## Abstract

We study the performance of various binary and nonbinary modulation methods applied to coherent laser communication through the turbulent atmosphere. We compare the spectral efficiencies and SNR efficiencies of complex modulations, and consider options for atmospheric compensation, including phase correction and diversity combining techniques. Our analysis shows that high communication rates require receivers with good sensitivity along with some technique to mitigate the effect of atmospheric fading.

©2010 Optical Society of America

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### Equations (7)

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(1)
$$\begin{array}{l}{\sigma}_{\chi}^{2}={\mathrm{log}}_{e}\left(1+{\sigma}_{\beta}^{2}\right)\\ {\sigma}_{\varphi}^{2}={C}_{j}{\left(\frac{D}{{r}_{0}}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.},\end{array}$$
(2)
$${p}_{\gamma}\left({\gamma}_{MRC}\right)={\left(\frac{1+r}{\overline{\gamma}}\right)}^{\frac{L+1}{2}}{\left(\frac{1}{Lr}\right)}^{\frac{L-1}{2}}\mathrm{exp}\left(-Lr\right)\mathrm{exp}\left[-\frac{\left(1+r\right){\gamma}_{MRC}}{\overline{\gamma}}\right]{I}_{L-1}\left[2\sqrt{\frac{L\left(1+r\right)r{\gamma}_{MRC}}{\overline{\gamma}}}\right],$$
(3)
$$\begin{array}{c}M\left(s\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}d\gamma \mathrm{exp}\left(s\gamma \right)\text{\hspace{0.17em}}{p}_{\gamma}\left(\gamma \right)\text{\hspace{0.17em}}}\\ \phantom{\rule{3.5em}{0ex}}={\left[\frac{1+r}{1+r-s\text{\hspace{0.17em}}\overline{\gamma}}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{exp}\left(\frac{r\text{\hspace{0.17em}}s\text{\hspace{0.17em}}\overline{\gamma}}{1+r-s\text{\hspace{0.17em}}\overline{\gamma}}\right)\right]}^{L}\text{\hspace{0.17em}}.\end{array}$$
(4)
$${p}_{s}\left(E|\gamma \right)\approx \frac{a}{\pi}\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}d\varphi \mathrm{exp}\left(-\frac{b\gamma}{{\mathrm{sin}}^{2}\varphi}\right)}.$$
(5)
$${p}_{S}\left(E\right)=\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{\infty}d\gamma}{p}_{S}\left(E\text{\hspace{0.17em}}|\text{\hspace{0.17em}}\gamma \right){p}_{\gamma}\left(\gamma \right),$$
(6)
$$\begin{array}{c}{p}_{s}\left(E\right)=\frac{a}{\pi}\text{\hspace{0.17em} \hspace{0.17em}}{\displaystyle {\int}_{0}^{\infty}d\gamma \text{\hspace{0.17em} \hspace{0.17em}}{\displaystyle {\int}_{0}^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}d\varphi \mathrm{exp}\left(-\frac{b\gamma}{{\mathrm{sin}}^{2}\varphi}\right)}}{p}_{\gamma}\left(\gamma \right)\\ =\frac{a}{\pi}\text{\hspace{0.17em} \hspace{0.17em}}{\displaystyle {\int}_{0}^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}d\varphi \text{\hspace{0.17em} \hspace{0.17em}}M\left(-\frac{b}{{\mathrm{sin}}^{2}\varphi}\right)},\end{array}$$
(7)
$$S=\frac{{R}_{b}}{B}=\frac{{R}_{s}{R}_{c}{\mathrm{log}}_{2}M}{B},$$