Random Media and Boundaries

Random Media and Boundaries presents a theory of boundary-value transport problems based on the Bethe-Salpeter equation. The topics discussed include: layer problems, a fixed scatterer imbedded in a bounded medium, construction of an optical scattering matrix for a complete system, and optical wave propagation in a turbulent medium. This book will be of interest to graduate students and researchers.

1 Operator Representation of a Random Medium.- 1.1 Single Random Quantity.- 1.2 N Discrete Random Quantities.- 1.3 Random Function in Space.- 1.4 Multi-Component Random Medium.- 2 Waves in a Homogeneously Random Medium.- 2.1 Operator Representation of Statistical Green’s Functions in a Medium of Independent Particles.- 2.2 Green’s Functions of First and Second Orders.- 2.3 Bethe-Salpeter Equation in a General Random Medium.- 2.3.1 Optical Condition.- 2.3.2 Optical Expressions and Transport Equation.- 2.4 Random Layer with Free Boundaries.- 2.4.1 Boundary Condition.- 2.4.2 Addition of Scattering Matrices.- 2.5 Eigenfunction Expansions and Diffusion Approximation.- 2.5.1 Symmetries of the Eigenfunctions and Eigenvalues.- 2.5.2 Diffusion Eigenfunctions.- 2.5.3 Mode Expansions in a Homogeneous Random Medium.- 2.5.4 Orthogonality and Power Carried by the Mode Waves.- 3 Random Rough Boundaries.- 3.1 Rough Surface (One-Sided Boundary).- 3.1.1 Reflection Coefficient.- 3.1.2 Scattering Matrix of a Small Boss on a Smooth Boundary.- 3.1.3 Optical Conditions.- 3.1.4 Surface Wave and Power Equation.- 3.1.5 Derivation of Surface Impedance.- 3.1.6 Integral Equation for the Reflection Coefficient.- 3.1.7 Tangent Plane Method ― Case of a Large-Scale Rough Surface.- 3.2 Statistical Green’s Functions of First and Second Orders.- 3.2.1 Incoherent Scattering Matrix.- 3.2.2 Optical Condition.- 3.2.3 M and K for a Surface of Randomly Distributed Bosses.- 3.2.4 Continuation to the Outside Space.- 3.2.5 Rough Surface as an Effective Random Medium.- 3.2.6 Optical Expressions and Cross Sections per Unit Area.- 3.2.7 Optical Relations.- 3.2.8 Examples.- 3.3 Transmissible (Two-Sided) Rough Boundary.- 3.3.1 Basic Equations.- 3.3.2 Evaluation of the Surface Green’s Function.- 3.3.3 Power Equations.- 3.3.4 Statistical Green’s Functions of First and Second Orders.- 3.3.5 Continuation to the Outside Spaces.- 3.3.6 Scattering Cross Sections and Optical Relations.- 3.3.7 Case of a Slightly Random Boundary.- 4 System of Random Media and Rough Boundaries.- 4.1 Bethe-Salpeter Equation for the Entire System and Scattering Matrices.- 4.1.1 Statistical Green’s Functions.- 4.1.2 Optical Relations and a Dispersive Medium.- 4.1.3 Case of Three Random Layers.- 4.1.4 Scattering Matrices and Solutions.- 4.2 Effective Boundary Scattering Matrices in a Random Medium and Construction of Solutions.- 4.2.1 Case of Three Random Layers with Two Rough Boundaries.- 4.2.2 I(12+23) and Boundary Scattering Matrices.- 4.2.3 Another Expression for I(q+12+23).- 4.2.4 Addition Formulas of Scattering Matrices Utilized.- 4.2.5 Optical Relations of Random Layers.- 5 Optical Cross Sections of a Random Layer.- 5.1 Construction of the Cross Sections.- 5.1.1 Case Involving No Boundary Scattering.- 5.1.2 Case Involving Boundary Scattering.- 5.1.3 Optical Relations and Reciprocity.- 5.2 Application of the Diffusion Approximation.- 5.2.1 Boundary Condition of the Diffusion Equation.- 5.2.2 Boundary-Value Solution of the Diffusion Equation.- 5.2.3 Case of a Random Layer with Smooth Boundaries.- 5.2.4 Reciprocity.- 5.2.5 Boundary Condition when Media are Random on Both Sides of the Boundary.- 6 Fixed Scatterer.- 6.1 Basic Equations.- 6.2 Power Equations and Optical Relations.- 6.3 Optical Cross Section and Shadowing Effect.- 6.4 Observation of a Fixed Scatterer Embedded in a Semi-Infinite Random Layer.- 7 Forward Scattering Approximation.- 7.1 Moment Equations of a Light Wave in a Turbulent Medium 185 7.1.1 Turbulent Medium of Kolmogorov Spectrum.- 7.2 Solutions of the Moment Equations.- 7.2.1 Basic Equations.- 7.2.2 Phase Screen.- 7.2.3 Formal Generalization to a Turbulent Continuum.- 7.2.4 Exact Solutions of All Orders in a Special Case.- 7.3 High Order Intensity Moments and the Cluster Approximation.- 7.3.1 Intensity Fluctuation.- 7.3.2 Cluster Approximation.- 7.4 Probability Distribution Function of Intensity.- 7.4.1 Rice-Nakagami Distribution.- 7.4.2 Integral Representation of the Probability Density Function for Given , n = 1, 2, 3.- 7.4.3 as the Characteristic Function of ln(I).- 7.4.4 Intensity Distribution for a Beam Wave Purely in the State of Wandering.- 7.4.5 Intensity Distribution Function in a Turbulent Medium.- 7.4.6 Comparison of Theories to Experimental Results.- 7.5 Two-Scale Method.- 7.5.1 Exact Integral Representation of M22(1).- 7.5.2 Two-Scale Representation as an Effective Approximation.- 7.5.3 Two-Frequency Intensity Correlation.- 7.5.4 Third Order Intensity Correlation Function.- 7.5.5 Summary and Discussion.- References.