for the case of 0.488 pm A+ laser by BASSet al. [1968]. They observed that at certain levels of laser power, the output was composed only of two strong modes. The frequency difference between these modes was nd where n depended nearly linearly on the magnitude of the electric field in the cavity. Bass, De Mars and Statz considered the laser output in terms of a train of pulses, and compared their experimental observations with the Fox and Smith formula for the width of a n pulse given earlier. Assuming a homogeneously broadened line and o = coo, they demonstrate that their results exhibit very poor agreement with this formula. However, the formula they use does not make allowance for inhomogeneous broadening of the A+ line, nor is it obvious that their definition of pulse length is compatible with that of Fox and Smith, so their calculations are not particularly useful. CARRUTHERSand BEIBER[1968] attempted to measure accurately the velocity of pulses in the cavity of a He-Ne 0.63 pm laser. They found that the speed of the pulses was a matter of 0.03 % less than that of light in the cavity. However, the errors in this experiment might possible account for such a discrepancy. FROVA et al. [1969] measured the pulse repetition frequency and pulse width in a self-locked He-Ne laser as a function of mean output power I . They found that both repetition frequency and pulse width depended approximately as I-*. This appears to confirm predictions made by McCall and Hahn for propagation of a 232 pulse in the medium, rather than a n-pulse as suggested by Smith. At this time, the experimental and theoretical situation with respect to the propagation of such pulses in a laser resonator remains unclear.

5 4.

Conclusions

The theory of forced locking seems at this time to be in an advanced state. The theories specifically derived to account for amplitude modulation are not as rigorous as their frequency modulation counterparts. The weakness in such theories is, however, more than negated by the information gained from the theories of Harris and McDuff in the limit where the difference between A and the modulation frequency is zero. The analytic solution of the detailed theory involving non-linear coupling and saturation remains to be achieved. However, it seems unlikely that any essentially different behaviour would be predicted from such

VI

REFERENCES

233

a solution compared with those derived from the numerical evaluations that have already been carried out. Experimentally all theoretical predictions seem to have been satisfactorily bourne out. It would not be surprising, though, if in time new locking techniques are developed. There seems no reason to suppose that those used so far are necessarily exhaustive. The techniques now to hand seem to be reliable and reproducible so that the requirements of forced locking are well known and realisable. In the case of self-locking the situation has now been reached when a self-locked output may readily be obtained from commercially obtainable lasers. However, the optimum conditions for self-locking have not really been established. The theoretical analysis is capable of extension in the direction of the ‘n-pulse’ or ‘2n-pulse’ analysis. It would appear that the most likely use of these techniques will be in the production of high intensity picosecond pulses. The “supermode” and other single frequency techniques clearly lend themselves to the problemsof frequency standardization, interferometry andmetrology, while the scanning pulse output from locked transverse modes may well prove useful in image identification and data storage processes. References ADLER,R., 1947, R o c . I.R.E. 34, 351. E. O., B. J. MCMURTRY andM. K. OSHMAK,1965, IEEE J. of Quant. AMMANN, Elect. QE-I, 263. J. A. and E. COURTENS,1968, IEEE J. of Quant. Elect. QE-4.411. ARMSTRONG, 1966, IEEE J. of Quant. Elect. QE-1, 169. ARECCHI,F. T. and R. BONIFACIO, AUSTON,D. H., 1968a, IEEE J. of Quant. Elect. QE-4, 420. AUSTON,D. H., 1968b, IEEE J. of Quant. Elect. QE-4, 471. 1968, J. Appl. Phys. 39, 4864. BAMBINI,A. and P. BURLAMACCHI, 1908, Appl. Phys. Letters 12, 17. BASS,M., G. DE MARSand H. STATZ, BRIDGES,T. J. and W. W. RIGROD,1965, IEEE J. of Quant. Elect. QE-1, 303. CADDES,D. E., L. M. OSTERINKand R. TARG,1968, Appl. Phys. Letters 12, 74. J . A. and T. BIEBER,1968, J. Appl. Phys. 40, 426. CARRUTHERS, V. P., I. M. BETEROVand V. LISITSYX, 1968, J. of Quant. Elect. CHEROTAEV, QE-4, 788. CROWELL,M. H., 1965, IEEE J. of Quant Elect. QE-1, 12. DI DOMENICO Jr., M., 1964, J . Appl. Phys. 35. 2870. 1965, Appl. Phys. Letters 6, 150. DI DOMENICO Jr., M.and V. CZARNIEWSKI, FOSTER,L. C., M. D. EWYand C. B. CRUMLY,1966, Appl. Phys. Letters 6 , 6. Fox, A. G. and P. W. SMITH,1967, Phys. Rev. Letters 18, 826. FROVA, A,, M. A. DUGAY,C. G. B. GARRETT and S. L. MCCALL,1969, J. Appl. Phys. 40, 3969. FUJIOKA, T. and T. UCHIYAMA, 1969, Roc. IEEE 57, 362. GAIIIIY,0. L. and E. M. SCHAEFER,1966, Appl. Phys. Letters 9, 281. and W. E. LAMB Jr., 1968, Phys. Rev. 169, 340. GYORFFY,B., M. BORENSTEIN

234

M O D E LOCKlNG I N GAS LASERS

[v

HAHN,E. L., 1950, Phys. Rev. 80, 580. HAKEN, H. and M. PAUTHIER, 1968, IEEE J . of Quant. Elect. QE-4, 464. HARGROVE, L. E., K. L. FORK andM. A. POLLACK, 1964, Appl. Phys. Letters 5 , 4 . HARRIS,S. E., 1963, Appl. Phys. Letters 2, 47. HARRIS, S. E. and 0. P. MCDUFF,1964, Appl. Phys. Letters 5, 205. HARRIS,S. E. and 0. P. MCDUFF,1965, IEEE J. of Quant. Elect. QE-1, 245. HARRIS, S.E. and €3. J . MCMURTY. 1965, Appl. Phys. Letters 7, 265. HARRIS,S. E., M. K. OSHMAN, B. J. MCMUKTRY and E. 0. AMMANN, 1966, Appl. Phys. Letters 6, 184. HARRIS, S. E. and R. TARG,1964, Appl. Phys. Letters 5, 202. HENNEBERGER, W. C. and H. J. SCHULTE, 1966, J. Appl. Phys. 37, 2189. HIRANO, J. and J. KIMURA, 1969, IEEE J. of Quant. Elect. QE-5, 219. HONG,G. W. and J. R. WHINNERY, 1969, IEEE J. of Quant. Elect. QE-5, 367. HUGGETT, G. K., 1968, Appl. Phys. Letters 13, 186. JONES, D. G. C., 1968, Appl. Phys. Letters 13, 301. JONES, D. G. C., M. D. SAYERS and L. ALLEN,1969, J. Phys. A2,95. KOHIYAMA, K., T. FUJIOKA and M. KORAYASHI, 1968, Proc. IEEE 56, 333. LAMBJr., W. E., 1960, in: Lectures in Theoretical Physics (Interscience, New York) p. 435. LAM^ Jr., W. E., 1964, Phys. Rev. 134, A1429. MASSEY, G. A., 1966, Appl. Opt. 5, 999. MASSEY,G. A., M. K. OSHMAN and K. TARG,1965, Appl. Phys. Letters 6, 10. XCCALL,S. L. and E. L. HAHN,1969, Phys. Rev. 183, 457. MCCLURE, R. E., 1966, Appl. Phys. Letters 7, 148. MCDUFF,0. P. and S. E. HARRIS,1967, IEEE J . of Quant. Elect. QE-3, 101. KASH.F. I 0. For 6123 < 0 it is the reverse. The peculiarity of the solutions 1 and 3 lies in the fact that the half-axes ratio of the ellipses depends strongly on frequency near resonance. Thus, for the normal wave (n = nl) far from resonance as for wave 2 in the whole resonance region, the half-axes ratio of the ellipse reaches at most a value of the order of 6123(w/c)n~2nl M 5 x 10-3. Conversely for the turning point where fi = (B)" the same ratio is equal to

270

CRYSTAL OPTICS W I T H S P A T I A L D I S P E R S I O N

[VI. § 4

which attains an order of magnitude of 0.1 when

As in addition the half-axes ratio is ii,/ii, times larger for the anomalous wave (i.e. the wave 3) than for the normal wave, a rapid increase (by more than an order of magnitude) of the ellipticity of the light traversing a plane parallel plate as the frequency of the light approaches the resonance could be an evidence for the excitation of an anomalous wave. Even in the cases where the anomalous wave is not excited or is strongly attenuated, the behaviour of the half-axes ratio of the polarization ellipse as a function of frequency in the resonance region could permit us to find the dependence of 6,on o and compare it with the theoretically computed one (see Fig. 3). We would like to point out a possibility of experimentally observing the effects described above. Let us consider, for the sake of simplicity, a one-axis gyrotropic crystal and assume that light of frequency u) propagates perpendicular to the optical axis. Without spatial dispersion two waves can propagate in such a crystal, one with a vector E parallel to the optical axis (extraordinary wave) and another with E perpendicular to it (ordinary wave). Let us consider, for instance, waves which are polarized along the optical axis in the vicinity of the resonance. Then, taking spatial dispersion into account and for B > 0, one finds that the normal waves able to propagate in the crystal and for which the long axis of the ellipse is parallel to the optical axis, are two in number. Thus, when light polarized along the optical axis falls on the plate with a frequency near the resonance frequency for fitl, the intensity of the light traversing will experience oscillations depending on the plate thickness. These oscillations happen because, when superposing two elliptically polarized oscillations 1 and 3 with the same frequency and different refractive indices, as A cos(wt-onl Zlc) and B cos (wt-wn3 Z/c), the time-averaged square of the amplitude of the resulting oscillation is equal to 1

I = +(A2+B2)+AB cos 2n(nl-n,) n. In this respect the situation examined is quite similar to that found with non-gyrotropic crystals, as we could neglect here the excitation of wave 2 in a first approximation.

VI,

5 41

S P A T I A L D I S P E R S I O N E F F E C T S IN CRYSTALS

271

4.3. GYROTROPIC BI-REFRINGENCE AND NEW WAVES I N ANTIFER-

ROMAGNETIC NON-GYROTROPIC CRYSTALS

In crystals having magnetic structures equation (2.16), obtained with the symmetry principle of the kinetic coefficients, relates the components of the tensor eij for structures which permute when time reversal is applied. It is thus necessary to find out, when using (2.16),how the tensor E~~ depends on the quantities characteristic of the magnetic structure of the medium. This analysis does not fit into the frame of this work, however, and we shall use (2.16) in the future only for non-magnetic media and when Bext= 0. As for the magnetic media, some information on the dependence of the tensor E~~ on k can be gained if one considers that in the absence of damping and when w and k are real, tensor sii(u,k)is Hermitean, i.e. E ~ ~ ( c ok, ) = E;

(w, k).

(4.18)

Taking only terms linear in k, we obtain Eij(W,k)

= Eij(O)+iYijZ(W)kl,

(4.19)

so that it follows immediately from (4.18) that &ij(W)

= E;(.(w),

(4.20a) (4.20b)

II

If yijZand y i j z are the real and imaginary parts of the tensor y i j l , we find from (4.20b) that (4.21) In non-magnetic media y:jI 3 0. We could have arrived at this conclusion from (2.16); introducing (4.19) in (2.16) with Bext= 0 and taking into account (4.21) one finds that (2.16) takes the form

As k is arbitrary, this proves the conclusion drawn above. In addition, it follows from (4.21) that the tensor ei3 is symmetric, when there is no attenuation. In general the tensor y:;, # 0 for magnetic structures. As a consequence, a so-called gyrotropic bi-refringence appears. This is an optical effect on which attention was drawn in BROWN et al. [1965] and which

272

CRYSTALOPTICS WITH S P A T I A L D I S P E R S I O N

[VI,

§ 4

is caused by spatial dispersion. I t can be accounted for with the terms linear in k in the expansion of the tensor cij(co, k). Let us dwell on this effect in greater detail. In view of considering also the vicinity of an absorption line (in contrast with BROWNet al. [1965]), we shall use instead of (4.19) the expansion of the tensor &G1(0,k) in the following form derived from (4.1)

The symmetry properties of the tensor 6:,, are well known from AGRANOVICH and GINZBURG [1962, 1963, 1965a, 1965b1, LANDAU. and LIFSCHITZ [1957]. As regards the tensor 6& which is symmetric with respect to a permutation of indices i and j , the number of its independent components cannot exceed 18. For crystals belonging to classes displaying sufficient symmetry, the number of these independent components can be still smaller. For instance, in the Cr20, crystal, belonging to magnetic point group S'm', the number of non-zero independent components of the tensor is two, whereas the total number of its non-zero components is 8, according to BROWN et al. [1965] and DZYALOSHINSKI J [1958]. These non-zero components of are, from BROWN et al. [1965] and BIRSS[1967],

&i1

II

I f

611,

=

-422

a,,

=

,,a,

I,

II

I f

It

-8212 = -6221; = -a2,, = 6;21. =

II

For the MnTiO, crystal, belonging to the magnetic point group 3, the number of independent components of 6ij1 is 8, whereas the non-zero components are, from BROWN et al. [1965],

In the coordinate system, where the axis z is parallel to k,and where as a consequence D, = 0,the equations relating the components D,and D, to each other have, in contrast with (4.2), the form

S P A T I A L DISPERSION E F F E C T S I N C R Y S T A L S

1

(3- E ; ~ ( o ) -E~:(o)D,+

+ 6g3k1Dl+d&kDz

r

3 -E;'(o)

273

- EL'(O)D, = i&,,kD,

(4.23)

+ 6&3k) + &3kD1 = -i6;23kD1.

A further simplification of the system of eqs. (4.23) is possible if one chooses the axes x and y parallel to the main axes of the rank two tensor E ; ~ ( o ) , a,j? = 1, 2 when &G1= &gl= 0. We will not write down the equation for the quantity fi2 in the general case, but will limit ourselves to antiferromagnetic structures not possessing natural optical activity. In these crystals the tensor 6ijZ= 0 whereas, in general, the tensor S& # 0. In this simple case (4.23) takes the form 1

(G -: :&

f

6ii3k)Dl = - S&kD,

s

(4.24)

D,

= - 6;i3kDl,

so that the refractive index 5 satisfies the equation

From (4.24) one sees, in agreement with BROWN et al. [1965], that the presence of the tensor 6ijz modifies the direction of the optical axes of the crystal slightly. When 6& = 0 and E;: # E;', in conformity with (4.241, the light waves are polarized either along the x-axis (whereby fi2 = E : . 3 nil) or along the y-axis (whereby fi2 = 8;' = 4,).However, if diiz # 0, for instance, for a wave with fi M nol, the component D, is already different from zero, so that (4.26)

In addition, from (4.23) and (4.25), the refractive index of the waves transforms when the direction of light propagation is reversed (this is equivalent to the transformation + -&jl). This effect in BROWN et al. [1965] is called gyrotropic bi-refringence. Another effect not considered in BROWN et al. [1965] is also linked to the existence of the tensor 13:;~and is described in AGRANOVICH L1969bI.

274

CRYSTAL OPTICS WITH SPATIAL DISPERSION

[VI, I 4

It is the possibility that supplementary waves appear in the vicinity of absorption lines in magnetic crystals which do not demonstrate natural optical activity (i.e. when 61j, = 0 ) . Let us consider the vicinity of an absorption line in an anisotropic crystal, i.e. the frequency region o M wo where n&(wo) = co when absorption is neglected. As we have fi2 >> n& in this frequency region for waves which transform to a wave with fi2 = nil when d;;, = 0, the dispersion relation (4.25) becomes simpler and takes the form (4.27)

If we now square both parts of this equation, we obtain an equation of the form (4.4),which has three real solutions to the left of the turning point and only one to the right. Let us designate the three solutions to the left of the singular point by the indexes 11, I, 111,in order of increasing values + then i: we shall have .ti2 > nol for solutions I and I11 and fi2 < no, for solution 11. This shows that when > 0 and fi > 0 only solutions I and I11 will satisfy (4.27),whereas when S& < 0 only the solution I1 will do so. Depending on the sign of 8&, the waves dispersion law can be represented schematically as is done on Figs. 5, 6. The horizontal line there corresponds to the solution of (4.25) which has the value 6 = no2for &it = 0. Spatial dispersion has practically no influence on the value of this solution in the frequency region considered.

"2

'0

' W

Fig. 5. ri as afunctionof l = (o-w,)/wo near the frequencyo = w,in antiferromagnetic non-gyrotropic (di,l = 0) crystals whose absorptionisneglectedand parameter dy,3 > 0.

VI, § 4;

S P A T I A L D I S P E R S I O N E F F E C T S I N CRYSTALS

375

It"

I

c

WO

w

Fig. 6. 7? as a function of it = ( w - o D ) / w o near the frequency o = wD in antiferromagnetic non-gyrotropic (a;,( = 0) crystals whose absorption is neglected and parameter 6Y13 < 0.

Comparing Figures 3 and 4 with Figures 5 and 6, one could get the impression that we are dealing here with a situation fully similar to that encountered with non-magnetic gyrotropic crystals. In fact, such an impression would be misleading for two reasons. One is that normal waves are not linearly polarized in non-magnetic gyrotropic crystals. The other is that in anisotropic gyrotropic crystals the frequency range where anomalous waves are possible is narrower than here. This is because in our case dealing with antiferromagnetic bodies this frequency region is determined practically by the contributions to the dispersion relation of terms proportional to k and not to k2,just as in isotropic gyrotropic crystals. We would like to point out, that when the light is propagating along the symmetry axis of the crystal, i.e. when Ell

= E22 =

&(a),

Sy6

=

a&,

equation (4.25) degenerates in a system of two equations 1

1

fi2

E(O)

1 fi2

1 E(W)

w + ;fi(s;;,-S;m)

=

0,

(4.28a)

w + ;fi(S;;3+d;;3)

=

0.

(4.28b)

These equations determine the values of fi for the linearly polarized waves whose vectors D are orthogonal to each other. As we did when analyzing (4.27), it is easy to convince oneself that,

276

C R Y S T A L O P T I C S W I T H S P A T I A L DISPERSIOlV

CVI. §

4

depending on the signs of

the total number of independent normal waves propagated in the direction considered will be four (if a > 0, b > 0 ) , three (if a > 0 , p < O or if a < 0, B > 0) or two (if a < 0, < 0). We did not take the attenuation of the waves into account when we examined the dispersion of normal waves in magnetic media. It is perfectly clear in this connection, that supplementary waves can only appear in the neighbourhood of an absorption line if the halfwidth d of this line is small enough. One can derive from qualitative considerations that this half-width is adequately small when *'4

c

L W O t

where w o is the frequency of the absorption line and

tm=

IW -01 ~

0 0

is the dimensionless frequency corresponding to the turning point of the dispersion curves for crystals with a magnetic structure (Figs. 5 and 6). According t o (4.6), and also according to AGRANOVICH and GINZBURG [1965a] and GINZBURG[1958], the turning point corresponds to a frequency given by (4.29) near the resonance of n& when where 6" is a quantity equal to nil # n& (see (4.27)). If however nil = n i 2 ,one should take instead of 6" the quantities (a1and each in its turn (see (4.28a) and (4.2%)). in the optical part of the spectrum The quantity (wo/c)6' for non-magnetic isotropic gyrotropic media. This value comes from theoretical estimates of the components of the tensor 6ij, as well as from the survey of experimental data (AGKANOVICH and GINZBCRG j1965aI Section 111. For magnetically ordered crystals the maximum values of the components of the tensor d:;, should, in general, differ from those of the tensor 61j, by a factor of the order of I / f i w o ,I being the exchange energy and wo the frequency of the absorption line considered. Thus when the absorption line lies in the visible or in the ultraviolet part of the spectrum, the components ( ~ ~ / c ) must d ~ ~ be , of the order of 10-3x

-

VI, S 41

S P A T I A L DISPERSION E F F E C T S I N C R Y S T A L S

-

-

277

I/kw, = If, however, (w/c)d" lo4, one finds with A w 0.2 and E, 1 (see (4.14), where we assume that = &,--A/t)from (4.29) that the value 151, M 10-3 corresponds to the turning point. In this case the wave dispersion calculations performed above remain valid if the line width satisfies the inequality

< 2 x 10-3~,, with oo= 2 x lo4 cm-l. d

which gives A < 40 cm-l If now I (w/c)d"I M 10-6 we get d < 8 cm-l, the other conditions remaining the same. The microscopic theory of the dielectric tensor E ~ , ( wk) , in antiferromagnetic materials has been developed in HORNREICH and SHTRICKMAN [1968]. Since in Hornreich and Shtrickman the same method is used as in AGRANOVICH and GINZBURG[1965a] for calculating this tensor, we shall only give the results of this calculation here. I t was shown in HOKNREICII and SHTRICKMAN ;1968] that to an accuracy up to the order of the fine structure constant e2 - 1

.-

fie

- __

137

the main contribution to the imaginary part of tensor y i j t is given by the quadrupole electric polarization (see (4.19)). Thus the following expression was obtained for tensor y i i z ( m ) :

+ (0,l

R:ls,)G,l

R6W)I.

(4.30)

Here o,are the eigenfrequencies of the crystal, el?, the dipole moment operator of the electrons belonging to the a-th node of the elementary crystal cell ( a = 1, 2, . . .), 10,) and (S,l the wave functions of a molecule or ion in the basic or the s-th excited state and Nl the number of elementary cells per unit volume of crystal. From (4.30) it follows that

when o >> w , .

It was shown in HORNREICH and SHTRICKMAN [1968] for the Cr,O, crystal that in the region of long waves (A = 5000A) the quantity I# lylll w/cl is of the order of 4 x lo-'. It had been assumed for this

278

CRYSTAL OPTICS WITH SPATIAL D I S P E R S I O S

[VI

that the main contribution to the tensor yiil was being given by the transition at the frequency w, = 250 cm-I. If one is interested in the frequency region w m w s ,then one gets values of the order of 4 x 10-3 instead of 4 x lo-’ for the above mentioned quantity.

Conclusion I n conclusion, we should stress that we have not treated all the effects of spatial dispersion which one could find in crystal optics. So, for instance, in the case of longitudinal waves, one will get a group velocity different from zero only if one takes spatial dispersion into account. Then the angle between the group velocity vector

and the wave vector k can be larger than &c and in particular be equal to ?G (in classical crystal optics this angle is always smaller than +T). Also, in classical optics, the product nK is always positive and in nonabsorbing media ,ii2= (n+iK)2 is always a real quantity. When spatial dispersion is present, the magnitude of the refractive index n and the absorption index K are not subject to such restrictions (see AGRANOVICH and GINZBURG [1965a]). Crystal optics with spatial dispersion accounts for a whole series of new effects and peculiarities with respect to classical crystal optics. We are far from having studied all these points well enough theoretically, not to mention experimentally. We should also remark that the microscopic approach to crystal optics on the present level is closely associated with the theory of excitons, in which spatial dispersion is an organic and quite essential element. So we have still better reasons to abandon the usual scheme in the phenomenological (macroscopic) approach of crystal optics and to develop crystal optics on the basis of spatial dispersion before going over to classical crystal optics. At the end of AGRANOVICH and GINZBURG [1965a], written five years ago, we expressed the hope that “a broad application of crystal optics including spatial dispersion and its appropriate combination with the microtheory of excitons would greatly facilitate further investigations

VI

’

REFERENCES

279

on optical and other properties of crystals and permit progress at a quick pace”. Clearly we were too optimistic. However, we have no reason to believe that our hopes were completely in vain. I t is true that fewer and fewer theoretical papers have appeared, in which simple and general cuIisequencesof phenomenological crystal optics with spatial dispersion were derived on the basis of a detailed analysis of concrete models in the frame of the exciton theory. Still we would like to stress once more the necessity of further efforts in the experimental study of spatial dispersion effects in crystal optics. We believe that, when designing the experiments and analyzing their results, it will be fruitful to make broad use of crystal optics with spatial dispersion in the spirit in which it was derived in AGHANOVICH and GINZBURG[1965a] and in the present article.

References AGRANOVICH, V. M., 1968, Theory of Excitons (Nauka, Moscow). AGRAKOVICH, V. M.,1969a, Fiz. Tverd. Tela 11, 1650. AGRANOVICH, V. M., 1969b, Report to the 2 All-Union Conf. on Solid Body Theory, Moscow. 1962. Usp. Fiz. Xauk 76, 643; 77, 663 AGRANOVICH, V. M. and V. L. GINZBURG, (English transl.: Soviet Phys. Usp. 5 (1962) 323, 5 (1963) 676). AGRANOVICH, V. M. and V. L. GINZBURG, 1963, Fortschr. Physik 11, 163. AGRANOVICH,V. M. andV. L. GINZBURG, 1966a, Crystaloptics under Consideration of Spatial Dispersion and Theory of Excitons (Nauka, Moscow). (English transl. : Spatial dispersion in crystal optics and the theory of excitons, Interscience Publ. 1966.) AGRANOVICH, V. M. and V. L. GINZBURG, 1965b, Fortschr. Physik 13, 175. AGRANOVICH, V. M., N. E. KAMENOCRADCKIJ and Yu. V. KONOBEEV. 1969, Fiz. Tverd. Tela 11, 1445. BIRSS,R. R., 1967, Phil. Mag. 15, 687. BROWKJR., W. F., S. SHTRIKMAN and D. TREVES, 1965, J.App1. Phys. 34,1233. DZYALOSHINSKIJ, I. E., 1958, J. Phys. Chem. Solids 4, 241. GINZBURG, V. L., 1968, Zh. Eksperim. i Teor. Fiz. 34, 1693. GINZBURG, V. L., A. A. RUKHADZE and V. P. SILIN,1961, Fiz. Tverd. Tels 3, 1835, 2890.

GINZBURG, V. L., A. A. RUKHADZE and V. P. SILIN, 1962, J. Physik Chem. Solids 23, 85. GROSS,E . F. and A . A. KAPLYANCKIJ, 1960, Dokl. Akd. Xauk SSSK 132, 93. GROSS,E. F. and A.A. KAPLYANCKIJ, 1961, Dokl. Akad. Xauk S S S H , 139, 75. HELLWEGE, K. H., 1951, 2. Physik 129, 626. HOKNREICH, R. M. and S. SHTRICKMAN, 1968, Phys. Rev. 171,1066. KRINCHIK, G. S. andM. V. CHETKIN, Zh. Eksperim. iTeor. Fiz. 41, 673.

280

CRYSTAL OPTICS WITH SPATIAL DISPERSION

[VI

LAKDAU, L. D. and E. M. LIFSCHITZ,1957, Electrodynamics of Continuous Media (Gostexizdat, SSSR). (English transl. : Electrodynamics of Continuous Media, Pergamon Press, Oxford and Addison-Wesley, Reading, Mass. 1960). LORENTZ, A. A., 1936, Collected Papers, Vol. 2, 79; Vol. 3,314. MOLCHANOV, A. G., 1966, Fiz. Tverd. Tela 8, 1156. SYE,J., 1956, Physical Properties of Crystals (Oxford University Press, London). PEKAR,S. I., 1957, Zh. Eksperim. i Teor. Fiz. 33, 1022. SILIK,V. P. and A. A. RUKHADZE, 1961, Electromagnetic Properties of Plasmas and Similar Media (Gosatomizdat, SSSK).

Note added in proof Spatial dispersion in crystal optics can be investigated also by means of X-rays and light scattering with creation of excitons. Such and way is rather obvious in principle (see for instance AGRANOVICH GINZBURG [ 1965al 9 15.1). However it became effective only recently with the help of lasers (see Wright [1969]). It happened that just combinational (Raman) scattering of light permitted for the first time to observe ”new” wave in gyrotropic crystal (for this question and some other connected with combinational light scattering in crystals, see AGRANOVICH and GINZBURG [ 19711). Additional References AGRANOVICH, V. M. and V. L. GINZBURG, 1971, Soviet Phys. JETP, in press. WRIGHT,G. B., editor, 1969, Light Scattering Spectra of Solids (SpringerVerlag, New York).

