Springer Series in Optical Sciences Volume 11 Edited by David L

Springer Series in Optical Sciences Volume 11 Edited by David L. MacAdam Springer Series in Optical Sciences Edited by David L. MacAdam Editorial Board: J. M. Enoch D. L. MacAdam A. L. Schawlow T. Tamir

Volume Solid-State Laser Engineering By W. Koechner Volume 2 Table ofLaser Lines in Gases and Vapors 2nd Edition By R. Beck, W. Englisch, and K. Gürs Volume 3 Tunable Lasers and Applications Editors: A. Mooradian, T. Jaeger, and P. Stokseth Volume 4 Nonlinear Laser Spectroscopy By V. S. Letokhov and V. P. Chebotayev Volume 5 Optics and Lasers An Engineering Physics Approach By M. Young Volume 6 Photoelectron Statistics With Applications to Spectroscopy and Optical Communication By B. Saleh Volume 7 Laser Spectroscopy III Editors: J. L. Hall and J. L. Carlsten Volume 8 Frontiers in Visual Science Editors: S. J. Cool and E. L. Smith III Volume 9 High-Power Lasers and Applications Editors: K.-L. Kompa and H. Walther Volume 10 Detection of Optical and Infrared Radiation By R. H. Kingston Volume 11 Matrix Theory of Photoelasticity By P. S. Theocaris and E. E. Gdoutos Volume 12 The Monte Carlo Method in Atmospheric Optics By G. 1. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov Volume 13 PhysiologicalOptics By Y. Le Grand and S. G. EI Hage Volume 14 Laser Crystals Physics and Properties By A. A. Kaminskii Volume 15 X-Ray Spectroscopy By B. K. Agarwal Volume 16 Holographie Interferometry From the Scope of Deformation Analysis of Opaque Bodies By W. Schumann and M. Dubas P. S. Theocaris E. E. Gdoutos

Matrix Theory of Photoelasticity

With 93 Figures

Springer-Verlag Berlin Heidelberg GmbH 1979 Professor PERICLES S. THEOCARIS, D.Se. Professor EMMANUEL E. GDOUTOS, Ph.D. Athens National Technical University, Athens, Greece

Editorial Board

JAY M. ENOCH, Ph. D.

Department of Opthalmology, J. Hillis MiIIer Health Center University of Florida, P.O. Box 733 Gainesville, FL 32610, USA

DAVID L. MACADAM, Ph. D.

68 Hammond Street, Rochester, NY 14615, USA

ART L. SCHAWLOW, Ph. D. Department of Physics, Stanford University Stanford, CA 94305, USA

THEODOR TA MIR, Ph. D.

981 East Lawn Drive, Teaneck, NJ 07666, USA

Library ofCongress Cataloging in Publication Data. Theocaris, Pericles S 1921-. Matrix theory of photoelasticity. (Springer series in optical sciences; v. 11) Bibliography: p. IncIudes·index. 1. Photoelasticity. I. Gdoutos, E. E., 1948-. joint author. H. Title. TA418.12.T48 620.1'1295 78-14275 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is eoneerned, speeifieally those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photoeopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, where eopies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. ISBN 978-3-662-15807-4 ISBN 978-3-540-35789-6 (eBook) DOI 10.1007/978-3-540-35789-6 © by Springer-Verlag Berlin Heidelberg 1979 Originally published by Springer-Verlag Berlin Heidelberg New York in 1979. Softcover reprint of the hardcover 1 st edition 1979 The use of registered names, trademarks, ete. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and thefore free for general use.

