## Abstract

By means of a numerical study we show particle-size distributions retrieved with the Chin–Shifrin, Phillips–Twomey, and singular value decomposition methods. Synthesized intensity data are generated using Mie theory, corresponding to unimodal normal, gamma,
and lognormal distributions of spherical particles, covering the size parameter range from 1 to 250. Our results show the advantages and disadvantages of each method, as well as the range of applicability for the Fraunhofer approximation as compared to rigorous Mie theory.

© 2007 Optical Society of America

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### Equations (16)

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(1)
$${f}_{G}\left(\alpha \right)=\{\begin{array}{cc}N\text{\hspace{0.17em}}\frac{{\left(\alpha -{\alpha}_{0}\right)}^{(1/b)-3}}{{\left(bt\right)}^{(1/b)-2}{\Gamma}^{\left[(1/b)-2\right]}}\text{\hspace{0.17em} exp}\left[\frac{{\alpha}_{\text{0}}-\alpha}{bt}\right]\hfill & \text{for \hspace{0.17em}}\alpha >{\alpha}_{\text{0}}\hfill \\ 0\hfill & \text{for \hspace{0.17em}}\alpha \le {\alpha}_{\text{0}}\end{array}\text{,}$$
(2)
$${f}_{L}\left(\alpha \right)=\frac{N}{\sigma \left(\alpha -{\alpha}_{0}\right)\sqrt{2\pi}}\text{\hspace{0.17em} exp}\left[-{\left(\frac{\text{ln}\left(\frac{\alpha -{\alpha}_{\text{0}}}{{A}_{0}}\right)}{\sigma \sqrt{2}}\right)}^{2}\right],$$
(3)
$$\mu =\text{exp}\left[\text{ln}\left({A}_{0}\right)+\frac{{\sigma}^{2}}{2}\right].$$
(4)
$${f}_{N}\left(\alpha \right)=\frac{N}{\sigma \sqrt{2\pi}}\text{\hspace{0.17em} exp}\left[-\frac{1}{2}{\left(\frac{\alpha -\mu}{\sigma}\right)}^{2}\right],$$
(5)
$$I\left(\theta \right)={\displaystyle {\int}_{0}^{\infty}I\left(\theta ,\alpha \right)f\left(\alpha \right)}\mathrm{d}\alpha ,$$
(7)
$$I\left(\theta \right)=\frac{{I}_{0}}{{k}^{2}{F}^{2}}\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{\infty}\frac{{\alpha}^{2}{{J}_{1}}^{2}\left(\alpha \theta \right)}{{\theta}^{2}}}\text{\hspace{0.17em}}f\left(\alpha \right)\mathrm{d}\alpha \text{,}$$
(8)
$$f\left(\alpha \right)=\frac{-2\pi {k}^{3}{F}^{2}}{{\alpha}^{2}}\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{\infty}\left(\alpha \theta \right){J}_{1}}\left(\alpha \theta \right){Y}_{1}\left(\alpha \theta \right)\frac{\mathrm{d}}{\mathrm{d}\theta}\left[{\theta}^{3}\text{\hspace{0.17em}}\frac{I\left(\theta \right)}{{I}_{0}}\right]\mathrm{d}\theta \text{,}$$
(10)
$$Q={A}^{-1}=V{P}^{-1}{U}^{T}.$$
(12)
$$I\left(\theta \right)={\displaystyle {\int}_{0}^{\infty}\frac{{I}_{0}\left({i}_{1}+{i}_{2}\right)}{2{k}^{2}{R}^{2}}}\text{\hspace{0.17em}}f\left(\alpha \right)\mathrm{d}\alpha ,$$
(13)
$${\Vert \mathrm{\u03f5}\Vert}^{\mathbf{2}}+{\gamma}^{\mathbf{2}}{\Vert Bx\Vert}^{\mathbf{2}}\to \mathrm{min}.$$
(14)
$$x={\left({A}^{T}A+\gamma H\right)}^{-\mathbf{1}}{A}^{T}y,$$
(15)
$${\gamma}_{0}=\frac{Tr\left({A}^{T}A\right)}{Tr\left(H\right)},$$
(16)
$$s=\frac{1}{{N}_{s}}\sqrt{{\displaystyle \sum _{i=1}^{{N}_{s}}\frac{{\left({f}_{i}-{x}_{i}\right)}^{2}}{{{f}_{i}}^{2}}}}\text{,}$$