## Abstract

The small-angle approximation to the radiative transport equation
is applied to particle suspensions that emulate ocean water. A
particle size distribution is constructed from polystyrene and glass
spheres with the best available data for particle size distributions in
the ocean. A volume scattering function is calculated from the Mie
theory for the particles in water and in oil. The refractive-index
ratios of particles in water and particles in oil are 1.19 and 1.01,
respectively. The ratio 1.19 is comparable to minerals and
nonliving diatoms in ocean water, and the ratio 1.01 is comparable to
the lower limit for microbes in water. The point-spread functions
are measured as a function of optical thickness for both water and oil
mixtures and compared with the point-spread functions generated from
the small-angle approximation. Our results show that, under
conditions that emulate ocean water, the small-angle approximation is
valid only for small optical thicknesses. Specifically, the
approximation is valid only for optical thicknesses less than
3.

© 2001 Optical Society of America

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### Equations (11)

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(1)
$$f\left(\mathrm{\theta},r\right)=2\mathrm{\pi}{\int}_{0}^{\infty}{J}_{0}\left(2\mathrm{\pi}\mathrm{\theta}\mathrm{\psi}\right)F\left(\mathrm{\psi},r\right)\mathrm{\psi}\mathrm{d}\mathrm{\psi},F\left(\mathrm{\psi},r\right)=2\mathrm{\pi}{\int}_{0}^{{\mathrm{\theta}}_{max}}{J}_{0}\left(2\mathrm{\pi}\mathrm{\theta}\mathrm{\psi}\right)f\left(\mathrm{\theta},r\right)\mathrm{\theta}\mathrm{d}\mathrm{\theta},$$
(2)
$$\mathrm{\sigma}\left(\mathrm{\theta}\right)=2\mathrm{\pi}{\int}_{0}^{\infty}{J}_{0}\left(2\mathrm{\pi}\mathrm{\theta}\mathrm{\psi}\right)\mathrm{\Sigma}\left(\mathrm{\psi}\right)\mathrm{\psi}\mathrm{d}\mathrm{\psi},\mathrm{\Sigma}\left(\mathrm{\psi}\right)=2\mathrm{\pi}{\int}_{0}^{{\mathrm{\theta}}_{max}}{J}_{0}\left(2\mathrm{\pi}\mathrm{\theta}\mathrm{\psi}\right)\mathrm{\sigma}\left(\mathrm{\theta}\right)\mathrm{\theta}\mathrm{d}\mathrm{\theta}.$$
(3)
$$\underset{4\mathrm{\pi}}{\iint}\mathrm{\sigma}\left(\mathrm{\theta}\right)\mathrm{d}\mathrm{\Omega}=1.$$
(4)
$$F\left(\mathrm{\psi},r\right)=exp\left[-\mathrm{\xi}r+{s}_{f}r\mathrm{\Sigma}\left(\mathrm{\psi}\right)\right],$$
(5)
$$\mathrm{\eta}=2\mathrm{\pi}{\int}_{0}^{{\mathrm{\theta}}_{max}}\mathrm{\sigma}\left(\mathrm{\theta}\right)sin\left(\mathrm{\theta}\right)\mathrm{d}\mathrm{\theta}.$$
(6)
$$F\left(\mathrm{\psi},r\right)=exp\left\{-\mathrm{\tau}\left[1-\mathrm{\eta}\mathrm{\Sigma}\left(\mathrm{\psi}\right)\right]\right\}.$$
(7)
$$\mathrm{\sigma}\left(\mathrm{\theta}\right)=\frac{{\mathrm{\theta}}_{0}}{2\mathrm{\pi}{\left(\mathrm{\theta}_{0}{}^{2}+{\mathrm{\theta}}^{2}\right)}^{3/2}},$$
(8)
$$\mathrm{\sigma}\left(\mathrm{\theta}\right)=\frac{1}{{\mathrm{\theta}}^{3/2}{\left(\mathrm{\theta}_{0}{}^{2}+{\mathrm{\theta}}^{2}\right)}^{1/2}}.$$
(9)
$$\mathrm{\sigma}\left(\mathrm{\theta}\right)=\frac{{E}_{2}\left(\mathrm{\theta}\right)/{E}_{2}\prime \left(0\right)-{E}_{1}\left(\mathrm{\theta}\right)/{E}_{1}\prime \left(0\right)}{\left({\mathrm{\tau}}_{2}-{\mathrm{\tau}}_{1}\right)\frac{a}{{R}^{2}}{cos}^{3}\left[\mathrm{arctan}\left(y/R\right)\right]}.$$
(10)
$${\mathrm{\theta}}_{max}=\mathrm{arcsin}\left\{\frac{{n}_{1}}{{n}_{2}}sin\left[\mathrm{arctan}\left(\frac{d}{R}\right)\right]\right\}$$
(11)
$$f\left(\mathrm{\theta},L\right)=\frac{P\left(\mathrm{\theta},L\right)}{{P}_{0}\mathrm{\Omega}}=\frac{E\left(\mathrm{\theta},L\right)}{{E}_{0}\frac{a}{{R}^{2}}{cos}^{3}\left[\mathrm{arctan}\left(y/R\right)\right]}.$$