## Abstract

We have developed a hit-and-miss Monte Carlo geometric ray-tracing program to compute the scattering phase matrix for concentrically stratified spheres. Using typical refractive indices for water and aerosols in the calculations, numerous rainbow features appear in the phase matrix that deviate from the results calculated from homogeneous spheres. In the context of geometric ray tracing, rainbows and glory are identified by means of their ray paths, which provide physical explanation for the features produced by the “exact” Lorenz–Mie theory. The computed results for the phase matrix, the single-scattering albedo, and the asymmetry factor for a size parameter of $\sim 600$ compared closely with those evaluated from the “exact” theory.

© 2010 Optical Society of America

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### Equations (16)

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(1)
$$\mathbf{P}=\left[\begin{array}{cccc}{P}_{11}& {P}_{12}& 0& 0\\ {P}_{12}& {P}_{22}& 0& 0\\ 0& 0& {P}_{33}& -{P}_{43}\\ 0& 0& {P}_{43}& {P}_{44}\end{array}\right].$$
(2)
$${P}_{ij}=\sum _{\gamma}\sum _{n}\frac{{p}_{ij,n}}{{p}_{11,n}\mathrm{sin}{\theta}_{n}},$$
(3)
$${p}_{11}=\frac{1}{2}({M}_{2}+{M}_{3}+{M}_{4}+{M}_{1}),$$
(4)
$${p}_{12}=\frac{1}{2}({M}_{2}-{M}_{3}+{M}_{4}-{M}_{1}),$$
(5)
$${p}_{22}=\frac{1}{2}({M}_{2}-{M}_{3}-{M}_{4}+{M}_{1}),$$
(6)
$${p}_{33}={S}_{21}+{S}_{34},$$
(7)
$${p}_{44}={S}_{21}-{S}_{34},$$
(8)
$${p}_{43}={D}_{21}+{D}_{34},$$
(9)
$${M}_{k}=|{S}_{k}{|}^{2},$$
(10)
$${S}_{kl}={S}_{lk}=({S}_{l}{S}_{k}^{*}+{S}_{k}{S}_{l}^{*})/2,$$
(11)
$${D}_{kl}=-{D}_{lk}=({S}_{l}{S}_{k}^{*}-{S}_{k}{S}_{l}^{*})i/2.$$
(12)
$${\mathbf{S}}_{n}=\{\begin{array}{ll}{\mathbf{R}}_{1}& \text{for}n=1\\ {\mathbf{P}}_{t}{\mathbf{P}}_{s}\prod _{q}[{\mathbf{T}}_{n}{\mathbf{P}}_{n}\prod _{k=n-1}^{2}({\mathbf{R}}_{k}{\mathbf{P}}_{k}){\mathbf{T}}_{1}{\mathbf{P}}_{1}{]}_{q}{\mathbf{P}}_{e}& \text{for}n\ge 2\end{array},$$
(13)
$${\mathbf{P}}_{k}=\left[\begin{array}{cc}\mathrm{cos}{\varphi}_{k}& \mathrm{sin}{\varphi}_{k}\\ -\mathrm{sin}{\varphi}_{k}& \mathrm{cos}{\varphi}_{k}\end{array}\right],\phantom{\rule[-0.0ex]{2em}{0.0ex}}{\mathbf{R}}_{k}=\left[\begin{array}{cc}{R}_{lk}& 0\\ 0& {R}_{rk}\end{array}\right],\phantom{\rule[-0.0ex]{2em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{\mathbf{T}}_{k}=\left[\begin{array}{cc}{T}_{lk}& 0\\ 0& {T}_{rk}\end{array}\right].$$
(14)
$$n(a)=C{a}^{(1-3{v}_{e}){v}_{e}}\mathrm{exp}(-\frac{a}{{a}_{e}{v}_{e}}),$$
(15)
$${a}_{e}=\frac{\int a\pi {a}^{2}n(a)\mathrm{d}a}{\int \pi {a}^{2}n(a)\mathrm{d}a},$$
(16)
$${v}_{e}=\frac{\int (a-{a}_{e}{)}^{2}\pi {a}^{2}n(a)\mathrm{d}a}{{a}_{e}^{2}\int \pi {a}^{2}n(a)\mathrm{d}a}.$$