VII A P P L I C A T I O N S OF OPTICAL M E T H O D S I N T H E DIFFRACTION THEORY O F ELASTIC WAVES BY

K. GNIADEK The Warsaw Technical University, Poland AND

J. PETYKIEWICZ Institute of Physics

of

the Polish Academy of Sciences. Poland

CONTENTS

PAGE

. . . . . KIRCHHOFF’S DIFFRACTION THEORY . . . . . HUYGENS’ PRINCIPLE F O R ELASTIC MEDIA . . .

3

1.

3

2.

Q

3.

3

4.

PROPERTIES O F THE SCREEN AND THE PROBLEM O F E D G E INTEGRALS . . . . . . . . . . . 290

4

5.

THE YOUNG-RUBINOWICZ INTERPRETATION I N KIRCHHOFF’S DIFFRACTION THEORY O F ELASTIC WAVES . . . . . . . . . . . . . . . . . . . 292

Q

6.

T E N S O R P O T E N T I A L O F E L A S T I C WAVES

293

3

7.

FRESNEL DIFFRACTION O F ELASTIC

297

Q

8.

FRAUNHOFER DIFFRACTION O F ELASTIC WAVES 304

4

9.

CONCLUSION.

INTRODUCTION . . . . . . . . . . . .

REFERENCES.

. . . . WAVES .

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

283 284 286

309 309

Q 1. Introduction Generally speaking the diffraction of elastic waves is considered as a boundary problem. So understood problems of diffraction of elastic waves passing obstacles with sharp edges were considered by many authors. We shall mention only some of them, e.g. MAUE [1953], FILIPPOV 119561, DE HOOP[1958], MILES jl9601, ASG and KNOPOFF [1964], MAL, AKGand KNOPOFF [1968]. All of them considered only simple cases, mostly the diffraction of plane waves passing obstacles with rectilinear edges. Only the last of the papers cited above is devoted to the problem of the diffraction of elastic waves passing a rigid disc. The difficulty lies in the fact that - contrary to the electromagnetic and acoustic problems in isotropic media where we have to consider one wave velocity only - at propagation of disturbances in the elastic, and even in the isotropic case, we have to consider two velocities, that of longitudinal and of transverse waves. This creates serious difficulties even in the most simple cases. The results obtained by the authors cited above are highly complicated in form, difficult to interprete physically. I t appears, however, that if we restrict ourselves to certain regions then the results describing the state of the field in its points - based on Kirchhoff’s theory of the diffraction of elastic waves - can be obtained in a much simpler way. This problem was considered by KNOPOFF [1956], DE HOOP[1958], GNIADEK[1967,1969] and PETYKIEWICZ [1966a, b]. Kirchhoff’s theory of diffraction is based on certain heuristic premisses and its rightness may be judged only by comparing its resultswith those of exact solutions to simplest solvable cases, or by comparing its results with experimental data. Kirchhoff’s solution does not solve the boundary problem. However it solves the so-called saltus problem (cf. RUBINOWICZ [ 19661, HONL, MAUE and WESTPFALL [1961]). I t appears that the result of Kirchhoff’s theory of diffraction in optical cases - when the dimensions of diffraction bodies are large compared with the wavelength and when 288

284

APPLICATIONS OF OPTICAL METHODS

[VII,

g 2

we are interested in points lying at a considerable distance from the diffracting edge - is in agreement with experimental data. Especially good agreement is obtained near the boundary of geometrical shadow. This is why instrumental optics uses chiefly Kirchhoff’s theory of diffraction. It is anticipated that the same refers to elastic waves, although no experimental data have been reported so far. In this paper we consider - in addition to diffraction of elastic waves within the framework of Kirchhoff’s theory - also the possibility of the Young-Rubinowicz interpretation for this class of problems. In 5 2 we discuss Kirchhoff’s theory of the diffraction of elastic waves. 9 3 is devoted to Huygens’ principle applied to elastic media. § 4 is devoted to Kirchhoff’s screen in elastic media and to the problem of edge integrals. $ 5 contains confirmation of the validity of the Young-Rubinowicz interpretation in Kirchhoff’s theory of elastic waves. In $ 6 we give an expression of the tensor potential for arbitrary elastic waves and discuss its properties. 9 7 contains: application of methods of stationary phase in elastic case, Fresnel diffraction of elastic-sphericallongitudinal-, as well as transverse waves. 9 8 is devoted to Fraunhofer diffraction of elastic waves.

Q 2. Kirchhoff’s diffraction theory The Kirchhoff method utilizes Huygens’ principle which in short may be characterized as a relation permitting us to express the state of the wave-field v at a point P by the valuesbf this fieid and its deriatives at points Q of the surface F surrounding the considered point, viz .

where 8 is a vector function depending linearly on the field v and its derivatives at points Q. If we are interested in the field at a point P lying in the half-space on the non-illuminated side of the diffraction screen, then as the closed surface F surrounding this point we may take the surface formed by (see Fig. 1) : (1) the aperture A, (2) a portion S of the non-illuminated side of the screen, and (3) a portion C of a large sphere of radius R, centered at P which, together with A and S, forms a closed surface. If we assume, after Kirchhoff, that on A the wave-field is equal to the incident wave, and on the non-illuminated side of the screen S the field

KIRCHHOFF'S

DIFFRACTION

THEORY

286

A Fig. 1. The closed surface F surrounding the point P. This surface consists of: -4 - t h e aperture, S - a portion of the non-illuminated side of the screen, C - a part of a large sphere centered at P with radius R.

and its derivatives vanish, then under the so-called radiation conditions (Sommerfeld conditions), we get the formula for the field at point P v(P) = -JJABdf. 1 4n Radiation conditions, given by KC'PRADZE [1963] for the case of elastic waves, take the form lim v I = 0 r-rm

r-rm

or

(2-3)

for longitudinal wave (1 = 1 ) and transverse wave ( = 2) independently and are fulfilled at infinity for all directions. The solution (2.2) does not satisfy conditions imposed upon the solution on screen and should not be regarded as the solution of the boundary problem (BOUWKAMP [1954]). One can show, however, that the problem of Kirchhoff's theory of diffraction reduces to the saltus problem. By virtue of Huygens' principle, the field at the point P is expressed by the integral (2.1) over the surface surrounding this point. This surface can be deformed at will. I t can be changed from its original shape (Fig. 2 ) , a continuous curve, to the shape indicated by the dashed line, embracing the diffracting screen and the source of vibrations L. Any such deformation is possible, assuming that radiation conditions are satisfied, which ensures vanishing of the integral over the surface elements lying in infinity. The Kirchhoff classical assumptions that at the illuminated side of the screen the wave field is equal to the in-

286

A P P L I C A T I O N S O F OPTICAL M E T H O D S

[VII,

§ 3

cident field, and the wave field and its first derivatives vanish at the non-illuminated side, may be replaced by the assumption (RUBINOWICZ [1966] p. 22) that when passing the screen the field and its derivatives undergo a jump in values with the values of the incident field, which leads to the saltus problem (KOTTLER L19231). The formulation of Kirchhoff’s diffraction theory of elastic waves as a saltus problem is dealt with by DE HOOP[1958].

, .LI r -

/

‘

\ /

/

Fig. 2. The initial surface F undergoes the deformation into two parts: S - t h e surface enclosing the screen and small sphere around the singular point L.

Thus to construct Kirchhoff’s diffraction theory of elastic waves one needs the knowledge of 1. Huygens’ principle for elastic media, 2. Radiation conditions, 3. Properties of the screen. These problems have so far not been clearly formulated in the literature which has led t o some controversies. PETYKIEWICZ ;1966a] scrutinized once more the known results of KUPRADZE[1963], KNOPOFF[1956], BOILLET[1957] and DE HOOP[1958] andsolvedsome of the controversies. He has shown that depending on the use of a special formulation of Huygens’ principle given by KUPRADZE [1963] one obtains a different formulation of Kirchhoff’s diffraction theory of elastic waves (see 9 3). I n subsequent paragraphs we discuss the role of an arbitrary constant appearing in the formulation of Huygens‘ principle given by V. D. KUPRADZE and its connection with the problem of so-called edge integrals ( KNOPOFF[ 19561).

Q 3. Huygens’ principle for elastic media Huygens’ principle for elastic media may be given as a relation expressing the state of the field at point P by the values of the field and

VII,

§ 31

HUYGENS’

PRINCIPLE FOR ELASTIC MEDIA

287

its derivatives at points Q of the surface surrounding the considered point. The known formulations of Huygens’ principle (KUPRADZE [19631) or of Huygens-Kirchhoff principle, concerning fields of arbitrary time dependence, given by KNOPOFF [1956], BOILLET[1957] and DE HOOP [1958], differ greatly in form. However, as was shown by PETYKIEWICZ [1966a] for the case of monochromatic fields, they all constitute a special realization of a general formulation of Huygens’ principle given by Kupradze, which only apparently depends on some arbitrary const ant. Kupradze formulates Huygens’ principle as follows: (3.1) where the tensor r is a function of two points - that of integration Q and that of observation P. It is defined by the relation

rk= curl curl (dlk)

-grad div (dlk)

7,)

(3.2)

where dk is the vector with components Ski,,,S S, (Kronecker delta). Putting k = 1, we obtain a vector constituting the first tensor column, and, correspondingly a t k = 2 the second column, and a t k = 3 the third tensor column. The operator P is determined by

P

a

= (a+p) - +pfi div+a(fixcurl),

(3.3)

an

-

where differentiation concerns the point Q, a/an = n grad and constants a and p satisfy a+p = A+p.

(3.4)

In (3.2) and (3.3) we use the following notations:

k , - length of wave vector of longitudinal vibrations, k , - length of wave vector of transverse vibrations, R and p - Lame’s constants, n - normal unit vector, external to the integration surface at point Q, + I - length of vector PQ. The physical sense of quantities appearing in (3.2) is as follows.

288

APPLICATIOSS O F OPTICAL METHODS

[VII,

83

Each column of tensor represents the solution to the equation of an isotropic elastic medium 1

-v = 3 grad

k,

1

div v - - curl curl v+ F k:

with the force F = 6(r)s, where s is a unit vector of the appropriate axis of the chosen system of coordinates. The first column corresponds to the unit vector 1, the second to the unit vectorj, and the third to the [1963] pseudo-stress. unit vector k.Vector P v was called by KUPRADZE He reasoned that, at a = p and /? = 1, P v passes into stresses acting on the element of the surface df whose normal is n. The expression (3.1) is a most general form of Huygens’ principle for an isotropic elastic medium. Its general character lies in the possibility of various choices of constants a and /? connected with equation (3.4). Thus, the assumption of a = p and /? = L gives a form of this principle expressing the field at point P by displacements and stresses at points Q of the surrounding surface. This form of Huygens’ principle is obtained from the results of DE HOOP[1958]. On the other hand assumption of a = - p and fi = A+2p gives the form of Huygens’ principle obtained from KNOPOFF’S[1956] results, and, indirectly, from BOILLET’S [1957] results. Then this form expresses the field of vibrations at the point P by displacements and pseudostresses at points Q of the surface surrounding the considered point. (For closer calculational details concerning these statements see PETYKIEWICZ [1966a].) Kupradze’s formula (3.1)expressing Huygens’ pinciple for an elastic isotropic medium was derived under the assumption that the field .appearing in it satisfies appropriate conditions of regularity. The equation (3.4)imposed on constants a and B shows, upon elimination of one of them, that Huygens’ principle will none the less still depend on one arbitrary constant. I t can, however, be shown (PETYKIEWICZ [1966a]) that the formula (3.1) can be transformed into

where the operator K is defined by

R = ( 1 + 2 p ) n div-p(n x curl).

(3.6)

In (3.5) Curl, ‘is the tensor rotation. From (3.5) one can see immediately that when the field v satisfies

VII.

8 31

289

HUYGENS’ PRINCIPLE FOR ELASTIC MEDIA

the conditions of uniqueness and regularity at all points of the surface F, the first integral is identically zero, and the second is independent of the arbitrary constant and is of the form of the Huygens’ principle. The passage from (3.1) to (3.5) has the following motivation. The integrand in (3.1) B = (r Pv-v Pr), (3.7)

-

-

can be split into two components determined by

B, = ( a s p ) {r -v-

av - [z --n

1

div v f n x c u r l v

ar

[% - - n D i v r + n x C u r l r

I)

(3.8)

B, = r - R v - v - R r , where Curb is the tensor rotation and Diva the tensor divergence. The expression B, is the integrand of the second integral of (3.6) whereas B, can be transformed into B, = (a+p)n - Curl [ v x r ]

(3.9)

which is the integrand of the first integral of (3.6). To obtain (3.9) it suffices to consider the x-th component of the vector B,. Its form is

&= (afp)

(rl- [g--n

1

div v f n x c u r l v

--n div r , + n x c u r l

(3.10)

where rlis the vector whose components are elements of the first column of the tensor r.Hence, taking into account the vector relations

-

-

div [a(b c ) ] = (b * c ) div a-a grad (b * c ) grad (b c ) = (b grad)a+ (a grad)b+b x curl a+a x curl b div (ax 6 )= b * curl a-a - curl b a

and the identity

r,. [(v.grad)n]-v*

[(r,.grad)n] = -(r1xv) wcurln,

we can write eq. (3.10) as

B,

= (a+p)n * curl ( V X F1).

(3.11)

290

APPLICATIONS O F OPTICAL METHODS

[VII.

§ 4

Q 4. Properties of the screep and the problem of edge integrals I n order to express the field a t the point P behind the screen by integration over the diffracting aperture A, one must, as has been mentioned in § 3, assume that the screen has properties ensuring the vanishing of the integrand in integral (3.1) in all its non-illuminated points, which is equivalent to v = 0 and f% = 0 a t all these points. I t would therefore seem that in the elastic case there may appear a number of different Kirchhoff screens depending on the operator P . KSOPOFF [1956], who was the first to consider the problem of diffraction of elastic waves within the framework of Kirchhoff's diffraction theory, assumed, that as in the scalar case, on the screen v = 0 and av/an = 0. A closer analysis of this problem (PETYKIEWICZ [1966aj) showed that the conditions imposed by Knopoff are fully equivalent to the more general conditions: v = 0 and P v = 0 a t an arbitrary operator p. Such is the case at least at points of the screen not localized at the edge. When the integration point Q crosses the edge of the aperture, the conditions P v = 0 and v = 0 give a discontinuity which has a character different from the discontinuity obtained under the conditions v = 0 and avian = 0. The character of the field's discontinuity near the edge essentially bears upon the value of the field a t the point P behind the screen. If one assumes that on the screen we have Pv = 0 and v = 0, then the assumption of P v = Pv, and v = v, in the points of the diffracting aperture gives the value of the field a t the point P in the form

This results directly from the formula (3.1). If the function v(Q) satisfies the regularity conditions in all points of the surface F, which is usually assumed when deriving Huygens' principle, then the first of the integrals vanishes and we obtain a formulation which does not depend on an arbitrary constant. If, however, we admit the occurrence on the integration surface F of some lines of discontinuity in the field v(Q),as is the case in Kirchhoff's theory of diffraction then the field a t the point of observation P will be v(P) = - T x

(r.Rv,-v,.~I')df,

(4.2)

VII.

§ 41

PROPERTIES OF THE SCREEN

29 1

where v2-v, is the jump in the value of the field when we go beyond the discontinuity curve D. The passage from formula (3.5) to formula (4.2) proceeds as in the electromagnetic case (RUBINOWICZ [ 19651). The reasoning is as follows. The discontinuity line D separates the integration surface into two parts, F, and F,. Applying the Stokes theorem to the first of the integrals taken over the surfaces F, and F,, correspondingly, we obtain two curvilinear integrals along the discontinuity line which differ in the direction of rotation. The direction of rotation is such that it associates itself by the rule of the right-hand screw with the normal unit vector in the points of the surface of the considered plane. Here the normal vector is the outward normal to the closed surface F,+F,. Taking this into account as well as the fact that boundary values of the field on both sides of the curve are different, we obtain (4.2). Formula (4.2) is the basis of Kirchhoff‘s theory of the diffraction of elastic waves. Within the framework of this theory the field at the point of observation P localized behind the screen is expressed by an integral over a surface of the opening A, and as can be seen from formula (4.2), essentially depends on the character of discontinuity appearing on the diffracting edge. The latter is connected, as shown above, with the conditions dominating at the edge and does not depend on the properties of the screen. That the edge may behave unlike the screen itself can be illustrated by the case of a free screen (5% = 0) with the fixed edge (v = 0). Only for a = - p the first of the integrals in (4.2) vanishes. Thus, the edge then influences the value of the field at point P through its shape only. Of course, such is in the case only when K v = 0 and v = 0 on the screen, and K v = Kv,, and v = v,, on the aperture. In this case the integrand in the second integral of (4.2)is exactly like the integrand in Huygens’ principle given by KNOPOFF[1956]. KNOPOFF,applying the Huygens’ principle to problems of diffraction on a half-plane screen having properties v = 0 and avian = 0 added some edge integrals to the surface integral extended over the aperture, claiming that they are the consequence of the field’s discontinuity appearing on A+S. Unfortunately, he did not give grounds for their appearance, confining himself to consideration of the electromagnetic analogy. One can show that the edge integrals are a particular case of the curvilinear integral in the formula (4.2) for a = p + l and the edge D being a straight line. This corresponds to the case of discontinuity on the integration surface determined by the conditions LV = 0 and

292

Cvrr. § 6

APPLICATIONS O F OPTICAL METHODS

v = 0 on the screen, and fiv = zv0 as well as v = v, on the aperture. The operator f, is the particular case of the operator P at cc = p + l and B = 0. Q 5. The Young-Rubinowicz interpretation in Kirchhoff’s diffraction theory of elastic waves Kirchhoff’s solution poses one interesting analytic feature, which permits very fine physical interpretation. As in the scalar case (RUBINOWICZ [1917, 1962a, 19651, MIYAMOTOand WOLF [1962]), the electromagnetic case (RUBINOWICZ [1962b]) and the electron waves case (RUBINOWICZ [1963]) one can show (PETYKIEWICZ [1966b]) that the integrand in the formula for Huygens’ principle (3.1) can be represented as the product of a normal unit vector by the tensor V; i.e.

where

V

+

CurlQ( v x r) (Af2p) (rdiv, v-v Div, -p(vx CurlQr)+p(curlQ v x I?),

= (z+p)

r) (5.2)

the symbols Curl, and Div, denoting the appropriate tensor rotation and divergence. This tensor is a sourceless quantity with respect to the integration point Q, i.e. its divergence vanishes: Diva V = 0. Consequently we can represent this tensor in the form of a rotation of another tensor field v = Curl, w, (543) which permits us to write Huygens’ principle in the form R

*

Curl, Wdf.

(5.4)

By introducing the tensor potential into the Huygens’ principle integral, we can represent the results of Kirchhoff’s diffraction theory as the sum of curvilinear integrals 1 $ d s - W+ - C $ d s 4n 4n 1

v(P) = -

D

c,

*

W,

(5.5)

where the first of the integrals is taken along the diffracting edge, and the remaining integrals round the singular points of the potential W lying on the integration surface.

VII, § 61

293

TENSOR POTENTIAL O F ELASTIC WAVES

The possibility of representing the diffraction field by the formula (5.5) permits us, as in the scalar case (RUBINOWICZ [1917, 1957, 1962a1,

MIYAMOTOand WOLF [1962]) to interpret it as a superposition of socalled diffraction waves arising as a result of reflection of the incidence field from the edge, and of so-called geometrical waves. This constitutes the basis for interpretation of the phenomenon of diffraction of elastic waves from Young’s point of view. To prove that Div, V = 0 it suffices to show the vanishing of the divergence of each of the vectors V,, V,, V, which are appropriate columns of the considered tensor. Since none of the columns of the tensor V plays any special role, it suffices to show that one of them satisfies the condition of vanishing of divergence. Thus, if we consider the first of the vectors determined by the equality diVQrl)

vl = (.+/A)Curl ( V X rl)+(2+2p) (rldiVQV-V -p(vxcurla ~ l ) - p ( ~ l ~ c u rv), lQ

(54

then, applying the operation div, to it, we get

-

diva V , = {rl [ (A+2p) grad div v-p curl curl v] -v [ ( A f 2 p ) grad div Fl-p curl curl rl] = {-rl * v+v rl)pw2 = 0.

-

(5.7)

In the above we made use of the equivalence div ( a x b ) = 6 . c u r l a - a . c u r l b and of the fact that each of the columns of the tensor satisfies the equation of motion of elastic medium (cf. KUPRADZE[1963] p. 13) that is 1

1

kl

k2

FI= - 7 grad div F, +

curl curl r,

~

The following sections are devoted to finding the explicit form of the tensor potential and to more exact analysis of the Young-Rubinowicz interpretation of the diffraction of elastic waves within the famework of Kirchhoff’s diffraction theory.

Q 6. Tensor potential of elastic waves In agreement with (5.5) we can express the boundary diffraction wave by

294

APPLICATIONS O F OPTICAL MKTHODS

[VII,

3

ti

where D is the diffracting edge. Then to obtain the diffraction wave u, it is sufficient to know the tensor potential W(P, Q) of the incident wave. The formula for the tensor potential W(P, Q) for an arbitrary monochromatic elastic wave was derived by GNIADEK[1967a]. The components of the tensor W(P, Q) are given by the following expressions :

( j , k = 1 , 2, 3).

(6.2)

The integrationin (6.2)must be carried out with respect to rover a halfray beginning a t the point Q, the extension of which passes through P. The integrals in (6.2) are convergent, if u(Q)obeys at infinity the Sommerfeld radiation condition (2.3). From the expression (6.1) for u,(P) we obtain in the case of arbitrary diffracting edges the tensor potential W(P, Q) defined up to an additional tensor field Z vanishing when integrated over a closed curve. The matrix of the tensor Z can therefore be represented as a set of three vectors each of which is a gradient of a certain univalent scalar function. As the curl of such a field is zero, any tensor potential W (P, Q) we derive in such a way represents the tensor field appearing in Huygens principle (5.1). From (5.4) it follows that the tensor potential W(P, Q) must have, as a function of Q, a t least one singularity on any surface surrounding the point P if u (P)is regular within IT. Otherwise, according to the tensorial Stokes theorem (cf. e.g. KVBINOWKZ[1962b]), the integral in (5.4)would vanish. To obtain the expression for the tensor potential in a closed form,

VII,

S

6j

296

TENSOR POTEXTIAL OF ELASTIC WAVES

we must calculate the linear integrals appearing in (6.2). This integration can, in general, be done only approximately and only for a special class of incident waves (INGARDEN [1955], RUBINOWICZ [1962a]). In the scalar case the integrals of this type may be calculated exactly only for an isotropic or multipole spherical or plane incident wave. In the elastic case this corresponds, for instance, to the wave originating from a point-source of disturbance having spherical symmetry. In the case of a spherical wave of dilatation the displacement u1for an infinite medium is expressed by elk1 P u'(Q)= gadQ (6.3)

p'

where p is the distance of the point-source disturbance L from the integration point Q on surface F. Taking this formula for u'(Q) into account, we shall calculate the integral appearing in (6.2). Its form is

Differentiation of gra% with respect to the point Q in the integrand can, in our case, be replaced by differentiation with respect to the coordinates of the source L. Thus we obtain

P

] dr'.

(6.5)

Here we made use of the relation grad, [(exp iklp)/p] = -grad,[ (exp ikfl)/p].Sincein (6.5)we integrate with respect to the point Q and differentiate with respect to the point L, we can reverse the order of these operations, i.e. we first integrate and then differentiate. Thus we obtain

The expression in square brackets can be represented by a vector potential of the isotropic divergent spherical wave uo = exp (ik,p)/p. This potential first announced by RUBINOWICZ [1917, 19661, has the form

Making use of this formula we finally obtain the value of the integral I j :

296

APPLICATIONS O F OPTICAL METHODS

m, 8

6

Hence for the incident wave (6.3) we get the expression for the tensor potential (6.2) in the closed form

(i,k = 1, 2, 3).

(6.9)

Let us now consider a spherical shear wave propagating in an infinite medium from a point-source disturbance. The displacement in point Q (neglecting the time factor exp (-id)) is then given by eika P ut(Q)= exgrad* -,

(6.10)

P

where e is the constant unit vector. Taking into account (6.7), as in the case of a spherical dilatational wave, we calculate the integral appearing in the expression for the tensor potential (6.11)

x5

where denotes the operator of differentiation with respect to the coordinates of the source L

Lj =

(exgrad,)

(6.12)

j .

Consequently, for the components of the tensor potential W,, of the spherical shear wave, we obtain, taking into account div ut = 0, the following closed formula

(i,A

= 1, 2, 3).

(6.13)

VII,

8 71

297

FRESNEL DIFFRACTION O F ELASTIC WAVES

Note that the tensor potentials (6.9) and (6.13) have singularities caused by singular points of the vector potential W,(P; Q) for divergent spherical wave. In the case of vector potential W,(P, Q) (6.7), all points Q on the segment PL of the line passing through these points are singular points. Here r / r = - e / p and therefore the denominator r p + q vanishes in the expression for the vector potential (6.7). This singularity is of essential importance in the Kirchhoff theory of diffraction as interpreted from Young’s point of view.

Q 7. Fresnel diffraction of elastic waves In the following we shall consider, within the framework of Kirchhoff’s theory, the Fresnel diffraction of a monochromatic elastic wave excited by point-source disturbance. Let us assume that the incident wave is a longitudinal wave (6.1). Then, in view of (6.9), the boundary diffraction wave uDis given by the curvilinear integral 1

u,(P) = --1(2b2(grad-

eU”P

elk1 ) sl+ (s,xgrade) P

Y

P e%r

P

$1

P

- -(s,xgrad-)

+

-(grad,

* W&)

)

elk1P

elkas

P

r

P

Y

P

[2b2(grad

ds

xFkl) s,- -(s, x grad-)]

ds.

