Radiation Transfer: Statistical and Wave Aspects

Cover
Half Title
Title Page
Copyright Page
Contents
Foreword
Preface
Chapter 1: A Historical Overview
1.1 Phenomenological Approach to Radiation Transfer Theory
1.2 Reinterpretation of Transfer Theory
1.3 Diffraction Content of the Radiative Transport Equation. Nonclassical Radiometry
Chapter 2: Phenomenological Theory of Radiative Transfer
2.1 Basic Concepts of Radiometry. Radiation Transport in Homogeneous Media
2.1.1 Incoherent Beams and the Concept of Radiance
2.1.2 Spectral Radiance
2.1.3 Irradiance, Emittance, Radiant Intensity and Energy Density
2.1.4 Radiometric Quantities in Anisotropic, Weakly Inhomogeneous and Dispersive Media
2.1.5 Applicability of Radiometric Concepts
2.1.6 Sources of Incoherent Radiation
2.1.7 Relation of the Radiance with the Coherence Function of the Field
2.1.8 Quantum Approach and the Radiance of Thermal Radiation
2.2 Radiative Transfer in Scattering Media. The Scalar Model
2.2.1 Transport Equation in Nonscattering Media. The Ray Refractive Index
2.2.2 Relation of the Radiance with the Energy Density of Quasi-Plane Waves
2.2.3 Transformation of the Radiance on the Plane Interface
2.2.4 Quasi stationary Conservative Media. Conservation of the Adiabatic Invariant
2.2.5 Allowance for Absorption and Nonstationarity
2.2.6 Transfer Equation in a Scattering and Radiating Medium
2.2.7 Measurement of Transfer Equation Parameters. Analytical Models for the Single Scattering Cross Section
2.2.8 Sparse Scattering Media
2.2.9 Strongly Rarefied Discrete Media. The Effect of Scattering on Effective Medium Parameters
2.2.10 Corollaries of the Transfer Equation. Energy Balance Conditions
2.2.11 The Generalized Clausius Law
2.2.12 The Kirchhoff Law for Equilibrium Thermal Radiation and Thermal Sources in Transfer Theory
2.2.13 Approximation of the Total Frequency Redistribution
2.2.14 Useful Auxiliary Parameters
2.2.15 Necessary Applicability Conditions of the Radiative Transfer Equation in a Scattering Medium
2.3 Phenomenological Transfer Theory for Electromagnetic Radiation
2.3.1 Polarization of a Plane Electromagnetic Wave
2.3.2 Phenomenological Description of Electromagnetic Beams
2.3.3 Transfer Equations in Anisotropic Media
2.3.4 Electromagnetic Beam in a Homogeneous Isotropic and Weakly Anisotropic Media
2.3.5 Transfer Equation in an Inhomogeneous Nonstationary and Nonscattering Medium. Allowance for Polarization Vectors
2.3.6 Transfer Equation in a Scattering Medium
2.3.7 Two-Dimensional Description and the Choice of a Basis
2.3.8 The Structure of Coefficients in the Transfer Equation
2.3.9 Coefficients of the Transfer Equation in the Isotropic Case
2.3.10 Measurements of Parameters of the Radiation Transfer Equation
2.3.11 Radiative Transfer Equation in a Continuous Weakly Scattering Medium
2.3.12 Scattering at a Single Scatterer
2.3.13 Jones and Müller Scattering Matrices
2.3.14 Transfer Equation in a Sparse Discrete Scattering Medium
2.3.15 Stokes Parameters
2.3.16 Stokes Parameters of a Completely Polarized Wave and a Poincaré Sphere
2.3.17 Some Properties of Stokes Parameters
2.3.18 Transfer Equation for the Stokes Vector
2.3.19 Variation of Polarization in the Absence of Scattering. A Geometrical Interpretation
2.3.20 Transfer Equation for the Stokes Vector in a Scattering Medium
2.3.21 Variation of Polarization in Reflection and Refraction at a Plane Interface
