## Abstract

The small-angle approximation to the radiative transport equation is used extensively in imaging models in which the transport medium is optically thick. The small-angle approximation is generally considered valid when the particles are very large compared with the wavelength, when the refractive-index ratio of the particle to the medium is close to 1, and when the optical thickness is not too large. We report results showing the limits of the validity of the small-angle approximation as a function of particle size and concentration for a particle-to-medium fixed refractive-index ratio of 1.196. This refractive-index ratio is comparable with that of minerals or diatoms suspended in water.

© 2001 Optical Society of America

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### Equations (19)

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(1)
$$f(\theta ,r)=2\pi {\int}_{0}^{\infty}{J}_{0}(2\pi \theta \psi )F(\psi ,r)\psi \mathrm{d}\psi ,$$
(2)
$$F(\psi ,r)=2\pi {\int}_{0}^{{\theta}_{max}}{J}_{0}(2\pi \theta \psi )f(\theta ,r)\theta \mathrm{d}\theta ,$$
(3)
$$\sigma (\theta )=2\pi {\int}_{0}^{\infty}{J}_{0}(2\pi \theta \psi )\mathrm{\Sigma}(\psi )\psi \mathrm{d}\psi ,$$
(4)
$$\mathrm{\Sigma}(\psi )=2\pi {\int}_{0}^{{\theta}_{max}}{J}_{0}(2\pi \theta \psi )\sigma (\theta )\theta \mathrm{d}\theta .$$
(5)
$$\int {\int}_{4\pi}\sigma (\theta )\mathrm{d}\mathrm{\Omega}=1.$$
(6)
$$F(\psi ,r)=exp[-\xi r+{s}_{f}r\mathrm{\Sigma}(\psi )],$$
(7)
$$\eta =2\pi {\int}_{0}^{{\theta}_{max}}\sigma (\theta )sin(\theta )\mathrm{d}\theta .$$
(8)
$$F(\psi ,r)=exp\{-\tau [1-\eta \mathrm{\Sigma}(\psi )]\}.$$
(9)
$$\sigma (\theta )=\frac{\mathrm{d}I(\theta )}{s{\mathrm{\Phi}}_{n}\mathrm{d}V},$$
(10)
$$\mathrm{d}{P}_{\sigma}(\theta )=\mathrm{d}I(\theta )\mathrm{\Omega}exp[-\xi (L-x)].$$
(11)
$${\mathrm{\Phi}}_{n}=({P}_{0}/A)exp(-\xi x),$$
(12)
$$\mathrm{d}{P}_{\sigma}(\theta )=\sigma (\theta )s\mathrm{\Omega}{P}_{0}exp(-\xi L)\mathrm{d}x,$$
(13)
$${P}_{\sigma}(\theta )=\sigma (\theta )s\mathrm{\Omega}{P}_{0}exp(-\xi L){\int}_{0}^{L}\mathrm{d}x=\sigma (\theta )\tau {P}_{0}\mathrm{\Omega}exp(-\xi L),$$
(14)
$${P}_{r1}(\theta )={P}_{n}exp(-{\xi}_{1}L)+\sigma (\theta ){\tau}_{1}{P}_{0}\mathrm{\Omega}exp(-{\xi}_{1}L),$$
(15)
$${P}_{r2}(\theta )={P}_{n}exp(-{\xi}_{2}L)+\sigma (\theta ){\tau}_{2}{P}_{0}\mathrm{\Omega}exp(-{\xi}_{2}L),$$
(16)
$$\sigma (\theta )=\frac{\{[{P}_{r2}(\theta )]/[{P}_{r2}^{\prime}(0)]\}-\{[{P}_{r1}(\theta )]/[{P}_{r1}^{\prime}(0)]\}}{\mathrm{\Omega}({\tau}_{2}-{\tau}_{1})},$$
(17)
$$\sigma (\theta )=\frac{\{[{E}_{2}(\theta )]/[{E}_{2}^{\prime}(0)]\}-\{[{E}_{1}(\theta )/{E}_{1}^{\prime}(0)]\}}{({\tau}_{2}-{\tau}_{1})({a}_{1}/{R}^{2}){cos}^{3}[\mathrm{arctan}(y/R)]},$$
(18)
$${\theta}_{max}=\mathrm{arcsin}\{({n}_{1}/{n}_{2})sin[\mathrm{arctan}(d/R)]\},$$
(19)
$$f(\theta ,L)=\frac{[P(\theta ,L)]}{({P}_{0}\mathrm{\Omega})}=\frac{E(\theta ,L)}{{E}_{0}(a/{R}^{2}){cos}^{3}[\mathrm{arctan}(y/R)]}.$$