## Abstract

The results of a Monte Carlo study on light propagation in dense turbid media are presented. The calculations refer to the radiation emerging from a spherical scattering cell containing the diffusing medium (no diffusers outside the cell are considered) in whose center a point source is placed. Both the total scattered power emerging from the sphere and the impulse response were evaluated for a large range of optical depths and different types of diffuser. The results pertaining to both the radiance at the surface of the scattering cell and the impulse response are described by simple empirical relations. The results suggest a method of measuring the albedo and the asymmetry factor of the diffusing medium. A comparison with some results of the diffusion approximation is also presented.

© 1991 Optical Society of America

Full Article |

PDF Article
### Equations (33)

Equations on this page are rendered with MathJax. Learn more.

(1)
$${l}_{d}=\frac{1}{{\sigma}_{s}(1-g)},$$
(2)
$$\beta ({\varphi}_{i}\le \varphi \le {\varphi}_{i+1})=\frac{1}{2\pi (\text{cos}{\varphi}_{i}-\text{cos}{\varphi}_{i+1})}\frac{{L}_{i}}{{\displaystyle \sum _{i=1}^{I}}{L}_{i}},$$
(3)
$${f}_{1}({l}_{i}\le l<{l}_{i+1})=\frac{1}{{l}_{i+1}-{l}_{i}}\frac{{M}_{i}}{{\displaystyle \sum _{i=1}^{I}}{M}_{i}},$$
(4)
$${f}_{2}({K}_{i}\le K<{K}_{i+1})=\frac{1}{{K}_{i+1}-{K}_{i}}\frac{{N}_{i}}{{\displaystyle \sum _{i=1}^{I}}{N}_{i}},$$
(5)
$${P}_{dr}({\tau}_{s})={P}_{e}[1-\text{exp}(-{\tau}_{s})]\frac{A}{4\pi {R}^{2}}2\pi {\int}_{0}^{\alpha}\beta (\varphi )\hspace{0.17em}\text{sin}\varphi d\varphi ,$$
(6)
$${P}_{or}({\tau}_{s})={P}_{e}\hspace{0.17em}\text{exp}(-{\tau}_{s})\frac{A}{4\pi {R}^{2}},$$
(7)
$$g(t)=[1-\text{exp}(-{\tau}_{s})]\frac{{\tau}_{s}}{R}c{f}_{1}(l),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l=tc{\tau}_{s}/R.$$
(8)
$${P}_{d}(t)={\int}_{-\infty}^{\infty}{P}_{e}({t}^{\prime})g(t-{t}^{\prime})d{t}^{\prime},$$
(9)
$${P}_{0}(t)={P}_{e}(t-{t}_{0})\text{exp}(-{\tau}_{s}),$$
(10)
$${g}_{1}\left(\frac{t}{{t}_{0}}\right)=[1-\text{exp}(-{\tau}_{s})]{\tau}_{s}{f}_{1}(l),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l={\tau}_{s}\frac{t}{{t}_{0}}.$$
(11)
$$\overline{\beta}(\varphi )=\frac{n+1}{2\pi}{\text{cos}}^{n}\varphi ,$$
(12)
$${q}_{1}(l)=\text{exp}(-l).$$
(13)
$${q}_{2}(l)={\int}_{0}^{l}{q}_{1}({l}^{\prime}){q}_{1}(l-{l}^{\prime})d{l}^{\prime}=l\hspace{0.17em}\text{exp}(-l).$$
(14)
$${q}_{K}(l)={\int}_{0}^{l}{q}_{K-1}({l}^{\prime}){q}_{1}(l-{l}^{\prime})d{l}^{\prime}=\frac{{l}^{K-1}}{(K-1)!}\text{exp}(-l).$$
(15)
$${\u3008{l}^{m}\u3009}_{K}={\int}_{0}^{\infty}{l}^{m}{q}_{K}(l)dl=K(K+1)\dots (K+m-1),$$
