## Abstract

A diffusion model is developed for describing the propagation of optical pulses
through dense scattering media (clouds). The model invokes both 1-D and 3-D
diffusion and is valid even for clouds of only a few diffusion thicknesses. The
predictions of the model are compared to experimental pulse shapes obtained by
other workers for real clouds and scale model clouds and by Monte Carlo
simulations. Comparisons indicate that the ratio of the 3-D to the 1-D component
in the transmitted pulse increases with both the diffusion thickness of the
cloud and the field of view of the receiver. Knowledge of the impulse response
of a cloud of known length is sufficient to determine the diffusion length and
hence the average scattering coefficient of the cloud.

© 1988 Optical Society of America

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### Equations (20)

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(1)
$$J=-{D}_{3}\nabla N,$$
(2)
$${D}_{3}{\nabla}^{2}N=\partial N/\partial t.$$
(3)
$${D}_{n}{\nabla}^{2}N=\partial N/\partial t,$$
(4)
$${D}_{n}={b}^{\prime}c/n,$$
(5)
$${J}_{n}=-{D}_{n}\nabla N;\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}n=1,3.$$
(6)
$${D}_{n}{\partial}^{2}N/\partial {x}^{2}=\partial N/\partial t,$$
(7)
$${J}_{n}=-{D}_{n}\partial N/\partial x;\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}n=1,3.$$
(8)
$${\partial}^{2}{J}_{n}/\partial {x}^{2}+\partial /\partial t(\partial N/\partial x)=0.$$
(9)
$${D}_{n}{\partial}^{2}{J}_{n}/\partial {x}^{2}=\partial {J}_{n}/\partial t,$$
(10)
$$\frac{1}{{\text{R}}^{\prime}{\text{C}}^{\prime}}\frac{{\partial}^{2}I}{\partial {x}^{2}}=\frac{\partial I}{\partial t},$$
(11)
$$\frac{{(\mathrm{\Delta}x)}^{2}}{\text{RC}}\frac{{\mathrm{\Delta}}^{2}I}{{(\mathrm{\Delta}x)}^{2}}=\frac{\mathrm{\Delta}I}{\mathrm{\Delta}t},$$
(12)
$${D}_{1}=\frac{{(\mathrm{\Delta}x)}^{2}}{{(\text{RC})}_{1}};\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{D}_{3}=\frac{{(\mathrm{\Delta}x)}^{2}}{{(\text{RC})}_{3}}.$$
(13)
$${(\text{RC})}_{1}={b}^{\prime}/c;\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{(\text{RC})}_{3}=3{b}^{\prime}/c.$$
(14)
$${I}_{n}(t)=S(t)*{h}_{n}(t);\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}n=1,3.$$
(15)
$${I}_{n}(t)={P}_{1}{I}_{1}(t)+{P}_{3}{I}_{3}(t),$$
(16)
$${t}_{pn}=0.1n{\tau}_{D}^{2}{b}^{\prime}{c}^{-1}$$
(17)
$$=0.1n{\tau}_{D}L{c}^{-1}$$
(18)
$$=0.1n{({b}^{\prime})}^{-1}{c}^{-1}{L}^{2}.$$
(19)
$$\begin{array}{l}\u3008l\u3009=0.5{b}^{\prime}{c}^{-1}{\tau}_{D}^{1.96};\\ {\sigma}_{l}=0.64{b}^{\prime}{c}^{-1}{\tau}_{D}^{1.81}.\end{array}$$
(20)
$$\begin{array}{l}\u3008t\u3009=0.55{b}^{\prime}{c}^{-1}{\tau}_{D}^{1.9},\\ {\sigma}_{t}=0.43{b}^{\prime}{c}^{-1}{\tau}_{D}^{1.8}.\end{array}$$