2153/3130-543210 Preface

Photoelasticity as an experimental method for analyzing stress fields in mechanics was developed in the early thirties by the pioneering works of Mesnager in France and Coker and Filon in England. Almost concurrently, Föppl, Mesmer, and Oppel in Germany contributed significantly to what turned out to be an amazing development. Indeed, in the fifties and sixties a tremendous number of scientific papers and monographs appeared, all over the world, dealing with various aspects of the method and its applications in experimental stress analysis. All of these contributions were based on the so-called Neumann-Maxwell stress-opticallaw; they were developed by means of the classical methods of vector analysis and analytic geometry, using the conventionallight-vector concept. This way of treating problems of mechanics by photoelasticity indicated many shortcomings and drawbacks of this classical method, especially when three-dimensional problems of elasticity had to be treated and when complicated load and geometry situations existed. Meanwhile, the idea of using the Poincare sphere for representing any polarization profile in photoelastic applications was introduced by Robert in France and Aben in the USSR, in order to deal with problems of polarization oflight passing through aseries of optical elements (retarders andjor rotators). Although the Poincare-sphere presentation of any polarization profile con stitutes a powerful and elegant method, it exhibits the difficulty of requiring manipulations in three-dimensional space, on the surface of the unit sphere. However, other graphical methods have been developed to bypass this difficulty. By a parallel projection of the points of the sphere on its equatorial plane, Kuske developed the j-circle method; Aben and others used the stereographic projection connected with the Wulff net, which maps polarization states so as to preserve angles. Another type of mapping, based on the Lambert-Schmidt projection, which preserves areas is currently under development by Theocaris. Concurrently with these graphical methods, some analytical methods were developed, based on the Stokes and J ones vectors that are extensively used in crystallography. These methods are the powerful Mueller and Jones calculi. Finally, the analytical method based on quaternia, which uses the Pauli spin matrices should also be mentioned. All ofthese modern graphical and analytical methods have been developed by use of various techniques based either on conventional light-vector concept, using vector calculus, or on matrix methods. However, no unified method exists to date that incorporates all of these modern ideas. VI Preface

The purpose of this book is to present a novel and unified interpretation of all of the problems of two- and three-dimensional photoelasticity by use of the modern methods of description of polarized light, based on the concepts of the Poincare sphere, as weIl as on Mueller and Jones calculi. Although these powerful methods were conceived many years ago, only in the last two decades they have been applied in the solution of current problems of photoelasticity. The simplicity and elegance of these methods enable one to solve complicated problems in straightforward and comprehensive manners and to devise simple solutions of complex problems that, because of their nature, could not be solved by use of classical methods. Furthermore, although the Poincare-sphere representation can help enormously to visualize and formulate problems and theorems in a qualitative manner, geometrie mani pulations that involve spherical trigonometry and the simple and efficient procedures of matrix analysis, provide accurate results in the sequence. This book, as conceived here, is based on matrix theory, because this power ful calculus was judged to be the most suitable for the presentation, in a simple and unified manner, of all of the theorems related to these methods. Thus, apart from the fact that this monograph is the first to exhibit all of these new methods, based on the Poincare sphere, and on Jones and Mueller calculi, it reformulates already-known theorems ofphotoelasticity and derives others on the basis of the present approach. This book is intended to be an instrument of learning, rather than a review of contributions in these modern techniques related to photoelasticity. There fore, from the great number of papers and monographs published on photo elastidty, we have selected only those that seemed to suit the purpose of this book. Likewise, references and the bibliography at the end of the book are limited to papers that are closely related to the subjects treated in this monograph. Particular care was taken throughout the book to give clear, straightforward, and simple presentations of the various topics. The authors hope that the monograph will be used as a basis on wh ich the future researcher and student can develop his own new and fruitful ideas and promote research in formula tion of problems of photoelasticity by matrices, which, despite its significance, has been given little attention as yet. FinaIly, the authors are particularly indebted to Mr. Constantin Thireos, Assistantat the Laboratory, who prepared the groundwork ofthe bibliography contained at the end of the book and who also contributed in the preparation of the manuscript.

Athens, Greece P. S. THEOCARIS· E. E. GDOUTOS September, 1978 Contents

1. Introduction 1

2. Electromagnetic Theory of Light 6 2.1 The Nature of Light 6 2.2 Maxwell's Equations 8 2.3 Propagation of Eleetromagnetie Waves in Isotropie Media 9 2.4 Propagation of Eleetromagnetie Waves in Anisotropie Media 11 2.4.1 The Fresnel Ellipsoid . . . 13 2.4.2 Uniaxial and Biaxial Crystals 17 Referenees 19

3. Description of Polarized Light 20 3.1 Introduetion .... 20 3.2 Ordinary and Polarized Light 21 3.3 Forms of Polarized Light 22 3.4 Deseription of Polarized Light 24 3.4.1 Vectorial Representation 25 Linear Polarization . 25 Circular Polarization 26 Elliptical Polarization 26 3.4.2 The Poineare Sphere 31 3.4.3 The j Circle 33 3.4.4 The Stokes Vector 34 3.4.5 The Jones Veetor 36 3.5 Relationship Between the Methods of Charaeterization of a Polarized Light ...... 41 3.5.1 The Poineare Sphere and the j Circle 41 3.5.2 The Poincare Sphere and the Stokes Vector 42 3.5.3 The Stokes and Jones Vectors .... 42 3.6 Discussion of the Methods of Characterization of Polarized Light 43 Referenees ...... 44