(16)
$${l}_{\text{max}K}=K-1.$$
(17)
$${q}_{K}^{\prime}\left(\frac{l}{K}\right)=K{q}_{K}(l)$$
(18)
$$\u3008K\u3009=0.50{\tau}_{s}{\tau}_{d}.$$
(19)
$$h(\xi )=3{\tau}_{s}({\tau}_{d}+1){f}_{1}(l),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l=3{\tau}_{s}({\tau}_{d}+1)\xi ,$$
(20)
$$\begin{array}{lll}\overline{h}(\xi )\hfill & =C{\xi}^{-3.75}\hspace{0.17em}\text{exp}\left(-\frac{0.3465}{\xi}\right)\hfill & \text{if}\hspace{0.17em}\xi >{\xi}_{0},\hfill \\ \hspace{0.17em}\hfill & =0\hfill & \text{if}\hspace{0.17em}\xi \le {\xi}_{0},\hfill \end{array}$$
(21)
$${\int}_{{\xi}_{0}}^{\infty}\overline{h}(\xi )d\xi =[1-\text{exp}(-{\tau}_{s})].$$
(22)
$$\begin{array}{l}{\overline{g}}_{1}(t/{t}_{0})=[1-\text{exp}(-{\tau}_{s})]\frac{C}{3({\tau}_{d}+1)}{\left[\frac{{t}_{0}}{t}3({\tau}_{d}+1)\right]}^{3.75}\times \text{exp}\left[-1.04({\tau}_{d}+1)\frac{{t}_{0}}{t}\right],\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}t>{t}_{0},\\ =0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}t\le {t}_{0}.\end{array}$$
(23)
$$\frac{{t}_{\text{max}}}{{t}_{0}}=\frac{1.04}{3.75}({\tau}_{d}+1),$$
(24)
$$\frac{\u3008t\u3009}{{t}_{0}}=0.58({\tau}_{d}+1),$$
(25)
$$\frac{\mathrm{\Delta}{t}_{1/10}}{{t}_{0}}=0.97({\tau}_{d}+1),$$
(26)
$${\sigma}_{a}\rho ={\tau}_{a}\frac{l}{{\tau}_{s}}.$$
(27)
$${g}_{1}(t/{t}_{0})=[1-\text{exp}(-{\tau}_{s})]{\tau}_{s}{f}_{1}(l)\hspace{0.17em}\text{exp}(-{\tau}_{a}l/{\tau}_{s}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l={\tau}_{s}\frac{t}{{t}_{0}},$$
(28)
$${P}_{t}({\tau}_{s},{\tau}_{a})={P}_{c}\left\{\text{exp}-({\tau}_{s}+{\tau}_{a})+[1-\text{exp}(-{\tau}_{s})]{\int}_{{\tau}_{s}}^{\infty}{f}_{1}(l)\text{exp}-\left({\tau}_{a}\frac{l}{{\tau}_{s}}\right)dl\right\},$$
(29)
$$\begin{array}{l}{P}_{t}=4\pi {R}^{2}{F}_{d}\\ ={P}_{e}\{1+{[3{\tau}_{a}({\tau}_{d}+{\tau}_{a})]}^{0.5}\}\text{exp}\{-{[3{\tau}_{a}({\tau}_{d}+{\tau}_{a})]}^{0.5}\},\end{array}$$
(30)
$$\begin{array}{lll}G\left(R,\frac{t}{{t}_{0}}\right)\hfill & =\frac{2\sqrt{\pi}}{{(4\pi R)}^{2}}{\left(3{\tau}_{d}\frac{{t}_{0}}{t}\right)}^{3/2}\text{exp}\left(-{\tau}_{a}\frac{t}{{t}_{0}}-\frac{3}{4}{\tau}_{d}\frac{{t}_{0}}{t}\right),\hfill & t>0,\hfill \\ \hspace{0.17em}\hfill & =0,\hfill & t<0,\hfill \end{array}$$
(31)
$${P}_{K}(\mathbf{r})\approx {\left(\frac{3}{2\pi K}\right)}^{3/2}\text{exp}\left[-\frac{3}{2K}({x}^{2}+{y}^{2}+{z}^{2})\right],$$
(32)
$${x}^{2}+{y}^{2}+{z}^{2}={R}^{2}\frac{{\sigma}_{s}^{2}}{2}$$
(33)
$$P(l,{\tau}_{s})\approx A{\left(\frac{1}{l}\right)}^{3/2}\text{exp}\left(-\frac{3}{4}\frac{{\tau}_{s}^{2}}{l}\right),$$