## 3 Challenges in FSO Communication
### FSO System and Channel Models
#### 3.2.2 FSO Channel Models
Assuming a Gaussian spatial intensity profile characterized by a beam waist \(w_{z}\) at the Rx plane--located a distance \(z\) from the Tx--and a circular aperture with a radius \(r_{a}\), the fraction of collected power resulting from geometric spreading with a radial displacement \(\alpha_{d}\) from the detector's origin can be approximated using a Gaussian distribution as [3; 89]  
\[h_{\mathrm{P}}(\alpha_{d})\approx A_{0}\exp\left(-\frac{2\alpha_{d}^{2}}{w_{z_{ \mathrm{eq}}}^{2}}\right), \tag{13}\]  
where \(w_{z_{\mathrm{eq}}}^{2}=w_{z}^{2}\sqrt{\pi}\mathrm{erf}(v)/2v\exp(-v^{2})\), \(v=\sqrt{\pi}r_{a}/\sqrt{2}w_{z}\), \(w_{z_{\mathrm{eq}}}\) is the equivalent beamwidth, \(A_{0}=[\mathrm{erf}(v)]^{2}\), and \(\mathrm{erf}(\cdot)\) is the error function defined as [94]  
\[\mathrm{erf}(x)=(2/\sqrt{\pi})\int_{0}^{x}e^{-u^{2}}du. \tag{14}\]  
Assuming that the horizontal and elevation sway are independent and follow identical Gaussian distributions, the radial displacement \(\alpha_{d}\) will follow a Rayleigh distribution. Consequently, \(f_{h_{\mathrm{P}}}(h_{\mathrm{P}})\) can be expressed as  
\[f_{h_{\mathrm{P}}}(h_{\mathrm{P}})=\frac{\gamma_{\mathrm{P}}^{2}}{A_{0}^{ \gamma_{\mathrm{P}}^{2}}}h_{\mathrm{P}}^{\gamma_{\mathrm{P}}^{2}-1},\quad 0 \leq h_{\mathrm{P}}\leq A_{0}, \tag{15}\]  
where \(\gamma_{\mathrm{P}}=w_{z_{\mathrm{eq}}}/2\sigma_{s}\) denotes the ratio of the equivalent beam radius to twice the standard deviation (jitter) of the PE displacement, and \(\sigma_{s}^{2}\) represents the jitter variance at the Rx. Atmospheric Turbulence  
The intensity fluctuations over the FSO channel have been described using various statistical models in the literature, tailored for different turbulence regimes. The commonly used models encompass the LN distribution, ideal for weak turbulence and frequently applied due to its strong correlation with experimental data, the IT distribution for moderate to strong turbulence, and the K-distribution for strong turbulence. These models incorporate PDFs based on turbulence intensity to characterize the impact of turbulence on FSO signals. Other widely used models include the I-K distribution, NE, and \(\mathcal{M}\) distribution models. However, this work focuses on the LN, IT, and generic \(\mathcal{M}\) distribution models. Log-normal Distribution: Generally, the LN model is only appropriate for weak turbulence conditions and for link ranges shorter than 100 m [95]. Therefore, the intensity fluctuation PDF for weak turbulence, modeled using the LN distribution, is presented by [96]  
\[f_{h_{\mathrm{a}}}(h_{\mathrm{a}})=\frac{1}{2h_{\mathrm{a}}\sqrt{2\pi\sigma_{ x}^{2}}}\exp\left(-\frac{(\ln(h_{\mathrm{a}})+2\sigma_{x})^{2}}{8\sigma_{x}^{2}} \right), \tag{16}\]  
where \(\sigma_{x}^{2}\) denotes the log-amplitude variance specified for plane waves and spherical waves, respectively, and is expressed as [96]  
\[\sigma_{x}^{2}|_{\mathrm{plane}}= 0.307C_{n}^{2}k^{7/6}L^{11/6}, \tag{17a}\] \[\sigma_{x}^{2}|_{\mathrm{spherical}}= 0.124C_{n}^{2}k^{7/6}L^{11/6}, \tag{17b}\]  
where \(L\) denotes the distance, \(k=\frac{2\pi}{\lambda}\) denotes the optical wave number, and \(C_{n}^{2}\) represents the altitude-dependent index of the refraction structure parameter. Also, with respect to the log-irradiance variance, \(\sigma_{x}^{2}=\sigma_{l}^{2}/4\), where \(\sigma_{l}^{2}\) is the log-irradiance variance expressed as [96]  
\[\sigma_{l}^{2}|_{\mathrm{plane}}= 1.23C_{n}^{2}k^{7/6}L^{11/6}, \tag{18a}\] \[\sigma_{l}^{2}|_{\mathrm{spherical}}= 0.50C_{n}^{2}k^{7/6}L^{11/6}. \tag{18b}\]  
The C\({}_{n}^{2}\) parameter is crucial for quantifying the refractive index fluctuations in AT. It varies with the wavelength, atmospheric altitude, and temperature. Various \(C_{n}^{2}\) profile models have been presented in the literature, with the Hufnagel valley model being the most widely used for altitude and is defined as [6; 91; 94; 96; 97]  
\[