Here s, is the unit vector tangent to the element of the edge ds, whereas W, (see eq. (6.7)) is the vector potential of the isotropic divergent spherical wave. The product grad, * W , of the operator grad, and vector W, in (7.1) denotes the diad, i.e. the matrix with elements (grad, * W0)jk = (a/axj)lvokObserve that the diffraction wave (7.1) arises by the superposition of two kinds of elementary waves, du; and dub. The contribtions du; contain the phase factor exp [ik,(r+p)] and represent the elementary dilatation waves propagating with the velocity a = (1+2,u)/p’. On the other hand, the contributions dub containing the phase factor exp [i(k,p+K,r)] represent elementary shear waves propagating with the velocity b = PIP’. The wave motion determined by the contributions dub is continuous in all points in the space on the side of the

298

APPLICATIONS OF OPTICAL METHODS

[VII,

5

7

screen which does not contain the source. On the other hand, the elementary dilatation waves consist of the continuous and discontinuous part which is denoted by du,. This part can be written as

The above formula has namely the jump at the so-called **boundary of elastic shadow”. The boundary of the shadow (in the sense of geometrical optics) is represented by the part of the lateral surface of the cone cut by the screen. The vertex of the cone is localized in the source, L, whereas the lateral surface is formed by a set of half-rays originating in the source and passing through particular elements of the diffracting edge. It can easily be seen that for observation points lying on the boundary of the shadow we haver, = --el, which leads to the vanishing of the denominator in eq. (7.2). I n this connection as will be seen later, the part of the diffraction wave determined by the contributions (7.2) jumps at the boundary of the shadow. This jump compensates for the jump of the geometrical-elastic wave here. As a result, the total wave-motion is continuous in all points of the space over the screen and the source. To obtain an approximate formula for the diffraction boundary wave we shall utilize for the calculation of the curvilinear integral (7.1) the method of stationary phase (see e.g. RUBINOWICZ 11924, 19661). In this connection we shall assume that the incidence wave is excited by a source producing vibrations of sufficiently high frequency so that we can consider that the exponential factors exp [i(k,p+h,r)] and exp [ik,(p+r)] vary rapidly compared with other factors in the integrand in (7.1). Furthermore, we shall assume that the dimensions of the diffracting aperture are considerably larger than the length of the incident wave. Because of the rapid oscillation of the phase factors exp [i(k,p+k,r)] and exp [ik,(p+r)] the contributions to particular elements ds of the edge will generally cancel each other by interference. Significant contributions will stem only from the neighbourhoods of so-called “active” points Qzand Qt on the diffracting edge at which the phases E = k,(p+r) and 7 = k,p+k,r have a stationary value. Hence at Q = Q L or Q = Qt the derivatives of the phases 5 or 17 vanish correspondingly: (7.3a)

VII, § 71

F R E S N E L DIFFRACTION O F ELASTIC WAVES

299

(7.3b) For the physical-geometrical interpretation of the above relations we write them in the following form: d[k,(r+p)l ds d(klp+W ds

=

+ g) kl [cos ds)+cos (e,ds)] dp dr kl- + k,- = klcos (e, ds)+k,cos (r,ds). ds ds =

(I,

(7.4a) (7.4b)

Taking into account that cos (e, ds) = e ds/p ds and cos (r,ds) = r dslr ds we obtain for (7.4a) and (7.4b) the following expressions: (7.5a)

e_.d s - - - - *k , r d s

(7.6b) kl r The expressions (7.5a) and (7.5b) have simple geometrical interpretation (RUBINOWICZ [1924]). Namely, they represent two cones cDL and @, of different angular aperture (see Fig. 3). The angular aperture .

P

Fig. 3. Reflection cones. The incident dilatational wave is reflected into two cones. @< -the reflection cone of dilatational wave, O,, - the reflection cone of shear wave, B - the diffracting edge.

of the cone cDC is equal to double the angle between the vector 8 and the element d s of the edge in the active point QL.On the other hand, the angular aperture of the @, cone is according to (7.5b), determined by the formula /l= arc [cos (k,/kl) cos (e, ds)]. The vertices of the

300

APPLICATIONS O F OPTICAL METHODS

[VlI,

§ 7

cones @( and @, lie in active points Ql or Qt correspondingly. The directions of the axes of these cones are determined by the directions of the tangents to the edge in active points Q t or Qt. We shall call the cones @( and @, the reflection cones. In the first approximation of the method of stationary phase we can therefore assume that the boundary diffraction wave arises through a kind of “reflection” of the incidence wave on particular linear elements of the diffraction edge. The “reflected” waves form in each point of the edge two reflection cones, of which corresponds to the dilatational wave, and @, to the shear wave. We shall now give an approximate expression for the diffraction wave u,(P) (7.1) which holds both far and near the boundary of the shadow (GNIADEK[1969]). We can obtain such an expression utilizing the results of RVBISOWICZ [1924] for a scalar spherical divergent wave and a calculation made by KAKCZEWSKI and PETYKIEWICZ [1967] for electromagnetic waves. Thus we get to the formula for the elastic diffract ion wave

(7.6)

where summation is done over all active points Ql and Qt . R denotes the distance of the observation point P from the source of disturbance + = PL. By

ti‘ and 7:‘ are denoted respectively the values of the second derivatives d2[k,(r+p)]/ds2 and d2(k,p+k,r)/ds2 in active point Q1 or Qt. The plus sign in the exponential factor exp (-&z) should be taken in the L. R, is the unit vector in the direction of vector R

VII,

8 71

FRESNEL DIFFRACTION O F ELASTIC WAVES

30 1

case E;' > 0 or qi' > 0 whereas the minus sign is to be taken in the case ti' < o or 7;'< 0. We shall now show that the jump of the dilatational diffraction wave at the boundary of the shadow is compensated by the jump of geometrical-elastic wave taking place here. To this aim let us first observe that at the boundary of the shadow both vectors e and r have the same direction but opposite senses; thus we get

pfr-R=O

and

el=&.

17.71

Next we utilize relations holding in the direct neighbourhood of the boundary of the shadow (RUBINOWICZ [1966]):

where a is the angle between vectors m x el and r1x el. Vector m is the unit vector normal to the boundary of the shadow and directed outward. Taking into account the relations (7.7), (7.8) and (7.9) we obtain the formula for the discontinuous part (7.2) of the dilatational diffraction wave (7.6) in the close neighbourhood of the boundary of the shadow: ei( h - R * f * ) e1kt-R -\/(ak1n-'(r+p--RJ 1 e*ll.'dv]l . (7.10) 4 l Rd2 grad R cos a/+.. 1

l -[

Observe that under our assumption, both vectors m x el and r, x el have on the boundary of the shadow from the "illuminated" side the same directions and senses, whereas on the external side of the cone surface they have opposite senses. Hence we have cos a = + l inside the cone and cos a = -1 on the shadow boundary from the shaded side. Therefore, a s we approach the boundary of the shadow from the internal side (illuminated in the sense of geometrical optics), we obtain the following limit value of (7.10): (7.11)

On the other hand, as we approach the boundary of the shadow from its external "unilluminated" side we obtain, correspondingly,

302

APPLICATIONS O F OPTICAL METHODS

[VIL

I

7

(7.12)

Hence the jump in value of the dilatational diffracted wave on crossing the shadow boundary amounts to (7.13)

This jump is equal to that of the jump of the geometrical-elastic wave at the shadow boundary taken with the negative sign, so that the jump of the total wave on crossing the boundary of the shadow is zero. Then the total wave-motion consisting of the diffraction wave and the geometrical-elastic wave is continuous on the boundary of the shadow and thereby in all points of the space except the screen and the source. As we have mentioned above, the approximate formula for the diffraction wave (7.6) holds for observation points lying both near and far from the boundary of the shadow. If, however, we are interested in the displacement field in the point of the medium lying far from the boundary of the shadow, then the formula for the diffraction wave can be given a simpler form. Since in this case the upper limit w = d{2kln-'(r+pR ) } in Fresnel integrals in (7.6) takes high values, we can substitute for these integrals their asymptotic approximation (cf. e.g. RUBINOWICZ [1966]) which, consequently, leads to a simpler formula for the diffraction wave (GNIADEK [1969]). We can consider analogously the diffraction of a shear elastic wave originating from a point-source disturbance. The displacement ut in an infinite medium is (neglecting the time factor exp ( - i d ) ) given by (6.10). Applying the stationary phase method, as before we get an approximate formula for the diffraction wave for the incident shear wave (6.10):

VII, §

71

303

F R E S N E L DIFFRACTION OF ELASTIC WAVES

eiin dv dk,] t

+ 2 e*tin 2

1

2 b 2 ( ux I’(kl))sl- rad - (curl u * s,) k:g r

(7.15) where t , = e x e l . Here q = k,(r+pl and 6 = k2p+k,r. The summation in (7.15) is done over all active points Qt and Qz.The localization of these active points on the diffracting edge is determined by the conditions (7.16a) (7.16b) The diffraction wave (7.15) consists of the dilatational and shear waves. Contrary to the previously considered case, the discontinuity on the boundary of the shadow is now observed in a part of the shear diffraction wave (in which appears denominator rp+re). Finally, when passing the boundary of the shadow, part of shear diffraction wave jumps in value. This jump compensates the jump of the directly incident geometrical-elastic wave thus ensuring continuity of total wave-motion on the boundary of the shadow. As in (7.5a) and (7.5b) we can interprete the relations (7.16a) and (7.16b) geometrically as two “reflection cones” formed by rays “reflected” from the diffracting edge. The first cone determined by (7.16a) corresponds to the shear wave, whereas the second determined by eq. (7.16b) corresponds to the dilatational wave. I n the first approximation we can therefore assume that the diffraction wave arises through a kind of “reflection” of the incident shear wave on particular elements of the diffraction edge. This in general causes formation of two diffraction waves - shear and dilatational. The “rays” of these waves form two cones with different angles of aperture. If the observation points lie far from the boundary of the shadow, then, substituting in (7.15) for the Fresnel integrals their asymptotic approximations, we can obtain a simpler formula for the diffraction wave (GNIADEK[1969]).

304

APPLICATIONS O F OPTICAL METHODS

[VIL

§ 8

Q 8. Fraunhofer diffraction of elastic waves Let us assume that the distances p and r of the source L and observation point P from the screen increase without limit. Then we can proceed to the limit in the expression for the Fresnel diffraction wave (7.1) (cf. e.g. RUBINOWICZ [ 1 9 6 6 ] ) and obtain the formula for the Fraunhofer diffraction wave. To this aim we denote by aLand a, the unit vectors determining the direction of incidence of the wave and the direction of observation, respectively. Using these notations we can put in the integrand in ( 7 . 1 ) the following approximations - apart from exponential factors:

e = poaLr

r = reap.

(8.1)

Here r0 and po denote respectively the distances of the source L and of the observation point P from a certain fixed point Qolying in the neighbourhood of the diffracting edge. Next, we can use the following approximations for distances p and r in the exponential factors:

r = r,+Ra,, (8.2) where R is the vector determining the position of the element ds of the diffracting edge with respect to an arbitrary fixed point Qo. Taking account of the approximations ( 8 . 1 ) and ( 8 . 2 ) and neglecting the terms of order higher than l / r or l / p when passing the limit in the integrand in (7.1) we obtain p = po+Ra,,

Before discussing eq. (8.3) we shall show that the last integral in (8.3) represents longitudinal displacements i.e. those taking place only in the direction of the vector a, which determines the direction of propagation of the plane wave. For this purpose we introduce the following not ations a. = a,+aL, Po = aL-ap. (8.4) Since a, and aL are unit vectors, we have

VII.

§ 81

FRAUNHOFER DIFFRACTION OF ELASTIC WAVES

4 = 2(1+a,a,),

305

Poao= 0.

(8.6) Using the vector relation a x ( b x c ) = b(ac)-c(ab) and taking into account the relation ( 8 . 5 ) , we can write the last integral in (8.3) as a sum of two integrals

Because of the relation a, exp (ihlRao) = (l/ihl) grad [exp ( i k l ~ a 0 ) ] , the second integral on the right-hand of (8.5) vanishes by virtue of the theorem

I

grad p ds

=

0.

Taking account of this fact and of relation (8.0) we finally get, upon dividing both sides of (8.3) by the phase-amplitude factor exp {ihl(ro+p,,)}/p0~,the formula for the Fraunhofer diffraction wave:

which holds for an arbitrary spatial curve b, limiting the diffraction aperture. From this formula we see that the Fraunhofer wave-motion u p for an incident plane longitudinal wave consists of two kinds of plane waves. One wave causes displacements along the direction of vector a, and propagates with the velocity a = (1+2p)/p'. This is therefore a plane longitudinal diffraction wave. The other wave, causing displacement perpendicular to the vector a, propagates with the velocity b = p/p'. I t is therefore the transverse plane wave. Next, let us observe that the transverse wave is continuous for an arbitrary direction ap.On the other hand, part of the longitudinal diffraction wave is discontinuous in the direction a,, = -aL,for then the denominator in the vector factor in the second integral of (8.7) vanishes. However, in the case when the diffraction edge is a finite closed curve the diffractionwave remains finite despite vanishing of the denominator )] for a p= -aL,since at the same time, integral J exp [ i h , I p ( a , + ~ ~ ~ds

306

APPLICATIONS O F OPTICAL METHODS

TVII,

§ 8

vanishes linearly with ]a,+ a,].The discontinuity of the Fraunhofer diffraction wave (8.7) in the case a, -+ -aL is caused by the fact that the Fresnel diffraction wave (7.1) shows discontinuity a t the boundary of the shadow, i.e. for rl = -el. Note that, because of different velocities of propagation of the longitudinal and transverse waves (a > b ) , there appears a phase shift ~ ( l / b - l / ~between ) ~ ~ both kinds of diffraction waves. Therefore, if the disturbance in the source would act in a finite period of time, then the Fraunhofer diffraction wave would consist of two wave trains, of which one corresponds to the longitudinal and the other to the transverse wave. At observation points lying sufficiently far from the screen both trains would pass separately, That is, the displacement would be caused first by the longitudinal waves and after some time there would appear a displacement corresponding to the transverse waves. Similar effects are observed in seismology. It should be noted that in the case of the Fraunhofer diffraction phenomena, the total Kirchhoff wave-motion is given only by the diffraction wave, whereas the incident wave compensates for the discontinuity of the Fraunhofer diffraction wave appearing in the limit direction ap = - a L . As an example, let us now consider the diffraction of a longitudinal elastic wave passing through a rectangular aperture whose sides are of

7: Fig. 4. The diffracting aperture used in the analysis of the Fraunhofer diffraction.

VII,

5

8;

FRAUNAOFER DIFFRACTION

O F ELASTIC WAVES

307

length A and B (Fig. 4). In order to obtain in this case the expression for the diffraction wave, we must calculate contributions originating from one part of the sides, say A . The location of an arbitrary point on this side is determined by the vector R = Ro+sls, where 4,is the vector originating in the center of the rectangle and directed to the middle point of the side. Here s1 is the unit vector parallel t o the side A . To obtain the contribution to the diffraction wave uF originating from the side A , we calculate the integral +tA

e'klRq s1ds

= s1elk

elk,s,W ds

The integrals appearing in ( 8 . 7 ) will consist, in the considered case, of the sum of integrals of the type ( 8 . 8 ) corresponding to particular sides of the rectangle. Thus for the diffracted wave uF on a rectangular aperture we obtain

In this formula we use the following notation sin [ i k , Bs,a,] [ i k ,A s , a,] I(%) = s; A sin k,Ria, sin ~-~ - s , B s i n k l ~ , a o ik1As1 a0 i k , Bs1 a, '

k2 ap. kl In the case of observation in the direction a, ( 8 . 9 ) the following formula:

a, = a,+%,

yo = a,+

(8.10)

-

=

-aL we obtain from

ikl uF = - q ( a L- n ) A . B 2n

(8.11)

308

APPLICATIONS O F OPTICAL METHODS

[VIL

§ 8

where n is the unit vector normal to the plane of aperture. From the eq. (8.11) one can see that if the plane longitudinal wave falls perpendicularly on the aperture, then in the direction a p= -aL there appears only the longitudinal wave and we do not observe the transverse wave. Similarly, as in the case of diffraction of a longitudinal wave, we obtain passing to the limit at p -+ co and r + co in the expression for the Fresnel diffraction wave, the following formula for the Fraunhofer diffraction wave - for an incident plane transverse wave (GNIADEK [1967b]):

Here

t,

= exa,,

kl k2

yo = a , + - a p .

(8.12)

From the above formula one can see that the diffraction wave u, consists of transverse and longitudinal waves. In contrast to the previously considered case (cf. eq. (8.7)) here the discontinuity in the limit direction up = -aL is observed in part of the transverse diffraction wave. Thus in both cases the discontinuity in the direction ap = --a, is shown by that part of the diffraction wave having the same character as the incidence wave. We should note here that, passing to the limit in the expression (7.1) for the Fresnel diffraction wave determined by the approximations (8.1) and (8.2) can be performed only in the case of a finite diffraction aperture. For only then the integrals appearing in the formulae for the diffraction wave are convergent in an ordinary sense. If, however, the diffracting edge at least partly extends to infinity, then these integrals and consequently the diffraction wave assume the character of a delta Dirac function. I t should be added that then, when deriving the formulae (8.7) and (8.12) for the Fraunhofer wave, no limiting assumptions were made as to the shape of the diffracting edge. It may therefore be in general given by a spatial curve. In this case use of our formulae (8.7) and (8.12) considerably simplifies calculations as compared with direct application of the Fraunhofer surface integral. In the case when the diffracting edge is

VII]

REFERENCES

309

a spatial curve composed of straight lines, the value of the integrals appearing in the expression for the Fraunhofer diffraction wave is given by a sum of the type (8.8). Finally, it should be mentioned that the problem of Fraunhofer diffraction of plane elastic waves passing an infinitely long slit in a plane Kirchhoff screen was also considered by KNOPOFF[1956]. However, in his calculations the author used surface integrals and added to them certain linear integrals (cf. 3 4). $j9. Conclusion

Based on the results of the Kirchhoff theory of diffraction of scalar and electromagnetic waves it is shown that this theory is also applicable to approximative description of diffraction phenomena in elastic media. In particular, the Kirchhoff integral for elastic waves transforms into Kirchhoff solution of equations of acoustics in linear approximation, where the applicability of the Kirchhoff theory was experimentally verified. I t should be stressed that the Kirchhoff theory and its interpretation from the Young-Rubinowicz point of view in classical homogeneous isotropic elastic media, may be extended also to Cosserats media which are characterized by a nonsymmetric tensor of stresses (NOWACKI [1969], GNIADEK[1970]). The authors wish to express their thanks to Professor W. Rubinowicz for his kind interest in this work and for valuable suggestions.

References ANG,D. D. and L. KNOPOFF,1964, Roc. S a t . Acad. Sci. USA 52, 1075. BOILLET, P., 1957, Cahiers Phys. 11, 238 (in French). C. J., 1954, Repts. Progr. Phys. 17, 35. BOUWKAMP, FILIPPOV, A. F., 1956, Prikl. Mat. i Mech. 20, 688 (in Russian). K., 1967a, ActaPhys. Polon. 31, 443. GNIADEK, GNIADEK, K., 1967b, Acta Phys. Polon. 32, 687. GKIADEK, K., 1969, Acta Phys. Polon. 36, 331. GKIADEK, K., 1970, Acta Phys. Polon., in press. DE HOOP,A. T., 1958, Representative Theorems for Displacement in an Elastic Solid and their Application to Elastodynamic Diffraction Theory, Thesis, Delft (Excelsior, the Hague). 1961, Theorie der Beugung, HandHONL,H., A. W. MAUEund K. WESTPFAHL, buch der Physik Band X X V / l (Springer-Verlag) (in German). INGARDEN, R. S., 1966, ActaPhys. Polon. 14, 77.

310

APPLICATIONS O F OPTICAL XETHODS

iv11

KARCZEWSKI, B. and J . PETYKIEWICZ, 1967, Acta Phys. Polon. 31, 163. KROPOFF,L., 1956, J. Acoust. SOC.Am. 28, 217. KorTLEH, F., 1923, Ann. Phys. 7 0 , 405. KUPRADZE, V. I)., 1963, Progress in Solid Mechanics, Vol. I11 (Sorth-Holland Publ. Co., Amstcrdam). MAL, A. K., D. D. ANGand L. KROPOFF,1968, Proc. Cambridge Phil. SOC.64, 237. MACE,A. W., 1953, Z. Angew. Math. Mech. 33, 1 . MILES, J . W., 1960, Quart. Appl. Math. 18, 37. MIYAMOTO, K. a n d E WOLF, 1962, J . Opt. SOC.Am. 52, 615. h'OWACK1, w . , 1969, proc. Vibr. Prob. 10, 3. PETYKIEWICZ, J., 1966a, Acta Phys. Polon. 30, 223. PETYKIEWICZ, J., 1966b, Acta Phys. Polon. 30, 437. RUBINOWICZ, A., 1917, Ann. Phys. 4, 53, 257. RUBINOWICZ, A., 1924, Ann. Phys. 4, 73, 339. KUBIROWICZ, A,, 1957 (first ed.), 1966 (second ed.), Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer-Verlag, Berlin und PWNWarszawa) (in German). RCBINOWICZ, A,, 1962a, Acta Phys. Polon. 21, 61. Rubinowicz, A., 1962b, Acta Phys. Polon. 21, 451. RUBI~XOWICZ, A., 1963, Acta Phys. Polon. 23, 727. RUBINOWICZ, A,, 1965, Acta Phys. Polon. 27, 435 (see Progress in Optics, Vol. IV (North-Holland Publ. Co.. Amsterdam)).

VIII

EVALUATION, D E S I G N A N D E X T R A P O L A T I O N M E T H O D S FOR O P T I C A L S I G N A L S , B A S E D O N U S E O F THE PROLATE FUNCTIONS RY

B. ROY FRIEDEN Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. USA

CONTENTS

PAGE

5 5

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

1.

INTRODUCTIOX

2.

MATHEMATICAL PROPERTIES O F T HE LIXEAR PROLAGE FUNCTIONS . . . . . . . . . . . . . . 320

5 5

3.

NUMERICAL CALCULATION O F T HE A,, AND y , ( x ) 334

4.

MATHEMATICAL PROPERTIES OF T HE CIRCULAR PROLATE FUNCTIONS . . . . . . . . . . . . . . 337

5

5.

NUMERICAL CALCULATION OF THE A N , , AND ON,,,(Y) . . . . . . . . . . . . . . . . . . . . . . 347

5

6.

APPLICATIONS: Laser modes, Maximal “concentration” of signals, Extrapolation of image data, Extrapolation beyond the optical bandwith (post and realtime), Degrees of freedom in the image, Evaluation of wave-aberrations . . . . . . . . . . . . . . . . . 349

5

7.

ACKNOWLEDGEMENTS.

REFERENCES.

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

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

313

405 406

Q 1. Introduction On a visit, to Bell Laboratories some 12 years ago, C. E. Shannon posed a question that was proving difficult to answer: to what extent are functions which are confined to a finite bandwidth also “concentrated” in the time domain? The interests of three workers - H. 0. Pollak, H. J. Landau and D. Slepian - were aroused, and in the Spring of ’69 David Slepian discovered the bandlimited function that is maximally concentrated, in the mean-square sense, within a given time interval. This function is the prolate spheroidal function of zero order. The prolate functions had been known to exist since NIVEN[1880!, but their fresh examination by Pollak, Landau and Slepian led to the discovery of new properties; properties which have since been applied to phenomena as diverse as acoustics, antennae, diffraction, filters, and noise. Some of these applications, specifically those dealing with the performance characteristics of optical systems, will be studied in this chapter. 1.1. LIKEAR PROLATE FUNCTIOKS

In 1961, Slepian and Pollak published a summary of their investigations, to that date, on the prolate spheroidal functions. (At least for the purposes of this chapter, we have found it useful to call these functions “linear prolate functions”.) At this point it is useful to outline, in words, some of their discoveries. Section 2 will treat the subject in some detail. The linear prolate functions are a set of bandlimited functions which, like the trigonometric functions, are orthogonal and complete over a finiteinterval. However, unlike the trig (or any other) functions, they are also complete and orthogonal over the infinite interval. An additional property, which lends itself specifically to optical use, is that the finite Fourier transform of a linear prolate function is proportional to the same prolate function. Although other functions exist which are their own infinite Fourier transform (e.g., exp ( - n x 2 ) , 313

314

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

f 1

lxl-3 and sech nx, see BRACEWELL :1965:), only the prolate functions enjoy the property for the finite transform. I n fact, we shall see in 9 2 that this property uniquely defines the prolate functions (to a constant multiplier). 1 . 2 . CIRCULAR P m L x r E FUNCTIOSS

A set of functions which are the circular analogy to the linear prolates, termed the “generalized prolate spheroidal functions” by SLEPIAX [1964j and the “hyperspheroidal wavefunctions” by HEUKTLEY [ 19641, have also been developed (independently) by these authors. These radial functions have strictly analogous properties to the linear prolates, where in general the Hankel transform of general order N rcplaces the linear Fourier transform. We shall call these functions the “circular prolate functions”. As a final item of nomenclature, we shall use the generic term “prolate functions” to describe both the linear and the circular prolate functions as one entity. The mathematical properties of the circular prolates, which make them readily applicable to the analysis of optical imagery, are summarized in 0 4. Note, in particular, that in one limit the circular prolate functions go over into Zernike radial polynomials. The latter have, of course, already proven useful in the evaluation of aberrated lens systems (see, e.g., BORNand WOLFj19641). 1.3. NORMALIZATION AND KOTATIOh-

A word or two is necessary on normalization, since different authors have used different normalizations for these functions. For the linear prolate functions, we shall adopt the normalization and notation of SLEPIAN and POLLAK [1961]. Thus, y,(x) represents the linear prolate function of order n a t the coordinate x . Associated with each y , ( x ) is an eigenvalue denoted as A,,. The circular prolate function of indices N , n will be denoted as QN,,,(r).This function is proportional to function Q I ~ , ~ of( ~SLEPIAN ) [1964], as indicated in eq. (4.1). The eigenvalue associated with Q N , n ( ~ is ) denoted as identical to ?.N,n of SLEPIAN;1964]. All of the prolate functions and eigenvalues are also a function of a free parameter c, which we shall be able to associate with the spacebandwidth product for our optical applications (this is further discussed below). Thus, the complete notation for the prolate quantities shows the c-dependence, Vn(‘,

%)> An(c); @ N , n ( C ,

r ) , I*N,n(C)-

(1.1)

VIII.

0

13

316

INTRODUCTIOK

However, this is a rather cumbersome notation and, like Slepian, we shall often leave off t h e c-notation, with the understanding that all prolate quantities depend upon c. 1.4. THE SPACE-BAXDWIDTH PRODUCT AND PROLATE FUXCTIONS

We have noted that the prolate functions and eigenvalues are a direct function of the space-bandwidth product c. This quantity is proving to be a useful descriptor of optical performance (see, eg., LUKOSZ [1966]). Thus, in all optical analysis using the prolate functions

&w+Qobiect

image

&=r/Xf #

Fig. 1.1 Illustrating formation of the space-bandwidth product c from the optical geometry. (Cnit magnification is assumed, for simplicity.) With coherent object radiation c = x , ~ , , , while for incoherent object radiation c = x , . 28,.

the role played by the space-bandwidth product is always transparent. In general, the size of c will be seen to fix the degree to which a given optical effect may be optimized. This indicates, in turn, that bandwidth may be traded for spatial extent in order to achieve a required degree of optimization. 1.5. PAST USE IN BOUNDARY-VALUE PROBLEMS

Historically, the first uses of the prolate functions stemmed from the fact that they are one component of solutions to the scalar wave equation, when the latter is expressed in three-dimensional, prolate spheroidal coordinates. Thus, solutions to the wave equation have been developed for problems involving prolate spheroidal boundaries, e.g., when a perfectly conducting prolate spheroid is placed in an electromagnetic field. FLAMMER [1957] provides a rather complete summary of such applications (to that date). Later work, involving prolate spheroidal antelznae, has been published by UNZ [1966] and, most recently, by LYTLEand SCHULTZ [1969]. The latter authors give an upto-date summary of progress in the field. We mention these three-dimensional applications merely for com-

316

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

5

1

pleteness. They are extensive in number, and will not be further considered in this study. 1.6. OPTICAL APPLICATIOKS, A SYKOPSIS

The optical applications which we shall consider arise from the modern use of the prolates as a convenient set of one-dimensional, orthogonal functions. Thus, their connection with prolate spheroidal coordinates and the three-dimensional wave equation is not explicitly used. An analogous situation arises for the one-dimensional Legendre polynomials, which originate as the angular part of a three-dimensional solution to Laplace’s equation (see, e.g., PANOFSKY and PHILLIPS [1956]).

These applications have been on the general subject of optical systems analysis. Performance characteristics of the laser have been established, along with the ultimate (in various senses of the word) ability of lens and lensless systems to form high quality images (both with and without supplemental image restoring methods). Also, the affect of certain wave-aberrations upon image quality has been analytically determined. There is one physical phenomenon that unifies these applications - diffraction at a finite aperture - and it may well be said that this phenomenon is optimally analyzed by use of the prolate functions. In the following, we offer a short history of these applications, along with some of the more important results. These results are taken from fj 6, the section on applications per se. 1.6.1. Laser mode patterns

Such was the clarity and utility of Slepian and Pollak’s 1961 paper, that within the year BOYDand GORDON[1961] were able to interpret and use the mathematical results to solve perhaps the central problem of laser resonator theory, analytic determination of the mode patterns. These authors showed that the mode patterns on the rectangular endplates of a confocal laser are individual (linear) prolate functions in each coordinate direction. This single application led, eventually, to

MI

M2

Fig. 1.2. Laser mode problem, What are the equilibrium distributions of light amplitude across mirrors MI,M, after many passes to and fro of the light wave? What role is played by the shape (square, circular, etc.) of the mirrors?

