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

In 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.

1. Introduction

In the first part of this series [1], we derived the vector radiative transfer equation for a discrete random layer with non-scattering boundaries by invoking at the very outset the algebraic far-field approximation to the Foldy equations [2, 3, 4] applicable to sparsely distributed particles. In other words, we assumed from the very beginning that each particle is located in the far zones of all the other particles and that the observation point is also located in the far zone of any particle. The coherent field and the vector radiative transfer equation were obtained by applying the Twersky approximation [5] to the far-field order-of-scattering expansion of the total field and by taking the configuration average of the resulting equations under the assumption that the positions of the particles are statistically independent.

In this paper we analyze how the use of the far-field assumption in the derivation of the radiative transfer equation can we delayed as far as possible. We therefore use the exact integral Foldy equations of electromagnetic scattering [6, 7, 8] formulated in terms of the transition dyadic of an individual particle. For a discrete random medium with uncorrelated particle positions, these are employed in conjunction with the Twersky approximation 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].

2. Dyson and Bethe–Salpeter equations

We consider the same scattering geometry as in Ref. [1]. More precisely, a group of N identical, homogeneous, nonmagnetic particles with permittivity ε₂ are placed in a lossless, homogeneous, nonmagnetic, and isotropic medium with permittivity ε₁ and permeability μ₀. The wavenumbers in the background medium and the particle are $k_1=\omega \sqrt{{\epsilon }_1{\mu }_0}$ and, $k_2=\omega \sqrt{{\epsilon }_2{\mu }_0}$ respectively, where ω is the angular frequency. The particles are centered at R₁, R₂, …, R_N, the origins of the particles are confined to a macroscopically plane-parallel layer with non-scattering plane boundaries. The domain occupied by particle i is denoted by D_i, the domain populated by the particles is denoted by D, and the particulate medium is characterized by the particle number concentration n₀. The particles have the same orientation, and the coordinate systems of the particles are aligned with the global coordinate system. The incident field is a plane electromagnetic wave with a wavenumber k₁, propagation direction $\widehat{s}$, and amplitude ${\mathcal{E}}_0(\widehat{s})$ i.e.,

$$E_0(r)={\mathcal{E}}_0(\widehat{s}){\text{e}}^{\text{j}{k}_1\widehat{s}\cdot r},$$ (1)

with $j=\sqrt{-1}$

Throughout this paper we will use integral-operator notation to write integral equations in a compact form [6]. This short-hand notation is introduced in Appendix 1 along with the Fourier transforms of vector and dyadic functions. Moreover, the transition dyadic of an individual particle, which is the key ingredient of our analysis, is defined in Appendix 2 (see also the recent tutorial [8]).

The frequency-domain scattering by a group of fixed particles can be described by the vector Foldy equations [2, 3, 7, 8] which can be formulated with respect to either the electric fields or the dyadic Green’s functions. For the electric fields, the Foldy equations read

$$E=E_0+\sum _i{\overline{G}}_0{\overline{T}}_i{E}_{\text{exc}i},$$ (2)

$$E_{\text{exc}i}=E_0+\sum _{j\ne i}{\overline{G}}_0{\overline{T}}_j{E}_{\text{exc}j},$$ (3)

where E is the total field, E₀ is the incident field, E_exci is the field “exciting” particle i, ${\overline{\mathbf{\text{T}}}}_i$ is the transition dyadic of particle i, ${\overline{\mathbf{\text{G}}}}_0$ is the dyadic Green’s function in free space, and the summations run implicitly from 1 to N. The transition dyadic can be thought of as being a unique scattering ID of a particle in that it expresses the field scattered by the particle everywhere in space in terms of the field inside the particle [8]. Eq. (2) gives the total field, while Eq. (3) is an integral equation for the exciting fields. For the dyadic Green’s functions, the Foldy equations are [6]

$$\overline{G}={\overline{G}}_0+\sum _i{\overline{G}}_0{\overline{T}}_i{\overline{G}}_i,$$ (4)

$${\overline{G}}_i={\overline{G}}_0+\sum _{j\ne i}{\overline{G}}_0{\overline{T}}_j{\overline{G}}_j,$$ (5)

where $\overline{G}$ is the dyadic Green’s function of the entire group of particles, and ${\overline{G}}_i$ is the ith particle Green’s dyadic.

The scattering by a discrete random medium is governed by the Dyson equations for the coherent field 〈E(r)〉 and the average dyadic Green’s function $\langle \overline{G}\left(r,r\prime \right)\rangle$ as well as by the Bethe–Salpeter equation for the dyadic correlation function $\langle E(r)\otimes E^{\star }\left(r\prime \right)\rangle$ where ⊗ is the dyadic product sign and the asterisk stands for “complex conjugate”. The kernels of these integral equations are the dyadic mass operator $\overline{\mathbf{\text{M}}}$ and the tetradic scattering intensity operator $\overline{\overline{I}}$ respectively [6]. The derivation of these integral equations under the Twersky approximation is given below.

2.1. The Dyson equation for the coherent field

Consider the Foldy equations for the total field (2) and the exciting fields (3). Inserting the iterated solution of the exciting fields equation

$$\begin{gathered} E_{\text{exc}i}=E_0+\sum _{j\ne i}{\overline{G}}_0{\overline{T}}_j{E}_0 \\ +\sum _{j\ne i}\sum _{k\ne j}{\overline{G}}_0{\overline{T}}_j{\overline{G}}_0{\overline{T}}_k{E}_0+\cdots \end{gathered}$$ (6)

into the total field equation, and employing the Twersky approximation [5], gives the following order-of-scattering expansion for the total field:

$$\begin{array}{ll}E\hfill & =E_0+\sum _i{\overline{G}}_0{\overline{T}}_i{E}_0+\sum _i\sum _{j\ne i}{\overline{G}}_0{\overline{T}}_i{\overline{G}}_0{\overline{T}}_j{E}_0\hfill \\ \hfill & +\sum _i\sum _{j\ne i}\sum _{k\ne i,j}{\overline{G}}_0{\overline{T}}_i{\overline{G}}_0{\overline{T}}_j{\overline{G}}_0{\overline{T}}_k{E}_0+\cdots .\hfill \end{array}$$ (7)

Recall that the Twersky approximation is based on keeping only the self-avoiding multi-particle sequences and is valid in the limit N → ∞ [1, 5]. Hereafter, we assume that the Neumann series for the total field (7) converges. Denoting

$${\overline{T}}_i\left(r,r\prime \right)=r\stackrel{i}{°}r{\prime }^{\stackrel{\text{def}}{=}}\stackrel{i}{°},$$

we illustrate the order-of-scattering expansion for the total field as

(8)

Assuming full ergodicity of the N-particle ensemble [9, 10] and taking the conditional configuration average of Eq. (7) under the assumption that the positions of the particles are uncorrelated, yields

$$\begin{array}{ll}\langle E\rangle \hfill & =E_0+n_0{\int }_D{\overline{G}}_0{\overline{T}}_i{E}_0{\text{d}}^3{R}_i\hfill \\ \hfill & +n_0^2{\int }_D{\overline{G}}_0{\overline{T}}_i{\overline{G}}_0{\overline{T}}_j{E}_0{\text{d}}^3{R}_j{\text{d}}^3{R}_i\hfill \\ \hfill & +n_0^3{\int }_D{\overline{G}}_0{\overline{T}}_i{\overline{G}}_0{\overline{T}}_j{\overline{G}}_0{\overline{T}}_k{E}_0{\text{d}}^3{R}_k{\text{d}}^3{R}_j{\text{d}}^3{R}_i+\cdots .\hfill \end{array}$$ (9)

In explicit form, the sum of the first two terms in the series (9) is

$$E_1(r)=E_0(r)+n_0{\int }_D\left[{\int }_{Di}{\overline{G}}_0\left(r,r_1\right)\cdot {\overline{T}}_i\left(r_1,r_2\right)\cdot E_0\left(r_2\right){\text{d}}^3{r}_1{\text{d}}^3{r}_2\right]{\text{d}}^3{R}_i.$$ (10)

Extending the domain of definition of the transition dyadic ${\overline{T}}_i$ to the whole D according to

$${\overline{T}}_i\left(r_1,r_2\right)=\overline{0}\text{for}{r}_1\text{and}/\text{or}{r}_2\notin D_i,$$ (11)

where $\overline{0}$ is the zero dyad, and interchanging the order of the integrations in Eq. (10) gives

$$\begin{array}{ll}{E}_1(r)\hfill & =E_0(r)+n_0{\int }_D{\overline{G}}_0\left(r,r_1\right)\hfill \\ \hfill & \cdot \left[{\int }_D{\overline{T}}_i\left(r_1,r_2\right){\text{d}}^3{R}_i\right]\cdot E_0\left(r_2\right){\text{d}}^3{r}_1{\text{d}}^3{r}_2\hfill \\ \hfill & =E_0(r)+n_0{\int }_D{\overline{G}}_0\left(r,r_1\right)\cdot \overline{M}\left(r_1,r_2\right)\cdot E_0\left(r_2\right){\text{d}}^3{r}_1{\text{d}}^3{r}_2,\hfill \end{array}$$ (12)

where the so-called dyadic mass operator is defined by [6]

$$\overline{M}\left(r_1,r_2\right)=n_0{\int }_D{\overline{T}}_i\left(r_1,r_2\right){\text{d}}^3{R}_i.$$ (13)

Thus, the dyadic mass operator is the integral of the transition dyadic of a particle over the position of the particle times the particle number density. Applying the same procedure to all terms in the series (9), we obtain the Dyson equation for the coherent field:

$$\begin{array}{ll}\langle E\rangle \hfill & =E_0+n_0{\int }_D{\overline{G}}_0{\overline{T}}_i{E}_0{\text{d}}^3{R}_i\hfill \\ \hfill & +n_0^2{\int }_D{\overline{G}}_0{\overline{T}}_i{\overline{G}}_0{\overline{T}}_j{E}_0{\text{d}}^3{R}_j{\text{d}}^3{R}_i+\cdots \hfill \\ \hfill & =E_0+{\overline{G}}_0\overline{M}{E}_0+{\overline{G}}_0\overline{M}{\overline{G}}_0\overline{M}{E}_0+\cdots \hfill \\ \hfill & =E_0+{\overline{G}}_0\overline{M}\langle E\rangle .\hfill \end{array}$$ (14)

With the notation

the diagramatic derivation of the Dyson equation (14) is

(15)

where the dyadic mass operator is represented as

$$\overline{M}\left(r,r\prime \right)=\langle \sum _{i}r\stackrel{i}{°}r\prime \rangle \stackrel{\text{def}}{=}\langle \sum _i\stackrel{i}{°}\rangle =\odot$$

and, according to the assumption that the positions of the particles are uncorrelated, we used the digramatic computation rule

Employing the same arguments for the Foldy equations for the dyadic Green’s function as given by Eqs. (4) and (5), and defining

we find

(16)

which is a diagramatic representation of the Dyson equation for the configuration-averaged dyadic Green’s function:

$$\langle \overline{G}\rangle ={\overline{G}}_0+{\overline{G}}_0\overline{M}\langle \overline{G}\rangle .$$ (17)

2.2. The Bethe–Salpeter equation for the dyadic correlation function

The Dyson equation for the coherent field has been obtained by taking the configuration average of the total field under the Twersky approximation as well as Assuming that the positions of the particles are uncorrelated. In a similar fashion, an equation governing a second-order moment of the field, the so-called Bethe–Salpeter equation, can be obtained.

