## Abstract

The propagation of light through turbid media is of fundamental interest in a number of areas of optical science including atmospheric and oceanographic science, astrophysics and medicine amongst many others. The angular distribution of photons after a single scattering event is determined by the scattering phase function of the material the light is passing through. However, in many instances photons experience multiple scattering events and there is currently no equivalent function to describe the resulting angular distribution of photons. Here we present simple analytic formulas that describe the angular distribution of photons after multiple scattering events, based only on knowledge of the single scattering albedo and the single scattering phase function.

© 2011 OSA

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### Equations (8)

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(1)
$${g}_{1}=2\pi {\displaystyle \underset{0}{\overset{\pi}{\int}}\tilde{\beta}\left(\theta \right)\mathrm{sin}\theta \mathrm{cos}\theta d\theta}$$
(2)
$$\mathrm{cos}{\theta}_{n}=\mathrm{cos}{\theta}_{1}\mathrm{cos}{\theta}_{2}+\mathrm{sin}{\theta}_{1}\mathrm{sin}{\theta}_{2}\mathrm{cos}\psi $$
(3)
$$\tilde{\beta}\left(\theta \right)=\frac{1}{4\pi}\frac{1-{g}^{2}}{{\left(1+{g}^{2}-2g\mathrm{cos}\theta \right)}^{3/2}}$$
(4)
$$\tilde{\beta}\left(\theta \right)=\frac{1}{4\pi {\left(1-\delta \right)}^{2}{\delta}^{\nu}}\left(\left[\nu \left(1-\delta \right)-\left(1-{\delta}^{\nu}\right)\right]+\frac{4}{{u}^{2}}\left[\delta \left(1-{\delta}^{\nu}\right)-\nu \left(1-\delta \right)\right]\right)$$
(5)
$$\nu =\frac{3-\mu}{2},\text{\hspace{1em}}\delta =\frac{{u}^{2}}{3{\left({n}_{r}-1\right)}^{2}},\text{\hspace{1em}}u=2\phantom{\rule{.1em}{0ex}}\mathrm{sin}\phantom{\rule{.1em}{0ex}}\left(\theta /2\right)$$
(6)
$${g}_{ms}=\frac{{\displaystyle \sum _{i=1}^{n}{g}_{i}{\omega}^{i}}}{{\displaystyle \sum _{i=1}^{n}{\omega}^{i}}}=\frac{{g}_{1}\left(1-\omega \right)}{\left(1-{g}_{1}\omega \right)}$$
(7)
$${\tilde{\beta}}_{n}\left({\theta}_{n}\right)=4\pi \frac{{\displaystyle \sum _{{\theta}_{1},{\theta}_{2},\psi}{\tilde{\beta}}_{n-1}\left({\theta}_{1}\right)\mathrm{sin}\left({\theta}_{1}\right)\mathrm{\Delta}{\theta}_{1}\cdot {\tilde{\beta}}_{1}\left({\theta}_{2}\right)\mathrm{sin}\left({\theta}_{2}\right)}\mathrm{\Delta}{\theta}_{2}}{{\displaystyle \sum _{{\theta}_{1},{\theta}_{2},\psi}\mathrm{sin}\left({\theta}_{1}\right)\mathrm{\Delta}{\theta}_{1}\cdot \mathrm{sin}\left({\theta}_{2}\right)\mathrm{\Delta}{\theta}_{2}}}$$
(8)
$${\tilde{\beta}}_{ms}\left(\theta \right)=\frac{{\tilde{\beta}}_{1}\left(\theta \right)+\omega {\tilde{\beta}}_{2}\left(\theta \right)+{\omega}^{2}{\tilde{\beta}}_{3}\left(\theta \right)+\dots +{\omega}^{n-1}{\tilde{\beta}}_{n}\left(\theta \right)}{1+\omega +{\omega}^{2}+\dots +{\omega}^{n-1}}$$