VIII,

f 11

INTRODUCTION

317

analytic determination of the resonant mode patterns under extremely general geometries. This was shown, in rapid succession and for increasingly general geometries, by BOYDand KOCELNIK [1962], STREIFER and GAMO [1964], HEURTLEY[1964, 19651, STREIFER[I9651 and HEURTLEY and STREIFER [1965]. I t is now known that, for any endplate geometry, the mode patterns are superpositions of prolate functions providing the outlines of the endplates are either rectangular or circular. 1.6.2. A n apodization problem, maximizing the encircled energy

More recent applications of the prolate functions have been on the subjects of optimum image formation and optimum image processing. Interestingly, SLEPIAX[1965a] seems to have been the immediate catalyzing agent for these applications. He was able to solve the prob-

Fig. 1.3. Maximizing the encircled energy. For a constant source of radiation at point S, what apodizing mask in the exit pupil of lens L causes the maximum radiant energy to fall within the circle of radius Y,?

lem of maximizing the encircled energy in the point impulse diffraction pattern for a lens system. To wit, the optimum lens is diffractionlimited, with a pupil coating of amplitude C P ~ , ~ (The ~ ) . maximized encircled energy factor is simply Other applications of this type are as follows. 1.6.3. Post extrapolation beyond the optical bandwidth

Before the work of WOLTER [1961], it was commonly thought that, because of its bandlimited nature, an optical image (coherent, or incoherent) could not in principle be processed back into the original object scene. Once the possibility for exceeding the bandlimit was proven, HARRIS[1964] presented a method for actually carrying out the processing. However, the method requires the solution of an arbitrarily large matrix of linear equations, as the demands on accuracy in the restoration are indefinitely increased. BARNES[1966] was able to express the formally perfect restoration as an infinite series of linear

318

OPTICAL SIGNALS A N D PROLATE FUSCTIOSS

[VIII,

$ 1

prolate functions, thereby avoiding the need for inverting large matrices. This was the first application of the prolate functions to the image processing problem. The restoring method was limited in scope, however, to coherent imagery in the presence of diffraction-limited optics. I

-extrap.

+yobserved---+o-extrop.

I

experimental uncertainty

-m

of extrap.

Fig. 1.4. Post extrapolation problem. If the object spectrum O ( w ) is known with error over the optical passband Iw] 5 9,how does this error propogate into an extrapolation of 0 ( w )?

FRIEUEN 119671 then derived a formula for (in principle) perfect restoration of an arbitrarily degraded image in a n y state of coherence. Again, the prolate functions formed the basis for the restoration. All of these methods for exceeding the optical bandwidth in the restoration are severely degraded by image noise (RUSHFORTII and HARRIS:1968]). 1.6.4. Post extrapolation of a n image piece

KINO[1969] showed a method for extrapolating a given image piece arbitrarily beyond its borders, so that (in principle) the entire image can be recreated. The extrapolated image is a superposition of linear prolate functions. Basing a method of bandlimited image processing upon the use of such extrapolated image data, he showed that under certain conditions the restored object is as accurately retrieved as if the entire image were used as input to the restoring formula. 1.6.5. Real-time extrapolation beyond the optical bandwidth F R I E D E N 11969, 19701 has shown that, in the Fraunhofer approximation, real-time image formation can be perfect, in the sense of approximating a Dirac delta function point amplitude distribution. Pupil functions were found which, if applied t o a lens aperture or to an empty aperture, approximate the required Dirac response in any desired plane, or along the optical axis if desired. The price paid for these ideal results is twofold: a decreasing amount of illumination in the image, and an increased sensitivity to random pupil noise, as the point amplitude distri-

VIII,

1:

319

IKTRODUCTION

limited

limited object

Fig. 1.5. Real-time extrapolation problem. By properly coating the exit pupil of lens I.1,can the image on finite field r iyo be formed as if the Zavger lens L, were present?

bution is made to approach the Dirac delta function. The required pupil functions are series of prolate functions. 1.6.6. Degrees of freedom in a n image According to classical theory, a linear, bandlimited system has in principle a finite number of degrees of freedom per unit length of its output (or image, in the optical case). By contrast, TORALDO DI FRANCIA [1969] has shown that (1) in principle an image has an i(x)

Fig. 1.6. Degrees of freedom in an image i ( x ) , using its sampling points. Are N sampled values of i ( x ) sufficient to uniquely describe i continuously across the interval 1x1 I_ x,? What if noise is present in the sampled values of i ?

infinite number of degrees of freedom per unit length; and (2) in practice, the presence of the minutest amount of noise destroys nearly all degrees of freedom beyond the classical number. Continuing Toraldo’s approach, we find ( 3 6.7.5) an expression for the “effective” number of degrees of freedom that persist in the presence of a finite signal-to-noise-raio. 1.6.7. Analysis of image quality for generally aberrated optics

ITOH[1970] has shown that it is useful to expand the pupil function of a circular lens sytem in terms of circular prolate functions. The effect of aberrations upon the Strehl flux ratio and the encircled energy factor is then analytically known. Furthermore, the aberrations may be of any magnitude and evaluation may be made in any receiving plane location.

320

OPTICAL SIGNALS A N D PROLATE FUIICTIONS

[VIII,

5 2

1.7. DISCUSSION, AND SOME QUESTIONS O F A FUNDAMENTAL

NATURE

The work that has just been summarized represents real advances in our knowledge of the capabilities of image-forming systems. Were the prolate functions still undiscovered, for the most part these advances could not have been made. By contrast, we shall see in 5 6 that these results practically “fall out” of the theory of prolate functions. This one mathematical advance has great leverage for purposes of application. Section 6 will have the dual aims of providing useful exercises in the application of the mathematical tools developed in 5 2 and 5 4, and in presenting some recent advances in the theory of image formation and processing. Some of these results will seem, a t first, to violate the dictums of conventional Fourier theory. In 5 6 we shall also try to provide answers to the following questions. How variable is a bandlimited function? By how much can such a function vary between sampling points? Can any laser geometry incur a smaller diffraction loss than does the concentric laser? Is the smallest spot size for a confocal laser actually given by that for the TEM,,, mode? Can the optical transfer function for a fabricated lens system be calculated. from observation of a piece of the point spread diffraction pattern? If any lens system has a finite cutoff frequency, how is it possible for the lens system to produce a point spread diffraction pattern of arbitrary resemblance to a Dirac delta function? Can an arbitrary point amplitude response be produced by a lens? Can an empty aperture be modified with a non-focusing amplitude grating to produce delta function imagery? Can an upper bound be found for the Wiener error of smoothing or of restoration over a finite interval? How many degrees of freedom/length are there in a noisy image?

Q 2. Mathematical Properties of the Linear Prolate Functions The aim of this section is to derive, and display for later use, the optically-oriented properties of the linear prolate functions. The emphasis is on establishing rules of use, so that the derivations are as abbreviated as possible. SLEPIAN and POLLAK [1961] have, of course, already derived most of these results. The proof in section 2.7 and some results in sections 2.9, 2.11 and 2.12 seem to be new.

VIII,

5 21

32 1

MATHEMATICAL PROPERTIES

2.1. RELATION TO CLASSICAL FUNCTIONS

All properties of the linear prolate functions yn(x) will be seen to derive from eq. (2.2) below. At the outset, however, it is proper to identify the yn(x)with a classical set of functions, the “angular prolate spheroidal functions” Son(%). Many properties of these functions, including their numerical calculation, have been known for some time; see, e.g., FLAMMER [1957]. The functions Son(%)are the angular part of the solution to the scalar wave equation, when the latter is separated in prolate spheroidal coordinates. The yn(x)are scaled versions of these functions, (k/xo)’Son(x/xo) c = xoQ, Yn(X) = a112 ( ~ ~ l d t [ S o n ( t ) ] ~ ) ’ ’n = 0,1, 2, - (2.1)

-

We have suppressed the c-notation (see section 1.3). The Sonare here assumed to follow the normalization of FLAMMER [1957]. In each of the following identities, index n (or m,when it occurs) is any number in the range 0, 1, 2, . . .. 2.2. DEFINIXG RELATION

We examine the properties of a set of real functions y, of a real variable x which are defined by the relation

for all real w . The right-hand constant has been chosen so as to simplify later results. I n words, eq. ( 2 . 2 )states that they, are essentially invariant t o a finite Fourier transform. Functions y n are called “linear prolate functions”, and the 1, are called “linear prolate eigenvalues”. 2.3. ORTHOGOIGALITY AND COMPLETENESS ON THE FINITE INTERVAL

SLEPIAS[1964] found that any function satisfying the integral equation (2.2) must also satisfy a particular Sturm-Liouville differential equation. This is the key step in the entire derivation, since it implies (see, e.g., CHURCHILL[1941]) that the yn are orthogonal and complete over the interval 1x1 5 x o , with unit weighting factor:

and

322

OPTICAL S I G I A L S A I D PROLATE F U N C T I O I S

[VIII,

g 2

63

zl.ilyn(y)vn(x) = ~ ( Y - X ) ,

for

!XI

I XO, lyJ5 xo.

(2.3b)

n5.0

6 ( x ) is the Dirac delta function, and d,, is the Kronecker delta. Eq. (2.3b) will be shown later to provide the basis for Dirac delta imagery on the finite field. 2.4. INVARIAKCE TO THE INFINITE F O U R I E R TKAKSFOKM OPERA-

TION

Since the left-hand side of eq. (2.2) is a finite Fourier transform, the right-hand function yn must be bandlimited. I n fact, taking the infinite Fourier transform of both sides of (2.2) shows that

~lo.om 0.002 0.001

- 0.001

Fig. 2.1. y o ( % )tpl(x), , y8(z)y . 3 ( x )versus x / x , , with c = 0.6. (Copyright, 1961, The American Telephone and Telegraph Company, reprinted by permission.)

x/xo

Fig. 2.2. y o (% ).y l ( x ) ,tp%(x),y B ( x )versus x / x , , with c = 1.0. (Copyright, 1961, The American Telephone and Telegraph Company, reprinted by permission.)

XIXO

Fig. 2.3. y o ( % )yl(x), , y , ( x ) , y a ( x )versus x / x , , with c = 2.0. (Copyright, 1961, The Amcrican Telephone and Telegraph Company, reprinted by permission.)

0

0.5

1.0 1.5

2.0 2.5

3.0

3.5

4.0

4.5

5.0

55

6.0

6.5

7.0

7.5

6.0

x/xo

Fig. 2.4. y o ( % ) y, l ( x ) , y a ( x ) , y s ( x ) versus x / x , , with c = 4.0. (Copyright, 1961, The American Telephone and Telegraph Company, reprinted by permission.)

X/XO

Fig. 2.6. yo(c,x ) versus x / x , , with c = 0.5, 1.0, 2.0, 4.0. (Copyright, 1961, The American Telephone and Telegraph Company, reprinted by permission. )

324

O P T I C A L SIGNALS A N D P R O L A T E F U N C T I O N S

[vvrrr,

2

Comparison with eq. (2.2) shows that the yn obey a duality of invariance to the Fourier transform operation, invariance holding for both the infinite and the finite transform. Identity (2.4) will be used in 9 6.5 to invert an infinitely extrapolated spectrum back into the space domain. 2.5. THE 1, AS EIGENVALUES O F THE SIN ( X ) / X K E R N E L

Performing a finite Fourier transform operation d o on both sides of eq. (2.2), we have

This identity will be shown to physically represent the resonance condition for the confocal laser with rectangular, spherical endplates. 2.6. DUALITY O F ORTHOGONALITY

The combination of eqs. (2.2) and (2.3a) implies that the yn are orthogonal over the infinite interval, m

dx Yrn( x ) Yn ( x ) = bmn *

(2.6)

Comparison with property (2.3a) shows that the yn obey a duality of orthogonality with respect to finite and infinite intervals. 2.7. ORTHOGONALITY AND COMPLETEXESS ON THE I W I N I T E INTERVAL

Eqs. (2.4) and (2.6) have established that the yn are bandlimited, and are orthogonal on the infinite interval. Given a generally complex, bandlimited function f ( x ) , it is then tempting to try an expansion W

f(x) =

2 anyn(C, x ) , n-0

c

= x,Q,

(2.7)

for all x . The cutoff frequency for f ( x ) would have to be less than or equal to Q. Since f ( x ) is generally complex, the a, would have to be generally complex coefficients, We may establish the validity of representation (2.7), which hinges on whether the yn are complete on the infinite interval, in the following manner. Adopting the usual criterion of minimized mean-square error

VIII.

§ 21

MATHEMATICAL PROPERTIES

326

we find, by use of orthogonality property (2.6), that the a, must satisfy m

an

=J-,

yn(x).

(2.8)

Substituting these a, back into eq. (2.7), and interchanging the orders of summation and integration, W

f ( x ) = /-mdy f ( Y > all 2

5 Y n ( Y ) Yn(x)*

(2.8.1)

-0

I t will be shown later (see eq. (2.20)) that the sum in (2.8.1) converges to W

2 y , ( y ) y n ( x ) = 7t-l.Q n=O

sinc Q ( y - x ) ,

sinc (a)= sin (a)/u, (2.8.2)

for all x , y . Using the sinc function for the sum in (2.8.1), we come up with a generally valid identity, that a bandlimited function is its own sinc transform. This proves that the series (2.7) with coefficients given by (2.8) converges, in general, to the required bandlimited function f ( x ) . The y , are therefore complete on the infinite interval. 2.8. AN EXTRAPOLATION FORMULA FOR BANDLIMITED FUNCTIONS

Eq. (2.8.1) states that f ( x ) on the infinite interval is determinable from knowledge of f ( x ) on the infinite interval, a rather unexciting result. Note, however, that we can alternatively find a formula for f ( ~on) the infinite interval which depends only on values of f ( x ) within a finite interval. Multiplying eq. (2.7) by w m ( x ) ,integrating dx over the finite interval 1x1 5 x,, and using orthogonality property (2.3a), we find J -2,

Substitution of coefficients (2.9) back into eq. (2.7) produces the required result,

Eq. (2.10) is an extrapolation formula for the class of bandlimited functions. I t states that i f a bandlimited fwnction is known over a finite interval of extent 2x, it is known everywhere.* That extrapolation should

* Note that the finite interval 2x0 need not be symmetric about the origin. For a general X-zo 5 x 5 X + x , , eq. (2.10) becomes OD

f(%) = all2

x+z,

2 Ai’y,,(%-X) x-2, I

n-0

dx‘f(z’)y,(%’-X).

326

OPTICAL SIGNALS A N D PROLATE FUNCTIOSS

;vm,

5

2

indeed be possible for bandlimited functions stems from the fact that a bandlimited function is analytic (see, c g . , WOLTER:1961j), and if an analytic function is known over any finite interval it is determinable everywhere. Eq. (2.10) will be used as the basis for “bandunlimited” restoration, where information inside the optical passband is extrapolated outwards t o obtain a gain in post resolution. 2.8.1. Comparison with a familiar i n t e r p o ~ ~ t ~f oornm d a It is interesting to compare eq. (2.10) with the well-known sampling theorem (see, e.g., O’KEILL [1963]) m

f(x)=

2

f ( n n / Q ) sinc (Srx--nn).

(2.10a)

VZ---o3

dl12

Here f ( x ) at any x is determined from discrete values of f that are distributed throughout the infinite interval 1x1 5 co. The sampling theorem is, thus, an interpolation formula. By comparison, eq. (2.10) is an extrapolation formula, where the output f is determined from a continuum of f-values located in one region ;XI 5 xo of the infinite interval. Note that by either formula an infinite number of basis values for f must be known. 2.9. B E H A V I O R O F THE E I GEXVAL UES &(c)

The eigenvalues l,,(c), c = xoR, are an infinite set of real, positive numbers obeying (2.11) 1 2 1, > 1, > 1, > . . . > 0. That the 1, are positive and less than unity can be seen from eqs. (2.3a) and (2.6).For small n the A, fall off slowly with n until a critical value *,,it

=

2c/n

(2.12)

is reached, beyond which the 1, rapidly approach zero (never reaching it). Curves of 1(c) for various n are shown in Fig. 2.6, and tabulated values are in Table 2.1. Dividing eq. (2.3a) by eq. (2.6), we find that

J’_.. d x [ Y . C ) l y jm_dxLYn(X)12. %

1,

(2.13)

Hence, measures the relative amount of “energy” of function y, that is within interval Ixi 5 xo. According to eq. (2.12), then, once n > 2c/n

VIII,

5

31

MATHEMATICAL PROPERTIES

327

C

Fig. 2.6. Eigenvalues A,(c) of the linear prolate functions. (Copyright, 1966, The American Telephone and Telegraph Company, reprinted by permission.)

the relative amount of function y n within interval 1x1 5 xo becomes very small. This effect can be seen in Figs. 2.3-2.5. Another measure of ?,. can be obtained by differentiating each of eqs. (2.2) and (2.4) with respect to o p times, evaluating them a t o = 0, and forming the quotient :

Hence, A,, measures the relative contribution of the finite interval to the p t h moment of y , ( x ) . Note that p is arbitrary, so that all moments of a given y,(x) are constrained in this manner. The result appears t o be new. 2.10. THE MAXIMUM FRACTIONAL “ENERGY” WITHIN 1x1 5 x, OVER THE CLASS O F BAXDLIMITED FUNCTIOXS

Let us consider the problem that originally led Slepian and his coworkers to study the prolate functions. What is the maximum fraction of its energy that a bandlimited function can concentrate into a

w m

TABLE 2.1 Values of .A

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40

5.7258178E-01 6.2791274E-02 1.2374793E-03 9.2009770E-06 3.7179286E-08 9.4914367E-11 1.6715716E--13 2.154449 1E-16 2.1207239E-19 1.6466214E-22 1.0343492E-25 5.3650197E-29 2.3367231E-32 8.6674831E-36 2.77096 12E-39 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

W

2.

3.

4.

6.

6.

8.8056992E-01 3.5564063E-01 3.6867688E--02 1.1522328E-03 1.8881549E-05 I .9358522E-07 1.3660608E-09 7.0488855E--I 2 2.7767898E-14 8.6266788E-17 2,1680119E-19 4.4986673E-22 7.838245OE-26 1.1630367E-27 1.4873466E-30 1.6563614E-33 1.6207613E-36 0. 0. 0. 0. 0. 0. 0.

9.75828633-01 7.09963343-01 2.061386831-01 1.8203800E42 7.0814710E-04 1.6551244E--05 2.64101653-07 3.0737365E-09 2.7281307E-11 1.9085689E--13 1,07979063-1 5 5.04311563-18 1.97754363-20 6.6033063E-23 1.90029293-25 4.7620029E-28 1.0485031E-30 2.0444867E-33 3.55518803,-36 5.5475853E-39 0. 0. 0. 0. 0.

9.95886493-01 9.1210742E-01 6.1906484E-01 1.1021099E-01 8.8278764E-03 3.8129172E-04 1.0950871E-05 2.2786389E-07 3.6065493E-09 4.4938297E-11 4.5252285E-13 3.7603029E-16 2.6228187E-17 1.5676942E-19 7.9711081E-22 3.561908OE-24 1.3905716E-26 4.8210691E-29 1.4905449E-3 1 4.1352414E-34 1.0352225E-36 0. 0. 0. 0.

9.9935241E41 9.7986456E-01 7.9992193E-01 3.4356219E41 5.6015851E-02 4.1820948E-03 1.9330846E44 6.3691502E-06 1.5822998E-07 3.0917267E-09 4.8757393E-11 6.3402794E-13 6.9 17 3022E-16 6.4235507E-17 5.1393068E-19 3.57974633-21 2.1906130E-23 1.1869344E-26 6.7350388E-28 2.4864675E-30 9.7273165E-33 0. 0. 0. 0.

9.9990188E-01 9.9606164E-01 9.4017339E-01 6.4679196E-01 2.0734922E-01 2.7387166E-02 1.9560007E-03 9.4848766E-05 3.4367833E-06 9.7321160E-08 2.2 189806E-09 4.1662263E-11 6.5674786E--13 8.7803771E-16 1.0125783E-16 1.0163838E-18 8.9610464E-21 6.9960907E-23 4.8687463E-25 3.0405184E-27 1.7132439E-29 0. 0. 0. 0.

0.

'd

k $ ~1

5

0

5

' Z u,

c(

M

N

-


x ) == (An/xoNn)'

2

dyJ'r(x/xo)

(3.1)

r=O,l

is valid. The summation is over even (odd) values of r if n is even (odd). The Pr are Legendre polynomials, the LZl)n are tabulated coefficients that depend upon c, and the M ( n ) are truncation numbers chosen such that (typically) Idyl < (3.2) Also, N , are normalization constants formed as M(n)

N, = 2

2

(2r+l)-l(Q)?

(3.3)

r=0,1

The eigenvalues A, are computed as 2c

- [2"@( ( i n )!)Z/n!]zfor n even A n = ( 2c x - [2"@'~(&(n-1))!(+(n+1))!/3(n+l)!]~ for

x

(3.4)

n odd.

Tabulated values of the d$ may be found in Flammer, but these are

336

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

8 3

restricted to values of n 5 3, which is too small for most of the optical applications considered in this study. A more complete tabulation of coefficients has been made by STUCKEY and LAYTON[1964]. The range of parameters n = O ( 1 ) l O and c = 0.2 (0.25)lO.O is covered, with d r given to 11-place accuracy. If values of c 6 10 are still required, but now with a general range of x-values, the following formula (Flammer) may be used in place of eq. (3.1): M (n)

where

Kn

(4%) ! ) 2 4 3 - l

for n even = 3(n+1)![2n(+(n-1))!(&(n+1))! cd?]-l

[

n ! [2"(

(3.5)

for n odd.

Functions jr are spherical Bessel functions. Coefficients and eigenvalues A, are as above. Eigenvalues A, may be directly expressed in terms of c, in place of formula (3.4) which uses tabulated dg" values. Slepian showed that, for small values of c, 22"

(n!

3

]'

c2n+l

[

exp -

(212f1)C2

(2n- 1)'(2n+ 3)'

1.

(3.6)

3.2. VALUES OF c 2 10

Slepian's formulae (1.4), when combined with our normalization eq. (2.1), permit yn(c, x ) to be computed at all x , for cases c x 10 with n not too large. (Note that, although Slepian's functions in (1.4) are not the classical functions So,(%), his W,,n(x) may be used to replace So,(x) in our eq. (2.1). The normalized functions so obtained will still be our functions y n ( x ) . )Because of its useinlaser theory, we show the first of these equations in the normalized form (2.1):

and H , is a Hermite polynomial. An algorithm for computing coeffi-

VIII. § 41

337

CIRCULAR PROLATE F U N C T I O N S

cients A: is given by Slepian. Normalization constant iV, in (3.7a) obeys

X, = (n/c)i n ![1 +2-7 (k2)-1 (n4+ 2 d +

23n2+ 22n

+12)].

(3.7~)

Finally, eigenvalue A, in (3.7a) obeys

1, (c) = 1 -4324 (n!)-12snc"*e-20[ 1- (32c)-l (6n2-2n+ 3)].

(3.8)

3.3. AN ALTERNATIVE TO CLOSED FORMULAE?

The implicit point of view in all of the foregoing is that the I , and yo can be computed from closed formulae. An alternative, which avoids the need for extensive computer programming, is to compute I , and y, as solutions to a numerical eigenvalue problem. For this purpose, eq. (2.6) could be used to generate M linear equations in M unknowns y(c, x,), x, = x o ( m - l ) / M , m = 1, 2, . . ., M . The other unknown, I , would be found by setting the determinant of coefficients equal to zero. Each value I , obtained in this way would result in a different solution set yn(xm),m = 1, 2, . . ., M . This multiplicity of solutions corresponds to different values of n,i.e., to values n = 0, 1, 2, . . .. We do not know if anyone has tried this method yet. The method must fail once ~t is taken greater than some critical value.

Q 4. Mathematical Properties of the Circular Prolate Functions The circular prolate functions and eigenvalues were invented independently by SLEPIAN [1964] and HEURTLEY [1964]. We shall draw freely from these works, as required. The emphasis in this section is upon establishing rules of use for the functions and eigenvalues involved, although all results will be shown to follow logically from one defining relation, eq. (4.2) below. I n general, there is a one-to-one correspondance between the results of § 2 for the litwar prolates and the results of this section for the circular prolates. To encourage comparisons between the two sets of results, we have labelled corresponding equations with corresponding numbers, e.g., eq. (2.2) * eq. (4.2). 4.1. RELATION TO FUNCTIONS O F SLEPIAN AND HEURTLEY

For application to optical geometries having circular symmetry, we would like to define a set of functions @ N , n ( ~ r, ) which are, as close as possible, the circular analogy to the linear prolate functions. Accordingly, define "circular prolate functions"

338

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

5

4

Functions q+,,, are the “generalized prolate spheroidal functions” of SLEPIAN [1964]. Note that, except for the extra factor y-8, this equation is strictly analogous to eq. (2.1), where the classical functions S,,(t) take the place of ~ ~ , ~ Functions ( t ) . Q i ~ , ~may ( t ) be related to the “hyperspheroidal functions” YN,,(t) of HEURTLEY [1964] via eq. (4.1) and the correspondance

4

3 2 1

0 -1 -2

-3

r /re

Fig. 4.1. Some of Slepian’s generalized functions r p ~ , ~ (r/vo). c , (Copyright, 1964, The American Telephone and Telegraph Company, reprinted by permission.)

VIII.

4j

CIRCULAR PROLATE FUNCTIONS

339

Fig. 4.2. More generalized functions ~ N , ~ (r/ro). c , (Copyright, 1064, The American Telephone and Telegraph Company, reprinted by permission.)

In each of the following identities, all indices N , n, m are assumed to be chosen independently over the range 0,1, 2, . . .. 4.2. DEFINING RELATION: INVARIANCE T O THE F I K I T E H A N K E L

TRANSFORM

With normalization (4. l ) , Slepian’s fundamental relation (SLEPIAN

340

CPTICAL S I G N A L S A N D P R O L A T E F U N C T I O N S

m 1 ,

84

[1964], eq. (20)) is

fdr

rJN(wr)@N.n(r)= (-l)n(ro/Q)1N,n*@N,n(wro/Q~~

c = r0Q, w

>= 0.

(4.2)

is the Nth order Bessel function of the first kind. All properties of the @, 1 will be seen to follow from this relation. Note that two subscripts N , n are now required for @ and 1, because there is an infinite sequence of @,, A,, n = 0, 1 , . . . that safisty (4.2) for each Hankel transform order N = 0, 1, . . .. (By comparison, defining eq. (2.2) for y, has only one “order”, corresponding to the single Fourier transform.)

JN

4.3. ORTHOGONALITY AND COMPLETENESS OVER A F I N I T E RADIAL

INTERVAL

HEURTLEY [1964] found that any function @N,n satisfying eq. (4.2) must also satisfy a particular Sturm-Liouville differential equation. This implies (see e.g., CHURCHILL[1941]) that the @N,n are orthogonal and complete over the interval 0 5 r 5 ro: (4.3a) and 00

2 &,ln@N,n(r’) @N,n(r’)= d(r-r‘)/r

n=O

for

o 5 r, r‘ 5 r,.

