## 2 Proposed System Model  
In Figure 1, we consider a cooperative transmission from a source node (S) to a destination node (D) via a relay node (R) that amplifies the received signal using AF relaying and retransmits the data to the final destination. Let \(\gamma_{1}\) be the SNR of the first FSO hop from S to R, while \(\gamma_{2}\) denotes the signal-to-noise ratio (SNR) of the second FSO hop from R to D. Then, the end-to-end SNR can be given as \(\gamma=\frac{\gamma_{1}\gamma_{2}}{\gamma_{2}+C}\), where \(C\) is a fixed relay gain. Moreover, both links are assumed to follow \(\mathcal{F}\) turbulence, where the channel gain for the \(i\)-th link is modeled as \(h_{i}=h_{i}^{h}t_{i}^{a}h_{i}^{P}\), where \(h_{i}^{l}\), \(h_{i}^{a}\), and \(h_{i}^{P}\) are the deterministic propagation loss, atmospheric turbulence attenuation, and pointing error attenuation, respectively. More precisely, the PDF of the irradiance [6] can be represented as  
Figure 1: System model.  
\[f_{h_{i}^{2}}(x)=\frac{\Gamma(a_{i}+b_{i})a_{i}^{a_{i}}(b_{i}-1)^{b_{i}}x^{a_{i}-1 }}{\Gamma(a_{i})\Gamma(b_{i})(a_{i}x+b_{i}-1)^{a_{i}+b_{i}}}, \tag{1}\]  
where \(a_{i}\) and \(b_{i}\) are two key factors affecting the optical propagation parameters of the \(i\)-th link, such as the atmospheric refractive-index structure parameter, propagation path length, and inner and outer scales of turbulence. Then, \(a_{i}\) and \(b_{i}\)[6] can be expressed as  
\[a_{i}=\frac{1}{\exp\left(\sigma_{\ln\mathcal{S}_{i}}^{2}\right)-1}\quad\text{ and}\quad b_{i}=\frac{1}{\exp\left(\sigma_{\ln\mathcal{L}_{i}}^{2}\right)-1}+2, \tag{2}\]  
where \(\sigma_{\ln\mathcal{S}_{i}}^{2}\) and \(\sigma_{\ln\mathcal{L}_{i}}^{2}\) are the small-scale and large-scale log-irradiance variances, respectively. More specifically, the small-scale log-irradiance variance [6], \(\sigma_{\ln\mathcal{S}_{i}}^{2}\), can be expressed as  
\[\sigma_{\ln\mathcal{S}_{i}}^{2}=\frac{0.51\delta_{\text{SP}_{i}}^{2}\left(1+0. 69\delta_{\text{SP}_{i}}^{12/5}\right)^{-5/6}}{1+0.90d_{i}^{2}\left(\sigma_{i} /\delta_{\text{SP}}\right)^{12/5}+0.62d_{i}^{2}\sigma_{i}^{12/5}}, \tag{3}\]  
where \(\delta_{\text{SP}_{i}^{2}}\) is the spherical wave scintillation index [11], \(d_{i}\) is the equivalent aperture diameter, and \(\sigma_{i}^{2}\) represents the strength of irradiance fluctuations, such as turbulence. Moreover, the large-scale log-irradiance variance [6], \(\sigma_{\ln\mathcal{L}_{i}}^{2}\), can be expressed as  
\[\sigma_{\ln\mathcal{L}_{i}}^{2}=\sigma_{\ln\mathcal{L}_{i}}^{2}(l_{0})-\sigma _{\ln\mathcal{L}_{i}}^{2}(L_{0}), \tag{4}\]  
where \(\sigma_{\ln\mathcal{L}_{i}}^{2}(l_{0})\) and \(\sigma_{\ln\mathcal{L}_{i}}^{2}(L_{0})\) denote the large-scale log irradiance variances that consider inner-scale and outer-scale effects, respectively. In addition, the PDF of the pointing error impairment [7] is  
\[f_{h_{i}^{p}}(y)=\frac{z_{i}^{2}}{A_{i}^{z_{i}}}y^{z_{i}^{2}-1},\quad 0\leq y \leq A_{i}, \tag{5}\]  
where \(A_{i}\) represents the fraction of the received power. Here, \(z_{i}=\omega_{i}^{zeq}/\sigma_{i}^{z}\), where \(\omega_{i}^{zeq}\) and \(\sigma_{i}^{z}\) are the beam-width and the pointing error displacement standard deviation, respectively.