The derivation of the Bethe–Salpeter equation for the dyadic correlation function under the ladder approximation parallels that described in Ref. [1]. With S being the set of all N random scatterers (i) we consider two disjoint subsets A₀ and B₀ of S, i.e., A₀ ∩ B₀ = Ø, (ii) we fix a particle i, and consider two disjoint subsets A_i and B_i of S_i = S \ {i}, (iii) we fix two particles i and j, and consider two disjoint subsets A_ij and B_ij of S_ij = S \ {i, j}, and so on. The sum of all scattering paths going through the particles in the sets A₀, A_i ∪ {i}, A_ij ∪ {i, j}, etc. gives the direct field E, while the sum of all scattering paths going through the particles in the sets B, B_i ∪ {i}, etc. gives the complex conjugate field E^⋆. As the sets are disjoint, the dyadic correlation function $\langle E(r)\otimes E^{\star }\left(r\prime \right)\rangle$ computes as

$$\begin{array}{l}\langle E(r)\otimes E^{\star }\left(r\prime \right)\rangle \hfill \\ ={\Sigma }_{A_0}{\Sigma }_{B_0}{\left\{E(r)\right\}}_{A_0}\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_{B_0}\hfill \\ +n_0{\int }_D{\Sigma }_{A_i}{\Sigma }_{B_i}{\left\{E(r)\right\}}_{A_i}\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_{B_i}{\text{d}}^3{R}_i\hfill \\ +n_0^2{\int }_D{\Sigma }_{A_{ij}}{\Sigma }_{B_{ij}}{\left\{E(r)\right\}}_{A_{ij}}\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_{B_{ij}}{\text{d}}^3{R}_j{\text{d}}^3{R}_i+\cdots ,\hfill \end{array}$$ (18)

where {E(r)}_A is the configuration average of the fields corresponding to all self-avoiding paths $\mathcal{P}(A)$ going through the particles in the set A taken over the positions of the particles in the set A. The sums ∑_A0 ∑_B0 involve all possible realizations of the sets A₀ and B₀, and for each realization, only those pairs of self-avoiding paths $\left(\mathcal{P}\left(A_0\right),\mathcal{P}\left(B_0\right)\right)$ which do not appear in previous realizations of A₀ and B₀ are taken into account. In the following, we assume that for large N, we can approximate

$${\Sigma }_{A_0}{\Sigma }_{B_0}{\left\{E(r)\right\}}_{A_0}\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_{B_0}\approx {\left\{E(r)\right\}}_S\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_S.$$ (19)

Applying the above assumption to all terms in the series (18) yields

$$\begin{array}{l}\langle E(r)\otimes E^{\star }\left(r\prime \right)\rangle \hfill \\ ={\left\{E(r)\right\}}_S\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_S+n_0{\int }_D{\{E(r)\}}_{S_i}\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_{S_i}{\text{d}}^3{R}_i\hfill \\ +n_0^2{\int }_D{\{E(r)\}}_{S_{ij}}\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_{S_{ij}}{\text{d}}^3{R}_j{\text{d}}^3{R}_i+\cdots .\hfill \end{array}$$ (20)

Next we compute the configuration average of the field over the positions of the particles in the sets S_i, S_ij, etc., which are the subsets of S with one, two, and more fixed particles. Considering the average field {E(r)}S_i, we let $S_i^a$ and $S_i^b$ be two disjoint subsets of S_i, with the property that the paths connecting the observation point r and particle i go through all particles in the subset $S_i^a$ and the paths connecting the entrance point (the first particle struck by the incident field) and particle i go through all particles in the subset $S_i^b$. In computing {E(r)}S_i we make an assumption which is similar to that in Eq. (19): the configuration average of a field taken over the positions of the particles in a subset A of S is computed by extending the sum over the particles in the subset A to the whole set S. Diagramatically, we have

(21)

that is,

$${\left\{E(r)\right\}}_{S_i}={\int }_{D_i}\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r_2\right)\cdot \langle E\left(r_2\right)\rangle {\text{d}}^3{r}_1{\text{d}}^3{r}_2.$$ (22)

Using the dyadic identities

$$(\overline{A}\cdot a)\otimes (\overline{B}\cdot b)=(\overline{A}\otimes \overline{B})\cdot (a\otimes b),$$ (23)

$$(\overline{A}\cdot \overline{C})\otimes (\overline{B}\cdot \overline{D})=(\overline{A}\otimes \overline{B})\cdot (\overline{C}\otimes \overline{D}),$$ (24)

extending the domain of definition of the transition dyadic ${\overline{T}}_i$ to the whole D, and interchanging the order of integrations, we find that the first term in the series (20) is given by

$$\begin{array}{l}\begin{array}{l}{n}_0{\int }_D\{E(r){\}}_{S_i}\otimes {\left\{E^{\star }\left(r\prime \right)\right\}}_{S_i}{\text{d}}^3{R}_i\hfill \\ =n_0{\int }_D\{{\int }_{D_i}\left[\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r_2\right)\right]\otimes \left[\langle {\overline{G}}^{\star }\left(r\prime ,r_1^{\prime }\right)\rangle \cdot {\overline{T}}_i^{\star }\left(r_1^{\prime },r_2^{\prime }\right)\right]\hfill \\ \cdot \left[\langle E\left(r_2\right)\rangle \otimes \langle E^{\star }\left(r_2^{\prime }\right)\rangle \right]{\text{d}}^3{r}_1{\text{d}}^3{r}_2{\text{d}}^3{r}_1^{\prime }{\text{d}}^3{r}_2^{\prime }\}{\text{d}}^3{R}_i\hfill \end{array}\\ ={\int }_D\left[\langle \overline{G}\left(r,r_1\right)\rangle \otimes \langle {\overline{G}}^{\star }\left(r\prime ,r_1^{\prime }\right)\rangle \right]\cdot \overline{\overline{I}}\left(r_1,r_2,r_1^{\prime },r_2^{\prime }\right)\\ \cdot \left[\langle E\left(r_2\right)\rangle \otimes \langle E^{\star }\left(r_2^{\prime }\right)\rangle \right]{\text{d}}^3{r}_1{\text{d}}^3{r}_2{\text{d}}^3{r}_1^{\prime }{\text{d}}^3{r}_2^{\prime },\end{array}$$ (25)

where the scattering intensity operator is defined by

$$\overline{\overline{I}}\left(r_1,r_2,r_1^{\prime },r_2^{\prime }\right)=n_0{\int }_D{\overline{T}}_i\left(r_1,r_2\right)\otimes {\overline{T}}_i^{\star }\left(r_1^{\prime }{r}_2^{\prime }\right){\text{d}}^3{R}_i.$$ (26)

To compute the configuration average with two fixed particles we proceed analogously. In this case, we find

(27)

or explicitly,

$$\begin{array}{ll}{\left\{E(r)\right\}}_{S_{ij}}\hfill & ={\int }_{D_i}\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r_2\right)\cdot \langle \overline{G}\left(r_2,r_3\right)\rangle \cdot {\overline{T}}_j\left(r_3,r_4\right)\hfill \\ \hfill & \cdot \langle E(r_4\rangle ){\text{d}}^3{r}_1{\text{d}}^3{r}_2{\text{d}}^3{r}_3{\text{d}}^3{r}_4.\hfill \end{array}$$ (28)

We thus obtain

$$\begin{array}{l}{n}_0^2{\int }_D\{E(r){\}}_{S_{ij}}\otimes {\left\{E^{\star }\left(r^{\prime }\right)\right\}}_{S_{ij}}{\text{d}}^3{R}_i{\text{d}}^3{R}_j\hfill \\ ={\int }_D\left[\langle \overline{G}\left(r,r_1\right)\rangle \otimes \langle {\overline{G}}^{\star }\left(r^{\prime },r_1^{\prime }\right)\rangle \right]\cdot \overline{\overline{I}}\left(r_1,r_2,r_1^{\prime },r_2^{\prime }\right)\hfill \\ \cdot \left[\langle \overline{G}\left(r_2,r_3\right)\rangle \otimes \langle {\overline{G}}^{\star }\left(r_2^{\prime },r_3^{\prime }\right)\rangle \right]\cdot \overline{\overline{I}}\left(r_3,r_4,r_3^{\prime },r_4^{\prime }\right)\hfill \\ \cdot \left[\langle E\left(r_4\right)\rangle \otimes \langle E^{\star }\left(r_4^{\prime }\right)\rangle \right]{\text{d}}^3{r}_1{\text{d}}^3{r}_2{\text{d}}^3{r}_1^{\prime }{\text{d}}^3{r}_2^{\prime }{\text{d}}^3{r}_3{\text{d}}^3{r}_4{\text{d}}^3{r}_3^{\prime }{\text{d}}^3{r}_4^{\prime }\hfill \end{array}$$ (29)

Note that the ladder approximation for the dyadic correlation function fundamentally relies on the assumption that in Eq. (29), the order of the particles i and j in the expressions of {E}S_ij and {E^⋆}S_ij is the same, or equivalently, that the ordered set of connected particles {i,j} is associated with both the direct and the complex-conjugate field. The same is true for the other terms in Eq. (20). By keeping only such ladder contributions and summing them up, we deduce that Eq. (20) is the iterated solution of the integral equation

$$\begin{array}{ll}\langle E(r)\otimes E^{\star }\left(r\prime \right)\rangle \hfill & =\langle E(r)\rangle \otimes \langle E^{\star }\left(r\prime \right)\rangle \hfill \\ \hfill & +{\int }_D\left[\langle \overline{G}\left(r,r_1\right)\rangle \otimes \langle {\overline{G}}^{\star }\left(r\prime ,r_1^{\prime }\right)\rangle \right]\cdot \overline{\overline{I}}\left(r_1,r_2,r_1^{\prime },r_2^{\prime }\right)\hfill \\ \hfill & \cdot \langle E\left(r_2\right)\otimes E^{\star }\left(r_2^{\prime }\right)\rangle {\text{d}}^3{r}_1{\text{d}}^3{r}_2{\text{d}}^3{r}_1^{\prime }{\text{d}}^3{r}_2^{\prime },\hfill \end{array}$$ (30)

with the scattering intensity operator being given by Eq. (26). Eq. (30) is the Bethe–Salpeter equation for the dyadic correlation function.

Using the following diagramatic representation for the dyadic correlation function $\langle E(r)\otimes E^{\star }\left(r\prime \right)\rangle$

we illustrate the Bethe–Salpeter equation (30) and its iterated solution as

where, in the ladder approximation, the scattering intensity operator is

and for example, the following product rule for the upper and lower scattering path applies

3. Coherent field

The derivation of the Dyson equation (14) is based on the Twersky approximation. An alternative derivation makes use of the Foldy approximation for the exciting fields. To show this, we consider the Foldy equation for the total field (cf. Eq. (2))$E=E_0+{\sum }_i{\overline{G}}_0{\overline{T}}_i{E}_{\text{exc}i}$. The exciting field E_exci is a function of all particle positions, i.e., E_exci = E_exci(R₁,…,R_N), while the transition dyadic ${\overline{T}}_i$ is only a function of the ith particle, i.e, ${\overline{T}}_i={\overline{T}}_i\left(R_i\right)$. Taking the configuration average of this equation under the assumption that the positions of the particles are uncorrelated, we obtain

$$\langle E\rangle =E_0+n_0{\int }_D{\overline{G}}_0{\overline{T}}_i{\langle E_{\text{exc}i}\rangle }_i{\text{d}}^3{R}_i,$$ (31)

where 〈E_exci〉_i is the conditional probability of the exciting field with the position of particle i held fixed. Employing the Foldy approximation

$${\langle E_{\text{exc}i}\rangle }_i=\langle E\rangle ,$$ (32)

the integral equation (31) becomes

$$\langle E\rangle =E_0+n_0{\int }_D{\overline{G}}_0{\overline{T}}_i\langle E\rangle {\text{d}}^3{R}_i.$$ (33)

From Eq. (14), we infer that Eq. (33) is the Dyson equation with the dyadic mass operator as in Eq. (13).

The next step is to derive the Foldy integral equation for the coherent field (see [1, 11]). Now, the far-field approximation will come into play. In order to simplify the notation we set, as usual, E_c (r) = 〈E(r)〉and express the Dyson equation (33) in explicit form as

$$E_{\text{c}}(r)=E_0(r)+n_0{\int }_D\left[{\int }_{D_i}{\overline{G}}_0\left(r,r_1\right)\cdot {\overline{T}}_i\left(r_1,r_2\right)\cdot E_{\text{c}}\left(r_2\right){\text{d}}^3{r}_1{\text{d}}^3{r}_2\right]{\text{d}}^3{R}_i.$$ (34)

In Eq. (34) we used the fact that the support of ${\overline{T}}_i$ is D_i, so that the integration domain for r₁ and r₂ is D_i. To compute the integral over D_i, we choose the origin of the coordinate system at O_i, and let r = r_i + R_i, r₁ = r_1i + R_i, and r₂ = r_2i + R_i (Fig. 2). Then, using ${\overline{G}}_0\left(r,r_1\right)={\overline{G}}_0\left(r_i,r_{1i}\right)$ and ${\overline{T}}_i\left(r_1,r_2\right)=\overline{T}\left(r_1-R_i,r_2-R_i\right)=\overline{T}\left(r_{1i},r_{2i}\right)$ where $\overline{T}$ is the transition dyadic of a particle centered at the origin of the coordinate system, we deduce that the integral over D_i is the field scattered by particle i when excited by the coherent field, i.e.,

$$E_{\text{sct}i}(r)={\int }_{D_i}{\overline{G}}_0\left(r_i,r_{1i}\right)\cdot \overline{T}\left(r_{1i},r_{2i}\right)\cdot E_{\text{c}}\left(r_2\right){\text{d}}^3{r}_{1i}{\text{d}}^3{r}_{2i}.$$ (35)

Figure 2:

The geometry showing the relevant quantities for computing the integral (35).

