FUNDAMENTALS OF POLARIZED LIGHT

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1 FUNDAMENTALS OF POLARIZED LIGHT A STATISTICAL OPTICS APPROACH Christian Brosseau University of Brest, France A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS, INC. New York - Chichester. Weinheim. Brisbane Singapore - Toronto r

2 CONTENTS PREFACE xi PART 1 HISTORICAL SURVEY OF UNDERSTANDING OF POLARIZED LIGHT First Period: Early Ideas and Observations from Bartholinus to Stokes Second Period: The Electromagnetic Nature of Light Third Period: The Coherence and Quantum Properties of Light 15 Further Reading 27 PART 2 PRELIMINARIES TO A CLASSIC RADIATION FIELD THEORY The Basic Differential Equations and Boundary Conditions Maxwell's Equations Boundary Conditions Fresnel's Formulas for Reflection and Transmission of a Plane Wave at a Planar Boundary between Homogeneous Isotropic Media Wiener's Experiment Invariance Transformations Gauge Transformations Time Inversion Spatial Inversion Duality Transformations 55

3 Vi CONTENTS 2.3. Monochromatic Plane Wave Harmonic Plane-Wave Solution of the Wave Equation Reciprocity Principle Poynting's Theorem Normal Modes Representation of Electromagnetic Field 63 Further Reading 66 PART 3 POLARIZATION AND THE RADIATION FIELD Elementary Concepts and Definitions Temporal Fluctuations in Light Beams: Characteristic Timescales and Partial Polarization The Analytic Signal Representation The Instantaneous Ellipse for the Electric Field The Complex Polarization Ratio Representation of Polarized Light Statistical Description of Fluctuations of a Partially Polarized Optical Field Correlation Functions of Analytic Signals The Probability Density Function of the Radiation Field Second-Order Coherence Properties of a Stochastic Radiation Field Relations between Correlation Tensors and Poynting Vector Dynamical Equations for Second-Order Correlation Tensors Wiener-Khintchine Theorem and Cross-Spectral Correlation Tensor Blackbody Radiation in Equilibrium in an Enclosure Stokes Parameters and Coherency Matrix Formalism Stokes Parameters Coherency (Density) Matrix Formalism Spectral Decomposition Theorem Principle of Optical Equivalence Reduction Property Covariance Matrix of Instantaneous Stokes Parameters Digression: Generalized Stokes Parameters 118

4 CONTENTS VÜ 3.2. Geometrie Representations of Partially Polarized Light The Stokes Vector Space Convexity Structure of Sets of Polarization States Stereographic Projection Plane of Poincare Sphere Geometric Phase and Polarization Cycles Analogy between Polarization and Two-Level Systems: Bloch Equation Statistics of the Radiation Field Statistics of Amplitudes and Phases for a Gaussian Random Wavefield Statistics of the Complex Polarization Ratio for a Gaussian Random Wavefield Statistics of Stokes Parameters for a Gaussian Field Probability Density Functions of Stokes Parameters Cumulants of Stokes Parameters Probability Density Functions of Normalized Stokes Parameters Statistics of Time-Integrated Stokes Parameters Unpolarized Radiation: Stokes-Verdet-Barakat Conditions Polychromatic Radiation Wavefield Entropy of the Radiation Field Entropy of a Stochastic Plane Wavefield Temperature of Polarization Maximum Entropy Principle 171 Further Reading 176 PART 4 INTERACTION OF RADIATION WITH LINEAR MEDIA Jones and Mueller Polarization Transfer Matrix Methods Jones Calculus Eigenvalue-Eigenvector Decomposition Jones Formalism and 2x2 Coherency Matrix Formulation Gain Jones Matrices for Common Optical Devices 191

5 viii CONTENTS Physical Realizability Constraint Polar Decomposition of Jones Matrices Optical Propagation in Multilayered Media Polarization Ratio Transformations Jones Calculus and Entropy Transformations Mueller Calculus Mueller Formalism and 4x4 Coherency Matrix Formulation Gain Physical Realizability Conditions Mueller Matrices for Common Optical Devices Mueller Matrices and Their Parametrizations Within Poincare Space Mueller Calculus and Entropy Transformations Relationship between Jones and Mueller Matrices Polarization Transfer in Nondepolarizing Optical Linear Media Mueller Matrix Analysis of Light Depolarization by a Linear Optical Medium Digression: Extended Matrix Formalisms Polarization Effects at Dielectric Interfaces Mueller Matrix Formulation of Fresnel's Equations The Pile-of-Plates Polarizer Principle of Ellipsometry Polarized Light and Symmetry Transformations Spatial Symmetry Relations for a Far-Field Scattering Time-Reversal Invariance and Reciprocity Time-Reversal Invariance Reciprocal Jones Medium Constraint Reciprocal Mueller Medium Constraint Random Media Jones and Mueller Matrices for Temporally Random Media Multiple Scattering by a Spatially Random Medium Rayleigh Scattering Monte Carlo Simulations of Wave Propagation Through Three-Dimensional Inhomogeneous Media 283

6 CONTENTS IX Backscattering Enhancement from a Random Distribution of Scatterers 293 Further Reading 297 PART 5 APPLICATIONS TO SELECTED TOPICS Electromagnetic Propagation in Linear Anisotropic Media Plane-Wave Propagation in a Linear Medium with Permittivity Tensor The Differential Polarization Matrices Ellipse of Polarization in Anisotropic Media Evolution of Stokes Parameters in Anisotropic Media Application to Liquid Crystals Optical Polarizing Components Polarizers Reflection Polarizers Transmission Polarizers Polarization by Selective Absorption Compensators Fresnel Rhomb Adjustable Compensators Measurement of Stokes Parameters Methods for Measuring Stokes Parameters Stokes' Procedure Collett's Procedure Division-of-Amplitude Photopolarimetry Probability Density Functions of Stokes Parameters Measurement of Jones and Mueller Polarization Matrices Methods of Determination of Jones Matrix Jones Procedure Differential Polarized Spectroscopy Methods for Measuring Mueller Matrix Transmission Experiment Polarization Modulation Method 352 Further Reading 357

7 X CONTENTS APPENDIXES 361 Appendix A. Analogy of Second-Order Coherence Properties of Blackbody Radiation with Theory of Homogeneous and Isotropic Turbulence of an Incompressible Fluid 363 Appendix B. Derivation of Eq. (3.4.5) Using the Spectral Decomposition Theorem 366 Appendix C. Degree of Polarization of an Incoherent Mixture of Partially Polarized Light Beams 368 Appendix D. Set of Generalized Stokes Parameters 371 Appendix E. Short Historical Account of the Concept of Entropy of Radiation Fields 372 Appendix F. Singular Value Decomposition 374 Appendix G. Derivation of Bounds for Degree of Polarization 376 Appendix H. Application of Maximum Entropy Principle in Polarization Optics 378 Appendix I. Set of Coefficients D' k 380 Appendix J. Evaluation of Correlation Function in Eq. (4.4.34) 382 Appendix K. Diffusing-Wave Spectroscopy 387 Appendix L. Polarization of Photons 392 AUTHOR INDEX 397 SUBJECT INDEX 401

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