## 2 System Design  
### Atmospheric Turbulence and Gamma-Gamma Channel  
Random variations in pressure and temperature along the propagation path cause the refractive index (\(\eta\)) of the medium to vary abruptly, thus leading to scintillation losses. _Turbulence theory:_ The optical field radiation traversing in the atmosphere experiences the intensity and wavefront, as shown in Figure 3. Mathematically, the refractive index at any distance \(L\) in space is related to channel parameters in the following manner [25]  
\[\eta\left(L\right)=1+77.6\frac{\mathrm{P}(L)}{\mathrm{T}_{\mathrm{e}}(L)}\times 1 0^{-6} \tag{1}\]  
where P is the atmospheric pressure in millibars, \(\mathrm{T}_{\mathrm{e}}\) is the temperature in Kelvin. The variations in channel refractive index of the medium, also known as optical turbulence, can be mathematically related to temperature in-homogeneities by the parameter \(C_{\mathrm{H}}^{2}\), also known as the refractive index structure parameter. Although the value of \(C_{n}^{2}\) varies with altitude (h) and wind velocity (v) as well as shown in Equation (2) [25],  
\[C_{n}^{2}(\mathrm{h})=0.00594\Big{(}\frac{\mathrm{v}}{27}\Big{)}^{2}\Big{(}10^{- 5}\Big{)}^{10}\exp\Big{(}\frac{-\mathrm{h}}{1000}\Big{)}+2.7\times 10^{-16}\exp \Big{(}\frac{-\mathrm{h}}{1500}\Big{)}+A\Big{(}\frac{-\mathrm{h}}{100}\Big{)} \tag{2}\]  
where \(\mathrm{\AA}\) is the nominal value of \(C_{n}^{2}\) (0) at the ground in m\({}^{-2/3}\), and h is the altitude in meters. For the sake of analytical simplicity, \(C_{n}^{2}\) can be written as a function of temperature structure constant \(C_{T}^{2}\) as:  
\[C_{n}^{2}=\Big{(}79\times 10^{-6}\mathrm{P}/\mathrm{T}^{2}\Big{)}^{2}C_{T}^{2} \tag{3}\]  
As a general rule of channel characterization in FSO links, \(C_{n}^{2}\) defines the severity of turbulence strength. \(C_{n}^{2}\) in the typical range of \(10^{-12}\) m\({}^{-2/3}\) signifies strong turbulence conditions while \(C_{n}^{2}\) in the range of \(10^{-15}\) m\({}^{-2/3}\) and \(10^{-17}\) m\({}^{-2/3}\) signify moderate and weak turbulence regimes [1; 7]. Various channel models have been proposed in the recent past to mathematically quantify the impact of channel turbulence on the received signal. Out of the different proposed turbulence models, the gamma-gamma channel model is considered to be most reliable because this model exhibits close to ideal convergence between analytical and experimental results for all forms of turbulence regimes, i.e., strong, moderate and weak. This model considers the effect of small scale (scattering) and large scale (refraction) eddies of the propagating optical signal. The probability density function (PDF) of received light intensity (\(I\)) as determined from the gamma-gamma function can be represented as [1; 8; 9]. \[\mathrm{P}(I)=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha) \Gamma(\beta)}I^{(\frac{\alpha+\beta-2}{2})}k_{\alpha-\beta}\Big{(}2\sqrt{ \alpha\beta I}\Big{)};\;I>0 \tag{4}\]  
where, \(\alpha\) and \(\beta\) represent the effective number of large- and small-scale eddies, respectively. \(k_{n}\) (.) is the modified Bessel function of the 2 nd kind of order (\(\alpha-\beta\)), and \(\Gamma\) (.) represents the gamma function. The parameters \(\alpha\) and \(\beta\) can be individually written as a function of Rytov variance \(\sigma_{\mathrm{R}}\) as:  
\[\alpha=\left[\exp\left(\frac{0.49\sigma_{\mathrm{R}}^{2}}{\left(1+1.11\sigma_ {\mathrm{R}}^{\frac{2}{3}}\right)^{\frac{7}{6}}}\right)-1\right]^{-1} \tag{5}\]  
\[\beta=\left[\exp\left(\frac{0.51\sigma_{\mathrm{R}}^{2}}{\left(1+0.69\sigma_ {\mathrm{R}}^{\frac{2}{3}}\right)^{\frac{5}{6}}}\right)-1\right]^{-1} \tag{6}\]  
where, \(\sigma_{\mathrm{R}}^{2}=1.23C_{n}^{2}k^{7/6}L^{11/6}\), here, \(L\) is propagation channel length and \(k\) is propagation constant, \(2\pi/\lambda\).