An approximate representation for the coherent field E_c(r₂) in D_i, can be obtained in the framework of the characteristic waves method (Appendix 4). In this setting, the coherent field propagating along the incidence direction $\widehat{s}$ can be written as

$$E_{\text{c}}(r)=\sum _{n=1}^2{\text{e}}^{\text{j}{K}_n(\widehat{s})\widehat{s}\cdot r}{\mathfrak{E}}_{n\text{T}}(\widehat{s}),$$ (36)

where $K_n(\widehat{s})$ is the effective wavenumber, and ${\mathfrak{E}}_{n\text{T}}(\widehat{s})$ is the transverse characteristic wave polarization associated with $K_n(\widehat{s})$ [6]. Then, supposing that for a sparse concentration of particles, $K_n(\widehat{s})\approx k_1$, we approximate

$$\begin{array}{ll}{K}_n(\widehat{s})\widehat{s}\cdot r_2\hfill & =K_n(\widehat{s})\widehat{s}\cdot R_i+K_n(\widehat{s})\widehat{s}\cdot r_{2i}\hfill \\ \hfill & \approx K_n(\widehat{s})\widehat{s}\cdot R_i+k_1\widehat{s}\cdot r_{2i},\hfill \end{array}$$ (37)

implying

$$E_{\text{c}}\left(r_2\right)\approx {\text{e}}^{\text{j}{k}_1\widehat{s}\cdot r_{2i}}{E}_{\text{c}}\left(R_i\right),\widehat{s}\cdot E_{\text{C}}\left(R_i\right)=0.$$ (38)

Note that this result which states that the coherent field in D_i can be approximated analytically by a plane electromagnetic wave with amplitude E_c(R_i), wavenumber k₁, and propagation direction $\widehat{s}$ has also been established in Ref. [1], provided that the Twersky approximation is used in conjunction with the far-field Foldy equations. Also note that for sparse media, the assumption $K_n(\widehat{s})\approx k_1$ is typical of the characteristic waves method (Appendix 4). Furthermore, using the far-field representation for the dyadic Green’s function

$${\overline{G}}_0\left(r_i,r_{1i}\right)=\left(\overline{I}-{\widehat{r}}_i\otimes {\widehat{r}}_i\right)\frac{{\text{e}}^{\text{j}{k}_1{r}_i}}{4\pi r_i}{\text{e}}^{-\text{j}{k}_1{\widehat{r}}_i\cdot r_{1i}},$$ (39)

and the relation between the far-field scattering dyadic $\overline{A}$ and the Fourier transform of the transition dyadic ${\overline{T}}_p$ (cf. Eq. (129) of Appendix 3),

$$\overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot \mathcal{E}(\widehat{s})=\frac{1}{4\pi }\left(\overline{I}-{\widehat{r}}_i\otimes {\widehat{r}}_i\right)\cdot {\overline{T}}_p\left(k_1{\widehat{r}}_i,k_1\widehat{s}\right)\cdot \mathcal{E}(\widehat{s}),$$ (40)

where in general, $\mathcal{E}(\widehat{s})$ is a vector field orthogonal to the incidence direction $\widehat{s}$, i.e., $\widehat{s}\cdot \mathcal{E}(\widehat{s})=0$ we obtain

$$\begin{array}{ll}{E}_{\text{sct}i}(r)\hfill & ={\int }_{D_i}{\overline{G}}_0\left(r,r_1\right)\cdot {\overline{T}}_i\left(r_1,r_2\right)\cdot E_{\text{c}}\left(r_2\right){\text{d}}^3{r}_1{\text{d}}^3{r}_2\hfill \\ \hfill & =\frac{{\text{e}}^{\text{j}{k}_1{r}_i}}{r_i}\overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right).\hfill \end{array}$$ (41)

The above result is also valid for any vector field E(r2) which can be approximated analytically by a plane electromagnetic wave in D_i, that is, for r₂ ∈ D_i, we have $E\left(r_2\right)=\text{exp}\left(\text{j}{k}_1\widehat{s}\cdot r_{2i}\right)E\left(R_i\right)$ E(R_i) with $\widehat{s}\cdot E\left(R_i\right)=0$. Substituting Eq. (41) in Eq. (34), we obtain the Foldy integral equation for the coherent field

$$E_{\text{c}}(r)=E_0(r)+n_0{\int }_D{g}_0\left(r_i\right)\overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right){\text{d}}^3{R}_i,$$ (42)

where g₀(r_i) = exp(jk₁r_i)/r_i.

To solve the Foldy integral equation, we apply the Helmholtz operator $\Delta +k_1^2$. Using $\Delta E_0(r)+k_1^2{E}_0(r)=0$, where 0 is a zero vector, and

$$\Delta g_0\left(r_i\right)+k_1^2{g}_0\left(r_i\right)=-4\pi \delta \left(r-R_i\right),$$ (43)

we get

$$\Delta E_{\text{c}}(r)+\left[k_1^2\overline{I}+4\pi n_0\overline{A}(\widehat{s},\widehat{s})\right]\cdot E_{\text{c}}(r)=0.$$ (44)

Finally, for, n₀ ≪ 1 we approximate

$$k_1^2\overline{I}+4\pi n_0\overline{A}(\widehat{s},\widehat{s})\approx {\left[k_1\overline{I}+\frac{2\pi }{k_1}{n}_0\overline{A}(\widehat{s},\widehat{s})\right]}^2,$$ (45)

and infer that the solution to Eq. (44) is

$$E_{\text{c}}(r)=\text{exp}\left\{\text{j}\left[k_1\overline{I}+\frac{2\pi }{k_1}{n}_0\overline{A}(\widehat{s},\widehat{s})\right]s(r,-\widehat{s})\right\}\cdot E_{\text{c}}\left(r_A\right),$$ (46)

where $s=s(r,-\widehat{s})=\widehat{s}\cdot \left(r-r_A\right)$, and r_A is the point where the straight line with the direction vector $-\widehat{s}$ going through the observation point crosses the lower boundary of the layer. This result coincides with that obtained in Ref. [1].

In the scalar case, an alternative method for solving the Foldy integral equation (42) can be found in Ref. [11]. In the vector case, an adapted version of this more “cumbersome” solution method is given in Appendix 5. Instead of solving the Foldy integral equation, the coherent field can be computed in the framework of the characteristic wave method discussed in Appendix 4. In this approach, the dispersion equation for the effective wavenumber as well as a general representation for the coherent field are formulated in terms of the dyadic mass operator.

4. Second-order moment of the electromagnetic field

To obtain the vector radiative transfer equation we will use the ladder approximated Bethe–Salpeter equation for the dyadic correlation function, or more specifically, an iterated solution of this integral equation. Since we intend to use the far-field approximation from now on, we will first establish some auxiliary results involving integrals of the average dyadic Green’s function in the far-field region. Actually, for a fixed particle i, we will compute the integral

$${\int }_{D_i}\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r_2\right)\cdot E_{\text{c}}\left(r_2\right){\text{d}}^3{r}_1{\text{d}}^3{r}_2,$$ (47)

when the observation point r is in the far-field region of D_i.

4.1. Integrals of the average dyadic Green’s function in the far-field region

The Dyson equation for the average dyadic Green’s function is (cf. Eq. (22))

$$\langle \overline{G}\left(r,r\prime \right)\rangle ={\overline{G}}_0\left(r,r\prime \right)+n_0{\int }_D[{\int }_{D_j}{\overline{G}}_0\left(r,r_1\right)\cdot {\overline{T}}_j\left(r_1,r_2\right)\cdot \langle \overline{G}\left(r_2,r\prime \right)\rangle {\text{d}}^3{r}_1{\text{d}}^3{r}_2]{\text{d}}^3{R}_j.$$ (48)

For a fixed i, we right-multiply Eq. (48) by ${\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right)$ with $r\prime ,r″\in D_i$, and integrate over r′ and r′′. Taking into account that the support of ${\overline{T}}_i$ is D_i and that of ${\overline{T}}_j$ is D_j, and interchanging the order of integration, we obtain the integral equation

$$\begin{array}{l}{\int }_{D_i}^{}\langle \overline{G}\left(r,r\prime \right)\rangle \cdot {\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right){\text{d}}^{3}r\prime {\text{d}}^{3}r″\\ ={\int }_{D_i}{\overline{G}}_0\left(r,r\prime \right)\cdot {\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right){\text{d}}^{3}r\prime {\text{d}}^{3}r″\\ +n_0{\int }_D\{{\int }_{D_j}{\overline{G}}_0\left(r,r_1\right)\cdot {\overline{T}}_j\left(r_1,r_2\right)\\ \cdot \left[{\int }_{D_i}\langle \overline{G}\left(r_2,r\prime \right)\rangle \cdot {\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right){\text{d}}^{3}r\prime \text{d}\prime r″\right]{\text{d}}^3{r}_1{\text{d}}^3{r}_2\}{\text{d}}^3{R}_j.\end{array}$$ (49)

To solve Eq. (49) for $\langle \overline{G}\left(r,r\prime \right)\rangle$ when the source point r′ is in D_i and the observation point r is in the far-field region of D_i, we seek a solution in the form

$$\langle \overline{G}\left(r,r\prime \right)\rangle =\overline{X}\left(r,R_i\right)\cdot {\overline{G}}_0\left(r,r\prime \right).$$ (50)

Thus, we assume that the unknown dyadic $\overline{X}\left(r,R_i\right)$ depends on the observation point r and the particle position R_i, but not on the source point r′. Moreover, we suppose that $\overline{X}\left(r,R_i\right)$ is a transverse dyadic with respect to the direction ${\widehat{r}}_i$, that is, ${\widehat{r}}_i\cdot \overline{X}\left(r,R_i\right)=0$ with r_i = r − R_i. Inserting Eq. (50) in Eq. (49) gives

$$\begin{array}{l}\overline{X}\left(r,R_i\right)\cdot {\int }_{D_i}{\overline{G}}_0\left(r,r\prime \right)\cdot {\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right){\text{d}}^{3}r\prime {\text{d}}^{3}r″\\ ={\int }_{D_i}{\overline{G}}_0\left(r,r\prime \right)\cdot {\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right){\text{d}}^{3}r\prime {\text{d}}^{3}r″\\ +n_0{\int }_D\{{\int }_{D_j}{\overline{G}}_0\left(r,r_1\right)\cdot {\overline{T}}_j\left(r_1,r_2\right)\cdot \overline{X}\left(r_2,R_i\right)\\ \cdot \left[{\int }_{D_i}{\overline{G}}_0\left(r_2,r\prime \right)\cdot {\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right){\text{d}}^{3}r\prime {\text{d}}^{3}r″\right]d^3{r}_1{\text{d}}^3{r}_2\}{\text{d}}^3{R}_j,\end{array}$$ (51)

and we solve now the integral equation (51) for $\overline{X}\left(r,R_i\right)$. The geometry showing the relevant quantities is illustrated in Fig. 3. The integrals in Eq. (51) are computed as follows. First, we evaluate the second term on the right-hand side of Eq. (51). The integral over D_i, denoted by E_i(r₂), is computed by means of Eq. (41); the result is

$$\begin{array}{ll}{E}_i\left(r_2\right)\hfill & ={\int }_{D_i}{\overline{G}}_0\left(r_2,r\prime \right)\cdot {\overline{T}}_i\left(r\prime ,r″\right)\cdot E_{\text{c}}\left(r″\right){\text{d}}^{3}r\prime {\text{d}}^{3}r″\hfill \\ \hfill & =\frac{{\text{e}}^{\text{j}{k}_1{r}_{2ji}}}{r_{2ji}}\overline{A}\left({\widehat{r}}_{2ji},\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right),\hfill \end{array}$$ (52)

whence, approximating

$$\frac{{\text{e}}^{\text{j}{k}_1{r}_{2ji}}}{r_{2ji}}\approx {\text{e}}^{\text{j}{k}_1{\widehat{R}}_{ji}\cdot r_{2j}}\frac{{\text{e}}^{\text{j}{k}_1{R}_{ji}}}{R_{ji}}$$ (53)

and

$$\overline{A}\left({\widehat{r}}_{2ji},\widehat{s}\right)\approx \overline{A}\left({\widehat{R}}_{ji},\widehat{s}\right),$$ (54)

we find that in the coordinate system centered at O_j,

$$E_i\left(r_2\right)={\text{e}}^{\text{j}{k}_1{\widehat{R}}_{ji}\cdot r_{2j}}{E}_i\left(R_j\right)$$ (55)

is a locally plane electromagnetic wave with propagation direction ${\widehat{R}}_{ji}$ and amplitude

$$E_i\left(R_j\right)=\frac{{\text{e}}^{\text{j}{k}_1{R}_{ji}}}{R_{ji}}\overline{A}\left({\widehat{R}}_{ji},\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right).$$ (56)

Figure 3:

Geometries showing the relevant quantities in (a) solving Eq. (51), and (b) computing the integral (63) by the stationary phase method.