2.3.22 Multimode System of Transfer Equations in a Scattering Medium
Problems
Chapter 3: Statistical Wave Content of Radiative Transfer Theory
3.1 Radiance and Correlation Functions of the Free Wave Field
3.1.1 Statistical Approach versus Phenomenological Approach
3.1.2 Statistical Wave Radiative Correlation Theory: Problem Formulation
3.1.3 Statistically Uniform and Stationary Fields. Spectra of Uniform Fluctuations
3.1.4 Spectra of Nonuniform Fluctuations. The Wigner Function
3.1.5 Wigner Function as a Local Spectral Density: Advantages and Disadvantages
3.1.6 Properties of the Wigner Function
3.1.7 Wigner Function and Uncertainty Relations
3.1.8 Quasi-Uniform Fields and Their Spectra
3.1.9 Local Quasi-Uniform Coherence Function in the Geometrical Optics Approximation. The Field of Quasi-Uniform Sources in a Homogeneous Medium
3.1.10 Fluctuation Spectrum of the Wave Field Far from the Sources
3.1.11 Extension to Quasi-Uniform Fields
3.1.12 Frequency-Angular Spectrum and Radiance
3.1.13 Uncertainty Relations for the Radiance
3.1.14 Equilibrium Thermal Electromagnetic Radiation
3.1.15 Statistical Definition of Radiance for an Arbitrary Wave Field in a Free Space
3.2 Transfer Equation for Radiation in a Free Space
3.2.1 Derivation of the Transfer Equation for Free Radiation
3.2.2 Conditions of Field Quasi-Uniformity
3.2.3 Radiometric Description of the Radiation of Plane Sources
3.2.4 Exact Solution for the Diffraction in a Half-Space
3.2.5 Relation of Transfer Theory with Exact Wave Theory
3.2.6 Generalized Radiance of Plane Sources and Nonclassic Radiometry
3.2.7 Diffraction and Interference of Coherent Radiation
3.2.8 Spatially Incoherent Random Source
3.2.9 Quasi-Uniform Sources
3.2.10 Diffraction of a Random Field at an Aperture. Van Cittert-Zernike Theorem
3.2.11 Radiant Intensity and the Inverse Problem for Correlation of Sources
3.2.12 Generalized Radiance for Paraxial Optical Systems
3.2.13 Uncertainty Relations and the Resolving Capacity in Radiometric Measurements
3.2.14 Coherent (Antenna) Measurements and Optical Heterodyning
3.2.15 Incoherent Radiometry
3.2.16 Time Averaging and Temporal Ergodicity
3.2.17 Fourier Spectroscopy
3.2.18 Spatial Averaging and Spatial Ergodicity
3.2.19 Self-Averaging Quantities
3.2.20 Measurements of Radiance by a Lens
3.2.21 Spatial Fourier Transform
3.3 Elements of Multiple Scattering Theory. The Dyson and Bethe-Salpeter Equations
3.3.1 Operator Form of Field Equations
3.3.2 Series of Perturbation Theory and the Born Approximation
3.3.3 Equations for Field Moments in Scattering Media. Perturbation Theory in the Denominator
3.3.4 Operators of Effective Inhomogeneities
3.3.5 Cumulant Relations
3.3.6 Operator Expansions
3.3.7 Energy Conservation Law and the Optical Theorem for Effective Inhomogeneities
3.3.8 Field Coherent Function Deep Inside the Medium
3.3.9 Group Expansions of Operators of Effective Inhomogeneities
3.3.10 Applicability of the One-Group Approximations
3.3.11 One-Group Approximation for the Dyson Equation. The Effective Wave Number
3.4 Discrete Scattering Media in Electrodynamic Problems
3.4.1 Media with Discrete Scatterers
3.4.2 Effective Medium Parameters in the Quasi-Static Limit
3.4.3 Exact Solution of the Scattering at a Cloud of Point Dipoles
3.4.4 Distribution Functions of Discrete Particles and the Thermodynamic N/V Limit
3.4.5 The Series of Multiple Scattering at an Ensemble of Particles
