An overview of methods for deriving the radiative transfer theory from the Maxwell equations. II: Approach based on the Dyson and Bethe–Salpeter equations

Abstract

a b s t r a c tIn this paper, the vector radiative transfer equation is derived by means of the vector integral Foldy equations describing the electromagnetic scattering by a group of particles. By assuming that in a discrete random medium the positions of the particles are statistically independent and by applying the Twersky approximation to the order-of-scattering expansion of the total field, we derive the Dyson equation for the coherent field and the ladder approximated Bethe-Salpeter equation for the dyadic correlation function. Then, under the far-field assumption for sparsely distributed particles, the Dyson equation is reduced to the Foldy integral equation for the coherent field, while the iterated solution of the Bethe-Salpeter equation ultimately yields the vector radiative transfer equation. (A. Doicu). tion to derive the Dyson equation for the coherent field and the ladder-approximated Bethe-Salpeter equation for the dyadic correlation function. The coherent field and the vector radiative transfer equation are then obtained by simplifying the Dyson and Bethe-Salpeter equations via the far-field assumption. The final section discusses the similarities of and differences between the two approaches to arrive at the same radiative transfer equation.We have tried to make this paper maximally self-contained while keeping its size manageable. To this end, we assume that the reader is already familiar with Ref.[1] and use the same conceptual base and notation. Neither Ref.[1] nor this second part are intended for a complete novice in the field of electromagnetic scattering; for the basics, we refer to the tutorial [8] and introductory text [9] . Dyson and Bethe-Salpeter equationsWe consider the same scattering geometry as in Ref. [1] . More precisely, a group of N identical, homogeneous, nonmagnetic particles with permittivity ε 2 are placed in a lossless, homogeneous, nonmagnetic, and isotropic medium with permittivity ε 1 and permeability μ 0 . The wavenumbers in the background medium and the particle are k 1 = ω √ ε 1 μ 0 and k 2 = ω √ ε 2 μ 0 , respectively, where ω is the angular frequency. The particles are centered at R 1 , R 2 , ... , R N , the origins of the particles are confined to a macroscopihttps://doi.

An overview of methods for deriving the radiative transfer theory from the Maxwell equations. III: Effects of random rough boundaries and packing density

“…In Ref. [2] it has been shown that the expression for the mass operator derived under the Foldy approximation for the exciting fields is the same as that obtained under the Twersky approximation applied to the Foldy equation for the total field. Both approaches also assume that the positions of the particles are statistically independent.…”

“…In the ladder approximation for the Bethe-Salpeter equation, the (tetradic) scattering intensity operator in Eq. (77) is given by [2]…”

“…In Refs. [1,2] , we defined through an angular spectrum representation for C . The procedure was to derive a series expansion for C from which the angular spectrum representation was apparent; the resulting series expansion for has led to a transport-type equation for this quantity.…”

“…In the quasi-uniform field approximation it is assumed that the dyadic correlation function varies weakly with R and varies strongly with ρ. When this assumption is made and the evanescent waves are neglected, the Wigner transform is of the form (2) so that by applying the inverse Wigner transform, we obtain the angular spectrum representation…”

“…In the effective and quasi-crystalline approximations, it has been shown that in the low frequency limit, i.e., p → 0, we have M pL (p) = M pT (p) = M p , so that the mass operator is of the form M p (p ) = M p I [11] . Consequently, k 2 pT (0) = k 2 pL (0) , and the second integral in the expression for G (R ) vanishes. From Eq.…”