Random Media and Boundaries, Softcover reprint of the original 1st ed. 1993 Unified Theory, Two-Scale Method, and Applications Springer Series on Wave Phenomena Series, Vol. 14

For a system consisting of a random medium with rough boundaries, the governing (Bethe-Salpeter) equation for boundary-value transport problems can be written in a form such that the medium and the boundaries are treatedon an equal footing. This enables several expressions for the solution to be obtained by interchanging the roles of the medium and the boundaries, thus allowing the most convenient one to be selected according to the specific situation and the information required. This book presents a unified theory based on the Bethe-Salpeter equation with particular attention being paid to: boundary-value problems of transport, layer problems, a fixed scatterer imbedded in a bounded random medium, construction of an optical scattering matrix for a complete system, and optical wave propagation in a turbulent medium. The last topic is treated in terms of first moment equations combined with the cluster expansion and, second, the two-scale method based on the Lagrange variational principle.

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.