3.4.6 Cross Section of Scattering at an Ensemble of Particles. The Single Scattering Approximation
3.4.7 Dynamic Group Expansions and Mutual Influence Operators
3.4.8 Effective Inhomogeneity Operators for Sparse Discrete Media
3.4.9 One-Group Approximation and Weak Dispersion Formula for Correlated Particles
3.4.10 Operators of Effective Inhomogeneities for Media with Strong Fluctuations
3.4.11 Effective Parameters — Non-selfconsistent Approximation
3.4.12 Effective Parameters — Selfconsistent Approximation
3.4.13 Generalized Lorenz – Lorentz Formula
3.4.14 Applicability of Expansions of Effective Inhomogeneity Operators to Discrete Scatterers
3.5 Derivation of the Radiative Transfer Equation for a Scattering Medium
3.5.1 Historical Background
3.5.2 Starting Equations
3.5.3 Necessary Conditions for the Quasi-Uniformity of the Wave Field in a Scattering Medium
3.5.4 Geometrical Optics Asymptotics for the Second Moments of Radiation: Expansion of the Bethe-Salpeter Equation in Small Parameters
3.5.5 Equations of Successive Approximations
3.5.6 Transition to the Radiative Transfer Equation
3.5.7 Remark on the Optical Theorem
3.5.8 Transfer Equation and the Coherent (Average) Field
3.5.9 Sources of Thermal Radiation. The Fluctuation-Dissipation Theorem
3.5.10 Scattering Cross Section in the One Group Approximation
3.5.11 Scattering Cross Section and Sources of EM Radiation
3.5.12 Scattering Cross Section for Discrete Scatterers
3.5.13 Scattering from Moving Inhomogeneities
3.6 Correlation Functions of the Scattered Field. Diagram Interpretation of the Transfer Equation
3.6.1 Correlation Scattering Cross Sections. An Isolated Scatterer in a Free Space
3.6.2 Weak Isolated Scatterer
3.6.3 Scattering at Continuous Gaussian Fluctuations
3.6.4 Quasi-Homogeneous Weak Scatterer. Analogy with the van Cittert-Zernike Theorem
3.6.5 General Case of Non-Gaussian Fluctuations
3.6.6 Diagram Interpretation of the Radiative Transfer Equation
3.6.7 Radiative Transfer Equation and the Correlation Cross Section in Scattering Media
3.6.8 Algebraic and Diagram Forms of the Transfer Equation
3.6.9 Radiance and Correlation Functions of the Scattered Field
Problems
Chapter 4: Solution Techniques for Transfer Equations
4.1 Analytical Solutions: the Fourier and Invariant Embedding Methods
4.1.1 Rigorous Methods in Transfer Theory
4.1.2 An Exact Solution Obtained by the Fourier Method: Green’s Function for an Infinite Isotropically Scattering Medium
4.1.3 Method of Invariant Embedding
4.2 Radiative Transfer Equation in the Small-angle Approximation
4.2.1 Scattering of Narrow Beams at Large-Scale Inhomogeneities
4.2.2 Small-Angle Approximation and the Parabolic Equation
4.2.3 General Solution of the Radiative Transfer Equation in the Small-Angle Approximation
4.2.4 Diffusion in Angular Variables
4.2.5 Nonstationary Radiation and Pulse Scattering
4.2.6 Small-Angle Transfer Equation for a Medium with Continuous Fluctuations and Discrete Scatterers in the Straight-Ray Approximation
4.3 Diffusion Approximation
4.3.1 Elementary Derivation of Diffusion Approximation Equations
4.3.2 Diffusion Approximation for Green’s Function of an Unbounded Medium
4.3.3 Diffusion Approximation and the Model of Random Walks
4.3.4 Justification of Diffusion Approximation: Methods of Spherical Harmonics and Asymptotic Expansion. The Chain of Equations for Moments