For computing the integral over D_j, we approximate

$$\overline{X}\left(r_2,R_i\right)\approx \overline{X}\left(R_j,R_i\right),$$ (57)

and define

$$E_j\left(r_2\right)=\overline{X}\left(R_j,R_i\right)\cdot E_i\left(r_2\right)={\text{e}}^{\text{j}{k}_1{\widehat{R}}_{ji}\cdot r_{2j}}{E}_j\left(R_j\right),$$ (58)

with

$$E_j\left(R_j\right)=\overline{X}\left(R_j,R_i\right)\cdot E_i\left(R_j\right).$$ (59)

Thus, analogously to E_i(r₂), E_j(r₂) is also a plane electromagnetic wave with propagation direction ${\widehat{R}}_{ji}$ and an amplitude as in Eq. (59). Noting the orthogonality relation

$${\widehat{R}}_{ji}\cdot E_j\left(R_j\right)={\widehat{R}}_{ji}\cdot \left[\overline{X}\left(R_j,R_i\right)\cdot E_i\left({\widehat{R}}_j\right)\right]=0,$$ (60)

owing to ${\widehat{R}}_{ji}\cdot \overline{X}\left(R_j,R_i\right)=0$, and applying again Eq. (41) with E_j(r₂) in place of E_c(r₂), we find that the second term on the right-hand side of Eq. (51) is

$$n_0{\int }_D\frac{{\text{e}}^{\text{j}{k}_1{r}_j}}{r_j}\overline{A}\left({\widehat{r}}_j,{\widehat{R}}_{ji}\right)\cdot E_j\left(R_j\right){\text{d}}^3{R}_j.$$ (61)

For the integrals over D_i on the left-hand side of Eq. (51) and in the first term on the right-hand side of Eq. (51), we directly apply Eq. (41); taking account of Eqs. (56), (59), and (61), we end up with

$$\begin{array}{l}\overline{X}\left(r,R_i\right):\left[\frac{{\text{e}}^{\text{j}{k}_1{r}_i}}{r_i}\overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right)\right]\hfill \\ =\frac{{\text{e}}^{\text{j}{k}_1{r}_i}}{r_i}\overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right)+n_0{\int }_D\frac{{\text{e}}^{\text{j}{k}_1{r}_j}}{r_j}\overline{A}\left({\widehat{r}}_j,{\widehat{R}}_{ji}\right)\hfill \\ \cdot \overline{X}\left(R_j,R_i\right)\cdot \frac{{\text{e}}^{\text{j}{k}_1{R}_{ji}}}{R_{ji}}\overline{A}\left({\widehat{R}}_{ji},\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right){\text{d}}^3{R}_j.\hfill \end{array}$$ (62)

The integral in Eq. (62) is computed by the stationary phase method [11]. Referring to Fig. 3b, we obtain

$$\begin{array}{l}{\int }_D\frac{{\text{e}}^{\text{j}{k}_1{r}_j}}{r_j}\overline{A}\left({\widehat{r}}_j,{\widehat{R}}_{ji}\right)\cdot \overline{X}\left(R_j,R_i\right)\cdot \frac{{\text{e}}^{\text{j}{k}_1{R}_{ji}}}{R_{ji}}\overline{A}\left({\widehat{R}}_{ji},\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right){\text{d}}^3{R}_j\hfill \\ =\frac{{\text{e}}^{\text{j}{k}_1{r}_i}}{r_i}\text{j}\frac{2\pi }{k_1}\overline{A}\left({\widehat{r}}_i,{\widehat{r}}_i\right)\cdot \left[\frac{1}{{\widehat{r}}_i\cdot \widehat{z}}{\int }_{z_i}^z\overline{X}\left(R_j,R_i\right)\text{d}{z}_j\right]\cdot \overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right),\hfill \end{array}$$ (63)

and we arrive at the integral equation

$$\overline{X}\left(r,R_i\right)=\overline{I}+\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}\left({\widehat{r}}_i,{\widehat{r}}_i\right)\cdot \left[\frac{1}{{\widehat{r}}_i\cdot \widehat{z}}{\int }_{z_i}^z\overline{X}\left(R_j,R_i\right)\text{d}{z}_j\right].$$ (64)

It is not hard to see that the solution of this integral equation is

$$\overline{X}\left(r,R_i\right)=\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}\left({\widehat{r}}_i,{\widehat{r}}_i\right)|r-R_i|\right]=\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}\left({\widehat{r}}_i,{\widehat{r}}_i\right)r_i\right].$$ (65)

Indeed, for

$$\overline{W}=\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}\left({\widehat{r}}_i,{\widehat{r}}_i\right),$$ (66)

the results

$$\overline{X}\left(r,R_i\right)=\text{exp}\left(\overline{W}|r-R_i|\right)=\text{exp}\left({\overline{W}}_{r_i}\right),$$ (67)

$$\overline{X}\left(R_j,R_i\right)=\text{exp}\left(\overline{W}|R_j-R_i|\right)=\text{exp}({\overline{W}}_s),$$ (68)

together with the dyadic identity

$${\text{e}}^{\overline{W}{r}_i}=\overline{I}+\overline{W}\cdot \left[{\int }_0^{r_i}{\text{e}}^{{\overline{W}}_s}\text{d}s\right]$$ (69)

prove the assertion. In Eqs. (68) and (69), s is defined by s = |R_j −R_i|, and in deriving Eq. (69), we made the change of variable $z_j=z_i+s{\widehat{r}}_i\cdot \widehat{z}$. Hence, from Eqs. (50) and (65), we infer that the desired expression for the average dyadic Green’s function is

$$\langle \overline{G}\left(r,r^{\prime }\right)\rangle =\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}\left({\widehat{r}}_i,{\widehat{r}}_i\right)r_i\right]\cdot {\overline{G}}_0\left(r,r^{\prime }\right),r^{\prime }\in D_i.$$ (70)

Combining Eqs. (41) and (70), we find that the integral (47) is

$${\int }_{D_i}\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r_2\right)\cdot E_{\text{c}}\left(r_2\right){\text{d}}^3{r}_1{\text{d}}^3{r}_2=\frac{\overline{t}\left({\widehat{r}}_i,r_i\right)}{r_i}\cdot \overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right),$$ (71)

where

$$\overline{\mathbf{t}}\left({\widehat{r}}_i,r_i\right)={\text{e}}^{\text{j}{k}_1{r}_i}\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}\left({\widehat{r}}_i,{\widehat{r}}_i\right)r_i\right]$$ (72)

is the coherent transmission dyadic from particle i to the observation point r. An interesting remark is in order: while the integral in Eq. (41) represents the field scattered by particle i at r, the integral in Eq. (71) represent the field scattered by particle i at r via various particle sequences.

A further consequence of Eqs. (41) and (70) is the integral result

$$\begin{array}{l}{\int }_{D_i}\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r_2\right)\hfill \\ \cdot \left[{\int }_{D_j}\langle \overline{G}\left(r_2,r_3\right)\rangle \cdot {\overline{T}}_j\left(r_3,r_4\right)\cdot E_{\text{c}}\left(r_4\right){\text{d}}^3{r}_3{\text{d}}^3{r}_4\right]{\text{d}}^3{r}_1{\text{d}}^3{r}_2\hfill \\ =\frac{\overline{t}\left({\widehat{r}}_i,r_i\right)}{r_i}\cdot \overline{A}\left({\widehat{r}}_i,{\widehat{R}}_{ij}\right)\cdot \frac{\overline{t}\left({\widehat{R}}_{ij},R_{ij}\right)}{R_{ji}}\cdot \overline{A}\left({\widehat{R}}_{ij},\widehat{s}\right)\cdot E_{\text{c}}\left(R_j\right),\hfill \end{array}$$ (73)

which represents the field scattered from j to i to r via various particle sequences. Note that in deriving Eqs. (73), we used the approximations (53) and (54) as well as the orthogonality relation ${\widehat{R}}_{ij}\cdot \overline{t}\left({\widehat{R}}_{ij},R_{ij}\right)=0$, which follows from

$${\widehat{R}}_{ij}\cdot \text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}\left({\widehat{R}}_{ij},{\widehat{R}}_{ij}\right)R_{ij}\right]=0.$$ (74)

4.2. Vector radiative transfer equation

The scattering intensity operator given by Eq. (26) is the kernel of the ladder approximated Bethe–Salpeter equation for the dyadic correlation function. Using the dyadic identities

$$\begin{array}{ll}(\overline{A}\otimes \overline{B})\cdot (\overline{C}\otimes \overline{D})\hfill & =(\overline{A}\cdot \overline{C})\otimes (\overline{B}\cdot \overline{D}),\hfill \\ \hfill (\overline{A}\otimes \overline{B})\cdot \overline{C}& =\overline{A}\cdot \overline{C}\cdot {\overline{B}}^T,\hfill \end{array}$$

where T stands for “transposed”, we express the Bethe–Salpeter equation (30) as

$$\begin{array}{l}\langle E(r)\otimes E^{\star }\left(r^{\prime }\right)\rangle \hfill \\ =E_{\text{c}}(r)\otimes E_{\text{c}}^{\star }\left(r^{\prime }\right)+n_0{\int }_D\{{\int }_{D_i}\left[\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r_2\right)\right]\hfill \\ \cdot \langle E\left(r_2\right)\otimes E^{\star }\left(r_2^{\prime }\right)\rangle \cdot {\left[\langle {\overline{G}}^{\star }(r^{\prime },r_1^{\prime }\rangle \rangle \cdot {\overline{T}}_i^{\star }\left(r_1^{\prime },r_2^{\prime }\right)\right]}^T\hfill \\ \times {\text{d}}^3{r}_1{\text{d}}^3{r}_2{\text{d}}^3{r}_1^{\prime }{\text{d}}^{3}r{}_2^{\prime }\}{\text{d}}^3{R}_i,\hfill \end{array}$$ (75)

and further, upon introducing the dyadic

$${\overline{P}}_i\left(r,r^{\prime }\right)={\int }_{D_i}\langle \overline{G}\left(r,r_1\right)\rangle \cdot {\overline{T}}_i\left(r_1,r^{\prime }\right){\text{d}}^3{r}_1,$$ (76)

as

$$\begin{array}{l}\langle E(r)\otimes E^{\star }\left(r^{\prime }\right)\rangle \hfill \\ =E_{\text{c}}(r)\otimes E_{\text{c}}^{\star }\left(r^{\prime }\right)+n_0{\int }_D[{\int }_{D_i}{\overline{P}}_i\left(r,r_2\right)\cdot \langle E\left(r_2\right)\otimes E^{\star }\left(r_2^{\prime }\right)\rangle \hfill \\ \cdot {\overline{P}}_i^{†}\left(r^{\prime },r_2^{\prime }\right){\text{d}}^3{r}_2{\text{d}}^3{r}_2^{\prime }]{\text{d}}^3{R}_i,\hfill \end{array}$$ (77)

where † stands for conjugate transpose. Iterating Eq. (77) gives

$$\begin{array}{l}\langle E(r)\otimes E^{\star }\left(r^{\prime }\right)\rangle \hfill \\ =E_{\text{c}}(r)\otimes E_{\text{c}}^{\star }\left(r^{\prime }\right)+n_0{\int }_D\{{\int }_{D_i}{\overline{P}}_i\left(r,r_2\right)\cdot \left[E_{\text{c}}\left(r_2\right)\otimes E_{\text{c}}^{\star }\left(r_2^{\prime }\right)\right]\hfill \\ \cdot {\overline{P}}_i^{†}\left(r^{\prime },r_2^{\prime }\right){\text{d}}^3{r}_2{\text{d}}^{3}r{}_2^{\prime }\}{\text{d}}^3{R}_i\hfill \\ +n_0^2{\int }_D\{{\int }_{D_i}{\int }_{D_j}{\overline{P}}_i\left(r,r_2\right)\cdot {\overline{P}}_j\left(r_2,r_4\right)\cdot \left[E_{\text{c}}\left(r_4\right)\otimes E_{\text{c}}^{\star }\left(r_4^{\prime }\right)\right]\hfill \\ \cdot {\overline{P}}_j^{†}\left(r_2^{\prime },r_4^{\prime }\right)\cdot {\overline{P}}_i^{†}\left(r^{\prime },r_2^{\prime }\right){\text{d}}^3{r}_4{\text{d}}^3{r}_4^{\prime }{\text{d}}^3{r}_2{\text{d}}^{3}r{}_2^{\prime }\}{\text{d}}^3{R}_j{\text{d}}^3{R}_i+\cdots ,\hfill \end{array}$$ (78)

where the integration domain over r₂ and $r_2^{\prime }$ is D_i, while the integration domain over r₄ and $r_4^{\prime }$ is D_j.