(4.3b)

A word is in order on the lower bound for values r, I‘ in (4.3b). It can be seen from (4.2) that @N,n(0)= 0, if N > 0. Therefore, when r = r’ = 0, the left side of (4.3b) is identically zero whereas the right side has a singularity. Therefore, if N > 0, eq. (4.3b) does not hold for r and r’ simultaneously zero. (The impulsive right side of (4.3b) can therefore only be a “ring impulse”, and not a point impulse.) However, for N = 0 these difficulties do not arise, since @“,.(O) # 0 in general. 4.4. ISVARIANCE TO THE INFIXITE HANKEL TRANSFORM

Taking the Hankel transform of order N of both sides of (4.2) shows are bandlimited, and invariant to a Hankel transform: that the @N,n(r)

(4.4)

VIII,

8 41

341

CIRCULAR PROLATE FUNCTXONS

Comparison with eq. (4.2) shows the @N,n(r)to obey a dzlality of invariance to the Hankel transform of order N . AS EIGENVALUES O F A “REFLECTION” KERNEL

4.6. THE

Any function f ( r ) whose JN-transform (Hankel transform of order N ) cuts off at a frequency SZ obeys an identity

where kernel

m

(4.6a) Kernel KN is thus the circular analog to the sinc kernel of one-dimensional theory. It can easily be shown that KN “reflects” BesseI functions, i.e.,

jOw dr’ (0’r’)K~ ~‘JN

(7, 7’)

=

JN(~‘Y)

for 0 5 w’ 5 IR for o’> a.

(4.5b)

We find that the @N,n(r) and AN,n are eigenfunctions andeigenvalues of the reflection kernel, JOT’

dr’ r‘KN(r, 7‘)

@N,n(r’)= AN,n@N,n(r)

2 0.

(4.5c)

This may be shown by taking a finite JN-transform of eq. (4.2). Eqs. (4.6~)and (2.6) are analogous. Eq. (4.6~)will be seen to determine the mode patterns for the laser with circular endplates, where the assumption of amplitude invariance after a reflection is assumed (a property of physical reflection for kernel

KN?). 4.6. DUALITY O F ORTHOGONALITY

By combining eqs. (4.2) and (4.3a), orthogonality over the infinite, radial interval may be established:

obey a duality of Comparing this relation with eq. (4.3a), the QN,%(r) orthogonality.

342

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

5

4

4.7. ORTHOGOKALITY AND COMPLETEKESS OVER THE IKFIhTTE

RADIAL INTERVAL

Any complex function f ( r ) whose JN-transform cuts off at a frequency f2 (or less) may be represented by an expansion a2

rrOifN>O

2

= n=O u N , n @ N , n ( C ,

‘1,

(4.7)

= ‘OQ

rZOifN-0

where the aN,nsatisfy (4.8)

This may be derived in perfect analogy to the corresponding results of €j3.7. Convergence of summation (4.7) to f ( r ) is in the mean-square sense, over the infinite interval 0 5 r 5 co. (See discussion following (4.3b).) Eq. (4.7) states that any one function f(r) may be represented by an infinity of series, one for each value of N . 4.8. AN EXTRAPOLATING FORMULA FOR BAXDLIMITED FUNCTIONS

O F RADIUS

Alternatively, the coefficients in series (4.7) may be represented as

This results from use of orthogonality condition (4.3a). Combining eqs. (4.7) and (4.9),

This is a relation by which the behavior of f ( r ) on the finite interval r 5 ro is extrapolated to all values of r > 0. 0 4.9. BEHAVIOR O F T H E EIGENVALUES

Eigenvalues , ? N , n ( ~ )are an infinite set of real, positive numbers obeying 0

< AN,, 5

1,

with

AN,n

> AN,nll

and

AN,,,

> AN+l,,,.

(4.11)

Fig. 4.3 shows curves of I ( c ) for variousN, n. Table 4.1 shows numerical values of I N , n ( ~ from ) more extensive tables in SLEPIAN [1964].

VIII,

5

41

CIRCULAR PROLATE FUXCTIOWS

343

C

Fig. 4.3. Eigenvalues j . N , n ( c ) of the circular prolate functions. (Copyright, 1964, T h e American Telephone and Telegraph Company, reprinted by permission.)

An analogous result to eq. (2.12) is not yet known, i.e., functional relations such that A N , ~ ( c )-+ 0 rapidly once either N > Ncfitor n > ncrlt.

Ner1t(C)j[ncrlt(c)

(4.12)

Each eigenvalue AN, measures the relative “amount” of the function 5 r S r,,

C D ~ , ~that ( ~ lies ) within interval 0

This is a result of eqs. (4.3a) and (4.6). Each AN,nalsomeasures the relative amount of the P t h moment of DN,%(r)that is contributed from within interval 0 5 r 5 r,,

Thus, strangely enough, for each (N, n) pair the right-hand side of (4.14) is fixed, independent of power P. Eq. (4.14) is obtained by operating with (o-ld/do)PcuNupon each of eqs. (4.2) and (4.4), forming the quotient of equations, and then evaluating the quotient at o = 0. This result appears to be new.

W P Ip

TABLE 4.1 Some numerical values of AN,,(c) C

A

C

0.1 0.6 1.0 1.6 2.0 3.0 4.0 6.0 10.0

2.4968775-3 6.0586348-2 2.21 11487-1 4.2961906-1 6.29830461 8.8706036-1 9.7496117-1 9.9634230-1 9.9999967-1

0.6 1.0 2.0 3.0 4.0 6.0 6.0 10.0

9.49826584 1.3986168-2 1.6123183-1 4.8326866-1 7.8473605-1 9.3671678-1 9.8634266-1 9.9998314-1

1

2 3 4 6 6 7 8 9

10

1.08298164 6.72144863 6.6746424-2 2.67427861 6.7877067-1 8.3060712-1 9.4973860-1 9.8782700-1 9.9738664-1 9.9947801-1

A N = 1,n= 1

N=O,n=l 1

2 4 6

6 7 8 9 10

7.4072661-7 1.8649611 4 3.8313661-2 1.6818804-1 4.2912667-1 7.1473948-1 8.9618892-1 9.7041388-1 9.9279210-1

q

C

z 0

: 0

z fn

M

4

VIII,

5 41

CIRCULAR PROLATE FUNCTIONS

346

4.10. THE MAXIMUM FRACTIONAL “ENERGY” W I T H I N 0 5 r 5 ro OVER THE CLASS O F BANDLIMITED, RADIAL FUNCTIONS

Let us define the encircled energy e due to a bandlimited, generally complex function of radius f ( ~as) (4.15)

We ask what function f(r) maximizes e. The solution is found as follows. Expanding a function f ( r ) by use of series (4.7),substituting it into eq. (4.16) and using the two orthogonality conditions (4.3a), (4.6), loo

Q)

(4.16)

Since all the 2 0 and, by eqs. (4.11),the AN,n monotonically decreasein size with n, e is a maximum when u N , ~# 0, u ~=,u N~, ~= . . . = 0. Hence, the required function f ( r ) obeys f(r) = ~ N , O @ N , O ( ~ ) ,

~ N , O arbitrary,

and the corresponding e = AN,o.

But N is still a free parameter. According to eqs. (4.11),the ANsn also decrease monotonically with increasing N . Hence, from the last equation, emax = &,o, N=O (4.17a) and the absolute optimum f ( v ) obeys f(r)=~O,O@O,O(~)~

ao.0

z 0.

(4.17b)

These results will be seen to directly carry over into the optical problem of maximizing the encircled energy in the point impulse response. 4.11. EXPANSION O F SOME FAMILIAR FUNCTIONS I N TERMS O F THE @N, (r) (1

Using eqs. (4.2), (4.7) and (4.9), we find that

346

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

provided 0

[VIII,

4

5 o 5 L?. Note the particular value when o is zero, 00

2 (-l)n&,fn@O,n(0) n=O

1 = (ro/Q)

(4.19)

@o,n(r)

which must hold for all r 2 0. In a like manner, the reflection kernel K N ( ~ r’), of eq. (4.5a) may be expanded as m

(4.20) According to eq. (4.3b), functions d(r-r’)/r and K N ( r ,r‘) differ only through the presence of eigenvalues AN,n in the series for d(r-r‘)/r. 4.12. A CLASS O F FUNCTIONS W H O S E F I N I T E JN-TRANSFORV IS

ASYXPTOTIC T O A DIRAC DELTA FUKCTIOK O F RADIUS

As in the analogous case of 9 2.12, we now seek a function whose finite JN-transform is a Dirac delta function. I n this circular case, however, we shall be satisfied with producing 6 (r-r’)/r, i.e., proportionality to the delta function (thereby still obtaining an impulse-type of result). Following an analogous development to that of 9 2.12, we require a function V ( w ,r’) to have the property rn

J

dw o J N ( w r )V ( o ,r’) = d(r-r’)/r

for 0

5 r, r’ 5 ro.

(4.21)

0

For an r‘ > 0 the right-hand function is a ring-like delta function of r. Let us represent V ( o ,r’) by the series m

V ( w , r‘) = OdoSR

2 aN,n@N,n(wyO/L?) n-0

(4.22)

with coefficients aN,nto be determined. Substituting (4.22) back into eq. (4.21), coefficients aN,nare required to obey m

(f2/r0)2 ( - - l ) n ~ , k , p ~ , ~ @ ~=, ~d(r-r‘)/r (r)

for

o s r, r’

I ro.

n-0

(4.23) Defining relation (4.2) was used. Comparing eq. (4.23) with the completeness relation (4.3b), the aN,nare uniquely determined: uN,n

(-l)n(rO/L?)~~~n@N,n(r’).

(4.24)

Substitution of coefficients (4.24) back into expansion (4.22) produces

VIII,

5

61

347

NUMERICAL CALCULATION

the answer: 00

V ( 0 ,r')

3

V(*)(O,r')

=

(ro/Q)2 (-l)nA&@NJr')

@N,n(wro/Q).

n-0

(4.25a) This function has been required to obey 8

loQdwo J ~ ( w ) V ( ~ ) r') ( W= ,

2 ,l&~D~,~(r)

(4.25b)

n-0

The special case r' = 0 of a point impulse will be of interest in the optical applications. We noted, following eq. (4.3b), that the right-hand side of (4.25b) remains impulsive with r' = 0 only if N = 0, i.e., only if the zero-order Hankel transform is taken. The conclusion, then, is that functions V(*)(o,r') with N > 0 can only produce ring-like impulses.via the finite JN-transform, whereas the single function V(O)(a,r') produces ring-like impulses or the point impulse via that transform:

J

-

d o o J o ( o r ) V o ) ( wr') , = d(r-r')/r,

for 0 5 r, r'

5 ro.

(4.26)

0

In order to evaluate the series (4.26a) and (4.25b) we have to truncate them at finite n = M . The truncated function V$)(O,r') therefore obeys M

JoQdO o J N ( O Y ) V$"(O, r') = aM(r-r')/r + d(r-r')/~, M

with

dM(r-r')/r

(4.27)

2 &:n@N,n(r)@N,n(r')

n=O

and allowable values 7 , r' as discussed above.

Q 5. Numerical Calculation of the AN,- and

aNJr)

SLEPIAN [1964] has derived formulae for calculation of the A N , n ( ~ ) and Q ) ~ , ~ ( tc), over all values of c, N , n and t. Each such formula for p may be related to our @ by use of normalization eq. (4.1). HEURTLEY [1964] has also derived formulae which are relatable to @N,n(r),and are valid for small c (with r 5 ro), and for large c (with 7 5 roc-)). He also established that, in the limit c -+ 0 , the hyperspheroidal function goes over into a polynomial of finite power - the Zernike polynomial RZ ( t ). Slepian's qN, (0,t ) may likewise be related to Zernike polynomials, in particular by comparing the hypergeometric

348

OPTICAL SIGNALS A N D PROLATE F U N C T I O N S

[VIII,

8 5

forms for the two functions. The result is

By eq. (4.1), then, c B ~ , ~ ( rO), is directly proportional to Zernike polynomial RfwN ( r ) . Some optical ramifications of this correspondance are discussed in 3 6.8. For the reader's convenience, we shall list some of Slepian's formulae for AN,n and CBN,n. 6.1. SMALL VALUES

OF c m

Function vN,,(0, t ) is a Zernike polynomial given by eq. ( 6 . 1 ) , and the d?" are tabulated coefficients that depend on c. Normalization constants NN,n and eigenvalues AN,* also depend on coefficients d?", through

and

All j-summations should be truncated at values j for which d?" become negligable. For problems in which a general range r 2 0 of values are required,

J is the ordinary Bessel function and N N , - is as in (5.3). If coefficients d?" are unknown, eigenvalues AN,n may be computed directly from knowledge of c, through AN,n

2

= "/",

(6.6)

~ 1 1 1 ,8

61

APPLICATIONS

349

5.2. LARGE VALUES O F c

Slepian's formulae (67) permit Q ) ~ , ~ (t )c to , be computed for all t, if c is sufficiently large with N and n fixed. Normalization eq. (4.1) may then be used to transform to circular prolate functions at all Y. Because of its application to laser theory, we relate the first of formulae (67) of Slepian to C D ~ , ~ ( C ,r1 0S

r sro 0-

= (~~,,/r,iV~,,>f (~/ro)'"fir"e""'2ro~~~(Qr2/ro) (5.7a)

where constants (5.7b) and functions L ( f )are Laguerre polynomials of degree n (Szego's notation). Finally, eigenvalues AN,n may be computed as n22N+4n+3CN+2n+l e-2a

AN,,

=1-

rt! ( N + n ) !

Q 6. Applications Each section 6.2-6.8 below will describe a different application to optical systems. We have tried to present each application as an entity apart from the others, so that they may be read in any order. AU of the applications will be seen to derive from one physical phenomenon, that of scalar, Fraunhofer-Fresnel diffraction at a finite aperture. In general, the aperture will contain either a geometrical focussing device (lens or mirror) plus an amplitude grating, or only an amplitude grating. In 3 6.1, we shall simply state the laws governing image formation for these two aperture cases. The notation for the applications is also established, in this manner. 6.1. FRAUNHOFER-FRESNEL IMAGE FORMATION

6.1.1. Focussing device in the aperture

Suppose a point source S of monochromatic radiation illuminates an optical aperture A, as shown in Fig. 6.1. The aperture contains a lens or mirror which both focusses the incident light and modulates it by a general complex function U of position in the aperture. We would like to know the resulting scalar amplitude u of radiation at any field point S' located in the image space.

360

O P T I C A L S I G N A L S A N D P R O L A T E FUIGCTIONS

A

[VIII,

3

6

I

Fig. 6.1. Geometry for a focussing device in the aperture.

As shown in Fig. 6.1, let the rectangular coordinates in the aperture be xl, yl, with image coordinates x 2 , yz, z2 measured from the gawss point G. The position of G with respect to an origin a t principle point H' is denoted as r G . I n the Fraunhofer-Fresnel approximation to scalar diffraction theory (see e.g., BORNand WOLF[1964]), field amplitude u obeys us,= aJS,

dxl dy, u(x19yl) exp [j (

~ (x2,+y2,)z2 t )

+j (WG) (x1x2+~1~2)l, aG = j (hG)-' exp (- jkrG).

(6.1)

As usual, k = 2n/j., where A is the wavelength of light. For z2 = 0, u is seen to depend upon U through a finite Fourier transform, which is highly suggestive of the defining equations (2.2) and (4.2) for prolate functions. 6.1.2 No focussing device in the aperture

Observing Fig. 6.2, we now locate the source point S upon the optical axis OA and at infinite distance from aperture A. I n the absence of a

wave front

Fig. 6.2. Geometry for no focussing device in the aperture.

V III,

5

61

APPLICATIONS

361

focussing device in A there is no longer a well-defined gauss point, and so our image coordinates x,, yz, L have as origin the center 0 of the aperture. In the Fraunhofer-Fresnel approximation, the amplitude u at a general point S’ is now Ns = aL/S,

hldy,

u(x19 yl) exp {(jk/2~)[(x,--x,)2+(y,-Y,)21},

aL = j(lL)-l exp ( - j k L ) .

(6.2)

Or, u is the finite “chirp” transform of U . 6.2. ANALYSIS O F LASER MODES

The following analysis is drawn from the works of the authors listed in 5 1.6.1. We limit this study to the role played by the prolate functions in the.laser mode phenomenon; a comprehensive analysis of the phenomenon is outside the scope of this study (for a fuller treatment, see, e.g., CHANG[1969]). Our general approach will be to proceed from the laser geometry that is simplest to analyze in terms of prolate functions - the confocal, square geometry (defined below) - to more general geometries. This was also the historical development of the subject. 6.2.1. Confocal laser with identical endplates The optical geometry for a confocal laser is shown schematically in Fig. 6.3. M, and M, are spherical mirrors, or endplates, with a common radius of curvature b and a separation b. The mirrors thereby share a common focal point, i.e. are “confocal”. Imaginary reference planes P, and P, are drawn through the vertices of M, and M,, at right angles to the optical axis OA. Because the mirrors are confocal, planes P, and P, are geometrical conjugates. This means that a point p on M, that

Fig. 6.3. A confocal laser. Points p and p’ are optical conjugates.

352

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII.

8 6

is sufficiently close to the OA is imaged by M, back onto MI (on the opposite side of the OA), and hence

rc

= b.

(6.3)

A light wave leaving M, will reflect back and forth between the two mirrors in the manner of an interferometer or a "resonator". With nothing but air between the mirrors, the light wave would of course quickly lose all of its energy because of scattering, diffraction beyond the mirror edges, etc. However, a laser has an amplifying medium in the space between the mirrors, such that each passing wave of light is amplified in energy by a sufficient amount to produce an equilibrium distribution of light on each of the mirrors. The aim of this analysis will be to find these distributions of light amplitude. The basis for determination of the laser mode patterns is the assumption that, after many bounces back and forth between the mirrors, an equilibrium light distribution will be attained which is proportional to the equilibrium distribution that is approached in the absence of the active medium. The latter is a certain function times a gain factor of zero (because of the loss phenomena discussed above). The tacit assumption is that the presence of an amplifying medium between the mirrors merely acts to make the gain factor non-zero (while leaving the light distribution function invariant). We therefore assume the existance of perfect vacuum between mirrors M, and M,, and seek the equilibrium distributions. Imagining a wave to be bouncing back and forth between the mirrors, it is plausible to expect that after many bounces the proportion of light amplitude that is lost with each bounce approaches a constant. This constant attenuation factor would arise from two loss phenomena: diffraction of light past the mirror edges, and transmission throagh the mirrors. For simplicity, we shall assume the latter loss to be made negligable by the presence of highly reflective mirrors. Attenuation is, then, only due to diffraction loss, and our solution to the problem should exhibit this effect. At equilibrium, it is also plausible to expect the patterns on the two mirrors to be identical in form. That is, denoting the distribution of amplitude across MIby U ( " ' ) ( Xyl), ~ , where m is the bounce number, and that across M, by G ( ~ + ~ ) ( X y,,), (at the following bounce), f p + l )= (TU(m).

IJ is the

(6.4)

attenuation factor, which also contains a phase part (the phase

WII.

§ 61

APPLICATIONS

363

change per bounce). But dm+l) and Vm) are also related by diffraction effect (6.1), which simplifies to

p

= j-lilb

exp (jkb)

(6.6) because of (6.3) and the fact that zg = 0 (M, and Ma are gaussian conjugates). By combining eqs. (6.4) and (6.5) we require equilibrium distribution U to obey

Superscript m is now superfluous, and has been dropped from the notation. The equilibrium distribution U is therefore the solution to an integral equation. The form of (6.6) reminds us of defining eqs. (2.2) and (4.2) for the prolate functions. To proceed any further with the analysis, the shape of mirror M, must be specified. 6.2.1.1. Case of square endplates

By assuming mirrors M, and Ma to be square, with side Zx,, and assuming the solution U to be separable in the form (6.7) U(x3 Y) = Fi(x)Fa(Y) the integral equation (6.6) separates into two symmetrk:requirements, (6.8a) where CT

= (a,u,)fj exp (-jkb).

(6.8b)

Finally, by defining a bandwidth

L2 = kxo/b,

(6.9a)

a space-bandwidth constant c = xoo,

(6.9b)

and a new coordinate w = ky/b,

(6.9~)

each of eqs. (6.8) take the form

dx F , ( x ) ejw = cri ( 2 n ~ o / ~ ) + F I ( w ~ 0 / i 2 ) .(6.10)

364

[VIII,

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

5

8

The similarity to eq. (2.2) is now inescapable. Functions F i , i = 1, 2 may therefore be identified with functions yn,and u$with j”Ai. The laser equilibrium patterns are therefore chosen from among

U ( x ,y )

3

Umn(x,y ) = ym(x) y,(y),

m,

= 0, 1, 2,

...

(6.lla)

while the attenuation coefficients are corresponding values of = jm+n+l e-lkb (Am 1.n)

’*

(6.11b)

Eqs. (6.7) and (6.8b) were used. By taking the modulus of eq. (6.11b), and noting from eqs. (2.11) that all A, 5 1, we see that u truly represents an attenuating factor, with a minimum attenuation of A,. Thus, the mode pattern y o ( x ) y,(y) causes the minimum diffraction loss, over all possible patterns. In fact, we shall later see that 1, is the minimzlm possible diffraction loss over all possible open resonator geometries using square endplates. The confocal geometry is thus a minimum-loss geometry. Fig. 2.6 shows the manner by which attenuation A, varies with c. Thus, by eqs. (6.9a, b), for an allowed diffraction-loss in mode Un,n there is an allowable trade-off between optical quality of the endplates (as measured by their cut-off frequency Q) and size of the endplates. Moreover, by eq. (2.12) the diffraction loss in mode U,,n is negligable (A, --f 1) once c > inn, indicating that it does not pay to increase the space-bandwidth much beyond a value $nn for purposes of minimizing diffraction loss. The phase of coefficient umn is useful to consider, because it determines what modes m,n will resonate. Since the laser resonator is an optical interferometcr, there will be a maximum of illumination on the mirrors, or resonance, only if the round-trip phase shift is a multiple q of 2n. Noticing from (6.4) that the phase shift per mirror bounce is given by the phase of u in (6.11b), and that two bounces make up a round trip, the condition for existance of modes m, n is that 2nq = 2/-M+4n(m+n+l)l.

(6.12)

The notation for the mode Umn(x,y) which resonates for a given number q, is conventionally TEM,,, , for “transverse electromagnetic mode” of orders m,n, q. Eq. (6.12) shows that for a given q, called the longitudinal mode number, there will ordinarily be many transverse modes m, n which simultaneously resonate. It is interesting to establish the form of the mode patterns once c > 10, a common occurrance. Approximation (3.7a, b) shows that

VIII,

61

366

APPLICATIONS

the patterns become Hermite-weighted Gaussian functions, a wellknown result. From this approximation it is also possible to define a spot-size on the mirrors. Taking this to be the point at which mode TEM,, falls off to l / e of its central value, eqs. (3.7) and (6.9a) yield a spot of extent 2 (Ab/n)*. (6.13) This is independent of x,. Thus, once the mirror extent x, is large enough that c 2 10, further enlargement of the mirror does not affect the spot size. At this point we emphasize the arbitrariness of "spot size" as given above. Note that the single mode TEMooQ was presumed present. Of course, a superposition of modes would result in a different spot size, even if the l/e criterion is still used. We shall show in 5 6.6.10 that the spot size for a laser is, in fact, arbitrarily small, as one particular superposition of modes is approached. 6.2.1.2. Case of circular endplates Returning to the general eq. (6.6) for resonance, we now assume M, and M, to be circular with radius r, . What are the resonant modes and diffraction-loss characteristics of this case? It is natural to now transform from rectangular to polar coordinates. Eq. (6.6) becomes

Jr]oro

dr, rl dOl U ( r , , 0,) exp {j (k/b)rlr2cos (02-f3,)}

= paU(r,,

0,).

(6.13)

Using the handy expansion (GROBNERand HOFREITER [1966]) 00

2

eJAcmB =

j"JN(A)ej""

(6.14)

N---m

in (6.13), and then trying to separate out the r and 0 dependence by the requirement on the solution

U i r , e)

= R(r)ejNB

(6.16)

leads to the following requirement on the r-dependence,

s,'"

dr, rl JN(krlr,/6)

I?(.,)

=

( 2 ~ ) - l j -p~R ( r 2 ) .

(6.16)

That is, the &dependence of solution (6.15) will balance in (6.13), if (6.16) can be satisfied. Eq. (6.16) looks suspiciously like defining relation (4.2) for the circular prolate functions.

366

OPTICAL SIGNALS A N D PROLATE F U N C T I O N S

[VIII,

5 6

Defining a bandwidth Q = kr,/b,

(6.17a)

a space-bandwidth product

c

= roQ = kr2,/b,

(6.17 b)

and using the change of variable o = kra/b,

(6.17~)

eq. (6.16) becomes

1;

dr, rl JN(Wrl) R(r,) = (ro/~)j-(~+l)eJ~baR(OrO/ICl). (6.18)

The definition (6.5) of ,u was used. Comparison of eqs. (6.18) and (4.2) establishes the correspondances (6.19a) (6.19b) Then, by eq. (6.15) the laser mode patterns are chosen from among (6.20) Eq. (6.19b) shows that the diffraction loss per bounce is precisely lNjJa function of space-bandwidth product c. From results (4.11), the minimum diffraction loss is At,o, occurring for the lowest-order mode Uo,o(r,6). Fig. 4.3 shows the variation of lo,o with c. For values of c 2 10, approximation (5.7a) for @ N , nshows (i) that the laser mode patterns become gaussian functions modified by Laguerre polynomials. The spot size for the lowest-order mode, U o , o ( ~ o),, is then (on the basis of l/e attenuation, as discussed) 2 (bA/n)*.

(6.21)

Eq. (6.17a) was used. The reader might want to compare diffraction loss in the two cases of square and circular endplates. This can be done by comparing Figs. 2.6 and 4.3 for corresponding values of (m, n, c) and (N, n, c ) , respectively. 6.2.2. Cortfocal Easer with non-identical ed#Eates Again referring to Fig. 6.3, we now assume the mirrors to be of dif-

WII.

5 61

367

APPLICATIONS

ferent lateral extent. The cases where both mirrors are rectangular, and where both are circular, will be considered. Since the two endplates are no longer identical, it is not reasonable to use an equilibrium condition (0.4) of proportional mode patterns over the two mirrors. Instead, it is reasonable to expect the same mirror to maintain proportional mode patterns for any two successive bounces of the light wave. Tracing the wave from M, to M, and back to M,, using diffraction law (0.1) at each step, we establish the equilibrium requirement dY, U'")(X,, rs) K(x,+x,; Yl+Y,)

U(m+e)(% r1) = P - q l M 1+a

= 4 U ( m )(XI, where the kernel K obeys

(0.22a)

y,),

I n order to evaluate the integrals we must assume specific shapes for the mirrors. 6.2.2.1. Case of differing, rectangular endplates Let M, and M, be rectangular, with sides 2x0, 2y0, and 2Z0, 2y0, respectively. Then eq. (6.22b) may be analytically evaluated, (0.23) Define geometric-mesa bandwidths

sz, = @/b)

lz, = (W) (rJo)4

(@0)+3

(6.24a)

geometric-mean space parameters =

(X$O)'J

aY

=

(YOYO)',

(6.24b)

and space-bandwidth products C, = a21z,,

C, = aYQ,.

(6.24~)

Substituting kernel (0.23) into eq. (6.22a), and using the new variables

.;

4 = go% . = go%. '

r; = k0Yl . r; = koYk

go

= @O/XOP

ko

= fiO/YO)*

(6.24d)

358

OPTICAL SIGNALS AND PROLATE PUXCTIONS

[VXXI.

6

the equilibrium requirement becomes

= --a2

exp (2jkb)U(x;/go, x;/h,).

(6.25)

(Redundant superscript m has been dropped from the notation.) Upon comparing this result with identity (2.5), we can identify

U(x;/go,~2'0)

= Wm(Cz,

4 )~ n ( c v 9Y;)

(6.26a)

and -uz

exp ( 2 j k b ) = lm(c,)ln(cu)(-l)m+n.

