## Abstract

An algorithm is presented based on an evolution strategy to retrieve a particle size
distribution from angular light-scattering data. The analyzed intensity patterns are generated
using the Mie theory, and the algorithm retrieves a series of known normal, gamma,
and lognormal distributions by using the Fraunhofer approximation. The distributions
scan the interval of modal size parameters
$100\le \overline{\alpha}\le 150$. The numerical results show that
the evolution strategy can be successfully applied to solve this kind of inverse problem,
obtaining a more accurate solution than, for example, the Chin–Shifrin inversion method,
and avoiding the use of *a priori* information concerning the domain of the distribution,
commonly necessary for reconstructing the particle size distribution when this analytical
inversion method is used.

© 2007 Optical Society of America

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### Equations (11)

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(1)
$$I\left(\theta \right)={\displaystyle {\int}_{0}^{\infty}\text{\hspace{0.17em}}I\left(\theta ,\alpha ,m\right)}f\left(\alpha \right)\mathrm{d}\alpha .$$
(2)
$$I\left(\theta \right)=\frac{{I}_{0}}{{k}^{2}{F}^{2}}{\displaystyle \text{\hspace{0.17em}}{\int}_{0}^{\infty}\text{\hspace{0.17em}}\frac{{\alpha}^{2}{{J}_{1}}^{2}\left(\alpha \theta \right)}{{\theta}^{2}}\text{\hspace{0.17em}}}f\left(\alpha \right)\mathrm{d}\alpha ,$$
(3)
$$f\left(\alpha \right)=\frac{-2\pi {k}^{3}{F}^{2}}{{\alpha}^{2}}{\displaystyle \text{\hspace{0.17em}}{\int}_{0}^{\infty}\text{\hspace{0.17em}}\left(\alpha \theta \right)}{J}_{1}\left(\alpha \theta \right){Y}_{1}\left(\alpha \theta \right)\frac{\mathrm{d}}{\mathrm{d}\theta}\left[{\theta}^{3}\text{\hspace{0.17em}}\frac{I\left(\theta \right)}{{I}_{0}}\right]\mathrm{d}\theta \text{,}$$
(4)
$${\stackrel{\u20d7}{x}}_{\ell}=\left[{x}_{\ell \mathrm{,}p},{\xi}_{\ell \mathrm{,}p}\right]\text{, \hspace{1em} where , \hspace{0.17em}}p=1\mathrm{\text{,}}\dots \mathrm{,}5.$$
(5)
$${x}_{\ell \mathrm{,}1}\equiv \text{mean \hspace{0.17em} size \hspace{0.17em} parameter},$$
(6)
$${x}_{\ell \mathrm{,}2}\equiv \text{standard \hspace{0.17em} deviation,}$$
(7)
$${x}_{\ell \mathrm{,}3}\equiv \text{first \hspace{0.17em} size \hspace{0.17em} parameter},$$
(8)
$${x}_{\ell \mathrm{,}4}\equiv \text{second \hspace{0.17em} size \hspace{0.17em} parameter,}$$
(9)
$${x}_{\ell \mathrm{,}5}\equiv \text{smallest \hspace{0.17em} size \hspace{0.17em} parameter},$$
(10)
$$\text{fitness}=\sqrt{{\displaystyle \sum _{i=1}^{j}{\left[{I}_{{\ell}_{i}}-{I}_{re{f}_{i}}\right]}^{2}}},$$
(11)
$$s=\frac{1}{{N}_{s}}\text{\hspace{0.17em}}\sqrt{{\displaystyle \sum _{i=1}^{{N}_{s}}\frac{{\left({f}_{re{f}_{i}}-{f}_{{\ell}_{i}}\right)}^{2}}{{{f}_{re{f}_{i}}}^{2}}}},$$