## Abstract

Some extinction laws for radiation transmitted through inhomogeneous random media were discussed by Kostinski [J. Opt. Soc. Am. A **18**, 1929 (2001)] by means of a complicated use of concepts of statistical theory of fluids. We show that these extinction laws are readily obtained in terms of classical probability theory. The validity of exponential extinction laws for large observation distances (as compared with the size of inhomogeneities of a medium) is proven and emphasized. It is shown that Kostinski’s results turn out to be applicable to small observation distances only, for which the concept of extinction law is hardly applicable.

© 2002 Optical Society of America

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### Equations (12)

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(1)
$$T=exp(-\tau )=exp\left[-\sigma {\int}_{0}^{L}c(l)\mathrm{d}l\right],$$
(2)
$$\u3008exp(-\tau )\u3009\ge exp(-\u3008\tau \u3009),$$
(3)
$$\u3008T\u3009=\u3008exp(-\tau )\u3009=exp\sum _{n=1}^{\infty}(-1{)}^{n}{\kappa}_{n}/n!=exp\sum _{n=1}^{\infty}(-\sigma {)}^{n}\int {g}_{n}\mathrm{d}{l}_{1}\mathrm{d}{l}_{2}\dots \mathrm{d}{l}_{n}/n!,$$
(4)
$$\u3008T\u3009=exp(-\u3008\tau \u3009)=exp\left[-\sigma {\int}_{0}^{L}\u3008c(l)\u3009\mathrm{d}l\right].$$
(5)
$$\u3008T\u3009=exp\left[-\sigma {\int}_{0}^{L}\tilde{c}(l)\mathrm{d}l\right],$$
(6)
$$\tilde{c}(l)=\u3008c(l)\u3009+\sum _{n=2}^{\infty}(-\sigma {)}^{n-1}{b}_{n}(l)/n!.$$
(7)
$$c(\mathbf{r})=\sum _{j=1}^{N}{c}_{j}(\mathbf{r}),$$
(8)
$${s}_{j}=\int \int \left\{exp\left[-\sigma {\int}_{-\infty}^{\infty}{c}_{j}(x,y,z)\mathrm{d}z\right]-1\right\}\mathrm{d}x\mathrm{d}y$$
(9)
$$\u3008T\u3009=exp\u3008-s\u3008C\u3009L\u3009,$$
(10)
$$\u3008T\u3009={\int}_{0}^{\infty}p(c)exp(-c\sigma L)\mathrm{d}c={\mathrm{\Lambda}}_{p(c)}(\sigma L).$$
(11)
$$p(c)=\u3008c{\u3009}^{-1}exp(-c/\u3008c\u3009).$$
(12)
$$\u3008T\u3009=1/(1+\u3008c\u3009\sigma L).$$