(6.26b)

The factor (-l)m+narises out of use of eq. (2.5) with argument ( y + x ) on the left-hand side, corresponding to arguments (xi+.;) and (yify;) on the left-hand side of (6.26). The fact that * Yn(-Y)

(6.27)

= (-l)nYn(Y)

was also used. Since diffraction-loss parameter uz has resulted from two bounces of the light wave, the net diffraction loss per bounce is the square-root of u2, and (6.2613) establishes that 0

mn

= jm+n+le-1k b [1 , (c,

)3>,(C*)]k

(6.28)

Comparing this result with (6.1l b ) for identical, square endplates, we see identical phase dependencies, indicating that the same modes are excited in the two cases; and we observe the same diffraction loss per bounce when (6.29) c, = c, = c, 1.e. a, = a, = x o . ( I n geometrical terms, the two rectangular endplates and the square endplate should all have the same area.) From (6.26a), we have mode patterns U n r n ( ~ Y, ) = Yrn(Cz> go%) ~ n ( c u h,

o~)

(6.30a)

on mirror M,. By symmetry, the mode patterns on M, must be Nrnn(X9

Y)

==

Yrn(Cz, %/go)

~ n ( ~ ylho). u ,

(6.30b)

Comparing eqs. (6.30) with eq. (6.11a) for the mode patterns for identical endplates, we note that the presence of unequal, rectangular end*

Obtained by matching real (imaginary) parts of defining eq. (2.2) with n even (odd).

V III,

5

61

369

APPLICATIONS

plates merely causes a stretching or contraction of the argument of the basic prolate dependence. 6.2.2.2. Case of differing, circular endplates Now suppose M, and M, to be circular, with radii yo and f o , respectively. Evaluation of eq. (6.22b) now produces a kernel 00

K(rl, el; r 2 , 0,) = 2n(fo/i20)2 2 exp [jn(n+O,--,)] ?l---8

x~

n [ ( ~ o / ~ (fo/ro)*r21. o ) h

(6.31)

Expansion (6.14) and the identity

J-nW

=

(6.32)

(-1)"Jn(X)

were used. Parameter 0,is a geometric-mean bandwidth, 00

= ( k / b )(rOfO)f,

(6.33)

and function K , in (6.31) is defined at eq. (4.5a). Casting (6.22a) into polar coordinates, substituting in kernel (6.31), and trying a solution of the form (6.15), the radial part of the solution must obey

This looks very much like identity (4.52) for the circular prolate functions. To foster the comparison, we use new variables

wi = (Fo/ro)bi,

i = 1, 2,

(6.35)

and define geometric-mean radius a . =

(ro~o)+

(6.36a)

and space-bandwidth co = aoDo.

(6.36b)

Comparison with eq. ( 4 . 5 ~then ) allows us to identify -$ exp

[j (N71+2Rb)] with A N , n ( ~ O )

(6.37a)

and

R[(rO/fO)*wi] with

@ N , n ( ~ o rwi),

i = 1, 2.

(6.37b)

880

OPTICAL S I G N A L S A N D P R O L A T E F U N C T I O N S

[VIII, § 6

Therefore, attenuation at a single bounce is

and the mode patterns on mirror M, are (6.38b) (Notice that when the square-root of eq. (6.37a) was taken, to produce (6.38a), an extra factor of (-l)a was picked up, From defining eq. of the eigenvalue always has a (4.2) for @N,n(r),the square-root factor (- 1)" attached.) Comparison with results (6.19b), (6.20) for identical, circular endplates shows that (a) the phase change per bounce is the same, indicating that the same modes will occur in each case; (b) the diffractionloss per bounce is the same when the geometric-mean radius equals the common radius; and (c), the mode patterns (6.38b) are simply a scaled version of modes (6.20). By symmetry with mode patterns (6.38b), the mode patterns on M, must be (6.38~) 4 7 , 0) = @N,n"~O/~O)+rl exp

we).

6.2.3. General laser geometry We now set out to show that, because of the Fraunhofer-Fresnel effect (6.2), even for the most general laser geometry the mode patterns must be proportional to prolate functions. The proviso is that the mirror outlines be either rectangular or circular. A general resonator geometry is shown in Fig. 6.4. The endplate surfaces are generally irregular and non-identical. Reference planes P, and P, are drawn perpendicular to the laser axis, and pass through the intersections of the endplates with the axis. W,(x,, y,) and Wa(xs,y a ) describe the deviations of the mirrors from the reference

---

OA

Fig. 6.4. A general endplate configuration.

~111,§

61

361

APPLICATIONS

planes. We suppose the deformations to be everywhere small, obeying

V,,

ilb.

(6.39)

For deformations of this size it is possible to express the amplitude u on mirror M, in terms of the amplitude U on MI,as

+(Yz--Y1)2/2b+ WI(x1,Yl) +w,(%,

Y2)Il.

(6.40)

Eq. (6.2) was used, and compensation was made for the fact that source and field points are now separated by a variable distance b--Wi(xi, Yi)-Wa(zaa ~ 2 ) . Suppose that the light wave has attained its equilibrium mode patterns on mirrors M, and M,, and that zc and U in (0.40) represent these patterns. (Eq. (6.40) must hold at each wave passage, including those that follow the establishment of equilibrium.) By defining a joint amplitude function

v = u exp Cjk[(x;+Y;)/2b+W,I},

(6.41)

eq. (6.40) may be simplified to

YA

u~(x2,

= uo(xar YA /JM,

dY1 v(x1, YJ exp (-j

(w) (x,x2+Y1Y2)} (6.42a)

where u,(x, y) = j (ilb)-l exp {-jk[b-- (xa+y2)/2b-W,(x, y ) ] } .

(6.42b)

Eq. (6.42a) shows that the mode pattern u(x2, ya)/uois a bandlimited function. We next use the fact that the prolate functions are complete sets of functions for bandlimited functions. Using eqs. (2.7) and (6.42), we can therefore represent 00

%(xz, ~

2 = ) uo(x2,

YJ 2

a m n ~ m ( x 2 Yn(Y2) )

(6.43)

m, n-0

if mirrors M, and M, are rectangular. This proportionality to prolate functions is what we set out to prove. Coefficients umnare characteristic of the diffraction loss and phase shift for each of the modes y m y nexcited. Use of equilibrium condition (6.22a) would establish the permissable values of am,,.

382

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

,VIII,

5

8

By analogous arguments, the mode patterns U on mirror ;MIare also a superposition of functions ym(xl) yn(yl). If M, and M2are circular, use of polar coordinates and identity (6.14) converts eq. (6.42a) into the form 00

UP^, 0 2 ) = N

c

o ( r 2 , ~ N=-w

(-jtNexp

(jNe2)Jr0drl rl 0

~ N ( k r l r z / b~ )

(rl),

(6.44a) where

W ( r )= J

r2n

dB V ( r , 0) exp (-jN0).

(6.44b)

0

The integral in eq. (6.44a) may be regarded as a function whose Nthorder Hankel transform cuts off. Then, the prolate series (4.7) may be used to represent the integral, and c o r n ~ ( 7 2 ,0 2 )

= uo(rz,

4)N=-w 2 n-02 b N , n @ N , n ( r 2 )

~ X P(jN02).

(6.45)

Coefficients bN,n are characteristic of the diffraction loss and phase shift for each of the modes @ N , n ( rexp ) (jNO) excited. By symmetry, the mode patterns U on mirror M, are also of the form (6.45). 6.2.4. Discussion The theory developed in €j2 and 9 4 has been based on the use of two complementary coordinates, one having units of space ( x or r ) and the other with units of frequency ( w ) . The maximum value of each space-frequency pair forms the space-bandwidth product c. For these reasons we have found it expedient to form frequency-coordinates w, = kx/r,,

(02 =

ky/r,

(6.46)

from either the pupil or image plane coordinates, in each of the laser mode problems considered above. The maximum values of w1 and w2 combined with the maximum values of the space coordinates are then used to form the two (one for each dimension) characteristic spaccbandwidth products of the laser geometry at hand. This general technique, embodying transformation (6.46), will be used in all of the prolate applications that arc based on the Fraunhofer-Fresnel laws (6.1) or (6.2). The space-bandwidth product c for the various laser geometries has

VIII, 5

6;

APPLICATIONS

363

been seen to uniquely determine the diffraction losses, i.e. the optical efficiency, of the various geometries. The bandwidth part of c has ordinarily been precisely the optical cut-off frequency for coherent radiation. The interesting exception to this rule was provided by eqs. (6.24a) and (6.33), where the bandwidth was the geometric mean between the bandwidths of the two endplates. Finally, we have seen that the general laser geometry with rectangular or circular endplates has mode patterns which must be a superposition of prolate functions (times another function). This follows from the fact that each mode pattern must be bandlimited, and the prolate functions are complete for describing bandlimited functions. 6.3.PROBLEMS O F OPTIMAL “CONCENTRATIOS” : O F ESCIRCLED ENERGY, POINT AMPLITUDE VARIABILITY, AND E R R O R O F RESTORATION O R SMOOTKIKG

In this section we apply the theorems (2.17) and (4.17) on functional concentration to some problems in systems optics. 6.3.1. Application to apodization theory: Maximizing the enclosed energy

A figure of merit that is commonly used in the design and evaluation of lens systems is “encircled energy” (see, e.g., JACQUINOT and DOSSIER [1964]). Observing Fig. 6.1, suppose a circle of radius rm t o be drawn in a receiving plane z, = constant, with its center lying on the z,-axis. Then the amount of radiant energy falling within this circle, relative to the total amount in the plane, is a fraction

(6.47)

called the “encircled energy factor”. Polar coordinates (r, 0 ) have replaced rectangular coordinates (xz, y,) shown in the figure. By criterion (6.47). a good lens system is characterized by a large b at a required radius rm. For a figure of merit to be effectively used, its optimum value should be known. Thus, we seek the maximum value of &‘(rm)for each r m , over the class of optical systems. LANSRAUX and BOIVIN[1961] and BARAKAT [1962] have found approximate solutions to this problem. SLEPIAX[1965] found the analytic solution, shown next.

364

OPTICAL SIGNALS AND PROLATE FUNCTIONS

[VIII,

5 6

6.3.1.1. Derivation Field amplitude u is assumed to be formed in compliance with the Fraunhofer-Fresnel law (6.1), r

n

(6.48a) where a composite aperture function W has been defined as W(X1, y1; 22) = U(x1, Y l ) exp tik(2rS)-1(~2,+r3z21. (6.48b) According to eq. (6.48a), zc is a bandlimited function. Assuming a circular aperture A of radius r,, we transform to polar coordinates, using a reduced radial coordinate

in the aperture. Eq. (6.48a) becomes

(6.49b) with W(p, 4 ; z 2 ) = U(p, 4) exp ( j w 2 / 2 k ) .

(6.49~)

Using the bandlimited nature of zc, we may expand it as

u(r, e)

m

=

m

2 2:

N--00

aN,n@N,n(r)

exp

(jm),c = yrnpo.

(6.50)

n-O

(See the derivation of eq. (6.45).) Substitution of series (6.50) into eq. (6.47) produces the simple result (6.51)

The two orthogonality relations (4.3a), (4.6) were used. At this point we seek the C Z ~that , ~ maximize 8 in (6.61). But, this is precisely the problem of 5 4.10, with solution - 0 for N , n # 0, ao,o# 0. Thus, the maximum possible value of 8 obeys aN,n

B ( r m ) = i20.0(4,

c = YrnPOl

(6.62)

(6.63a)

~111,

8 61

APPLICATIONS

386

and this is attained when the amplitude response is N r , 0) = ~o,o@o,o(~),

(6.63b)

a purely radial function. But, is this solution physically attainable; does a pupil function U exist whose response is given by (6.63b)? Substituting the solution (6.63b) into eq. (6.49b), we see that function W must not be a function of 8,. Performing the +integration, (6.64a) By the reciprocating property of the Hankel transform, rm

and by mathematical identity (4.4), Wb;

2.2)

= U ( f )exp c i Z z P 2 / 2 4 = @o,o(f~m/fo).

(6.66)

(The constant multiplier of @o,o may be ignored.) 6.3.1.2. Discussion Eq. (6.55) states that the required pupil

U(P>= @ o , o b y m / ~ o )~ X P( - j z 2 p z / W

(6.66)

maximizes 8 ( r m )in the arbitrary receiving plane position 2,. In the gauss plane z, = 0, U takes the simple form @o,o. Note that because @o,o is everywhere finite, U can be normalized to represent a passive pupil coating. Therefore, the solution (6.53a, b) to the problem is physically realizable. Once more we see that the figure of merit for an optical geometry, 8 given by (6.63a), is a direct function of the space-bandwidth product. Fig. 4.3 shows the dependence of Ao,o upon c. Note the intuitive results that d increases with optical cut-off frequency po (for fixed rm);and, symmetrically, 8 increases with Y, for fixed po. 6.3.1.3. Case of rectangular aperture

If the aperture is rectangular, and if the “enclosed” energy within a rectangular region in the receiving plane is t o be maximized, the results are highly analogous to those above. Now eq. (6.47) has rectangular regions of integration, and the function %(%a, y,,) substituted into (6.47)

366

[VIII,

OPTICAL SIGNALS A N D PROLATE PUNCTIONS

5 6

is a double expansion of linear prolate functions, as in cq. (6.43). The result is a maximum

which occurs for a pupil function

Figs. 2.5, 2.6 show the variation with c of A, and yo(x). I n the case of a slit pupil and a one-dimensional energy interval in the receiving plane, the solutions are single components of eqs. (6.57). 6.3.1.4. Kote on energy concentration in the confocal laser

It is interesting to compare the results (6.11) for the modes and diffraction-efficiency for a confocal, rectangular laser geometry with results (6.57) for the maximum concentration of energy within a rectangular region. For the TEMo,o mode, the results coincide. Therefore, the confocal geometry achieves the minimum possible diffraction loss over all possible endplate separations. 6.3.2. Variability of the point amplitude response and other bandlimited functions For simplicity, let us consider one-dimensional, bandlimited functions. For example, eq. (6.1) in one dimension has amplitude u obey 8.

dB U ( B )elSzr

u(x2)=

(6.58a)

-80

in the gauss plane, where

B is the reduced B = kX1/rG.

coordinate (6.58b)

The ability of function U ( X ) to vary within a fixed interval u a+2x0 may be measured by quantities Vn(X0)

=

s,

a+2+0

dx Iu"(x)lPIS_m_ dx Iun(x)I2,n

= 0, 1, 2 , .

. ..

5 x 5. (6.59)

Each n designates a derivative order. Notice that v0 measures the energy concentration within the interval; while v1 measures the relative amount of sharpness that occurs within the interval. Now, from eq. (6.58a) the nth derivative of u is still a bandlimited

VIII,

5

61

367

APPLICATIONS

Fig. 6.6. A function u ( x ) with high variability v1 and v s on the finite interval.

function (the integration limits remain fixed). Therefore, the results of 5 2.10 may be used, according to which each

v* 5 I . , ( C ) ,

c

= XoQ,

(6.60)

where l2 is the cut-off frequency for the function u. All the variations are bounded by the same number! In its dependence on the product xof2, variation v, complies with intuition: e.g., since I., increases with f2 the results (6.60) state that the higher the cut-off frequency the more variable the function can be. In particular, let us examine the variability between sampling points. In this case, 2x0 = nil2 so that c = in, (6.16) a fixed number. We have therefore established the fact that, between its sampling points, a bandlimited function can vary no more than V,

= il,(*n) M 0.8.

(6.62)

6.3.3. Application to filtering theory: maximum Possible error of restora-

tion or smoothing over a finite interval Xoisy, bandlimited functions are commonly subjected to a linear filtering, in order to smooth out the noise or to restore hidden details. Calling the difference between a filtered signal and its ideal value the noise function n ( x ) , if n ( x ) is bandlimited its square, [n(x)I2,is also bandlimited. Therefore, the results of 5 2.10 apply, and as the result of a single filtering operation the squared error over a finite interval a 5 x 5 a+2x0 is related to the total squared error through

308

CPTICAL SIGNALS AND PROLATE FUNCTIONS

[ ~ I L5 6

What is usually of more interest is the mean of the squared error over many such filtering operations. Then, by (6.63) the mean-squared error obeys (4) W ( E 2 ) . (6.64)

s

Now the quantity (e2) is normally assumed known, as in the theory of Wiener filtering. Result (6.64) therefore permits knowledge of an upper bound to the mean-squared error over any fin-ite region of the filtered signal, without the need for extra knowledge of the statistics of signal and noise. Rather, it must be known that the noise is bandlimited. 6.4. THE EXTRAPOLATION O F IMAGE DATA

The signals that emenate from optical instruments are ordinarily bandlimited, the consequence of a finite aperture in such instruments. It has been known for sometime now that bandlimited signals may be exactly interpolated by use of the sampling theorem (2.10a) (see, e.g., BARAKAT[1964], FRIEDEN [1966]). But in addition, we saw from eq. (2.10) that bandlimited signals may also be extrapolated. 6.4.1. Application to the incoherent image Applying this principle to the case of an incoherently radiating, onedimensional object scene, its image i ( x ) obeys

i ( ~ )= l h all0

N

2 A;’

N+al n-0

Lo 5

yn(x)

dx’ i(~’)y n ( ~ ’ ) ,c

= x,Q.

(6.65)

Notice that the right-hand side of this formula presumes knowledge of a piece (any piece) of the image. The mathematical operations on the right-hand side serve to predict what the image would be everywhere, even at points beyond the edge of the image piece. This extrapolation of the image reminds us of the well-known effect in holography, whereby a piece of the hologram is sufficient to recreate the image scene. However, in holography the object radiates coherently and the amplitude and phase are preserved everywhere on the hologram, which at least suggests the plausibility of holographic extrapolation. By contrast, with our assumption of an incoherently radiating object scene information about phase is totally lacking in the image +). Despite this lack of information, eq. (6.65) states that extrapolation is still possible, an interesting result. Naturally, the inevitable presence of noise in any measured image

VIII,

8

61

369

APPLICATIONS

rextrap.

+ observed

,

I

extrap.

7

I

-L

L

Fig. 6.6. Illustrating the growth of error in the extrapolationas distance from the image piece increases.

piece would preclude use of (6.66) for extrapolation much beyond the edge of the image piece. The remarkable dual orthogonality properties (2.3a) and (2.6) for yn(x) now permit us to compute the effect of such noise upon the extrapolation. 6.4.1.1. Effect of noise upon the extrapolation

With the measured image i(x) differing from the true image io(x)by the noise r t ( x ) , the use of (6.66) produces a noise error e ( x ) that obeys

(Note that, in any practical use of (6.65), N must be finite.) We now assume that extrapolation is attempted over a wide range of x-values, so that a measure of the acquired error will be N

CD

E~ =/-w

dx [e(x)I2=

2

W

m, n-O

A;'A;'a,,,unS

-m

(6.67a)

dx y,,,(x) y n ( x )

N

=

2 A,-":

(6.67b)

t%-0

by orthogonality (2.6). Then the expected square error attempts at extrapolation will be

over many

( E ~ )

N

(9)=

2 A,ya:) n-0

where, by (6.66)

(6.68a.)

370

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

5

6

Finally, if we make the assumption ( n ( x ) n(x’)) = u2 S ( x - x f )

(6.69)

of white, uncorrelated noise, orthogonality property (2.3a) yields (a:)

= u2izn,

(6.70a)

so that N

(E2)

= u2

2 A;?

(6.70b)

n=O

By inequalities (2.11), ( E ~ >increases monotonically with N ; and by eq. (2.12) (9) increases rapidly with N once N exceeds 2c/n. The relation of N to the required degree of extrapolation may be found as follows. Being a bandlimited function, i,(x) has degrees of freedom spaced a distance n/Qapart (the sampling interval, see eq. (2,lOa)).Therefore, to represent io(x) over a distance 2L would require 2LQ/n+1 = ( 2 4 ~(L/xo) ) 1 degrees of freedom. But formula (6.65) with N finite represents io(x)with N + 1 degree of freedom. Therefore, in effect

+

N = (2c/n)(2L/2x0) = ncrlt(2L/2xo).

(6.71)

Or, with fixed c, a value of N represents Itcrlt times the distance 2L over which i o ( x ) is to be adequately represented by series (6.65), in units of the distance 2x0 over which image data is acquired. We see from (6.71) that once 2L exceeds 2x,, N exceeds ncritand hence, by (2.12) and (6.70b), ( 9 ) starts to rapidly increase. Or, once the extrapolation interval exceeds the measurement interval, the error of extrapolation starts to sharply increase, a logical effect. Fig. 6.7 shows the variation of ( E ~ )with xo/L ( 3t i n Fig. 6.7), computed from eq. (6.70b) with u2 3 1. RUSHFORTH and HARRIS [1968] have shown the effects of various types of noise upon an extrapolation procedure that is related to (6.65). 6.4.2. Experimental determination of the optical transfer fmetion The point spread function s ( x ) , being a bandlimited function, may likewise be extrapolated using eq. (6.65). This suggests that the optical transfer function z(o)may be determined from observation of s ( x ) on a finite interval. Indeed, by taking the Fourier transform of eq. (6.65) (with i replaced by s), we find that m

T(O) =

2 j-nl.;~yn(oxo/Q) n-0

=.a

(6.72)

VIII,

5

61

APPLICATIONS

37 I

Fig. 6.7. Extrapolation error <E') versus enhancement factor 1. (After RUSHFORTH and HARRIS [1968] p. 644.)

(A multiplicative constant has been ignored.) Thus, by this formula it may be possible to base determination of T(O) only upon those values of s ( x ) which have a high signal-to-noise. The experimentally noisy values of s ( x ) would thereby automatically be extrapolated to from within the region of high signal-to-noise. 6.4.3. Bandlimited restoration from extrapolated image data

Suppose an incoherently radiating object scene o ( x ) to be imaged as i(x), by optics of point spread function s ( x ) . (For this case, s ( x ) is

372

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

g 6

simply ] U ~ ( Xwith ) ~ ~us given by eq. (6.1).) The problem we consider is that of restoring o ( x ) from observation of i ( x ) . In the presence of additive noise n ( x ) in the image, and presuming a linear theory to hold (see, e.g., O’NEILL [1963]), m

i(x) =

s_,

dx‘ o ( x ’ ) s ( x - x ’ ) + n ( x ) ;

(6.73a)

or, by taking the Fourier transform of both sides, I ( w ) = .(W) O ( u ) + N ( w ) .

(6.73b)

As a final item of notation, we denote the power spectra for object and noise as +,(w) and + N ( w ) , respectively, where +,(w)

= (l0(w)l2)

and

+N(o)

= (IN(w)j2).

(6.74)

Assuming, now, that all quantities i(x),s ( x ) , $ o ( w ) and $N(o) arc known (and without any further knowledge about o ( x ) ) , the optimum mean-square estimate of o ( x ) - call it 6 ( x ) - which can be obtained by a linear filtering of image i(x) is given as R

~ ( x= )

SD

dw 6 ( w ) elw,

where Q is the optical cut-off frequency (i.e., and estimate

t=0

(6.75a) for all

101 >= Q), (6.76b)

This is a well-known result of Wiener filtering theory (see, e.g., HELSTROM [1967]), which holds when noise and signal are statistically independent. From the Iast equation, this method presumes knowledge of I f w ) , which ordinarily would presume knowledge of i ( x ) over its entire, infinite domain. (That i ( x ) is infinitely extended is a consequence of the model (6.73a) for the imagery, where o ( x ’ ) is a generally extended function.) Suppose, however, that the experimenter can only measure i ( x ) over a limited interval 1x1 5 xo. This might occur if a portion of the image were accidentally torn away or obliterated in some fashion. Alternatively, it might be known that over certain portions of the image there is an exceedingly small signal-to-noise ratio, so that the experimenter would want to base the restoration on image data over a finite interval. Can the object scene be restored on this basis, and if so, would

VIII,

5 S]

373

APPLICATIONS

it intrinsically suffer additional error because of being based on less image data? We shall show that, provided the noise N(X) is bandlimited to interval IwI 5 f2, use of the finite data interval does not necessarily cause more expected error of restoration than by use of the infinite data interval. We note from eq. (6.73b) that, if ~ ( x is) bandlimited to IwI 5 f2, i ( x ) is also bandlimited. Therefore, its spectrumI(w) may be computed from knowledge of i ( x ) over 1x1 5 x, alone, via eq. (6.721. Substitution of (6.72) into formula (6.75) then produces the restoration, W

rn

(6.76a) where coefficients (6.76b) depend only upon the image piece, and weighting function (6.76~) In the absence of truncation of series (6.76a), restoration (6.76a) is exactly equivalent to the result (6.76a, b) which uses image data on the infinite interval. Both of these restorations would incur the same meansquare error over the extent of the object scene. 6.4.3.1. Effect of truncation on restoration method

To permit an analytic approach, assume q(w) in (6.76a) is a slowly varying function of w. Then, by use of prolate identity (2.2), (6.76a) becomes - w

.*(XI -

= (2nQ/xo)’ q(w)

2 L’in

~n(x),

(6.77)

n-0

where q(w)is an average value of q on the finite interval. With truncation of (6.77) at a finite value N of N, the data extrapolation is imperfect and (6.77) departs from restoration (6.75) for data on the infinite interval. It is natural to expect this error to be dependent upon the extent of x over which restoration 6 ( x ) is to be produced: so that we shall compare two cases, (a) where 1x1 x, (restoration interval equals data interval), and (b) where 1x1 5 00. Case (b) will evidently have the greater error due to truncation.

374

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

f 6

Calling the truncation of (6.77) ~ N ( x ) ,the departure of C?N(X) from the ideal restoration (6.75) is given by the departure of 6 ~ ( x from ) the infinite series (6.77), or -

m

e ( x ) = (2nQ/xo)' q ( 0 )

2

L 1 i n wn(x)-

(6.78)

n=N+1

By use of the two orthogonality relations (2.3a), (2.6), we determine the two errors of restoration

Evidentally, with fixed N , error (6.78b) is the more serious, because of the effect An + 0 once n > nCrit,given by (2.12). Also note that each error ( E ; ) , ( E ~ )can be made arbitrarily small througn selection of N since, by (6.77) and orthogonality conditions (2.3a), (2.6), (6.79) and (6.80) fixed numbers. 6.4.3.2. Numerical simulation The theory in this section has been developed by RINO[1969]. This author also tested the theory in a numerical simulation, shown in Fig. 6.8. For this study, the object is the sum of two gaussian functions, the image is formed as if the optics are diffraction-limited, and the two power spectra are presumed constant in eq. (6.76~).The restoring formula (6.76a) is used with 9 terms. The results are interesting, for the restoration is effectively extrapolated about 3 sampling points beyond each edge of the image-data interval. (The optical cut-off SZ was 8, so that the sampling point distance is in on the X-scales shown.) Counting

VIII,

5

61

376

APPLICATIONS

-

OBJECT

1.60*

0.80*

0.00

.

.

1

X

1

RESTORATION

- 0 . 4 0 L

-

.

-4.00 -2.67 -1.33

.

. . .

0.00

1.33

.

. . 2.67 4.00

X

Fig. 6.8. Example of a restoration based on image information on a finite interval. (After RINO[I9691 p. 661.1

the number of sampling points within the image-data interval, we see about a doubling of sampling points caused by the extrapolation, and this is despite the presence of considerable noise in the image. 6.5. EXTRAPOLATION BEYOND THE BANDWIDTH

I t was noted that, with no /&her information about the object than its image i ( x ) , the point spread function s ( x ) , and power spectra $, dN,the optimum mean-square estimate of o ( x ) is given by a balzdlimited restoration (6.75). This is strictly true when o and n are independent, gaussian random variables at each x .