From Eqs. (71) and (73), we have the integral results

$${\int }_{D_i}{\overline{P}}_i\left(r,r_2\right)\cdot E_{\text{c}}\left(r_2\right){\text{d}}^3{r}_2=\overline{\text{V}}\left(r_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right)$$ (79)

and

$$\begin{array}{l}{\int }_{D_i}{\int }_{D_j}{\overline{P}}_i\left(r,r_2\right)\cdot {\overline{P}}_j\left(r_2,r_4\right)\cdot E_{\text{c}}\left(r_4\right){\text{d}}^3{r}_4{\text{d}}^3{r}_2\hfill \\ =\overline{V}\left(r_i,{\widehat{R}}_{ij}\right)\cdot \overline{V}\left(R_{ij},\widehat{s}\right)\cdot E_{\text{c}}\left(R_j\right),\hfill \end{array}$$ (80)

where

$$\overline{V}\left(r_i,\widehat{s}\right)=\frac{\overline{t}\left({\widehat{r}}_i,r_i\right)}{r_i}\cdot \overline{A}\left({\widehat{r}}_i,\widehat{s}\right).$$ (81)

Applying now the dyadic identity

$$\overline{A}\cdot (a\otimes b)\cdot {\overline{B}}^T=(\overline{A}\cdot a)\otimes (\overline{B}\cdot b),$$

setting r = r′ in Eq. (78), switching to the (ladder) coherency dyadic ${\overline{C}}_{\text{L}}(r)=\langle E(r)\otimes E^{\star }(r)\rangle$ and its coherent part ${\overline{C}}_{\text{c}}(r)=E_{\text{c}}(r)\otimes E_{\text{c}}^{\star }(r)$, and using Eqs. (79) and (80), we arrive at

$$\begin{array}{ll}{\overline{C}}_{\text{L}}(r)\hfill & ={\overline{C}}_{\text{c}}(r)+n_0{\int }_D\overline{V}\left(r_i,\widehat{s}\right)\cdot {\overline{C}}_{\text{c}}\left(R_i\right)\cdot {\overline{V}}^{†}\left(r_i,\widehat{s}\right){\text{d}}^3{R}_i\hfill \\ \hfill & +n_0^2{\int }_D\overline{V}\left(r_i,{\widehat{R}}_{ij}\right)\cdot \overline{V}\left(R_{ij},\widehat{s}\right)\cdot {\overline{C}}_{\text{c}}\left(R_j\right)\cdot {\overline{V}}^{†}\left(R_{ij},\widehat{s}\right)\hfill \\ \hfill & \cdot {\overline{V}}^{†}\left(r_i,{\widehat{R}}_{ij}\right){\text{d}}^3{R}_i{\text{d}}^3{R}_j+\cdots .\hfill \end{array}$$ (82)

This expansion was also derived as Eq. (145) in Ref. [1] and was the key point of the further development. The complete derivation of the radiative transfer equation from the expansion (82) was given in Section 11.3 of Ref. [1] and will not be repeated here. Essentially, from Eq. (82), a series expansion for the (ladder) specific coherency dyadic ${\overline{\Sigma }}_{\text{L}}$, defined through the angular spectrum representation

$${\overline{C}}_{\text{L}}(r)=\int {\overline{\Sigma }}_{\text{L}}(r,-\widehat{p}){\text{d}}^2\widehat{p},$$ (83)

was obtained. This series expansion is the expanded form of an integral equation for ${\overline{\Sigma }}_{\text{L}}$, which can be transformed into an integral equation for the diffuse specific coherency dyadic ${\overline{\Sigma }}_{\text{dL}}$, defined by

$${\overline{\mathbf{\Sigma }}}_{\text{L}}(r,-\widehat{p})={\overline{\Sigma }}_{\text{dL}}(r,-\widehat{p})+\delta (\widehat{p}+\widehat{s}){\overline{C}}_{\text{c}}(r).$$ (84)

Finally, differentiating the integral equation for ${\overline{\Sigma }}_{\text{dL}}$ the vector radiative transfer equation

$$\begin{array}{ll}\frac{{\text{dI}}_{\text{d}}(r,\widehat{q})}{\text{d}s}\hfill & =-n_0\text{K}(\widehat{q}){\text{I}}_{\text{d}}(r,\widehat{q})+n_0\text{Z}(\widehat{q},\widehat{s}){\text{I}}_{\text{c}}(r)\hfill \\ \hfill & +n_0\int Z\left(\widehat{q},{\widehat{q}}^{\prime }\right){\text{I}}_{\text{d}}\left(r,{\widehat{q}}^{\prime }\right){\text{d}}^2{\widehat{q}}^{\prime },\hfill \end{array}$$ (85)

is derived. Here, I_d is the diffuse specific intensity column vector, I_c is the Stokes column vector of the coherent field, while Z and K are the phase and the extinction matrix of a nonspherical particle in a fixed orientation.

5. Conclusions

As in Part I [1], in this paper we have derived the vector radiative transfer equation for a discrete random layer with non-scattering boundaries and a sparse concentration of randomly positioned particles. This time, however, we maximally delayed the use of the far-field assumption for sparsely distributed particles by using the exact integral Foldy equations. Specifically, we proceed as follows.

First, by Assuming that the discrete random medium in question if fully ergodic while the positions of the constituent particles are statistically independent, and by applying the Twersky approximation to the iterated solution of the integral Foldy equations, we derived the Dyson equation for the configuration-averaged coherent field and the ladder approximated Bethe–Salpeter equation for the configuration-averaged dyadic correlation function. Furthermore, we proved that the Dyson equation can be also obtained by means of the Foldy approximation for the exciting fields. Strictly speaking, these results do not rely on the electromagnetic far-field assumption and can thus be considered rather general. One can argue however that the particle number density should be sufficiently low to ensure the presumed statistical independence and uniformity of particle positions.

Second, we showed that under the far-field approximation, the Dyson equation reduces to the Foldy integral equation for the coherent field. To this end, we used the far-field representation of the dyadic Green’s function and the relation between the far-field scattering dyadic and the Fourier transform of the transition dyadic. We discussed two methods for solving the Foldy integral equation: an approach based on the application of the Helmholtz operator to the integral equation, and an approach similar to that used by Ishimaru in the scalar case [11]. Next, we showed that the same expression for the coherent field can be obtained in the framework of the characteristic waves method. However, in this case, the additional assumptions that the coherent field propagates along the incidence direction and that the effective wavenumber is close to that of the background medium had to be imposed.

Third, we derived the vector radiative transfer equation by considering an iterated solution of the Bethe–Salpeter equation for the dyadic correlation function and by computing each term in the series under the far-field approximation. In fact, we estimated various integrals involving the average dyadic Green’s function in the far-field region. Switching to the coherency dyadic, we found that the resulting series representation coincides with that obtained in Ref. [1]. A direct consequence of this series expansion is an integral equation for the diffuse specific coherency dyadic implying the vector radiative transfer equation for the diffuse specific intensity column vector.

In the final analysis, the same assumptions and approximations as those used in Ref. [1] have been invoked, albeit in a different order, and the same vector radiative transfer equation has been arrived at. However, two important comments are called for.

First, it should be pointed out that a key point of the analysis in this paper is the assumption (38) stating that the coherent field can be locally approximated by a plane electromagnetic wave having the same wavenumber as that of the background medium. This assumption implies the relation (41) which is used many times during the derivation. To justify the assumption (38) we used two arguments which, strictly speaking, are not necessarily consistent with the present approach: one is a result of the characteristic wave method, wherein the assumptions mentioned above are made, and the other one is a result of the analysis performed in Ref. [1], wherein the Twersky approximation is used in conjunction with the far-field Foldy equations.

Second, it is relatively straightforward to justify the use of the ladder approximation in the computation of second moments in the field after having made the far-field assumption. Indeed, in that case different multi-particle contributions to the total field at an observation point are transverse waves that can be characterized by the corresponding cumulative phases. It can then be argued that upon configurational averaging, the extreme sensitivity of the respective complex exponential phase factors on particle positions will zero out the contributions of all second-moment diagrams except those of the ladder diagrams (see, e.g., Section 8.11 of Ref. [4] or Section 18.2 of Ref. [9]). Strictly speaking, this argument cannot be made in the case of densely packed particles to which the far-field assumption can be inapplicable.

We thus have to conclude that as compared to the derivation given in Ref. [1], the present derivation is less self-consistent.

Finally we note that our primary objective has been to trace the first-principles origin of the radiative transfer theory. Therefore, we have not discussed numerical methods for solving the vector radiative transfer equation (or its simplified scalar version) and specific practical applications in various fields of science and engineering. This further information can be found in Refs. [4, 9, 13, 14] and numerous publications cited therein.

Figure 1:

(a) A group of particles confined to a macroscopically plane-parallel layer with non-scattering boundaries, and (b) the distance $s(r,-\widehat{s})=\widehat{s}\cdot \left(r-r_A\right)$, where r_A is the point where the straight line with the direction vector $-\widehat{s}$ going through the observation point P crosses the lower boundary of the layer.

Appendix 1. Integral-operator notation and Fourier transforms of vector and dyadic functions

Considering the dyadic function $\overline{A}\left(r,r^{\prime }\right)$, the vector function x(r), and the dyadic function $\overline{X}\left(r,r^{\prime }\right)$, we define the vector function $\left(\overline{A}x\right)$(r) and the dyadic function $(\overline{A}\overline{X})$(r,r′) through the linear transformations

$$\left(\overline{A}x\right)(r)\stackrel{\text{def}}{=}\int \overline{A}\left(r,r_1\right)\cdot x\left(r_1\right){\text{d}}^3{r}_1,$$ (86)

and

$$(\overline{A}\overline{X})\left(r,r^{\prime }\right)\stackrel{\text{def}}{=}\int \overline{A}\left(r,r_1\right)\cdot \overline{X}\left(r_1,r^{\prime }\right){\text{d}}^3{r}_1,$$ (87)

respectively, the entire three-dimensional space ${ℝ}^3$ serving as the integration domain. In general, we have

$$(\overline{A}\overline{B}\overline{X})\left(r,r^{\prime }\right)=\int \overline{A}\left(r,r_1\right)\cdot \overline{B}\left(r_1,r_2\right)\cdot \overline{X}\left(r_2,r^{\prime }\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1.$$ (88)

The linear operator $\overline{A}$ acting on vectors and dyadics is an integral operator with the kernel $\overline{A}\left(r,r^{\prime }\right)$. Although we use the same notation, we assume that it can be inferred from the context if a quantity is an operator or a dyadic function.

