## 3 Optical Wireless Satellite Channel Model
### Atmospheric Turbulence: Beam Wander, Beam Spreading, and Beam Scintillation
**Turbulence-induced beam scintillation:** When the size of turbulent eddies is of the _same order_ of the size of the optical beam, then the eddies act as lenses that focus and defocus the incoming beam. In this situation, the eddies lead to a redistribution of the signal energy that generates a temporal and spatial fluctuation of the irradiance at the receiver aperture. This phenomenon, which is known as _scintillation_, represent one of the major sources of degradation in the performance of an optical feeder link. Atmospheric turbulence also leads to loss of spatial coherence of an initially coherent optical beam, and may also produce depolarization of the light and temporal stretching of the optical pulse. The atmospheric scintillation is measured in terms of the scintillation index, which is the normalized variance of the intensity fluctuations, i.e.,  
\[\sigma_{I}^{2}\ \overset{\Delta}{=}\ \frac{\langle(I-I_{m})^{2}\rangle}{I_{m}^{2}} \ =\ \frac{\langle I^{2}\rangle-I_{m}^{2}}{I_{m}^{2}}=\frac{\langle I^{2} \rangle}{I_{m}^{2}}-1,\qquad I_{m}\ =\ \langle I\rangle, \tag{19}\]  
where \(I\) is the irradiance (intensity) in the detector plane and \(\langle\cdot\rangle\) denotes the ensemble average. The Gamma-Gamma distribution has been proposed to describe the turbulence-induce scintillation over a broad range of beam diameters. The Probability Density Function (PDF) of the Gamma-Gamma turbulence model and the scintillation index are given by:  
\[f_{I}(x)\ =\ \tfrac{2}{\Gamma(a)\Gamma(f)\chi}\left(\tfrac{a\beta x}{I_{m}} \right)^{\frac{a+\beta}{2}}K_{a-\beta}\left(2\sqrt{\tfrac{a\beta x}{I_{m}}} \right)\qquad x\geq 0,\qquad\qquad\sigma_{I}^{2}\ =\ \tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{a \beta}, \tag{20}\]  
respectively, where \(I_{m}\) denotes the mean irradiance, \(\Gamma(x)\) is the Gamma function, and \(K_{n}(x)\) is the modified Bessel function of the second kind. The parameters \(\alpha\ =\ 1/\sigma_{X}^{2}\) and \(\beta\ =\ 1/\sigma_{Y}^{2}\) of the Gamma-Gamma distribution in (20) are directly related to the atmospheric conditions, and for the untracked beam case are given:  
\[\sigma_{X}^{2}\ =\ 5.95(H-h_{0})^{2}\sec^{2}(\xi)\left(\frac{2W_{0}}{r_{0}} \right)^{5/3}\left(\frac{a_{\rm pe}}{W}\right)^{2}\exp\left\{\frac{0.49\, \sigma_{\rm Bu}^{2}}{\left[1+(1+\Theta)\,0.56\,\sigma_{\rm Bu}^{12/5}\right]^ {7/6}}\right\}-1, \tag{21}\]  
and  
\[\cong 0.54\left(H-h_{0}\right)^{2}\sec^{2}\left(\zeta\right)\left(\frac {\lambda}{2W_{0}}\right)^{2}\left(\frac{2W_{0}}{r_{0}}\right)^{5/3}, \tag{15}\]0  
The various parameters that appear in Equations (21) and (22) are defined as follows:  
\[\cong 0.54\left(H-h_{0}\right)^{2}\sec^{2}\left(\zeta\right)\left(\frac {\lambda}{2W_{0}}\right)^{2}\left(\frac{2W_{0}}{r_{0}}\right)^{5/3}, \tag{15}\]1  
and are the jitter-induced angular pointing error, the pointing error variance, slant path length, and scaling constant, respectively. Similarly, the diffractive beam radius at the receiver is given by:  
\[\cong 0.54\left(H-h_{0}\right)^{2}\sec^{2}\left(\zeta\right)\left(\frac {\lambda}{2W_{0}}\right)^{2}\left(\frac{2W_{0}}{r_{0}}\right)^{5/3}, \tag{15}\]2