4. Passage of Polarized Light Through Optical Elements 45 4.1 Introduction ...... 45 4.2 Definition of Optical Elements 48 4.2.1 Orthogonal F orms . . 48 VIII Contents

4.2.2 Polarizers ...... 49 4.2.3 Retarders ...... 49 4.3 The Passage of a Polarized Beam Through aPolarizer or a Retarder ...... 49 4.3.1 Vector Calculus 50 The Linear Polarizer 50 The Linear Retarder 50 4.3.2 The Jones Calculus . 51 4.3.3 An Independent Introduction of Jones Matrices 55 4.3.4 The Mueller Calculus 56 The Linear Polarizer 57 The Linear Retarder 61 4.3.5 The Poincare Sphere 63 The Linear Polarizer 65 The Linear Retarder 67 4.3.6 The j-Circle Method 72 4.4 Interrelation Between the Mueller and Jones Matrices 72 4.5 Some Useful Theorems 77 References ...... 80

5. Measurement of Elliptically Polarized Light 82 5.1 Introduction ...... 82 5.2 General Considerations ...... 83 5.3 Determination of the Elements of the Stokes and Jones Vectors 85 5.4 Photoelectric Methods ...... 88 5.4.1 General Considerations ...... 88 5.4.2 Special Techniques for Determining the State of Polarization ...... 90 5.5 Visual Methods for Measuring the State of Polarization 92 5.5.1 Measurement of Azimuth 92 5.5.2 Measurement of Ellipticity 95 Direct Methods .... 96 Compensators . . . . . 98 5.5.3 Determination of Handedness 99 5.6 Determination of the Matrix Elements of an Optical Device · 100 5.6.1 Determination of a Mueller Matrix · 101 5.6.2 Determination of a Jones Matrix 103 References · 104

6. The Photoelastic Phenomenon 105 6.1 Introduction ..... 105 6.2 The Photoelastic Law . . 106 6.2.1 Principal Birefringent Directions 106 6.2.2 Principal Refractive Indices · 108 References · 112 Contents IX

7. Two-Dimensional Photoelasticity · 113 7.1 Introduction ...... · 113 7.2 The Plane Polariscope · 114 7.2.1 The Jones Calculus for a Plane Polariscope · 114 7.2.2 The Mueller Calculus for a Plane Polariscope · 116 7.3 The Circular Polariscope ...... · 117 7.3.1 The Jones Calculus for a Circular Polariscope · 118 7.3.2 The Mueller Calculus for a Circular Polariscope · 120 7.3.3 The Poincare Sphere in the Circular Polariscope · 122 7.4 The Senarmont Compensation Method ...... · 123 7.4.1 The Jones Calculus in the Senarmont Method · 124 7.4.2 The Mueller Calculus in the Senarmont Method · 125 7.4.3 The Poincare Sphere in the Senarmont Method · 126 7.5 The Tardy Compensation Method ...... · 127 7.5.1 The Jones Calculus in the Tardy Method · 128 7.5.2 The Mueller Calculus in the Tardy Method · 129 7.5.3 The Poincare Sphere in the Tardy Method · 130 References · 131

8. Three-Dimensional Photoelasticity · 132 8.1 Introduction ...... · 132 8.2 The Equivalence Theorem . . · 137 8.3 Basic Equations of Three-Dimensional Photoelasticity · 138 8.3.1 The Stokes Vector · 138 8.3.2 The J ones Vector · 141 8.4 Neumann's Equations · 142 8.5 The Drucker-Mindlin Problem 144 8.6 Solution of the Problem by U se of the Equivalence Theorem ...... 145 8.6.1 Formulation of the Problem . 145 8.6.2 Solution of the Problem . 146 The Faraday Effect . . . . . 147 The Kerr Effect ...... 148 8.7 Solution of the Problem by Integrating the Equations of Three Dimensional Photoelasticity ...... 149 8.7.1 Application of the Peano-Baker Solution to the Drucker Mindlin Problem ...... 150 8.8 Another Approach to the Problem by Using the Characteristic Matrix ...... 152 8.9 The Poincare Sphere in Three-Dimensional Photoelasticity . . 155 8.10 The Electromagnetic Approach to the Three-Dimensional Photo- elastic Problem . 156 References ...... 163 X Contents