The inverse dyadic function ${\overline{A}}^{-1}\left(r,r^{\prime }\right)$ of the dyadic function $\overline{A}\left(r,r^{\prime }\right)$ is defined by the relation

$$\left(\overline{A}{\overline{A}}^{-1}\right)\left(r,r^{\prime }\right)=\left({\overline{A}}^{-1}\overline{A}\right)\left(r,r^{\prime }\right)=\delta \left(r-r^{\prime }\right)\overline{I},$$ (89)

or explicitly,

$$\begin{array}{ll}\int \overline{A}\left(r,r_1\right)\cdot {\overline{A}}^{-1}\left(r_1,r^{\prime }\right){\text{d}}^3{r}_1\hfill & =\int {\overline{A}}^{-1}\left(r,r_1\right)\cdot \overline{A}\left(r_1,r^{\prime }\right){\text{d}}^3{r}_1\hfill \\ \hfill & =\delta \left(r-r^{\prime }\right)\overline{I},\hfill \end{array}$$ (90)

where δ(r − r′) is the three-dimensional delta function and $\overline{I}$ is the identity dyadic. As a result, we find

$$\left({\overline{A}}^{-1}\overline{A}\overline{X}\right)\left(r,r^{\prime }\right)=\int \delta \left(r-r_1\right)\overline{X}\left(r_1,r^{\prime }\right){\text{d}}^3{r}_2=\overline{X}\left(r,r^{\prime }\right),$$ (91)

that is

$${\overline{A}}^{-1}\overline{A}\overline{X}=\overline{A}{\overline{A}}^{-1}\overline{X}=\overline{X}.$$ (92)

The Fourier transforms of the vector function x(r) and the dyadic function $\overline{X}\left(r,r^{\prime }\right)$ are defined respectively, by

$$\mathcal{F}(x)(p)=\int {\text{e}}^{-\text{j}p\cdot r}x(r){\text{d}}^{3}r,$$ (93)

$$\mathcal{F}(\overline{X})\left(p,p^{\prime }\right)=\int {\text{e}}^{-\text{j}p\cdot r}\overline{X}\left(r,r^{\prime }\right){\text{e}}^{\text{j}{p}^{\prime }\cdot r^{\prime }}{\text{d}}^3{r}^{\prime }{\text{d}}^{3}r,$$ (94)

and the inverse transformations are

$$x(r)=\frac{1}{{(2\pi )}^3}\int {\text{e}}^{{}^{\text{j}p\cdot r}}\mathcal{F}(x)(p){\text{d}}^{3}p,$$ (95)

$$\overline{X}(r,r\text{'})=\frac{1}{{(2\pi )}^6}\int {\text{e}}^{\text{j}p\cdot r}\mathcal{F}(\overline{X})\left(p,p^{\prime }\right){\text{e}}^{-\text{j}{p}^{\prime }\cdot r^{\prime }}{\text{d}}^3{p}^{\prime }{\text{d}}^{3}p.$$ (96)

The well-known representation for the Dirac delta function,

$$\delta \left(r-r^{\prime }\right)=\frac{1}{{(2\pi )}^3}\int {\text{e}}^{\text{j}p\cdot \left(r-r^{\prime }\right)}{\text{d}}^{3}p,$$ (97)

plays an important role in the calculation.

It is not hard to see that the Fourier transforms of $(\overline{A}x)(r)$ and $(\overline{A}\overline{X})\left(r,r^{\prime }\right)$ are, respectively,

$$\mathcal{F}(\overline{A}x)(p)=\frac{1}{{(2\pi )}^3}\int \mathcal{F}(\overline{A})\left(p,p^{\prime }\right)\cdot \mathcal{F}(x)\left(p^{\prime }\right){\text{d}}^3{p}^{\prime },$$ (98)

$$\mathcal{F}(\overline{A}\overline{X})\left(p,p^{\prime }\right)=\frac{1}{{(2\pi )}^3}\int \mathcal{F}(\overline{A})\left(p,p_1\right)\cdot \mathcal{F}(\overline{X})\left(p_1,p^{\prime }\right){\text{d}}^3{p}_1,$$ (99)

and that

$$\mathcal{F}\left(\overline{A}{\overline{A}}^{-1}\right)\left(p,p^{\prime }\right)=\mathcal{F}\left({\overline{A}}^{-1}\overline{A}\right)\left(p,p^{\prime }\right)={(2\pi )}^3\delta \left(p-p^{\prime }\right)\overline{I}.$$ (100)

Using the short-hand notation $\mathcal{F}(x)(p)={\text{x}}_p(p)$ and $\mathcal{F}(\overline{X})\left(p,p^{\prime }\right)={\overline{X}}_p\left(p,p^{\prime }\right)$, we summarize below the Fourier transforms of some special dyadic functions.

1. If $\overline{X}\left(r,r^{\prime }\right)$ is a translation invariant dyadic, i.e.,

   $$\overline{X}\left(r,r^{\prime }\right)=\overline{X}\left(r-r^{\prime }\right),$$ (101)

   we have

   $${\overline{X}}_p\left(p,p^{\prime }\right)={(2\pi )}^3\delta \left(p-p^{\prime }\right){\overline{X}}_p(p),$$ (102)

   where

   $$\begin{array}{ll}{\overline{X}}_p(p)\hfill & =\int \overline{X}\left(r-r^{\prime }\right){\text{e}}^{-\text{j}p\cdot \left(r-r^{\prime }\right)}{\text{d}}^3\left(r-r^{\prime }\right)\hfill \\ \hfill & =\int \overline{X}(R){\text{e}}^{-\text{j}p\cdot R}{\text{d}}^{3}R.\hfill \end{array}$$ (103)
   The Fourier transform of the vector function $a(r)=(\overline{X}b)(r)$ is

   $$a_p(p)={\overline{X}}_p(p)\cdot b_p(p),$$ (104)

   while the Fourier transform of the dyadic function $\overline{A}\left(r,r^{\prime }\right)=(\overline{X}\overline{B})\left(r,r^{\prime }\right)$ is

   $${\overline{A}}_p\left(p,p^{\prime }\right)={\overline{X}}_p(p)\cdot {\overline{B}}_p\left(p,p^{\prime }\right).$$ (105)
   Taking the Fourier transform of Eq. (90), using Eq. (99) and (cf. Eqs. (94) and (97))

   $$\mathcal{F}(\delta \overline{I})\left(p,p^{\prime }\right)={(2\pi )}^3\delta \left(p-p^{\prime }\right)\overline{I},$$ (106)

   we obtain

   $$\frac{1}{{(2\pi )}^3}\int {\overline{X}}_p\left(p,p_1\right)\cdot {\overline{X}}_p^{-1}\left(p_1,p^{\prime }\right){\text{d}}^3{p}_1={(2\pi )}^3\delta \left(p-p^{\prime }\right)\overline{I},$$ (107)

   whence, accounting for Eq. (102), we find

   $${\overline{X}}_p(p)\cdot {\overline{X}}_p^{-1}(p)=\overline{I}.$$ (108)

   Thus, in the Fourier space, ${\overline{X}}_p^{-1}$ is the inverse of ${\overline{X}}_p$.

2. A dyadic function such that

   $${\overline{X}}_i\left(r,r^{\prime }\right)=\overline{X}\left(r-R_i,r^{\prime }-R_i\right)$$ (109)

   is called translational. The Fourier transform of ${\overline{X}}_i$ is

   $${\overline{X}}_{ip}\left(p,p^{\prime }\right)={\text{e}}^{-\text{j}p\cdot R_i}{\overline{X}}_p\left(p,p^{\prime }\right){\text{e}}^{\text{j}{p}^{\prime }\cdot R_i},$$ (110)

   where ${\overline{X}}_p\left(p,p^{\prime }\right)$ is the Fourier transform of $\overline{X}$. The Fourier transform of the vector $a_i(r)=\left({\overline{X}}_{i}b\right)(r)$ is

   $$a_{ip}(p)=\frac{1}{{(2\pi )}^3}\int {\text{e}}^{-\text{j}\left(p-p_1\right)\cdot R_i}{\overline{X}}_p\left(p,p_1\right)\cdot b_p\left(p_1\right){\text{d}}^3{p}_1,$$ (111)

   while the Fourier transform of the dyadic function ${\overline{A}}_i\left(r,r^{\prime }\right)=\left({\overline{X}}_i\overline{B}\right)\left(r,r^{\prime }\right)$ is

   $${\overline{A}}_{ip}\left(p,p^{\prime }\right)=\frac{1}{{(2\pi )}^3}\int {\text{e}}^{-\text{j}\left(p-p_1\right)\cdot R_i}{\overline{X}}_p\left(p,p_1\right)\cdot {\overline{B}}_p\left(p_1,p^{\prime }\right){\text{d}}^3{p}_1.$$ (112)

Appendix 2. Transition dyadic

Consider a homogeneous particle embedded in a lossless, homogeneous and isotropic medium. The wavenumber in the host medium is k₁, while the wavenumber inside the particle is k₂ = mk₁, where m is the relative refractive index of the particle. The particle is centered at the origin of the coordinate system, and we denote by D₀ the domain occupied by the particle. The scattering potential of the particle $U_0(r)=k_1^2\left({\text{m}}^2-1\right)\Theta (r)$, with

$$\Theta (r)=\{\begin{array}{ll}1,\hfill & r\in D_0\hfill \\ 0,\hfill & r\notin D_0\hfill \end{array},$$

can be elevated to a dyadic, the so-called potential dyadic, defined by

$${\overline{U}}_0\left(r,r^{\prime }\right)=U_0(r)\delta \left(r-r^{\prime }\right)\overline{I},r,r^{\prime }\in {ℝ}^3.$$ (113)

In terms of ${\overline{U}}_0$, the integral equations for the total field and the dyadic Green’s function are

$$\begin{array}{ll}E(r)\hfill & =E_0(r)+\int {\overline{G}}_0\left(r,r_1\right)\cdot {\overline{U}}_0\left(r_1,r_2\right)\hfill \\ \hfill & \cdot E\left(r_2\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1,r\in {ℝ}^3,\hfill \end{array}$$ (114)

and

$$\begin{array}{ll}\overline{G}\left(r,r^{\prime }\right)\hfill & ={\overline{G}}_0\left(r,r^{\prime }\right)+\int {\overline{G}}_0\left(r,r_1\right)\cdot {\overline{U}}_0\left(r_1,r_2\right)\hfill \\ \hfill & \cdot \overline{G}\left(r_2,r^{\prime }\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1,r,r^{\prime }\in {ℝ}^3,\hfill \end{array}$$ (115)

respectively.

The transition dyadic $\overline{T}$ is defined through the relation

$$\begin{array}{ll}E(r)\hfill & =E_0(r)+\int {\overline{G}}_0\left(r,r_1\right)\hfill \\ \hfill & \cdot \overline{T}\left(r_1,r_2\right)\cdot E_0\left(r_2\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1,r\in {ℝ}^3;\hfill \end{array}$$ (116)

like for ${\overline{U}}_0$, the support of the transition dyadic $\overline{T}$ is D₀, that is,

$$\overline{T}\left(r_1,r_2\right)=\overline{0}\text{for}{r}_1\text{and}/\text{or}{r}_2\notin D_0.$$

The scattered field, given by

$$E_{\text{sct}}(r)=\int {\overline{G}}_0\left(r,r_1\right)\cdot {\overline{U}}_0\left(r_1,r_2\right)\cdot E\left(r_2\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1,r\in {ℝ}^3\backslash D_0,$$ (117)

and the internal field E_int can be expressed in terms of the transition dyadic as

$$E_{\text{sct}}(r)=\int {\overline{G}}_0\left(r,r_1\right)\cdot \overline{T}\left(r_1,r_2\right)\cdot E_0\left(r_2\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1,r\in {ℝ}^3\backslash D_0,$$ (118)

and

$$E_{\text{int}}(r)=\frac{1}{k_1^2\left({\text{m}}^2-1\right)}\int \overline{T}\left(r,r_1\right)\cdot E_0\left(r_1\right){\text{d}}^3{r}_1,r\in D_0,$$ (119)

respectively. Similarly, in terms of the transition dyadic, the integral equation for the dyadic Green’s function is

$$\overline{G}\left(r,r^{\prime }\right)={\overline{G}}_0\left(r,r^{\prime }\right)+\int {\overline{G}}_0\left(r,r_1\right)\cdot \overline{T}\left(r_1,r_2\right)\cdot {\overline{G}}_0\left(r_2,r^{\prime }\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1.$$ (120)

The transition dyadic satisfies the Lippmann–Schwinger equation

$$\overline{T}\left(r,r^{\prime }\right)={\overline{U}}_0\left(r,r^{\prime }\right)+\int {\overline{U}}_0\left(r,r_1\right)\cdot {\overline{G}}_0\left(r_1,r_2\right)\cdot \overline{T}\left(r_2,r^{\prime }\right){\text{d}}^3{r}_2{\text{d}}^3{r}_1,$$ (121)

so that once $\overline{T}$ is known, the scattered and internal fields can be computed by means of Eqs. (118) and (119), respectively.

