Multiple Scattering of Light in Ordered Particulate Media

Abstract

The methods to describe electromagnetic wave interaction with random and highly ordered particulate media as applied to solve problems of optics, photonics, and optoelectronics are presented. The approach to find spatial arrangement of particles forming the planar crystal with imperfect lattice is described. It is used to simulate light absorption by the solar cells and transmittance of antireflecion coatings, selective reflectors, multispectral filters based on periodic, quasiperiodic and aperiodic structures of monolayers; to solve the inverse scattering problem—retrieving the refractive index of particles forming the 3D photonic crystal. A number of scattering problem solutions for partially ordered particulate layers is considered. In particular: angular distribution of light scattered by monolayer, small-angle light scattering and transmission by polymer dispersed liquid crystal film, quenching effect for coherent component of transmitted light, and the spatial optical noise. Features in scattering and transmittance by correlated particles in liquating glasses are explained.

Appendix. Expansion of Fields in Terms of Scattering Orders

Let us write the main equations of the theory of multiple scattering of waves (TMSW) in the form:

$$\begin{aligned} {\psi _\mathbf{r}}&= {\psi _\mathbf{r}^i} + \sum \limits _{j = 1}^N {{t_{\mathbf{r}j}}{\psi _j^e,}} \end{aligned}$$

(A.1)

$$\begin{aligned} {\psi _j^e}&= {\psi _j^i} + \sum \limits _{k = 1,k \ne j}^N {{t_{jk}}{\psi _k^e,}} \end{aligned}$$

(A.2)

Field ${\psi }_{\mathbf{r}}$, i.e. the solution of Eq. (A.1), can be found by putting the (A.2) into (A.1):

$$\begin{aligned} {\psi _\mathbf{r}}&= {\psi _\mathbf{r}^i} + \sum \limits _{j = 1}^N {{t_{\mathbf{r}j}}\left( {{\psi _j^i} + \sum \limits _{k = 1,k \ne j}^N {{t_{jk}}{\psi _k^e}} } \right) } = \nonumber \\&= {\psi _\mathbf{r}^i} + \sum \limits _{j = 1}^N {t_{\mathbf{r}j}}{\psi _j^i} + \sum \limits _{j = 1}^N \sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}}^N {{t_{\mathbf{r}j}}{t_{jk}}{\psi _k^i + }} \sum \limits _{j = 1}^N \sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}}^N \sum \limits _{\begin{array}{c} l = 1 \\ l \ne k \end{array}}^N {t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{\psi _l^i}\\&+ \sum \limits _{j = 1}^N \sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}}^N \sum \limits _{\begin{array}{c} l = 1 \\ l \ne k \end{array}}^N \sum \limits _{\begin{array}{c} m = 1 \\ m \ne l \end{array}}^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{t_{lm}}{\psi _m^i} + \cdots .}\nonumber \end{aligned}$$

(A.3)

The first term, ${\psi }_\mathbf{r}^{i}$, on the right side of (A.3) describes the field of incident wave in the observation point r. The second and third terms represent the sum of N and $N(N-1)$ contributions of singly $t_{\mathbf{r}j}{{{\psi }_{j}^{i}}}$ and doubly $t_{\mathbf{r}j}t_{jk}{{{\psi }_{k}^{i }}}$ scattered waves, respectively. The fourth term describes the $N(N-1)^{2}$ contributions of triple scattering. It concludes the terms with $l=j$. Thus, this sum can be divided into two ones which describe three scattering events only by different particles, $t_{\mathbf{r}j}t_{jk}t_{kl}{{\psi }_{l}^{i}}$ ($k\ne j$, $l\ne k$, $l\ne j)$, and three scattering events at double passing by wave the same particle, $t_{\mathbf{r}j}t_{jk}t_{kj}{{\psi }_{j}^{i}}$ ($l=j\ne k)$, i.e. “forward-backward” scattering:

$$\begin{aligned} \sum \limits _{j = 1}^N \sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N \sum \limits _{\begin{array}{c} l = 1 \\ l \ne k \end{array}} ^N {t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{\psi _l^i} = \sum \limits _{j = 1}^N \sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N \sum \limits _{\begin{array}{c} l = 1 \\ l \ne k,l \ne j \end{array}} ^N {t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{\psi _l^i} + \sum \limits _{j = 1}^N \sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kj}}{\psi _j^i.}} \end{aligned}$$

(A.4)

The fifth term in (A.3) describes the sum of $N(N-1)^{3}$ events of fourfold scattering. It can be divided into the sums describing scattering events only on different particles and at passing by wave the particles more than one time:

$$\begin{aligned}&\sum \limits _{j = 1}^N \sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N \sum \limits _{\begin{array}{c} l = 1 \\ l \ne k \end{array}} ^N {\sum \limits _{\begin{array}{c} m = 1 \\ m \ne l \end{array}} ^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{t_{lm}}{\psi _m^i} = \sum \limits _{j = 1}^N {\sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N {\sum \limits _{\begin{array}{c} \,\,\,\,\,\,l = 1 \\ l \ne k,l \ne j \end{array}} ^N {\sum \limits _{\begin{array}{c} \,\,\,\,\,\,\,\,\,\,\,\,m = 1 \\ m \ne l,m \ne k,m \ne j \end{array}} ^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{t_{lm}}{\psi _m^i} + \sum \limits _{j = 1}^N {\sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N {\sum \limits _{\begin{array}{c} \,\,\,\,\,\,l = 1 \\ l \ne k,l \ne j \end{array}} ^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{t_{lk}}{\psi _k^i} + } } } } } } } } } \nonumber \\&+ \sum \limits _{j = 1}^N {\sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N {\sum \limits _{\begin{array}{c} \,\,\,\,\,\,l = 1 \\ l \ne k,l \ne j \end{array}} ^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kl}}{t_{lj}}{\psi _j^i} + \sum \limits _{j = 1}^N {\sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N {\sum \limits _{\begin{array}{c} \,\,\,\,\,\,\,m = 1 \\ m \ne k,m \ne j \end{array}} ^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kj}}{t_{jm}}{\psi _m^i} + \sum \limits _{j = 1}^N {\sum \limits _{\begin{array}{c} k = 1 \\ k \ne j \end{array}} ^N {{t_{\mathbf{r}j}}{t_{jk}}{t_{kj}}{t_{jk}}{\psi _k^i}} } } } } } } } \end{aligned}$$

(A.5)

On the right side of (A.5) the first term describes $N(N-1)(N-2)(N-3)$ events of fourfold scattering on different particles. The second, third, and fourth ones represent the $N(N-1)(N-2)$, and fifth one describes the $N(N-1)$ events of fourfold scattering at passing by wave the particles more than one time.