9. Seattered-Light Photoelasticity ... . · 164 9.1 Introduetion ...... · 164 9.2 Polarization of Light by Seattering · 167 9.3 Polarization by Seattering in a Photoelastie Medium · 169 9.4 Interpretation of Seattered-Light Fringe Patterns · 171 9.4.1 General Considerations ...... · 171 9.4.2 Constant Prineipal-Stress Direetions · 172 9.5 Determination of the Principal-Stress Direetions · 173 9.6 Determination of the Stress-Optieal Retardation · 176 9.7 The General Three-Dimensional Photoelastie Medium · 178 9.7.1 General Coneepts ...... · 178 9.7.2 Polarization by Seattering Used as Analyzer · 178 9.7.3 Polarization by Seattering Used as Polarizer · 181 9.8 Applieation to the Plane Problem · 182 Referenees ...... · 183

10. Interferometrie Photoelastieity · 184 10.1 Introduetion ..... · 184 10.2 Stress-Optieal Relations . · 186 10.3 Intensity Variation of a Light Ray in a Transparent Material · 190 10.4 Optieal Systems Used in Interferometrie Photoelasticity · 192 10.5 Analysis of Interferometrie Systems ...... · 195 10.6 Physieal Explanation of the Interferometrie Patterns .200 Referenees ...... 206

11. Holographie Photoelastieity .208 11.1 Introduetion .208 11.2 Basie Holographie Equations .210 11.3 Physieal Explanation of Holography · 216 11.4 Holographie Interferometry . . . . .221 11.4.1 Introduetion ...... · 221 11.4.2 Real-Time Holographie Interferometry .222 11.4.3 Double-Exposure Holographie Interferometry .226 11.4.4 Interpretation of the Fringe Patterns .227 11.5 Holographie Photoelastieity ...... 228 11.5.1 Introduetion ...... 228 11.5.2 Combined Isoehromatie-Isopaehie Patterns .232 11.5.3 Determination of Isoclinies ...... · 236 11.5.4 Separation of Isoehromaties and Isopaehies .238 Referenees .241

12. The Method of Birefringent Coatings .243 12.1 Introduetion ...... 243 12.2 Interpretation of the Birefringent-Coating Fringe Patterns .246 Contents XI

12.3 Accuracy of the Strains Measured by the Method of Birefringent Coatings ...... 250 12.3.1 Reinforcing Effect of Birefringent Coatings . 251 12.3.2 Edge Effect of Birefringent Coatings . 253 12.3.3 Strain Gradient and Curvature Effects . 256 12.3.4 Effect of Different Coating-body Poisson Ratios . 258 12.4 Photoelastic Strain Gauges . 258 References ...... 260

13. Graphical and Numerical Methods in Polarization Optics, Based on the Poincare Sphere and the Jones Calculus . 262 13.1 Introduction ...... 262 13.2 The Wulff-Net Method . 265 13.2.1 . The Poincare Sphere . 265 13.2.2 The Wulff Net . . . . 266 13.2.3 Applications of Wulff Net to Problems of Polarization Optics ...... 269 Passage of Polarized Light Through a Retarder (Birefringent Plate) ...... 269 Passage of Polarized Light Through aPolarizer . 271 13.2.4 Applications of Wulff Net to Problems of Two-Dimen- sional Photoelasticity . 273 13.3 The j-Circle Method ...... 275 13.3.1 ThejCircle ...... 275 13.3.2 Applications to Problems of Polarization Optics . 277 13.3.3 Applications to Photoelastieity . 279 Circular Polariscope ...... 279 The Senarmont Compensation Method . 280 The Tardy Compensation Method . 281 13.4 The Method of Quaternions ...... 282 13.4.1 Definition of Quaternions ...... 282 13.4.2 Representation of a Linear Retarder by Quaternions . 283 13.4.3 Applications to Photoelasticity . 285 The Plane Polariscope ...... 285 The Circular Polariscope ...... 286 13.4.4 A Geometrie Method Based on Quaternions . 287 References . 289

Bibliography . 291

Author Index . 337

Subject Index .345