Appendix 3. Relationship between the far-field scattering dyadic and the Fourier transform of the transition dyadic

As in Appendix 2, we consider the scattering by a homogeneous particle centered at the origin of the coordinate system. For an incident plane electromagnetic wave

$$E_0(r)={\mathcal{E}}_0(\widehat{s}){\text{e}}^{\text{j}{k}_1\cdot r},$$ (122)

with $k_1=k_1\widehat{s}$ and ${\mathcal{E}}_0(\widehat{s})={\mathcal{E}}_{0\theta }\widehat{\theta }(\widehat{s})+{\mathcal{E}}_{0\phi }\widehat{\mathit{\phi }}(\widehat{s})$, the scattered field is given by

$$\begin{array}{ll}{E}_{\text{sct}}(r)\hfill & =\frac{1}{{(2\pi )}^3}\int {\overline{G}}_0\left(r,r_1\right)\hfill \\ \hfill & \cdot \left[\int {\text{e}}^{\text{j}{p}_1\cdot r_1}{\overline{T}}_p\left(p_1,k_1\right)\cdot {\mathcal{E}}_0(\widehat{s}){\text{d}}^3{p}_1\right]{\text{d}}^3{r}_1,\hfill \end{array}$$ (123)

where

$${\overline{T}}_p\left(p_1,p_2\right)=\int {\text{e}}^{-\text{j}{p}_1\cdot r_1}\overline{T}\left(r_1,r_2\right){\text{e}}^{\text{j}{p}_2\cdot r_2}{\text{d}}^3{r}_1{\text{d}}^3{r}_2,$$ (124)

is the Fourier transform of the transition dyadic. In the far-field region, the dyadic Green’s function becomes

$${\overline{G}}_0\left(r,r_1\right)=(\overline{I}-\widehat{r}\otimes \widehat{r})\frac{{\text{e}}^{\text{j}{k}_{1}r}}{4\pi r}{\text{e}}^{-\text{j}{k}_1\widehat{r}\cdot r_1},$$ (125)

in which case, Eq. (123) yields

$$E_{\text{sct}}(r)=\frac{{\text{e}}^{\text{j}{k}_{1}r}}{4\pi r}(\overline{I}-\widehat{r}\otimes \widehat{r})\cdot {\overline{T}}_p\left(k_1\widehat{r},k_1\widehat{s}\right)\cdot {\mathcal{E}}_0(\widehat{s}).$$ (126)

On the other hand, the scattered field can be expressed in terms of the far-field scattering dyadic

$$\overline{A}(\widehat{r},\widehat{s})=\sum _{\eta ,\mu =\theta ,\phi }{[\text{S}(\widehat{r},\widehat{s})]}_{\eta \mu }\widehat{\eta }(\widehat{r})\otimes \widehat{\mu }(\widehat{s}),$$ (127)

where $\text{S}(\widehat{r},\widehat{\text{s}})$ is the amplitude scattering matrix, according to

$$E_{\text{sct}}(r)=\frac{{\text{e}}^{\text{j}{k}_{1}r}}{r}\overline{A}(\widehat{r},\widehat{s})\cdot {\mathcal{E}}_0(\widehat{s}),r\to \infty .$$ (128)

From Eqs. (126) and (128), the relation between the far-field scattering dyadic and the transition dyadic is

$$\overline{A}(\widehat{r},\widehat{s})\cdot {\mathcal{E}}_0(\widehat{s})=\frac{1}{4\pi }(\overline{I}-\widehat{r}\otimes \widehat{r})\cdot {\overline{T}}_p\left(k_1\widehat{r},k_1\widehat{s}\right)\cdot {\mathcal{E}}_0(\widehat{s}).$$ (129)

Putting successively ${\mathcal{E}}_0(\widehat{s})={\mathcal{E}}_{0\theta }\widehat{\theta }(\widehat{s})$ and ${\mathcal{E}}_0(\widehat{s})={\mathcal{E}}_{0\phi }\widehat{\mathit{\phi }}(\widehat{s})$ in Eq. (129), we obtain

$$\overline{A}(\widehat{r},\widehat{s})=\frac{1}{4\pi }(\overline{I}-\widehat{r}\otimes \widehat{r})\cdot {\overline{T}}_p\left(k_1\widehat{r},k_1\widehat{s}\right)\cdot (\overline{I}-\widehat{s}\otimes \widehat{s}),$$ (130)

or equivalently,

$$\overline{A}(\widehat{r},\widehat{s})=\frac{1}{4\pi }{\overline{T}}_{p\text{T}}\left(k_1\widehat{r},k_1\widehat{s}\right),$$ (131)

where ${\overline{T}}_{p\text{T}}$ is the transverse component of the dyadic ${\overline{T}}_p$ involving only the dyads $\widehat{\mathit{\eta }}(\widehat{r})\otimes \widehat{\mu }(\widehat{s}),$ with. η, μ = θ, φ. In matrix form, we use the representations (127) and

$${\overline{T}}_{p\text{T}}\left(k_1\widehat{r},k_1\widehat{s}\right)=\sum _{\eta ,\mu =\theta ,\phi }{\left[{\text{T}}_{p\text{T}}\left(k_1\widehat{r},k_1\widehat{s}\right)\right]}_{\eta \mu }\widehat{\eta }(\widehat{r})\otimes \widehat{\mu }(\widehat{s}),$$ (132)

to obtain

$$\text{S}(\widehat{r},\widehat{s})=\frac{1}{4\pi }{\text{T}}_{p\text{T}}\left(k_1\widehat{r},k_1\widehat{s}\right).$$ (133)

Appendix 4. Characteristic waves method

Applying the operator $\nabla \times \nabla \times -k_1^2$ to the Dyson equation for the coherent field $\langle E\rangle =E_0+{\overline{G}}_0\overline{M}\langle E\rangle$, taking the Fourier transform of the resulting equation, using the computation rule F(∇×f)(p) = jp×F (f)(p), where F(f)(p) is the Fourier transform of f = f (r) at p, Assuming that for a statistical homogeneous medium, the dyadic mass operator is translation invariant, i.e., $\overline{M}\left(r,r^{\prime }\right)=\overline{M}\left(r-r^{\prime }\right)$, which implies (see Appendix 1) $\mathcal{F}(\overline{M})\left(p,p^{\prime }\right)={(2\pi )}^3\delta \left(p-p^{\prime }\right){\overline{M}}_p(p)$, we find [6]

$$\left[\left(p^2-k_1^2\right)(\overline{I}-\widehat{p}\otimes \widehat{p})-k_1^2\widehat{p}\otimes \widehat{p}-{\overline{M}}_p(p)\right]\cdot E_p(p)=0,$$ (134)

with $\mathcal{F}(\langle E\rangle )(p)=E_p(p)$ This equation, which can be interpreted as an eigenvalue equation, shows that E_p is nonzero only if

$$\text{det}\left[\left(p^2-k_1^2\right)(\overline{I}-\widehat{p}\otimes \widehat{p})-k_1^2\widehat{p}\otimes \widehat{p}-{\overline{M}}_p(p)\right]=0.$$ (135)

Here, the notation det$\left(\overline{X}\right)$ should be understood as det(X), where X is the matrix associated with the dyadic $\overline{X}$. Putting $p=K\widehat{p}$, the eigenvalues $K=K(\widehat{p})$ satisfying the characteristic equation

$$\text{det}\left[\left(K^2-k_1^2\right)(\overline{I}-\widehat{p}\otimes \widehat{p})-k_1^2\widehat{p}\otimes \widehat{p}-{\overline{M}}_p(K\widehat{p})\right]=0,$$ (136)

are the values of the effective wavenumber (propagation constant) for the specified direction of propagation $\widehat{p}$, while $E_p(K\widehat{p})$ satisfying Eq. (134) are the corresponding eigenvectors. In general, the dispersion equation (136) can have several solutions. In order to insure that the radiation condition at infinity is satisfied, only the solutions with a positive imaginary part are considered.

For the incidence propagation direction $\widehat{S}$, let $K_n=K_n(\widehat{s})$ be an eigenvalue solving the dispersion equation (136) with, $\widehat{p}=\widehat{s}$, and let ${\mathfrak{E}}_n(\widehat{s})$ be the corresponding eigenvector satisfying the eigenvalue equation (134) with $p=K_n(\widehat{s})\widehat{s}$ (that is, with $p=K_n(\widehat{s})$ and $\widehat{p}=\widehat{s}$). Then,

$$E_p(p)={(2\pi )}^3\sum _n\delta \left[p-K_n(\widehat{s})\widehat{s}\right]{\mathfrak{E}}_n(\widehat{s})$$ (137)

satisfies Eq. (134), and consequently, the corresponding coherent field is

$$\langle E(r)\rangle =\sum _n{E}_n(r,\widehat{s}),$$ (138)

$$E_n(r,\widehat{s})={\text{e}}^{j{K}_n(\widehat{s})\widehat{s}\cdot r}{\mathfrak{E}}_n(\widehat{s}).$$ (139)

The wave $E_n(r,\widehat{s})$ propagating in the direction $\widehat{s}$ and satisfying the vector Helmholtz equation with wavenunber K_n is the characteristic wave, while the eigenvector ${\mathfrak{E}}_n(\widehat{s})$, which is defined up to a multiplicative constant, is the characteristic wave polarization associated with K_n.

For a discrete random medium with sparsely distributed particles, the approximation K_n ≈ k₁, implying ${\overline{M}}_p\left(K_n\widehat{s}\right)\approx {\overline{M}}_p\left(k_1\widehat{s}\right)$, can be assumed. Moreover, in this case, the far-field approximation, according to which the fields in the far zone are transverse, applies. Decomposing ${\mathfrak{E}}_n(\widehat{s})$ into a longitudinal and a transverse component ${\mathfrak{E}}_{n\text{L}}(\widehat{s})$ and ${\mathfrak{E}}_{n\text{T}}(\widehat{s})$, respectively, that is, ${\mathfrak{E}}_n(\widehat{s})={\mathfrak{E}}_{n\text{L}}(\widehat{s})+{\mathfrak{E}}_{n\text{T}}(\widehat{s})$, with ${\mathfrak{E}}_{n\text{L}}(\widehat{s})={\mathfrak{E}}_{ns}(\widehat{s})\widehat{s}$ and ${\mathfrak{E}}_{n\text{T}}(\widehat{s})={\mathfrak{E}}_{n\theta }(\widehat{s})\widehat{\theta }(\widehat{s})+{\mathfrak{E}}_{n\phi }(\widehat{s})\widehat{\mathit{\phi }}(\widehat{s})$, the far-field approximation implies ${\mathfrak{E}}_{n\text{L}}(\widehat{s})=0$. This results may also follow from the fact that in Eq. (134), $k_1^2$ is much larger than the $(\widehat{p}\otimes \widehat{p})-,(\widehat{p}\otimes \widehat{\theta }(\widehat{p}))-$, and $(\widehat{p}\otimes \widehat{\mathit{\phi }}(\widehat{p}))$-components of the dyadic $\overline{M_p}\left(k_1\widehat{s}\right);$ in a matrix-form representation, the first equation in Eq. (134) yields ${ℭ}_{ns}(\widehat{s})=0$ [6]. Consequently, the matrix equation for the transverse part ${\mathfrak{E}}_{n\text{T}}(\widehat{s})$ is

$$\left[\left(K_n^2-k_1^2\right){\text{I}}_2-{\text{M}}_{p\text{T}}\left(k_1\widehat{\text{s}}\right)\right]\left[\begin{array}{c}{\mathfrak{E}}_{n\theta }(\widehat{s})\\ {\mathfrak{E}}_{n\phi }(\widehat{s})\end{array}\right]=0,$$ (140)

where I₂ is the two-dimensional identity matrix, 0 is the two-dimensional zero vector, and ${\text{M}}_{p\text{T}}\left(k_1\widehat{s}\right)$ is the matrix associated with the transverse component ${\overline{M}}_{p\text{T}}(p)$ of the dyadic ${\overline{M}}_p(p)$. As shown in Ref. [6], for the direction of propagation $\widehat{s}$ there are two effective wavenumbers $K_1(\widehat{s})$ and $K_2(\widehat{s})$, and accordingly, two transverse characteristic waves with polarizations ${\mathfrak{E}}_{1\text{T}}(\widehat{\text{s}})$ and ${\mathfrak{E}}_{1\text{T}}(\widehat{s})$. Furthermore, under the assumption K_n ≈ k₁, we have $K_n^2-k_1^2\approx 2k_1\left(K_n-k_1\right)$, and the eigenvalue equation (140) yields

$$K_n\left[\begin{array}{c}{\mathfrak{E}}_{n\theta }(\widehat{s})\\ {\mathfrak{E}}_{n\phi }(\widehat{s})\end{array}\right]=\left[k_1{\text{I}}_2+\frac{1}{2k_1}{\text{M}}_{p\text{T}}\left(k_1\widehat{s}\right)\right]\left[\begin{array}{c}{\mathfrak{E}}_{n\theta }(\widehat{s})\\ {\mathfrak{E}}_{n\phi }(\widehat{s})\end{array}\right].$$ (141)

