## 2 Fisher-Snedecor \(\mathcal{F}\) Composite Fading Model  
The Fisher-Snedecor (F-S) \(\mathcal{F}\) turbulence-induced (TI) fading model attracts significant interest in the field of FSO communications [13; 14; 15; 16; 27; 33]. The F-S \(\mathcal{F}\) TI fading distribution can be expressed as the product of independent and identically distributed (i.i.d) Gamma (G) and normalized inverse Gamma (I-G) random variables (RVs) [13; 33]:  
\[Z_{\mathcal{F}S}=X_{G}Y_{IG}=X_{G}\frac{1}{Y_{G}} \tag{1}\]where \(X_{G}\) and \(Y_{IG}\) are G and normalized I-G RVs, respectively. Since the normalized I-G RV can be mathematically expressed as \(Y_{IG}=\frac{1}{Y_{G}}\), the probability density functions (PDFs) of \(X_{G}\) and \(Y_{G}\) are given as:  
\[p_{X_{G}}(x_{G})=\frac{(m_{G_{1}}/\Omega_{G_{1}})^{m_{G_{1}}}}{\Gamma(m_{G_{1}} )}(x_{G})^{m_{G_{1}}-1}e^{-\frac{m_{G_{1}}}{m_{G_{1}}}(x_{G})} \tag{2}\]  
\[p_{Y_{G}}(y_{G})=\frac{\big{(}(m_{G_{2}}-1)/\Omega_{G_{2}}\big{)}^{m_{G_{2}}}}{ \Gamma(m_{G_{2}})}(y_{G})^{m_{G_{2}}-1}e^{-\frac{(m_{G_{2}}-1)}{\Omega_{G_{2}} }(y_{G})} \tag{3}\]  
whose shape parameters are \(m_{G_{1}}\) and \(m_{G_{2}}\), respectively, whereas the mean powers are \(\Omega_{G_{1}}\) and \(\Omega_{G_{2}}\), respectively. The \(\Gamma(\cdot)\) is the Gamma function ([39], Equation (8.310.1)). By applying a transformation of RVs, \(\gamma_{X_{G}}={X_{G}}^{2}\) and \(\gamma_{y_{G}}={Y_{G}}^{2}\), the PDFs of the squared G and the normalised squared I-G RVs are, respectively:  
\[p_{\gamma_{X_{G}}}(\gamma_{X_{G}})=\bigg{|}\frac{dx_{G}}{d\gamma_{X_{G}}} \bigg{|}p_{X_{G}}({\gamma_{X_{G}}}^{1/2})=\frac{(m_{G_{1}}/\Omega_{G_{1}})^{m_ {G_{1}}}}{2\Gamma(m_{G_{1}})}({\gamma_{X_{G}}})^{\frac{1}{2}m_{G_{1}}-1}e^{- \frac{m_{G_{1}}}{m_{G_{1}}}({\gamma_{X_{G}}})^{\frac{1}{2}}} \tag{4}\]  
\[p_{\gamma_{Y_{G}}}(\gamma_{Y_{G}})=\bigg{|}\frac{dy_{G}}{d\gamma_{Y_{G}}} \bigg{|}p_{Y_{G}}({\gamma_{Y_{G}}}^{1/2})=\frac{\big{(}(m_{G_{2}}-1)/\Omega_{ G_{2}}\big{)}^{m_{G_{2}}}}{2\Gamma(m_{G_{2}})}({\gamma_{Y_{G}}})^{\frac{1}{2}m_{G_{2} }-1}e^{-\frac{(m_{G_{2}}-1)}{\Omega_{G_{2}}}({\gamma_{Y_{G}}})^{\frac{1}{2}}} \tag{5}\]  
The PDF of \(\gamma_{\mathcal{F}S}=\frac{\gamma_{Y_{G}}}{\gamma_{y_{G}}}\) can be obtained as:  
\[p_{\gamma_{\mathcal{F}S}}(\gamma)=\int_{0}^{\infty}\bigg{|}\frac{d\gamma_{X_{ G}}}{d\gamma_{\mathcal{F}S}}\bigg{|}p_{\gamma_{X_{G}}}(\gamma\times\gamma_{y_{G}})p_{ \gamma_{Y_{G}}}(\gamma_{y_{G}})d\gamma_{y_{G}} \tag{6}\]  
where \(\big{|}\frac{d\gamma_{X_{G}}}{d\gamma_{\mathcal{F}S}}\big{|}=\gamma_{y_{G}}\). From (4) to (6), and by applying ([39], Equation (3.326.2)) and ([39], Equation (8.384.1)), respectively, the PDF of \(\gamma_{\mathcal{F}S}\) can be written as:  
\[p_{\gamma_{\mathcal{F}S}}(\gamma) = \frac{(\frac{m_{G_{1}}}{\Omega_{G_{1}}})^{m_{G_{1}}}(\frac{m_{G_{2 }}-1}{\Omega_{G_{2}}})^{m_{G_{2}}}(\Omega_{G_{2}}\Omega_{G_{1}})^{m_{G_{1}}+m_{ G_{2}}}}{(\Omega_{G_{2}}m_{G_{1}}\gamma^{\frac{1}{2}+\Omega_{G_{1}}(m_{G_{The PDF of the received signal-to-noise ratio (SNR) for the considered FSO RIS-ACs transmission link can be expressed as ([24], Equation (17)):  
\[p_{\gamma}(\gamma)=\int_{0}^{\infty}\frac{1}{r}p_{\gamma_{h_{1}}}(\frac{\gamma}{ r})p_{\gamma_{h_{2}}}(r)dr \tag{9}\]  
where \(p_{\gamma_{h_{1}}}\) and \(p_{\gamma_{h_{2}}}\) are SNR's PDFs of the links \(h_{1}\) and \(h_{2}\), respectively, which in the case of intensity modulation-direct detection (IM-DD) with the on-off keying (OOK) technique for Fisher-Snedecor \(\mathcal{F}\) TI fading distribution can be written as ([13], Equation (20)):  
\[p_{\gamma_{h_{i}}}(\gamma_{i}) = \frac{a_{i}{}^{a_{i}}(b_{i}-1)^{b_{i}}\bar{\gamma}_{i}^{b_{i}/2} \gamma_{i}^{a_{i}/2-1}}{2\beta(a_{i},b_{i})(a_{i}\bar{\gamma}_{i}^{1/2}+(b_{i} -1)\bar{\gamma}_{i}^{1/2})^{a_{i}+b_{i}}},\ i=1,2; \tag{10}\]  
where \(\bar{\gamma}_{i}\) is the average SNR, whereas \(a_{i}\) and \(b_{i}\) are Fisher-Snedecor \(\mathcal{F}\) small-scale and large-scale cells related to TI fading severity conditions, respectively.