## Abstract

The probability density function (PDF) of aperture-averaged irradiance fluctuations is calculated from wave-optics simulations of a laser after propagating through atmospheric turbulence to investigate the evolution of the distribution as the aperture diameter is increased. The simulation data distribution is compared to theoretical gamma-gamma and lognormal PDF models under a variety of scintillation regimes from weak to strong. Results show that under weak scintillation conditions both the gamma- gamma and lognormal PDF models provide a good fit to the simulation data for all aperture sizes studied. Our results indicate that in moderate scintillation the gamma-gamma PDF provides a better fit to the simulation data than the lognormal PDF for all aperture sizes studied. In the strong scintillation regime, the simulation data distribution is gamma gamma for aperture sizes much smaller than the coherence radius ${\rho}_{0}$ and lognormal for aperture sizes on the order of ${\rho}_{0}$ and larger. Examples of how these results affect the bit-error rate of an on–off keyed free space optical communication link are presented.

© 2009 Optical Society of America

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### Equations (9)

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(1)
$$p(I)=\frac{2(\alpha \beta {)}^{(\alpha +\beta )/2}}{\mathrm{\Gamma}(\alpha )\mathrm{\Gamma}(\beta )I}(I{)}^{(\alpha +\beta )/2}{K}_{\alpha -\beta}(2\sqrt{\alpha \beta I}),\phantom{\rule[-0.0ex]{2em}{0.0ex}}I>0,$$
(2)
$$\alpha =\frac{1}{{\sigma}_{X}^{2}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\beta =\frac{1}{{\sigma}_{Y}^{2}}.$$
(3)
$${\sigma}_{X}^{2}=\mathrm{exp}({\sigma}_{\mathrm{ln}X}^{2})-1,\phantom{\rule[-0.0ex]{2em}{0.0ex}}{\sigma}_{Y}^{2}=\mathrm{exp}({\sigma}_{\mathrm{ln}Y}^{2})-1,$$
(4)
$$p(I)=\frac{1}{I\sqrt{2\pi {\sigma}_{\mathrm{ln}I}^{2}}}\mathrm{exp}\{-\frac{[\mathrm{ln}(I)+\frac{1}{2}{\sigma}_{\mathrm{ln}I}^{2}{]}^{2}}{2{\sigma}_{\mathrm{ln}I}^{2}}\},\phantom{\rule[-0.0ex]{2em}{0.0ex}}I>0,$$
(5)
$${\sigma}_{\mathrm{ln}I}^{2}=\mathrm{ln}({\sigma}_{I}^{2}+1),$$
(6)
$${\sigma}_{I}^{2}=\frac{\u3008{I}^{2}\u3009-\u3008I{\u3009}^{2}}{\u3008I{\u3009}^{2}},$$
(7)
$$U(x,y,z)={A}_{0}\frac{{W}_{0}}{W(z)}\mathrm{exp}[-\frac{(x+y{)}^{2}}{{W}^{2}(z)}]\times P(x,y),$$
(8)
$${\sigma}_{I}^{2}=(1+\frac{1}{\alpha})(1+\frac{1}{\beta})-1.$$
(9)
$$\mathrm{BER}(\mathrm{OOK})=\frac{1}{2}{\int}_{0}^{\infty}p(I)\text{erfc}\left(\frac{\u3008\mathrm{SNR}\u3009I}{2\sqrt{2}}\right)\mathrm{d}s,$$