376

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIXI,

8 6

The bandlimited nature of (6.75) is basically due to the existance of a sharp cutoff in the optical transfer function, for IwI 2 Q.

z(w)= 0

(6.81)

Combining eqs. (6.80) and (6.73b), the image seems to contain information about the object only within the optical bandpass 101 5 9. But, we can ask (at this point rhetorically) whether it might be possible to infer values of O(w) outside the bandpass from the behavior of the estimate 6 (w) inside the bandpass. Indeed, from extrapolating eq. (2.10), this would be possible were 0 (w) a bandlimited function. Accordingly, we presume the spatial object o ( x ) to have finite extent, or o ( x ) = 0 for 1x1 2 a. The spectrum O(w) is thereby bandlimited (or “space-limited’’ by the choice of units), and eq. (2.10) may be used to extend O(w) outside the bandwidth, W

O(W) = all0

2 A;’y,,(o)/ n-0

n

-n

do’ O(w’) y n ( w ‘ ) , c = &.

(6.82a)

Since this formula produces O(w)at aZ2 w,its Fourier transform may be taken, to form the exact restoration

/

W

2 j-nA;,-pyn(x9/a)

~ ( x= ) IzlSo

n-0

n

-n

do‘ O(w’) Y,,(w’).

(6.82b)

Prolate identity (2.4) was used. This result would be exact if 0 (0) were known with perfect accuracy over the optical bandpass region 101 5 9.However, the best we can do is to estimate O(o)on this interval via eq. (6.75b). Therefore, we define a restoration 00

6(x) = lzlsz

2 j-”A?y,(xS/a) n-0

n

s_*

do‘ 6(w‘)y,(w’)

(6.82~)

with 8(w ) given by (6.75b). Restoration (6.82~)has the desired property of approaching perfection, and arbitrarily exceeding the optical bandwidth, as the image noise approaches zero. 6.5.1. Effect of noise and truncation

In any practical use of eq. (6.82c), the series must be truncated at a value N of n. Calling the truncated (6.82~)~ N ( x ) and , the truncated (6.8213) O N ( X ) , the total error due to truncation and noise will be

e(x) = O ( X ) - ~ N ( X )

= [o(x)-o~(x)]+[oiv(x)-o^~(x)]

(6.83a)

VIXI,

5

61

377

APPLICATIONS

or

(6.83b)

(6.83~) (6.83d) Forming the total m.s. error over the object extent <E2>bt

(6.84a)

q a d x (le(x)I2>,

the use of orthogonality relation (2.3a) and eq. (6.83b) produce m

C,,

= xOPo6,,

m=O

(6.94b) and similarly for U M ( y ;y ) . Solution (6.94a) is essentially the product of two one-dimensional solutions (6.92a). However, we see that whereas solution (6.92a) is purely real, indicating the need for only zero and x phase shifts in the pupil, solution (6.94a) is generally complex wherever either z2 # 0, X # 0, or Y # 0. In this case, a continuum of phase shift is required. 6.6.3. Generally circular field Given a two-dimensional, finite object

~ ( r0), = 0 for r

> $yo,

(6.95a)

we now want a response function u ( r )obeying

u ( r ) = C d(r-R)/r,

for jr-R]

5 r,, R 2 0,

(6.95b)

with imagery at equal conjugates. Eq. (6.95b) describes a ringlike impulse response, for R # 0. Suppose, in addition, that the receiving plane is to be a general distance z2 from the gauss plane. Assuming that the solution is a pupil V(p) having radial symmetry, with coordinate p = (@2+y2))a, eq. (6.1) becomes

The combination of eqs. (6.95b) and (6.96) is equivalent to problem (4.21), which had the solution (4.25a). Therefore, the required pupil is M

V ~ P= )2 (-l)mA&@o,m(W

@ o , m b ~ o / ~ o )~

X P{-j (z2/2k)p2}, (6.97)

m=O

c = Yap,. This pupil is purely real, for imagery in the gauss plane. Imagery outside the gauss plane would require a continuum of phase change in the pupil.

VIII,

4 61

383

APPLICATIONS

6.6.4. Ultimate response from a lensless aperture With these (ostensibly) encouraging results, we question whether the aperture really needs a geometrical focussing device (i.e., the presence of a finite rG) to produce the delta-function imagery. Accordingly, we now use eq. (6.2) in combination with requirement (6.90). Using the transformed coordinate B = kx,/L, (6.98a) we want (in one dimension) a pupil W(B)that satisfies

I-, Bc.

Jk2’/2L

d/? [W(/?)ejLb“2k] e-jbz = C d(x), 1x1 5 x,.

(6.98b)

Comparison with problem (2.21), and use of solution (2.27),shows that pupil w M ( B ) = u,(B)e-jLbzlzk (6.99a) produces an amplitude response

c 6M (x)e-Jkz’/2L2 c d ( x )

for 1x1 5 x,.

(6.99b)

The last identity results from the “sifting” property of d(x). Hence, even in the absence of a lens, d,(x)-type imagery can be produced in a required receiving plane. Generalization of this result to two dimensions is easily obtained, and we shall not show it here. 6.6.5. Ultimate depth of focus The preceding problems have been concerned with d,(x)-type imagery in a direction that is transverse to the optical axis. Can the amplitude response along the optical axis, call it u(z,), likewise be made proportional to dM(z2)?If so, an arbitrarily narrow depth of focus (within a finite longitudinal field Jz] 5 2,) would be attained. Using eq. (6.1) with transformed coordinates

/? = kx,/rG,

y = kx,/r,,

(6.1OOa)

we now want a pupil L(B, y ) that satisfies 70

80

dB dy L(B,y ) exp {j ( z 2 / 2 k )(B2++r2)} = C W2), lz21 5 2 0 .

1%I-&

(6.100b)

By requiring separation of the pupil as L(B,7)= L(B)M ( y ) ,

(6.101)

384

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

fc

6

response

L

Fig. 6.11. Geometry for attaining an arbitrarilynarrow focal depth, within a required field depth z,,.

problem (6.100b) separates into two problems of the type 8,

dj3 L(/3)ej(z*a/2k)Ba = C 6(z,),

Iz,~

5 2,.

(6.102)

s-Po

We may transform the integral in (6.102) into a finite Fourier transform, by requiring that L(j3) be of the form (6.103a) and by the change of variable

B’

=

(2k)-1(/3-4bE).

(6.103b)

Requirement (6.102) becomes

s

PO2/4k

dj3’ F(8’) elear = ( C / 2 k )6 ( z 2 ) exp (-j~~Z2/4k}. (6.104)

-Poa14k

This is the basic problem that we have repeatedly solved above. Function F(j3‘) is therefore UM(j3’),provided we substitute j3%/4kfor Po, and a field-of-view parameter z, for x,. After re-use of (6.103a), we find that M

L(B)= L M ( j 3 ) = IBI

2

m(even)=O

(-l)fm~;%n(0) Ym[zo(l--2B2/B:)13

c1 = pfzo14k.

(6.106a)

The other pupil component M ( y ) obeys an analogous expression, with C2

= y:Zo/4k.

(6.10Sb)

Use of (6.101) then determines the net pupil. We note from eq. (6.104) that each coordinate direction gives rise to an amplitude response proportional to 6,(za). The net pupil L(B, y ) must therefore give rise to a response that goes as dM(,Q2. This is still, however, an arbitrarily sharp function. The problem of normalization for 6M(zz)pis also not serious, because for finite M this function is everywhere finite.

VIII,

8 61

385

APPLICATIONS

Space-bandwidth parameters cl, c2 may be related to the classical focal depth d = 4Af#2, where f# is the system flnumber. Supposing, for the moment, that Po = yo, and noting that

Po = (k/rG)xpupil halfwidth

= n/Af#,

(6.106)

from the definitions (6.106) of cl, c2 c1 = c2 = +n(z,/d).

(6.107)

Thus, c1 and c2 now measure the longitudinal field extent z,, in units of classical focal depth d, over which response u(z2)oc &(z~)~. The behavior of pupil component &(/I) is discussed below. 6.6.6. Behavior of the solutions

In this section we shall examine the functional behavior of pupils and LM(P),and the amplitude response 6&). The method of evaluation is empirical, based on their evaluation for various M and c upon the electronic computer. Numerical formulae (3.1)-(3.4) were used, with the tabulation of coefficients @' of STUCKEYand LAYTON "641. We note that most of the solutionspresented above depend upon these functions.

U&)

\ W

08-

" JUncoated

386

OPTICAL SIGNALS AND PROLATE PUNCTIOXS

6.6.6.1. Point amplitude

[VIII,

8

6

~ M ( x )

For purposes of analyzing this function, it is convenient to examine the normalized quantity AM(%)

=dM(X)/dM(O).

Fig. 6.12 shows A M ( z )for the case c = 6.26, with M show the positive half since, by (6.27) and (6.92b), AM(-%) = AM(%).

(6.108) = 40.

We only (6.109)

From Fig. 6.12 and other plots of AM(%)we empirically conclude that: (1) There are exactly M zeros over the entire field 1x1 5 x,. (2) Let A M represent the total width of the central core. AM thereby measures the extent of super-resolution achieved by A M @ ) .Defining the superresolution d~ as the ratio of AM to the central core width 2n//?,for uncoated optics, i.e. d~ AM/?o/~z, (6.110) we empirically find that for M 2 20 6~

M

3cllCfn.

(6.111)

This simple relation shows that the core width in A M @ )m a y be made arbitrarily narrow by increasing truncation index M . It has been known for some time (see WILKINS[1950]) that core width may be made arbitrarily narrow, but normally at the expense of very large sidelobes immediately adjacent to the core, a situation which we see not to be true for A,(%) in Fig. 6.12. Fig. 6.12 shows core width reduction by d,, = 0.15, which is seen by comparing the solid curve A,,(%) with the dashed curve sinc (pox) for ordinary, diffraction-limited optics. (3) The functional form of A M ( % is ) quite close to that of an ordinary sinc function (dotted in Fig. 6.12) sine [ ( B O l d M b l ,

(6.112)

where A M ( z )is significantly large (in the figure, where IxI/x, 5 0.3). This means that pupil UM(/?) acts to the image as if i t synthetically magnifies the aperture to which it is applied by a factor l / d ~ which , is arbitrarily large according to eq. (6.111) ! We shallsee later that U M ( ~ ) extrapolates outwards any bandlimited pupil function to which UM is applied.

VIII,

5

61

387

APPLICATIONS

-0.4‘ 0

I

0.4 0.6 0.8 RELATIVE COORDINATE 8/8.

0.2

1.0

Fig. 0.13. The “extrapolating pupil” P,,(/?). Note the transition from a uniform cosine curve a t low ,5’/8,(nearthe pupil center) into an increasingly oscillatory function at the margin. (After FRIEDEN [I9891 p. 802.)

6.6.6.2. Pupil function U M ( ~ )

Figure 6.13 shows the normalized pupil PM(B) = UM(B)/UM(BO)

(6.113)

for the case M = 40, c = 6.25. This pupil causes point amplitude A,,(x) previously shown. Fig. 6.13 only shows the positive half b 5 B, of the pupil since, by (6.27) and (6.92a),

UIM(-B)

= UM(B).

(6.114)

From the behavior of Pd0(B)and other plots P M ( ~we) conclude that: (1) There are exactly M zeros to PM over the entire pupil. Normalization (6.113) allows PM@) to physically represent a passive coating applied to the optical pupil. This coating function must therefore have alternating zones of zero and n phase shift, +M of each in total. From the standpoint of fabrication, this would not be nearly as bad as trying to make pupil UM@;X )which, from (6.94b), would require a continuzm of phase across its extent. (2) For a central region of I‘M(@),to a very good approximation PM(B)

= (-l)fMcos

(BxOpM).

(6.115)

Approximation (6.115) holds, e.g., for values /I//?, $ 0. 3 in Fig. 6.13. Beyond this region PM (B) becomes increasingly oscillatory, and attains its maximum value at 181 = Po.

388

O P T I C A L SIGNALS A N D P R O L A T E F U N C T I O N S

PUPIL COORDINATE

[VIII,

5

6

p

Fig. 6.14. Pupil W,,(B) for narrowing the depth of focus by a required amount. (After FRIEDEN [1970].)

6.6.6.3. Pupil function LM(/?)

Figure 6.14 shows the normalized pupil WM(j3)

=LM(P)/LM(Bo)r

(6.116)

for the case M = 20, c = 1. The positive axis j3 5 Po alone is shown because, by (6.27) and (6.105a),

LM(-p)

=LM(B).

(6.117)

This pupil would cause a longitudinal amplitude response C 6 (zJ. We have found that (1) There are exactly 2M+1 zeros to W ~ ( j 3over ) the entire pupil, or, twice as many as for pupil I'M(/?). By definition (6.116), W M ( ~ ) may be physically realized as a passive pupil coating; and by the preceding the coating has alternating zones of zero and n phase shift, M + l of each in total. (2) For a central region of W M ( ( ~to) ,a very good approximation W M ( j 3 ) oc (-l)*MIBl cos [(Z0/2k6M) (j3"+j33].

(6.118)

Beyond this region the pupil becomes increasingly oscillatory, attaining its maximum value at IflI = Po. 6.6.7. Strehl flux ratio awd sensitivity to pupil wave ewoy

By the orthogonality of the prolate functions, it is possible to

VIII,

§ 61

389

APPLICATIONS

analytically relate the Strehl flux ratio, and the r.m.s. signal-to-noise ratio in the point amplitude response, to the power spectrum of random phase error in the pupil. The analyses show a rapid dropoff in the Strehl, and in signal-to-noise, once the attained super-resolution drops significantly below the Rayleigh resolution; unless an exceedingly small field of view x, is used (the order of a few Airy cores in extent). A t the extreme limit M = co,the image i s perfect b d it cannot be seen, a paradoxical situation. Curves of Strehl and signal-to-noise are shown in FRIEDEN[1969] and FRIEDEN [1970]; small-field applications to telescopy and microscopy are also discussed in the former work. 6.6.8. Real-time extrapolation of functions using P@il P M ( ~ ) We saw that the effect on the image of coating with I'M(/?) a clear, diffraction-limited pupil of finite extent 2p0, is to simulate the presence of a clear, diffraction-limited pupil of arbitrary extent ~/?,/SM (aside from total illumination in the image). In a sense, coating Pm@) extrapolates outward the unit pupil that pre-exists in the diffractionlimited case. What, then, would be the effect on the imagery if P M ( ~ ) were applied to a general pupil G(p) that pre-exists in the optical aperture? Would this pupil appear to be extrapolated outwards? We now show that the answer is yes, provided G ( p )is a piece of a bandlimited function. has been seen to satisfy (aside from a constant multiCoating PM(/?) plier) 80

eja@-@ ) = S ~ ( y - x ) , for Iy-xl dp P M ( ~

5 x,.

(6.119)

J-Po

Multiplying both sides by a general function g ( x ) , and integrating dx from -Qxo to gx, produces

The right-hand side may be denoted asgM(y), since S ~ ( y - x ) + S(y--x) as M -+ co according to (2.24) and (6.92b). The left-hand side may be cast as (6.121) where

390

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII,

f 6

This essentially completes the proof. If g ( x ) is a completely general function, (6.120) has

Lo 80

G ( B ) I ~= ~ ~g 'M ( y ) , for I Y I 5 +xo.

~B[PM(B)

(6.123)

T h i s equation i s the Fraunhofer law for creation of a n amplitude response by a pupil function PM(B)e(B).Hence, the general response g M ( y ) may be optically produced, over a limited field IyI 5 +xo: the required pupil is the composite pupil PM(,!~) c(/3).The real-time synthesis of general diffraction patterns is possible. But further, if g ( x ) has the special property that gM(y)

g ( x ) = 0 for 1x1 > 8 x o ,

(6.124)

i.e. its spectrum G(B) is a bandlimited function, then by (6.122) C(B) = G(B), and identity (6.123) becomes

Lo 10

dB IIPM(B) G ( B ) P b Y= gM(Y), IYI 5 ! P o .

(6.125)

Compare this with the Fraunhofer law (assuming it held true) for the infinite pupil G (/?)

s_,

00

dB G(B)elbY= d Y ) .

(6.126)

For M sufficiently large we hadgM(y) --f g ( y ) . The asymptotic equality of eqs. (6.125) and (6.126) then shows that the finitely extended pupil P M ( ~G(B) ) simzclates the Fraunhofer behavior of the indefinitely extended pupil G.* Hence, in this sense pupil coating PM(B)acts as an extrapolator for the class of bandlimited functions G ( / ? ) . 6.6.8.1. Numerical simulation

As an example of amplitude synthesis, suppose we wish to optically produce a rectangular response

The infinite pupil G,(B) = (x0/4n)sinc ($Bx0)

(6.128)

* The implication is that both the pupil and the theary are synthetically extended outwards by the effect (6.126). This effect, itself based on Fraunhofer theory, must break down for M sufficiently large.

VIII,

5

61

APPLICATIONS

391

would suffice (if the Fraunhofer law held for infinite pupils), but any optical pupil must be finite, obeying I/?I 5 Po. By eqs. (6.125), (6.126) we should simply coat the truncated pupil PM(/?) G,(/?) upon the aperture of a diffraction-limited optical system. Within a field of view 1x1 5 &xo the observed amplitude would approximate g,(x) with arbitrary precision.

Fig. 6.16. Numerical simulation of the synthesis of a rectangular amplitude response. [1970].) Pupil coating P,,(P)G,(B) causes the response g,,(z) shown. (After FRIEDEN

The composite pupil P,,(P) G,(/?) is shown in Fig. 6.15, along with its finite Fourier transform g,,(x), 1x1 5 &x0. Parameters /lo = x, = 1 were used. We see that g,,(x) does approximate the required rectangle fairly well. For comparison, pupil Gr(/3) and its finite Fourier transform & ( x ) are also shown. Of course, t , ( x ) looks nothing like the required g,(x), since G,(P) is truncated by the pupil edge before it can fall appreciably below unity. 6.6.9. A thought-experiment: picture extrapolatiolz by use of a coherent

processor We saw in 3 6.4 that an incoherent image is capable of extrapolation a posteriori, i.e., by manipulation of the already-formed image. We now ask, in view of the remarkable properties of “extrapolating pupil” PM(/?), whether such image extrapolation could be done in real-time. In Fig. 6.16 we show optical apparatus which is meant to do just this, at least in theory. The apparatus is the familiar coherent processor due to MARECHALand CROCE [1953], with a slight alteration: letzs M

is larger in aperture than lens L. As is usual with this setup, lenses L and M are diffraction-limited and of focal length f.All physical entities are spaced a focal length f apart. Coherent light illuminates the input plane A, where the amplitude distribution i ( x ) P M ( x ) ,1x1 5 xo, is placed. What will the output amplitude distribution be in plane

A'? According t o the transfer theorem, the spectrum I ( w ) of i ( x ) cuts off at the optical cutoff frequency 52. Hence, i ( x ) for 1x1 5 x,, is a piece of a bandlimited function. According to eqs. (6.124)-(6.126), the finite Fourier transform of i ( x ) P M ( x )is I M ( o ) , which is asymptotic with M to I ( w ) for all ( w ] 5 Q. By the Fraunhofer-Fresnel law (6.l ) , the amplitude distribution in the Fraunhofer plane F for the processor will be the finite Fourier transform of i(x) P M ( x )which , is I M ( w )for [wl 5 52 according to the preceding paragraph. Let us now mask the amplitude distribution I M ( o ) in plane F for values Iw[ > 52. Since lens M is one focal length away from the distribution I w f w ) , lens M sees the entire spectrum of the image I@). Therefore, by law (6.1) in its output plane an image i ~ ( x ) asymptotic to i ( x ) will be produced. But, since lens M is greater in aperture than lens L, & ( x ) is produced over a greater range of x than is the range of the input i ( x ) . In this manner, the image is extrapolated. In view of the rapid falloff with M of Strehl flux ratio and signal-tonoise in the amplitude response (see 3 6.6.7), this coherent extrapolator would be difficult to implement at high M . This means that only small extrapolations could be easily achieved with good quality. Quite aside from engineering problems, the degree to which Fraunhofer theory holds when M is large also has to be questioned (see FRIEDEN [1970]). Evidentally, this is a case where one good experiment is worth a thousand conjectures.

VXII, §

61

393

APPLICATIONS

6.6.10. A laser saperposition mode with an arbitrarily small spot size

Comparing eqs. (4.3b) and (6.20), we see that a certain superposition of laser modes Uo,n(7) produces an impulsive amplitude distribution over the entire endplate. Specifically, if each mode Vo,n(r) were weighted by an amount @o,n(0)/ilo,n, with n = 0, 1, . . ., M , then an amplitude distribution M

2: ActQo,n ( 0 ) @o,n (7) = dM ( r )/ r 5

( r )17

(6.129)

n-0

would occur on the endplate. As M -+ 03, the width of the central core in d ~ ( r approaches ) zero. Evidentally, from the standpoint of spot size the attainment in practice of the mode superposition (6.129) would be quite desirable. Work is already progressing on forcing a different superposition than (6.129) (AUSTON[l968]). 6.7. DEGREES O F FREEDOM I N THE IMAGE

A measure of the variability of a function on a finite interval is the number of Fourier components required to represent the function. If N components are required, the function is said to have N degrees of freedom (d.0.f.) on the interval. More generally, a function is said to have N d.0.f. if N independent pieces of information are needed to construct the function on the interval. This simplistic approach breaks down, of course, in the presence of noise, where the function cannot be exactly constructed from measured data. (This point is considered in some detail later on.) The image is a function that is bandlimited and noisy. This study concerns itself with determining the d.0.f. for the (coherent or incoherent) image. 6.7.1. By counting sampling points within the interval

It is traditional to regard the sampled image values in eq. (2.10a),

i(nz/Q)3 in

(6.130)

as the d.0.f. for the image. Parameter D is the cutoff frequency in the spectrum of the (coherent, or incoherent) image. On this basis, the number of d.0.f. within a finite length 2x0 of the image is 2xo/(n/sz)= s = Z(x,Q)/n = 2c/n,

(6.131)

a finite number. (The last equality is to define a space-bandwidth product c . ) TORALDO DI FRANCIA [1969] has called S the “Shannon number”.

394

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[VIII, !j

6

6.7.2. B y including contributions from sampling points outside the interval But, eq. (6.131) is a misleading result. Although there are truly a finite number S of sampling points on the interval 1x1 5 x,, the image i ( x ) on this interval cannot be perfectly created from its sampled values in on the interval alone. (Eq. (2.10a) shows that an infinity of sampling points are needed.) Thus, the image i(x) for 1x1 5 x, has more freedom to vary than as determined by S values in. Indeed, on this basis i(x) has an infinite number of d.0.f. governing its ability to vary on the finite interval. 6.7.3. A n “effective number” Nevertheless, formula (6.131) is often a good approximation to the actual (or, practicable) number of degrees of freedom. I t is simple to see why, from eq. (2.10a). Once In1 exceeds a critical value Qx,/n only sidelobes of the basis functions sinc (Qx-nn) contribute to i ( x ) at all points within 1x1 5 x,. Because of the l / x f a l l off of the sidelobes, values in for In1 > Qx,/n therefore contribute ever-decreasingly, with Inl,to values i ( x ) lying inside the interval. Twice the critical number is therefore a measure of the “effective number” of d.0.f. for i ( x ) within the interval. This is again, by (6.131), the Shannon number S. We have seen that, although in principle an image has an infinite number of d.0.f. on the finite interval, there exists a finite subgroup of them that “effectively” form the image on the finite interval. At this point, it is necessary to establish a measure of the error that results from ignoring all degrees of freedom beyond the Shannon number. To be realistic, the image should also be presumed to suffer from noise. Because of their convenient properties of orthogonality and completeness, we shall use the prolate functions to analyze the problem. 6.7.4. Prolate degrees of freedom

The following derivation is due to TORALDO DI FRANCIA [1969]. Suppose that a space-limited, coherent object obeying o ( x ) = 0 for 1x1

> x,

(6.132)

is being imaged by a diffraction-limited lens. The image obeys the familiar convolution theorem

dx’ o ( x ‘ ) u ( x - x ’ ) ,

with u(x) = sin Qx/nx.

(6.133)

~111,

f 61

396

APPLICATIONS

By the orthogonality and completeness properties (2.3) for the linear prolate functions, we may represent a general o ( x ) as m

o(x) = l46%

2 a,y,(x),

c

= x,Q.

(6.134)

-0

Then by (6.133) a3

i(x) =

20

2 aDJFa0dx‘ y,(x’) sin Q ( x - x ’ ) / n ( x - x ‘ ) .

(6.135)

9-0

Use of identity (2.5) then establishes

z: m

i(x)=

%&Y,(X)-

(6.136)

2-0

Let us compare eqs. (6.134) and (6.136). Delegating one d.0.f. to each y,(x) function in the expansions, we note that there is mathematically no loss in degrees of freedom from object to image. Both o ( x ) and i(x) have an infinity of d.0.f. On the other hand, each d.0.f. a,y,(x) of o ( x ) is modulated by A, in the image; and, by (2.12), the A, rapidly approach zero once p exceeds 2cln. Comparison with (6.131) shows that this is, once more, the Shannon number. This gives another interpretation of S. From two different viewpoints, then, an image has an infinity of d.o.f., of which a finite number contribute strongly to the image, and the rest weakly. Suppose, now, the measured image suffers from additive noise. One would expect the noise to more seriously degrade the weak d.0.f. than the strong ones. But also, as the noise decreases (by some mechanism) toward zero more and more of the weak d.0.f. should be discernable in the image. In the following analysis, we find that there is an optimum number of d.0.f. for a given level of noise in the image. This number is optimum in the sense of minimizing the effect of noise upon the image as constructed from series (6.136). It is natural to call this optimum number the “effective number of d.0.f. in the presence of noise”. In general, this effective number of d.0.f. is not the same as the Shannon number. 6.7.5. Effective degrees of freedom in the presence of noise

Let us suppose that o ( x ) suffers additive noise n ( x ) on the finite interval, m

(6.137)

396

OPTICAL SIGNALS A X D PROLATE FUNCTIONS

-

10

XO

@II,

§ 6

X

Fig.6.17.With noise in any reading i ( x ) , how many degrees of freedom has i ( n ) on the interval In1

5 x,?

Then, by the analysis (6.133)-(6.136) the observed image is 00

iobb) =

2 (av+~v)~vYv(4. v-0

(6.138)

If the series for iob(x)is truncated at a value P of 9, any acquired iob(x) will be in error because of the truncation, and because of the noise. The error e ( x ) obeys P

00

e(x) =

2 advwvW-

P-0

c

(av+%)b&l(x)

(6.139)

P

cn

=

c

P==O

a d v Y v ( X )- 2 % A V V P ( 4

v=P+1

(6.140)

v-0

In (6.140), the first sum is the truncation error, the second is the noise error. It is apparent that as P increases the first sum decreases while the second increases (in magnitude). We seek a value of P that will minimize e ( x ) in some fashion. Assuming all values 1x1 5 x, t o be of equal use to the experimenter, we form the more convenient mean-square error (6.141) From (6.140) and the orthogonality condition (2.3a), =

2 PPP+1

+ 2 (.:>A;. P

00

(E2>

A;

(6.142)

p=o

(ni)

Coefficients (a:) and may be usefully replaced by their values for flat, uncorrelated object and noise statistics. By eqs. (2.3a) and (2.91, (a:) = crt/Av and = &/Av, (6.143)

(ni)

VIII,

5 61

397

APPLICATIONS

where 4,cr; are the variances of object and noise statistics. Substitution of eqs. (6.143) into (6.142) produces (6.144)

the required result. W e now define the “effective mmber” of d.0.f. in the observed image as the valae of P that minimizes the mean-sqaare error ( E ~ ) . We see, from (6.144), that an increase in P decreases the truncation error (first sum) while increasing the noise error (second sum), indicating that an optimum P might exist. To find it, we imagine the discrete sums to be replaced by integrals; since the &, change rather smoothly with p this would seem a fairly good approximation. I t is then possible to differentiate (6.144) with respect to the (now) continuous variable P, setting the result equal to zero to find the extremum. Doing so yields the requirement on P aP/AP+l

= ao/aN

*

(6.145)

Thus, the signal-to-noise u0/aNand the space-bandwidth c (through A ) jointZy fix the effective d.o.f., a reasonable result. We note that, since P is a discrete variable, for prescribed ao/aNthe equality in (6.145) cannot generally be satisfied. P would then be the number that comes closest to satisfying the equality. An interesting situation to analyze is where signal equals noise. The right-hand side of (6.146) is then unity. For any c, the best approximation to unity for a ratio Ap/A,+, occurs for P = 0 (see Table 2.1). Thus, when signal equals noise there are no effective d.0.f. in the image. This seems a reasonable result. In the particular case of small c (small Shannon number), ratio iZP/Ap+, is well-approximated by (SLEPIAN[1966] eq. (4.6))

iZp/Ap+l

= c-2(P+ 1)-1(2P+1)2(2P+3)2.

