## Abstract

A numerical method was developed using geometrical optics to predict far-field optical scattering from particles that are symmetric about the optic axis. The diffractive component of scattering is calculated and combined with the reflective and refractive components to give the total scattering pattern. The phase terms of the scattered light are calculated as well. Verification of the method was achieved by assuming a spherical particle and comparing the results to Mie scattering theory. Agreement with the Mie theory was excellent in the forward-scattering direction. However, small-amplitude oscillations near the rainbow regions were not observed using the numerical method. Numerical data from spheroidal particles and hemispherical particles are also presented. The use of hemispherical particles as a calibration standard for intensity-type optical particle-sizing instruments is discussed.

© 1991 Optical Society of America

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### Equations (17)

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(1)
$${S}_{1r}=ka{\u220a}_{1}{D}^{1/2}{e}^{i\sigma},$$
(2)
$${S}_{2r}=ka{\u220a}_{2}{D}^{1/2}{e}^{i\sigma},$$
(3)
$$\text{optical path length}=k[{X}_{1}+({n}_{2}Lp)+d].$$
(4)
$$\delta =2ka-k[{X}_{1}+({n}_{2}Lp)+d]$$
(5)
$$=2ka-k[(a-asin\tau )+{n}_{2}p(2asin{\tau}^{\prime})+(a-asin\tau )]$$
(6)
$$=2ka(sin\tau -{n}_{2}psin{\tau}^{\prime}),$$
(7)
$$\sigma =(\pi /2)+\delta +(\pi /2)\phantom{\rule{0.2em}{0ex}}({n}_{\text{a}}+{n}_{\text{b}}).$$
(8)
$${S}_{\text{D}}={(ka)}^{2}\frac{{J}_{1}(kasin\theta )}{kasin\theta},$$
(9)
$${S}_{1}=\sum _{p=0}^{\infty}({S}_{1r})+{S}_{\text{D}},$$
(10)
$${S}_{2}=\sum _{p=0}^{\infty}({S}_{2r})+{S}_{\text{D}}.$$
(11)
$$dl={({{R}_{1}}^{2}+{{R}_{2}}^{2}-2{R}_{1}{R}_{2}cos{\theta}_{d})}^{1/2},$$
(12)
$$\begin{array}{ll}{R}_{1}={({x}_{1}+{y}_{1})}^{1/2},\hfill & {\theta}_{1}=arctan({y}_{1}/{x}_{1}),\hfill \\ {R}_{2}={({x}_{2}+{y}_{2})}^{1/2},\hfill & {\theta}_{2}=arctan({y}_{2}/{x}_{2}),\hfill & {\theta}_{d}={\theta}_{2}-{\theta}_{1}.\end{array}$$
(13)
$$dp={R}_{a}sin{\theta}_{a}d\varphi ,$$
(14)
$$\begin{array}{cc}{R}_{a}=({R}_{1}+{R}_{1})/2,& {\theta}_{a}=({\theta}_{1}+{\theta}_{2})/2\phantom{\rule{0.2em}{0ex}}.\end{array}$$
(15)
$$\text{flux}={I}_{0}\cdot dA$$
(16)
$$={I}_{0}sin\left(\frac{{\beta}_{1}+{\beta}_{2}}{2}\right)\phantom{\rule{0.2em}{0ex}}dldp,$$
(17)
$$d{A}^{\prime}={r}^{2}sin\theta d{\theta}^{\prime}d\varphi ,$$