## 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 (73)

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(1)
100\le \overline{\alpha}\le 150
(4)
\lambda =0.6328\text{\hspace{0.17em} \mu m}
(5)
a\text{=25 \hspace{0.17em} \mu m}
(6)
\overline{\alpha}=100
(7)
\overline{\alpha}=150
(8)
I\left(\theta \right)
(9)
I\left(\theta \right)
(10)
f\left(\alpha \right)
(11)
$$I\left(\theta \right)={\displaystyle {\int}_{0}^{\infty}\text{\hspace{0.17em}}I\left(\theta ,\alpha ,m\right)}f\left(\alpha \right)\mathrm{d}\alpha .$$
(12)
I\left(\theta ,\alpha ,m\right)
(13)
f\left(\alpha \right)
(14)
f\left(\alpha \right)\mathrm{d}\alpha
(15)
\alpha +\mathrm{d}\alpha
(16)
I\left(\theta ,\alpha \right)
(17)
$$I\left(\theta \right)=\frac{{I}_{0}}{{k}^{2}{F}^{2}}{\displaystyle \text{\hspace{0.17em}}{\int}_{0}^{\infty}\text{\hspace{0.17em}}\frac{{\alpha}^{2}{{J}_{1}}^{2}\left(\alpha \theta \right)}{{\theta}^{2}}\text{\hspace{0.17em}}}f\left(\alpha \right)\mathrm{d}\alpha ,$$
(20)
0.0573\xb0\le \theta \le 10.886\xb0
(21)
\Delta \theta =0.0458\xb0
(23)
$$f\left(\alpha \right)=\frac{-2\pi {k}^{3}{F}^{2}}{{\alpha}^{2}}{\displaystyle \text{\hspace{0.17em}}{\int}_{0}^{\infty}\text{\hspace{0.17em}}\left(\alpha \theta \right)}{J}_{1}\left(\alpha \theta \right){Y}_{1}\left(\alpha \theta \right)\frac{\mathrm{d}}{\mathrm{d}\theta}\left[{\theta}^{3}\text{\hspace{0.17em}}\frac{I\left(\theta \right)}{{I}_{0}}\right]\mathrm{d}\theta \text{,}$$
(25)
{\theta}_{\mathrm{min}}
(26)
{\theta}_{\mathrm{max}}
(27)
{\theta}_{\mathrm{max}}
(28)
\left(\text{\mu}+\beta \right)
(29)
{\stackrel{\u20d7}{x}}_{\ell}
(30)
{x}_{\ell \text{,}p}
(31)
{\xi}_{\ell \text{,}p}
(32)
$${\stackrel{\u20d7}{x}}_{\ell}=\left[{x}_{\ell \mathrm{,}p},{\xi}_{\ell \mathrm{,}p}\right]\text{, \hspace{1em} where , \hspace{0.17em}}p=1\mathrm{\text{,}}\dots \mathrm{,}5.$$
(33)
$${x}_{\ell \mathrm{,}1}\equiv \text{mean \hspace{0.17em} size \hspace{0.17em} parameter},$$
(34)
$${x}_{\ell \mathrm{,}2}\equiv \text{standard \hspace{0.17em} deviation,}$$
(35)
$${x}_{\ell \mathrm{,}3}\equiv \text{first \hspace{0.17em} size \hspace{0.17em} parameter},$$
(36)
$${x}_{\ell \mathrm{,}4}\equiv \text{second \hspace{0.17em} size \hspace{0.17em} parameter,}$$
(37)
$${x}_{\ell \mathrm{,}5}\equiv \text{smallest \hspace{0.17em} size \hspace{0.17em} parameter},$$
(38)
{\stackrel{\u20d7}{x}}_{\ell}
(39)
{\stackrel{\u20d7}{x}}_{\ell}
(41)
{\stackrel{\u20d7}{x}}_{\ell}
(42)
{\stackrel{\u20d7}{x}}_{\ell}
(43)
f\left(\alpha \right)
(47)
$$\text{fitness}=\sqrt{{\displaystyle \sum _{i=1}^{j}{\left[{I}_{{\ell}_{i}}-{I}_{re{f}_{i}}\right]}^{2}}},$$
(56)
30\le \alpha \le 200
(57)
\overline{\alpha}=100
(58)
50\le \alpha \le 250
(59)
\overline{\alpha}=150
(60)
$$s=\frac{1}{{N}_{s}}\text{\hspace{0.17em}}\sqrt{{\displaystyle \sum _{i=1}^{{N}_{s}}\frac{{\left({f}_{re{f}_{i}}-{f}_{{\ell}_{i}}\right)}^{2}}{{{f}_{re{f}_{i}}}^{2}}}},$$
(63)
6.1934\times {10}^{-5}
(64)
1.4415\times {10}^{-5}