4.4 Iterative Solution to the Equation of Radiative Transfer
4.4.1 Integral Form and Iterative Series for the Transfer Equation
4.4.2 Modified Born Approximation
4.4.3 Coherence Function of the Scattered Field
4.4.4 Conditions for Convergence of Iteration Series
4.5 Two-Flux Approximation
4.5.1 General
4.5.2 Equations of Two-Flux Theory
4.5.3 Boundary Conditions
4.5.4 Estimation of Parameters of Two-Flux Approximation Equations
4.5.5 Optically Soft Scattering Layer
4.6 Numerical and Statistical Solution Techniques
4.6.1 Numerical Methods for Solving the Transfer Equation
4.6.2 Expansion in Terms of Base Functions
4.6.3 The Discrete Ordinate Method
4.6.4 Statistical Modeling Methods
Problems
Chapter 5: Radiative Transfer and Coherent Effects
5.1 Coherent Backscattering Enhancement. Weak Localization
5.1.1 Coherent Channels of Backscattering Enhancement
5.1.2 Backscattering Enhancement
5.1.3 Weak Localization and Radiative Transfer Theory
5.1.4 Diagram Interpretation of Backscattering Enhancement. Maximal Crossed Diagrams
5.1.5 Observation of Weak Localization and Other Results
5.2 Incoherent Mechanisms of Backscattering Enhancement
5.2.1 General Description of Physical Mechanisms in Backscattering Enhancement
5.2.2 Statistical and Dynamic Enhancement Mechanisms
5.2.3 Single Particle Mechanisms
5.2.4 Collective Mechanisms
5.2.5 Geometrical Mechanisms
5.2.6 Focusing Mechanism
5.2.7 Autoadaptive Mechanism
5.2.8 Shadow Mechanism
5.2.9 Phase Mechanisms
5.2.10 Classification of Backscattering Enhancement Mechanisms
5.3 New Effects in Statistical Radiative Transfer
5.3.1 Background
5.3.2 Complex Gaussian Fields
5.3.3 Sources of Non-Gaussian Scattered Field
5.3.4 Scalar Memory Effect
5.3.5 Time Reversed Memory Effect
5.3.6 Long Correlation Effects for Intensities
5.3.7 Polarization Effects
5.3.8 Correlation Effects of Imaging in Diffuse Scattering Media
5.4 Strong Localization and Nonlinear Transfer Theory
5.4.1 Effect of Strong Localization
5.4.2 Nonlinear Bethe-Salpeter Equation
5.4.3 Self consistent Equation for Diffusion Coefficient
5.4.4 Anderson Transition and Scattering Regimes in the Region of Localization
5.4.5 Diffusion with Memory and Localization Criteria
Problems
Chapter 6: Related Problems
6.1 Turbid Media with Random Parameters. Secondary Averaging of the Transfer Equation
6.1.1 Transfer Equation with Random Parameters
6.1.2 Bleaching Effect in Turbid Media with Random Parameters
6.1.3 Transfer Equation with Effective Parameters
6.1.4 Manifestations of the Bleaching Effect
6.2 Thermal Radiation of Scattering Media and Surfaces
6.2.1 “Warming” and “Cooling” Effects in Scattering Media
6.2.2 Reduction of the Brightness Temperature of the Thermal Radiation of Antarctic Glaciers
6.2.3 Thermal Radio Emission of Rough Surfaces
6.3 Acoustical Problems
6.3.1 General Wave Nature of Radiative Transfer Theory
6.3.2 Ambient Ocean Noise
6.3.3 Acoustic Radiance of a Horizontally Uniform Field
6.3.4 Scattering from the Ocean Surface
6.4 Transfer Equations for Normal Waves in Multimode Waveguides
6.5 Miscellaneous Problems
6.5.1 Nonlinear Problems of Radiative Transfer Theory
6.5.2 Radiative Transfer Equation and Kinetic Wave Equations in Weak Turbulence Theory
6.5.3 Neutron Transfer Theory
Appendices
Appendix A: The Method of Geometrical Optics
Appendix B: The Quasi-Isotropic Approximation
Appendix C: The Ray Refractive Index
References
Notation
Index