Putting $\text{A}=k_1{\text{I}}_2+\left(1/2k_1\right){\text{M}}_{pT}\left(k_1\widehat{s}\right)$, and ${\text{x}}_n={\left[{\mathfrak{E}}_{n\theta }(\widehat{s}){\mathfrak{E}}_{n\phi }(\widehat{s})\right]}^T$, so that Ax_n = K_nx_n, we see that ${\text{A}}^2{\text{x}}_n=K_n{\text{Ax}}_n=K_n^2{\text{x}}_n$, and that in general, ${\text{A}}^m{\text{x}}_n=K_n^m{\text{x}}_n$ for any m ≥ 0. This result implies exp(jK_ns)x = exp(jAs)x, that is,

$${\text{e}}^{\text{j}{K}_{n}s}\left[\begin{array}{c}{\mathfrak{E}}_{n\theta }(\widehat{s})\\ {\mathfrak{E}}_{n\phi }(\widehat{s})\end{array}\right]={\text{e}}^{\text{j}\left[k_1{\text{I}}_2+\frac{1}{2k_1}{\text{M}}_{p\text{T}}\left(k_1\widehat{s}\right)\right]s}\left[\begin{array}{c}{\mathfrak{E}}_{n\theta }(\widehat{s})\\ {\mathfrak{E}}_{n\phi }(\widehat{s})\end{array}\right],$$ (142)

with $s=s(r,-\widehat{s})=\widehat{s}\cdot \left(r-r_A\right)$. In dyadic representation, Eqs. (138), (139) and (142) imply

$$\begin{array}{ll}\langle E(r)\rangle \hfill & =\sum _{n=1}^2{\text{e}}^{\text{j}{K}_n(\widehat{s})s(r,-\widehat{s})}{\text{e}}^{\text{j}{K}_n(\widehat{s})\widehat{s}\cdot r_A}{\mathfrak{E}}_{n\text{T}}(\widehat{s})\hfill \\ \hfill & ={\text{e}}^{\text{j}\left[k_1\overline{\text{I}}+\frac{1}{2k_1}{\overline{M}}_{p\text{T}}\left(k_1\widehat{\text{s}}\right)\right]s(r,-\widehat{\text{s}})}\langle E\left(r_A\right)\rangle ,\hfill \end{array}$$ (143)

where $\langle E\left(r_A\right)\rangle ={\sum }_{n=1}^2{\text{e}}^{\text{j}{K}_n(\widehat{s})\widehat{s}\cdot r_A}{\mathfrak{E}}_{n\text{T}}(\widehat{s})$. The coherent field given by Eq. (143) is a superposition of two plane electromagnetic waves propagating along the incidence direction but with different wavenumbers. Thus, in general, the coherent field is not a plane electromagnetic wave. That would be the case only if the effective wavenumber with the smallest positive imaginary part were considered (the characteristic wave with a larger imaginary parts is more strongly attenuated in the medium, and so, can be neglected).

From the Dyson equation, the dyadic mass operator is given by Eq. (13) in the coordinate space, and by

$${\overline{M}}_p(p)=n_0{\overline{T}}_p(p,p)$$ (144)

in the Fourier space. In deriving Eq. (144), we used the fact that $\overline{M}$ is translation invariant, and that ${\overline{T}}_i$ is a translational dyadic, i.e., ${\overline{T}}_i\left(r,r^{\prime }\right)=\overline{T}\left(r-R_i,r^{\prime }-R_i\right)$. Taking into account the relationship between the far-field scattering dyadic and the Fourier transform of the transition dyadic given by Eq. (131), we obtain

$${\overline{M}}_{p\text{T}}\left(k_1\widehat{s}\right)=n_0{\overline{T}}_{p\text{T}}\left(k_1\widehat{s},k_1\widehat{s}\right)=4\pi n_0\overline{A}(\widehat{s},\widehat{s}).$$ (145)

Finally, substitution of Eq. (145) in Eq. (143) gives

$$\langle E(r)\rangle =\text{exp}\left\{\text{j}\left[k_1\overline{I}+\frac{2\pi }{k_1}{n}_0\overline{A}(\widehat{s},\widehat{s})\right]s(r,-\widehat{s})\right\}\cdot \langle E\left(r_A\right)\rangle ,$$ (146)

which is Eq. (46). In summary, the representation (146) has been obtained under the assumptions and approximations which are representative of a discrete random medium with a sparse concentration of particles: the far-field approximation, the assumption that the positions of the particles are statistically independent, and the Twersky approximation (the last two assumptions essentially, yield Eq. (144)). In addition, we supposed that the coherent field propagates along the incidence direction, and that the effective wavenumber is close to that of the background medium.

Appendix 5. Solution of the Foldy integral equation for the coherent field

To solve the Foldy integral equation for the coherent field,

$$E_{\text{c}}(r)=E_0(r)+n_0{\int }_D{g}_0\left(r_i\right)\overline{A}\left({\widehat{r}}_i,\widehat{s}\right)\cdot E_{\text{c}}\left(R_i\right){\text{d}}^3{R}_i,$$ (147)

we assume a particular form for E_c(r). Taking into account that the incident field can be expressed as E₀ (r) = exp[jk₁s(r)]E₀(r_A), where s(r) stands for $s(r,-\widehat{s})$ hereinafter, we suppose that E_c(r) has a similar dependency on E_c(r_A). More specifically, we set

$$E_{\text{c}}(r)=\text{exp}\left[\text{j}\overline{X}(\widehat{s})s(r)\right]\cdot E_{\text{c}}\left(r_A\right)$$ (148)

for some unknown dyadic $\overline{X}$ depending on the propagation direction of the incident wave $\widehat{s}$. Assuming E_c(r_A) = E₀(r_A), and representing $\overline{X}$ as

$$\overline{X}(\widehat{s})=k_1\overline{I}+\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s}),$$ (149)

gives

$$E_{\text{c}}(r)=\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s})s(r)\right]\cdot E_0(r),$$ (150)

where $\overline{W}(\widehat{s})$ is now the unknown dyadic to be determined. Setting r = R_i in Eq. (150) yields

$$E_{\text{c}}\left(R_i\right)=\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s})s\left(R_i\right)\right]\cdot E_0\left(R_i\right).$$ (151)

Substituting Eq. (151) in Eq. (147), making the change of variable R_i = r+p, which implies p = −r_i and d³R_i = d³p, using the partial results

$$E_0\left(R_i\right)={\text{e}}^{\text{j}{k}_1\widehat{s}\cdot p}{E}_0(r)$$ (152)

and

$$\begin{array}{l}\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s})s\left(R_i\right)\right]\hfill \\ =\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s})(p\cdot \widehat{s})\right]\cdot \text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s})s(r)\right],\hfill \end{array}$$ (153)

and finally, accounting of Eq. (150), we find

$$\begin{array}{ll}{E}_{\text{c}}(r)\hfill & =E_0(r)+n_0\{{\int }_D\frac{{\text{e}}^{\text{j}{k}_{1}p}}{p}{\text{e}}^{\text{j}{k}_1\widehat{s}\cdot p}\overline{A}(-\widehat{p},\widehat{s})\hfill \\ \hfill & \cdot \text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s})(p\cdot \widehat{s})\right]{\text{d}}^{3}p\}\cdot E_{\text{c}}(r).\hfill \end{array}$$ (154)

The asymptotic expansion of a plane wave in spherical waves [12]

$${\text{e}}^{\text{j}{k}_1\widehat{s}\cdot p}=\text{j}\frac{2\pi }{k_{1}p}\left[\delta (\widehat{s}+\widehat{p}){\text{e}}^{-\text{j}{k}_{1}p}-\delta (\widehat{s}-\widehat{p}){\text{e}}^{\text{j}{k}_{1}p}\right],$$ (155)

where $\delta (\widehat{s}-\widehat{p})$ is the solid-angle delta function, then yields

$$\begin{array}{ll}{E}_{\text{c}}(r)\hfill & =E_0(r)+\text{j}\frac{2\pi }{k_1}{n}_0\overline{A}(\widehat{s},\widehat{s})\hfill \\ \hfill & \cdot \left\{{\int }_0^{s(r)}\text{exp}\left[-\text{j}\frac{2\pi }{k_1}{n}_0\overline{W}(\widehat{s})p\right]\text{d}p\right\}\cdot E_{\text{c}}(r).\hfill \end{array}$$ (156)

Left-multiplying the above equation by $\widehat{s}$, using $\widehat{s}\cdot E_0(r)=0$ and $\widehat{s}\cdot \overline{A}(\widehat{s},\widehat{s})=0$ we obtain $\widehat{s}\cdot E_{\text{c}}(r)=0$ Left-multiplying now Eq. (150) by $\widehat{s}$, using $\widehat{s}\cdot E_{\text{c}}(r)=0$, and employing a series representation of the dyadic exponential, we find $\widehat{s}\cdot \overline{W}(\widehat{s})=0$. We further make the assumption that $\overline{W}(\widehat{s})\widehat{s}=0$, or equivalently, that $\overline{W}(\widehat{s})$ is a transverse dyadic, i.e.,

$$\overline{W}(\widehat{s})=\sum _{\eta ,\mu =\theta ,\phi }{[\text{W}(\widehat{s})]}_{\eta \mu }\widehat{\eta }(\widehat{s})\otimes \widehat{\mu }(\widehat{s}).$$ (157)

Because the coherent field is a transverse field, we write

$$E_{\text{c}}(r)=E_{\text{c}\theta }(r)\widehat{\theta }(\widehat{s})+E_{\text{c}\phi }(r)\widehat{\mathit{\phi }}(\widehat{s}),$$ (158)

and define the two-element column vector E_c(r) according to ${\text{E}}_{\text{c}}(r)={\left[E_{\text{c}\theta }(r)E_{\text{c}\phi }(r)\right]}^T$. As a result, the matrix equation associated with the dyadic equation (156) is

$$\begin{array}{ll}{\text{E}}_{\text{c}}(r)\hfill & ={\text{E}}_0(r)+\text{j}\frac{2\pi }{k_1}{n}_0\text{A}(\widehat{s},\widehat{s})\hfill \\ \hfill & \times \left\{{\int }_0^{s(r)}\text{exp}\left[-\text{j}\frac{2\pi }{k_1}{n}_{0}W(\widehat{s})p\right]\text{d}p\right\}{\text{E}}_{\text{c}}(r),\hfill \end{array}$$ (159)

while the matrix equation associated with the dyadic equation (150) is

$${\text{E}}_{\text{c}}(r)=\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\text{W}(\widehat{s})s(r)\right]{\text{E}}_0(r).$$ (160)

The integral in Eq. (159) is

$$\begin{array}{l}{\int }_0^{s(r)}\text{exp}\left[-\text{j}\frac{2\pi }{k_1}{n}_0\text{W}(\widehat{s})p\right]\text{d}p\hfill \\ ={\left(\text{j}\frac{2\pi }{k_1}{n}_0\right)}^{-1}{\text{W}}^{-1}(\widehat{s})\left\{{\text{I}}_2-\text{exp}\left[-\text{j}\frac{2\pi }{k_1}{n}_0\text{W}(\widehat{s})s(r)\right]\right\},\hfill \end{array}$$ (161)

and we obtain

$$\begin{array}{ll}{\text{E}}_{\text{c}}(r)\hfill & ={\text{E}}_0(r)+\text{A}(\widehat{s},\widehat{s}){\text{W}}^{-1}(\widehat{s})\hfill \\ \hfill & \times \left\{{\text{I}}_2-\text{exp}\left[-\text{j}\frac{2\pi }{k_1}{n}_0\text{W}(\widehat{s})s(r)\right]\right\}{\text{E}}_{\text{c}}(r).\hfill \end{array}$$ (162)

Finally, by means of Eq. (160), we get

$$\begin{array}{ll}\text{0}\hfill & =\left[{\text{I}}_2-\text{A}(\widehat{s},\widehat{s}){\text{W}}^{-1}(\widehat{s})\right]\hfill \\ \hfill & \times \left\{\text{exp}\left[\text{j}\frac{2\pi }{k_1}{n}_0\text{W}(\widehat{s})s(r)\right]-{\text{I}}_2\right\}{\text{E}}_0(\text{r}),\hfill \end{array}$$ (163)

and, since E₀ (r) is arbitrary, the solution of Eq. (163) is $\text{W}(\widehat{s})=\text{A}(\widehat{s},\widehat{s})$. Applying this result to Eqs. (148) and (149), we find that the coherent field is as in Eq. (46).