# Free Space Optical Communication: An Enabling Backhaul Technology for 6G Non-Terrestrial Networks
## 2 Fundamentals of FSO Signal Transmission
### Turbulence-Induced Fading for Airborne Nodes
\right., \tag{6}\]  
Various statistical models have been proposed in the literature to model the irradiance statistics, including Rician-Lognormal (RLN) distribution, Malaga (\(M\)) distribution, Double Generalized Gamma (DGG), distribution, \(K\) distribution, negative exponential distribution, \(I-K\) distribution, Nakagami-\(m\) distribution, Weibull distribution, and Rayleigh distribution [97]. The most commonly used model is the lognormal distribution. This is often employed for weak turbulence conditions or short-range FSO links, due to its good agreement with experimental measurements. Its physical interpretation is based on modeling the atmosphere as a sequence of successive layers, each composed of thin slabs. As the optical field passes through each slab, it encounters random variations in both amplitude (\(\alpha_{k}\)) and phase. Let \(z_{a}\) represent the amplitude of the turbulence coefficient. We can express this as \(z_{a}=\prod_{k=1}^{K}\alpha_{k}\), indicating that \(\ln(z_{a})=\sum_{k=1}^{K}\ln\alpha_{k}\) follows a Gaussian distribution. The intensity of an optical wave (\(I\)) is a measure of the energy per unit area carried by the wave. It is proportional to the square of the amplitude of the electric field,  
Figure 8: Effects of atmospheric turbulence. i.e., \(I=z_{a}^{2}\), leading to \(\,0.5\ln(I)=\sum_{k=1}^{K}\ln\alpha_{k}\). The Probability Density Function (PDF) of \(I\) can then be expressed as [97] [Section 9.2.1.2]  
\[f_{I}(I)=\frac{1}{I\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{\left(\ln(I)-\mu \right)^{2}}{2\sigma^{2}}\right),\qquad I>0 \tag{7}\]  
where \(\mu=2\mu_{x}\) and \(\sigma^{2}=4\sigma_{x}^{2}\). Here, \(\mu_{x}\) and \(\sigma_{x}^{2}\) denote, respectively, the mean and variance of the log-amplitude coefficient \(X=0.5\ln(I)\). The second moment of the fading coefficient represents the average change of the received power. Utilizing \(E\big{[}I^{k}\big{]}=\exp\big{(}k\mu+0.5k^{2}\sigma^{2}\big{)}\), the second moment can be written as \(E\big{[}I^{2}\big{]}=\exp\big{(}2\mu+0.5k^{2}\sigma^{2}\big{)}=\exp\big{(}4\mu _{x}+8\sigma_{x}^{2}\big{)}\). In order to keep the average power value unchanged, the fading amplitude is normalized, such that \(\mathrm{E}[I]=1\). This implies \(\mu_{x}=-\sigma_{x}^{2}\). The average change in the received power due to turbulence assuming a lognormal fading channel can be written as \(L_{\mathrm{Tur,LN}}=10\mathrm{log}_{10}\big{(}\exp\big{(}4\sigma_{x}^{2}\big{)} \big{)}=(40/\ln 10)\sigma_{x}^{2}\). Utilizing the relation between log-amplitude variance and a scintillation index of \(\sigma_{x}^{2}=0.25\ln\big{(}1+\sigma_{s}^{2}\big{)}\), the average variation in received power due to turbulence-induced fading can be formulated as  
\[L_{\mathrm{Tur,LN}}=10\,\mathrm{log}_{10}\Big{(}1+\sigma_{s}^{2}\Big{)}. \tag{8}\]  
There are scenarios involving moderate to strong turbulence, where the scintillation index exceeds unity, leading to more pronounced turbulence effects, where lognormal cannot fit the statistical distribution of turbulence-induced fading. Accordingly, alternative models capable of accommodating a broad range of turbulence conditions have been proposed [97] over the years. A widely adopted model is gamma-gamma fading. In accordance with modified [PERSON] theory, irradiance fluctuations can be modeled as the product of two random variables, one of which is generated by small-scale turbulent eddies (\(I_{x}\)) and the other by large-scale turbulent eddies (\(I_{y}\)). Experimental data have shown that each variable follows a gamma distribution. This results in the PDF of \(I=I_{x}I_{y}\), to follow the well-known gamma-gamma distribution [97] [Section 9.2.1.4] and given as  
\[f_{I}(I)=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma(\beta) }I^{((\alpha+\beta)/2)-1}\mathrm{K}_{\alpha-\beta}\Big{(}2\sqrt{\alpha\beta I }\Big{)}, \tag{9}\]  
where \(\Gamma(\cdot)\) and \(\mathrm{K}_{\mathrm{r}}(\cdot)\) are the gamma function and modified Bessel function of the second kind with order \(v\).