(6.146)

This expression may be substituted into (6.145) to explicitly relate P to the signal-to-noise. The limit of large U,/ON produces a particularly simple solution for P, (6.147)

the last equality resulting from (6.131). The power-law dependencies are interesting. Note in particular that P -+ 03 as ao/aN-+ a,which is reasonable, and that P > S once uo/uNexceeds (12S/ns)c. Eq. (6.147) might be regarded as a rule of thumb relation for counting degrees of

398

[VIII,

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

f 6

TABLE 6.1. Effective degrees of freedom P for various signal/noise values P

r.e.p

r.e.S

0 1

1 0.0221 0.0004 0.0001 4 x 10-6 10-0

1 0.0221 0.0134 0.0134 0.0134 0.0134

3 4 10 15

ao/ON

1 10

50 100 500 1000

freedom in the presence of noise. Where c < 1, (6.147)is a good approximation t o P. However, for c > 1 we have found that the growth of P with ao/aN is faster than the power indicated. In Table 6.1 we show results for intermediate ratios aO/aN,for the case c = 2. Table 2.1 was used to satisfy requirement (6.145). For c = 2, S = 1 according to (6.131). By comparison P grows monotonically with a,/a~, as one would expect. In particular, once a0/aN= 1000, the effective number of d.0.f. is 15 times the Shannon number! Quantity r.e. is the relative error (+/< dx [ i ( x ) I 2 )over the image extent, and is computed by means of eq. (6.144) and the easily derived

es0

(6.148)

Subscripts P, S of r.e. indicate use of (6.144) with these respective indices. I t is apparent from Table 6.1 that the use of P determined by (6.145) does offer significantly lower relative error than does use of S. 6.7.6. Degrees of freedom intrinsic to the “extrapolating

pzcpil”

We observed in 5 6.6.6.2 that the extrapolating pupil U M ( / ~has ) precisely M zeros within the aperture. I t will now be shown that, in the absence of noise, each zero of UM cazlses one degree of freedom in the

coherent image. Suppose the object o ( x ) is limited in extent and represented by its degrees of freedom through expansion (6.134). The image will be formed according to the convolution (6.133), but now with an a m plitude response due to the pupil U M ( / ~which ), is M

~M(x= )

2 1;‘ n-0

Vn(0)

wn(x)

(6.149)

VIII,

8 61

399

APPLICATIONS

according to eq. (6.92b). Substitution of the object expansion (0.134) and amplitude response (6.149) into convolution (0.133) yields

i(x) =

m

M

a0

9-0

n-0

-a0

z a, L: Ailyn(.,S

b‘ yn(x‘) y,(x’)

(6.150)

provided we make the approximation M

M

L: K1w ~ ( oWn(x-x‘) )

L: ~ ’ ~ n ( x ’ )

=

n=-O

1~n(x).

(6.151)

n-0

In the limit M + 00 both sides of (6.151) approach the same function, so that eq. (6.150) is true for M sufficiently large. Use of orthogonality relation (2.3a) in (6.150) then shows that d(x-x’),

M

i(4 = 2: @,Y,(4,

(6.162)

9 4

the required result. Comparison of eqs. (6.134) and (6.162) shows that pupil U M creates exactly M object components in the image. Recalling from 5 6.6.6.2 we now that M also represents the number of zeros in the pupil PM(/?), see that each zero in PM engenders a new degree of freedom in the image. As the number of zeros in PM is (in theory) at the user’s discretion, we once more see that the image can have an infinity of d.0.f. (in the absence of noise). 6.8. EVALUATION O F WAVE-ABERRATIONS

Optical systems are often judged on their ability to form a narrow point-spread diffraction pattern s(x,, y , ) in a required receiving plane position z, (see Fig. 6.1). By scalar theory, s

= Izc12.

(6.163)

Point amplitude response zc(xa, y , ) is generally broadened by the existence of aberrations in the optical pupil. The implicit affect of aberrations upon zc is simple to formally exhibit, but rather difficult to explicitly evaluate. 6.8.1. Formal solution Using polar coordinates (since lens systems usually have circular pupils), eq. (6.1) becomes M(ra3

6,) = /;s,”dp1

p1 do1 ~

b l01), exp [jpira cos ( ~ S - ~ J I * (6.164)

400

OPTICAL S I G N A L S A N D P R O L A T E F U N C T I O N S

[VIII,

5

6

where the frequency coordinate P1

= krllrG

(6.155)

has been used. The aberrations enter through pupil function F , which for a uniformly transmitting aperture obeys F ( p , 6 ) = exp [jkW(p,

41.

(6.156)

Assuming the optics to have axial symmetry, function W must have the form of a polynomial formed by taking all possible powers and crossproducts of (6.157) &, p2 and x,p cos 8 . xG is the x-component of rG (see Fig. 6.1). The coefficients of this polynomial define the Seidel aberrations. The connection between zc and

the aberrations is, then, difficult to explicitly establish from eq. (6.154). 6.8.2. Past work by Zernike and Nijboer

ZERNIKE[1934] contributed to solving the problem by expanding W as a sum of polynomials in p times exponentials in 6, the polynomials being orthogonal over the circular pupil. This expansion requires a redefining of the aberrations in terms of the Seidel expansion. Assuming that these Zernike aberrations are small, NIJBOER [1942] showed how to analytically compute the affect upon zc of any small number of the aberrations. The intensity at the gaussian focus takes on a particularly simple form. Perhaps most importantly, the problem of “aberration balancing” was solved by use of these aberrations: the problem being to find what amounts of lower-order aberrations should be present to balance the affect upon the Strehl flux ratio of a fixed amount of one higher-order aberration. The main drawback to the use of Zernike aberrations is that they must be small, and few in number, to allow their explicit affect upon zc to be analytically known. Is there another set of aberrations which do not suffer this constraint? 6.8.3. A new approach to the problem

ITOH[1970] has essentially solved this problem, in the following manner. Instead of expanding the aberration function W in terms of an orthogonal set of functions (Zernike and Nijboer’s approach), expand

VIII,8 61

401

APPLICATIONS

the pupil function F itself in this manner. In particular, let F be a superposition of circular prolate functions, m

E(p, 0)

=

m

2 2 b N , n @ N , n ( p ) exp (jNe), N-0 n-0

c=

rmpo.

(6.158)

(Note the similarity to expansion (6.45).) Coefficients bN,* are new aberrations. They may be numerically related to the Seidel, or other, aberrations that define W as follows. By use of orthogonality relation (4.3a), eq. (6.158) may be inverted to

By using relation (6.156) and numerically evaluating the integrals in (6.169), the new aberrations bN,n are computed from the old. 6.8.3.1. Analytic dependence of amplitude response upon aberrations bN.73

By substituting pupil expansion (6.158) into eq. (6.154), using identity (6.14) and rearranging orders of integration and summation,

(6.160)

Finally, use of prolate identity (4.2) in (6.160) establishes m ~ ( 7 2 02) ,

=

m

Z: jN exp (jNe2)n-0Z: ( - 1 ) n b N , n A ~ , n @ N , n ( P o r 2 / r m ) '

N-0

(6.161)

A constant multiplier has been ignored. 6.8.3.2. Analytic dependence of encircled energy b ( r m )upon aberra-

tions bN,n We have discussed the significance of the encircled energy &(rm)in design problems (see 5 6.3.1). Because of the remarkable dual orthogonality properties (4.3a). (4.6) for functions QN,,, B(r,) may be analytically related to the bN,n. Substitution of amplitude response (6.161) into eq. (6.153), which is then substituted into defining relation (6.47) for &(r,,,),yields m o o

8(rm) =

2 2 lbN,n12Ak,n N-O n-0 m

m

2 IbN.nlaAN,n

N-O n-O

(6.162)

402

OPTICAL SIGNALS A N D PROLATE FUNCTIONS

[vrrr, 8 6

6.8.3.3. Discussion

Eqs. (6.161) and (6.162) analytically relate u and Q(r,) to the new aberrations, regardless of their magnitude or number. This is a distinct advantage over use of the Zernike or Seidel aberrations. The intensity on the optical axis is particularly easy to calculate. By (6.161), 01

~ ( 0= )

2 (-l)"bO,n~~,n@o,n(O).

(6.163)

n=O

This results from definition (4.2) of functions @N,n, according to which = 0 unless N = 0. We also note that the receiving plane position z2 has been arbitrary in this analysis, having been regarded as an aberration of pupil function F (cf. eqs. (6.1) and (6.164)). Therefore, and 8 may be computed in any receiving plane by use of eqs. (6.161) and (6.162). The affect of the new aberrations bN,n upon the encircled energy is particularly transparent. From (6.162), the aberrations merely contribute additively, and only through their moduli. (Notice that in 5 6.3.1.1 we have computed the optimum "aberrations" bN,n for maximizing &(Y,).) Because the AN,n rapidly approach zero once N and n are large enough, eq. (6.162) shows that &(r,) is not affected by aberrations of higher order than these critical N and TZ values. For values of rm of the order of the Airy disc size, which is the usual case of interest, the critical N and n. are small. However, for values of rm of the order of 10 times the Airy disc the formula (6.162) converges so slowly in N , n that it is then of little use for computational purposes. A final disadvantage to the use of formulae (6.161), (6.162) is that one Seidel or Zernike aberration engenders an infinite number of aberrations bN,n. The integral (6.159) does not cut off in N , n when F is aberrated at all. By contrast, one Seidel aberration does give rise to a finite number of Zernike aberrations, and vice-versa. 6.8.4. Numerical results

When the optics are aberrated by orders of spherical aberration alone, W consists entirely of powers of p 2 (see eq. (6.157)). Then by (6.156) F(p, 0) = F ( p ) . The d0 integration in (6.159) may therefore be carried through, resulting in (6.164) JO

bN,n

=0

for N # 0.

WII, §

6J

403

APPLICATIONS

By eq. (6.162), the encircled energy simplifies to b(rrn) =

2 ~ b o , n ~ ~ G ,n-0 n2 / ~bo,n~~h,n-

(6.165)

n=O

Itoh examined the particular aberration case

F ( P )= exp @KO[ ( P / P ~ ) ~ (P/P~)~.II). -

(6.166)

W , thereby represents the Zernike third-order spherical aberration. Coefficients bo,n were found by numerically integrating eq. (6.164)by use of Simpson's rule. Substitution of the b0,=into formulae (6.163)and (6.165)permitted plots to be made of on-axis irradiance lzc(0)la and encircled energy 8 ( a ) against axial position of the receiving plane. Radius a is the position of the first zero in the Airy diffraction pattern, a = 3.83/p0.This is shown in Fig. 6.18, for various amounts of aberration W40. These plots show that for small aberration W 4 0 there is only one optimum axial position for a maximum in irradiance or in encircled energy. Moreover, as is evident in graph (A), the maximum in irradiance occurs precisely halfway between the marginal and paraxial foci; which should be true for small aberration. Graphs (B)-(E) show that as the aberration is increased, there appear multiple positions of maximum irradiance or encircled energy, and that the principle maximum shifts increasingly toward the paraxial focus. This is a wellknown effect. We may conclude, therefore, that at least from the numerical standpoint it is convenient to analyze the affects of aberrations upon the encircled energy by using the new aberrations bN,, . This is provided the encircling radius rm is less than about ten times the Airy disc radius, and provided the pupil is circular. 6.8.5. Rectangzclar pupil case

By analogous reasoning, the proper aberrations to use are a set w,, defined by expanding the pupil in rectangular coordinates as F(B,Y ) =

2 2 wrnnYrn(C1J B) vn(Ca,

m-0 n-0

7)s

1 9 0 x 3

~2

= ~o'oy.

(6.167)

Coordinates (B, y ) are defined in terms of the spatial pupil coordinates as in eq. (6.46).The pupil is thereby described by linear prolate functions, for this geometry. The amplitude response %(Xa, ra) and the

404

O P T I C A L SIGNALS A N D P R O L A T E F U N C T I O N S

A

[VIII,

5

6

WII.

§ 71

405

ACKNOWLEDGEMENTS

D

M

AL S

t

z

3

& Q

E

m

(L:

Q

MAR1 FC

XlAL

:us

Fig. 6.18. Normalized irradiance (---) and encircled energy (-) along the OA for five different amounts of third-order spherical aberration W,o: (A) 21, (B)4?., ( C ) 6?., (D) 101, (E) 20A. (After ITOH[1970] p. 14.)

enclosed energy in a rectangular region lxzl S X , 1y.J 5 Y will depend on aberrations w,,,,,as simply as in the analogous expressions (6.161) and (6.162). Once more, the aberrations may be of any magnitude and of any number. This problem does not seem to have been numerically investigated yet.

8 7 . Acknowledgements The author wishes to thank David Slepian for his many helpful remarks on technical and historical aspects of this subject. Many happy hours were spent by the author in communication with the fol-

406

OPTICAL SIGNALS AND PROLATE FUNCTIONS

[VIII

lowing workers, on the subject of aperture extrapolation: R. N. Bracewell, G. Toraldo di Francia, A. C. Schell, D. R. Rhodes and W. Swindell. H. A. Brown was most helpful in providing the author with a table of dT-coefficients and a Fortran program for generating them. All our numerical work with prolate functions has been based on their use. Finally, Y. Itoh was exceedingly cooperative in providing the author with the fruits of his research on wave aberrations.

References AUSTON, D. H., 1968, IEEE J. Quantum Electron. QE-4, 420. BARAKAT, R., 1962, J. Opt. sot. Am. 52, 264. BAKAKAT, R., 1964, J. Opt. SOC. Am. 54, 920. BARNES, C. W., 1966, J. Opt. SOC.Am. 56. 675. BIRAUD,Y., 1969, Astron. Astrophys. 1, 124. BORN,M. and E. WOLF,1964, Principles of Optics, 2nd. ed. (The Macmillan Co., New York). BOYD,G. D. and J. P. GORDON,1961, Bell System Tech. J. 40, 489. BOYD, G. D. and H. KOGELNIK, 1962, Bell System Tech. J. 41, 1347. BRACEWELL. R., 1965, The Fourier Transform and its Applications (McGrawHill, Inc., Kew York) pp. 357-365. 1967, IEEE Trans. AP-15, 376. BUCK,G. J. and J. J. GUSTINCIC, CHANG, W. S., 1969, Principles of Quantum Electronics: Lasers; Theory and Applications (Addison-Wesley Publishing Co., Reading, Massachusetts). CHuncHILL, R. V., 1941, Fourier Series and Boundary Value Problems (McGrawHill Book Co., New York) pp. 47-61. FLAYMER, C., 1957, Spheroidal Wave Functions (Stanford University Press, Stanford). FRIEDEN, B. R., 1966, J. Opt. SOC.Am. 56, 1365. FRIEDEN, B. R., 1967, J. Opt. SOC.Am. 57z 1013. FRIEDEN, B. R., 1969, Opt. Acta 16, 796. FRIEDEN, B. R., 1970, Appl. Opt., 9, 2494. GROBNER, W. and N. HOPREXTER, 1966, Integraltafel, zweiter Teil: Restimmte Integrale, 4th rev. ed. (Springer-Verlag, Vienna) p. 188. HARRIS, J. L., 1964, J. Opt. SOC.Am. 54, 931. HELSTROM, C . W., 1967, J. Opt. SOC. Am. 57, 297. HEURTLEY, J. C., 1964, Hyperspheroidal Functions - Optical Resonators with Circular Mirrors, in: Roc. Symp. on Quasi-Optics, New York, 1964, ed. J. Fox (Polytechnic Press, Brooklyn) p. 367. HEURTLEY, J. C., 1965, A Theoretical Study of Optical Resonator Modes and a New Class of Special Functions, the Hyperspheroidal Functions; Ph. D. dissertation, Electrical Engineering Dept., University of Rochester, Rochester, New York. HEURTLEY, J. C. and W. STREIFER,1965, J. Opt. SOC.Am. 55, 1472. ITOH, Y., 1970, J. Opt. SOC.Am. 6 0 , 10.

VIII]

REFERENCES

407

P. and B. DOSSIER, 1964, Apodisation, in: Progress in Optics, Vol. 111, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) Ch. 11, pp. 77,

JACQUINOT,

78.

LANSRAUX, G. and G. BOIVIN, 1961, Can. J. Phys. 39, 158. LUKOSZ, W., 1966, J. Opt. SOC.Am. 56, 1463. LYTLE, R. J. and F. V. SCHULTZ, 1969, IEEE Trans. Antennas and Propagation AP-11, 496.

MARECHAL, A. and P. CROCE,1953, Compt. Rend. 237, 706. NIJBOER, B. R. A., 1942, The Diffraction Theory of Aberrations, Thesis, University of Groningen, Netherlands. NIVEN,C., 1880, Phil. Trans. Roy. SOC.London, Ser. A, 171, 117. O’NEILL,E. L., 1963, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Massachusetts). PANOFSKY, W. K. and M. PHILLIPS, 1956, Classical Electricity and Magnetism (Addison-Wesley Publishing Co., Reading, Massachusetts) pp. 73, 74. RINO,C. L., 1969, J. Opt. SOC.Am. 59, 547. RUSHFORTH, C. K. and R. W. HARRIS, 1968, J. Opt. SOC.Am. 58,639. SLEPIAN, D. and H. 0. POLLAK, 1961, Bell System Tech. J. 4 0 , 4 3 . SLEPIAN, D., 1964, Bell System Tech. J . 43, 3009. SLEPIAN, D., 1966a, J. Opt. SOC.Am. 55, 1110. SLEPIAN, D., 196sb, J. Math. and Phys. 44, 99. STREIFER, W. and H. GAMO,1964, On the Schmidt Expansion for Optical Resonator Modes, in: Proc. Symp. on Quasi-Optics, New York, 1964, ed. J. Fox (Polytechnic Press, Brooklyn) p. 361. STREIFER, W., 1966, J. Opt. SOC. Am. 55, 868. STUCKEY, M. M. and L. L. LAYTON, 1964, Numerical Determination of Spheroidal Wave Function Eigenvalues and Expansion Coefficients, AML Report 164, U. S. Naval Research Laboratory, Washington, D. C. TORALDO DI FRANCIA, G., 1969, J. Opt. SOC.Am. 59,799. UNZ,H., 1966, Appl. Sci. Res. 12B,360. WILKINS, J. E., 1950, J. Opt. SOC.Am. 40,222. WOLTER, H., 1961, On Basic Analogies and Principal Differences Between OpticaI and Electronic Information, in: Progress in Optics,Vol. I, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) Ch. V, 4.6. ZERNIKE,F., 1934, Physica I , 689.

A U T H O R INDEX A ADLER,R., 181, 226, 233 AGRAXOWCH, V. M.. 240, 247, 248, 260262, 264, 266-262, 264, 266, 267, 272, 273, 2 7 6 2 7 9 ALLEN, L., 208, 217, 224, 228, 234 AMXANN.E. 0..189, 206, 233. 234 ANG, D. D., 283, 309, 310 ARECCHI,F. T... 230.. 233 ARMSTRONG. J. A., 230, 233 AUSTON,D. H.. 194. 209, 211, 216, 233. 393, 406

B A., 194, 226, 233 BAMBINI, BARAKAT, R., 363, 368, 406 BARNES, C. W., 317, 406 BASS, M., 232, 233 BETEROV, I. M., 217, 233 BIEBER, T., 232, 233 BIRAUD, Y.,379, 406 BIRSS,R. R., 269, 272, 279 BOILLET,P., 286288, 309 BOIVIN,G., 363, 406 BONIFACIO, K., 230, 233 BORENSTEIN, M., 229, 233 BORN, M., 314, 350, 406 B O U W K A M P , C. J., 285, 309 BOYD, G. D., 316, 317, 406 BRACEWELL, R., 311, 406 BRIDGES,T. J.. 214, 233 BROWN JR., W. F.. 259, 271-273, 279 BUCK,G. J., 379, 406 BURLAMACCHI, P., 194, 226, 233

C CADDES,D. E., 192, 209, 233 CARRUTHERS, J. A., 232, 233 CHANG,W. S., 351, 406 CHEBOTAEV,V. P.. 217, 233

CHETKIN. M.v., 243. 280 CHURCHILL, R.v., 321, 340, 406 COURTENS,E., 230, 233 CROCE,P., 391, 407 CROWELL, M. H., 197, 214, 233 CRUMLEY, c. B.. 186. 233 CZARNIEWSKI, V.. 233

D DE HOOP,A. T., 283. 286-288, 309 D ~ M G.,~ 219, ~ 232-234 ~ , DI DOMENICO JR., M., 196, 233 DOSSIER,B.. 363, 380, 406 DUGAY,M. A., 232, 233 DZYALOSHINSKIJ, I. E., 272, 279

E EWY,M.D., 185, 233 F FILIPPOW, A. F.. 283, 309 FLAMJIER,C., 316, 321. 334, 406 FORK,K. L., 184, 234 FOSTER, L. C., 186, 233 Fox, A. G., 217, 230, 233 FRIEDEN,B. R., 318, 368, 379, 386, 387389, 391, 392, 406 F R O V h , A., 232, 233 FUJIOKA, T.,215, 216, 233, 234

GADDY,0. L., 214, 233 Garno. H., 317, 407 GARRETT,C. G. B., 232, 233 GINZBURG, V. L., 240, 247, 248, 260. 261, 262, 264, 266-262, 264, 266, 267, 272, 276279 GNIADEX,K., 283, 294, 300, 302, 303, 308, 309 408

AUHTOR lNDEX

GORDON, J. P., 316, 406 GROBNER, W.. 366, 406 GROSS,E. F., 260, 279 GUSTINCIC, J. J.. 379, 406 GYORFFY,B.,229, 233

H HAHN,E. L., 230, 234 HAKEN,H., 212, 234 HARGROVE, L. E., 183, 234 HARRIS,J. L., 317, 379 HARRIS,R. W., 318, 370, 371, 379, 407 HARRIS,S. E., 187-189, 193, 200, 206, 207, 209, 234

HELLWEGE,K. H., 260, 279 HELSTROM, C. W.. 372. 406 HENNEBERGER, W. C., 193, 234 HEURTLEY, J. C., 314, 317, 337. 338, 340,

409

LAMBJR., W.E.,201, 212, 221, 229, 230, 233, 234

LANDAU,L. D., 249, 260, 269, 272, 280 LANSUVX,G., 363, 406 LAYTON, L. L., 336, 385, 407 LIFSCHITZ,E. M.. 249, 266, 289, 272, 280 LISITSYN, V., 217, 233 LORENTZ, A. A., 260, 280 LUKOSZ, W., 316, 406 LYTLE,R. J.. 316, 407

M MCCALL,S. L., 230, 232-234 MCCLURE,R. E., 214, 234 MCDUFF,0. P., 189, 193, 200, 207. 209. 234

HIRANO, J., 216, 234 HOFREITER, N., 366, 406 HONG,G. W., 191, 209, 234 H ~ N LH., , 283, 309 HORNREICH, R. M., 244, 269, 277, 279 HUGGETT, G. H., 196, 234

MCMURTRY, B. J., 189,206,207,233,234 MAL,A. K., 283, 310 MARECHAL. A.. 391, 407 MASSEY,G. A., 188, 192. 207, 234 MAUE. A. W., 283, 309, 310 MILES,J. W., 283, 310 MXYAMOTO, K.,292, 293, 310 MOLCHANOV, A. G., 262, 280

I

N

INGARDEN, R. S., 295, 309 ITOH,Y.,319, 400, 403, 406

NASH,F. R., 214, 216, 224, 234 NIVEN,C., 313, 407 NOWACKI, W., 309, 310 NIJBOER,B. R. A., 400, 407 NYE, J., 269, 279

347. 406

J JACQUINOT, P., 363, 380, 406 D. G. C., 217, 228, 234

JONES,

K

0

O’NEILL,E. L., 326, 372, 407 OSHMAN, M. K., 188, 189, 206, 207, 233, 234

OSTERINK, L. M., 192, 209, 233 KAMENOGRADCKIJ, N. E., 261, 279 KAPLYANCKIJ, A. A., 260, 279 KARCZEWSKI, B., 300, 310 KIMURA,J., 216, 234 KNOPOFF, L., 283, 286-288, 291, 309, 310 KOBAYASHI, M., 215, 234 KOGELNIK, H., 317, 406 KOHIYAMA, K., 216, 234 KONOBEEV, Yu. V., 261, 279 KOTTLER. F., 286, 310 KRINCHIK, G. S., 243, 280 KUPRADZE, V. D.. 286-288, 293, 310

P PANOFSKY, W. K., 316, 407 PAUTHIER, M., 212, 234 PEKAR. S. I.. 246, 264, 280 ~ T E R S O N , D. G., 212, 234 PETYKIEWICZ, J., 283, 286-288, 290, 292. 300, 310

PHILLIPS, M., 316, 407 POLLACK, M. A., 183, 234 POLLAK, H. 0..314, 320, 336, 407

410

A U T H O R INDEX

R

T

RAYLEXGH, Lord, 181. 234 RIGROD.W. W., 214. 233. 234 B N O , c. L., 318, 374, 376, 407 RUBINOWXCZ, A., 283, 286, 291-296, 298302. 304, 310 RUKHADZE, A. A.. 247. 249. 279, 280 RUSHFORTH, C. K., 318,370, 371,379, 407

TANG,C. L..196. 211, 218, 219, 221. 234 TARG,R., 187, 188, 1 9 2 , 2 0 7 , 2 0 9 , 2 3 3 , 2 3 4 TORALDO Dx FRANCXA, G., 319, 393, 394, 407 TREVES,D., 259, 271-273. 279

S SAYERS,M. D., 208, 217, 224, 228, 234 SCHAEFER. E. M., 214, 233 SCHIFP, L. I., 234 SCHULTE, H. J . , 193, 234 SCHULTZ, F. V., 315, 407 SCHWARZ, S. E., 217, 234 SKTRICKMAN. s.. 244, 269, 271-273, 277, 279 SXLXK, V. P., 247. 249, 279, 280 SLATER.J. C., 197, 234 SLEPIAN,D., 314, 317, 320, 321, 334, 336, 337, 338, 339, 342, 347, 363, 397, 407 SMITH,P. W., 193, 216-217, 230. 231, 233, 234 STATZ,H.. 196, 211, 218, 219, 221, 232234 STEIER.W. H., 182, 196, 211, 234 STENHOLM. S., 229, 234 STOVER,H. L., 182, 195, 211, 234 STREIFER.W., 317, 406, 407 STUCKEY. M. M., 336, 386, 407

U UCHIDA,T., 214, 234 UCHIYAMA, T., 216, 233 UEKI. A., 214, 234 UNZ,H., 316, 407

V V A N DER

POL,

n..181, 234

W WARTER, P.J., 230, 234 WESTPFAHL, I