Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion

Abstract

CONTENTS I. Introduction A. Length scales B. Weak localization, closed paths and the backscatter cone. C. Anderson localization D. Correlation of different diffusons II. Macroscopics: the diffusion approximation A. Transmission through a slab and Ohm's law B. Diffusion propagator for slabs III. Mesoscopics: the radiative transfer equation A. Specific intensity B. Slab geometry 1. Isotropic scattering 2. Anisotropic scattering and Rayleigh scattering 3. The transport mean free path and the absorption length C. Injection depth and the improved diffusion approximation IV. Microscopics: wave equations, t matrix, and cross sections A. Schrödinger and scalar wave equations B. The t matrix and resonant point scatterers C. The t matrix as a series of returns D. Point scatterer in three dimensions 1. Second-order Born approximation 2. Regularization of the return Green's function 3. Resonances 4. Comparison with Mie scattering for scalar waves E. Cross sections and the albedo V. Green's functions in disordered systems A. Diagrammatic expansion of the self-energy B. Self-consistency VI. Transport in infinite media: isotropic scattering A. Ladder approximation to the Bethe-Salpeter equation B. Diffusion from the stationary ladder equation C. Diffusion coefficient and the speed of transport VII. Transport in a semi-infinite medium A. Plane wave incident on a semi-infinite medium B. Air-glass-medium interface C. Solutions of the Schwarzschild-Milne equation VIII. Transport through a slab A. Diffuse transmission B. Electrical conductance and contact resistance IX. The enhanced backscatter cone A. Milne kernel at nonzero transverse momentum B. Shape of the backscatter cone C. Decay at large angles D. Behavior at small angles X. Exact solution of the Schwarzschild-Milne equation A. The homogeneous Milne equation B. The inhomogeneous Milne equation C. Enhanced backscatter cone D. Exact solution for internal reflections in diffusive media E. Exact solution for very anisotropic scattering XI. Large index mismatch A. Diffuse reflected intensity B. Limit intensity and injection depth C. Comparison with the improved diffusion approximation D. Backscatter cone XII. Semiballistic transport XIII. Imaging of objects immersed in opaque media A. Spheres B. Cylinders XIV. Interference of diffusons: Hikami vertices A. Calculation of the Hikami four-point vertex B. Six-point vertex: H 6 50 C. Beyond the second-order Born approximation 51 D. Corrections to the conductivity 51 XV. Short range, long range, and conductance correlations: C 1 , C 2 , and C 3 51 A. Angular resolved transmission: the speckle pattern 52 B. Total transmission correlation: C 2 52 C. Conductance correlation: C 3 53 XVI. Calculation of correlation functions 53 A. Summary of diffuse intensities 54 B. Calculation of the C 1 correlation 55 C. Calculation of the C 2 correlation 55 1. Influence of incoming beam profile 56 2. Reflection correlations 57 D. Conductance Fluctuations: C 3 57 E. Calculation of the C 3 correlation function 58 XVII. Third cumulant of the total transmis...

Second harmonic generation in suspensions of spherical particles

“…This implies statistical independence of these paths which is generally accepted to be true for kl * >> 1 [17]. The discussion of phase matching in the case of dilute suspension at the end of the previous section also applies (with evident modifications) in the case of concentrated suspension, requiring (a) complete disorder in particle positions and (b) large (>> λ) interparticle distance for our analysis to hold.…”

Walk dimension for light in complex disordered media

“…It is well known that the diffuse lifetime in finite systems should scale as τ D ∝ (L + 2z e ) 2 , where z e is the so-called "extrapolation length" [5]. Since z e = 2 3 t for three-dimensional systems with index matched boundaries (neglecting internal reflections) [31], when L t the contribution of z e to L e is negligible (L e L) and thus z = d w = 2. However, for our homogeneous sample with t = 40 µm and thicknesses in the range 230 ≤ L ≤ 450 µm, we expect a difference between z and d w .…”

Exact electrodynamics versus standard optics for a slab of cold dense gas

“…Closely related numerical techniques based on the analysis of the eigenstates of the coupled system of the light and the atoms [15,[27][28][29][30][31][32] or density matrices and quantum trajectories [33][34][35] are also widely used today. Other ideas drawn from the theory of radiative transfer [36,37] and multiple scattering [38,39], enhanced with numerics, also have potential to make inroads into the questions about light propagation in atomic media [40].…”