## 2 System and Channel Model
### Satellite-to-HAP Link  
In the satellite-to-HAP link, the HAP receiver uses equal gain combination (EGC) to combine signals, i.e., assigning equal weights to the signals on each receive aperture. The advantage of EGC over other merging techniques is that it does not require channel state information. The output SNR of the HAP with EGC can be expressed as [26]  
\[\gamma_{s-p}=\overline{\gamma}_{s-p}\left(\sum_{i=1}^{N_{E}}I_{i}\right)^{2} \tag{1}\]  
where \(\overline{\gamma}_{s-p}\) is the average SNR, \(N_{E}\) is the number of receive apertures, and \(I_{i}\) denotes the receive irradiance of the link at the \(i\)th receive aperture. In satellite-terrestrial laser communication, the M distribution channel model is universal for weak, moderate, and strong turbulence. The PDF for the M distribution is as follows [27]:  
\[f_{I_{a}}(I)=A\sum_{k=1}^{\beta}a_{k}I^{\frac{\alpha+k}{2}-1}K_{\alpha-k}\left( 2\sqrt{\frac{I}{B}}\right) \tag{2}\]  
where \(a_{k}=\left( \begin{matrix}\beta-1\\ k-1\end{matrix} \right)\frac{\left(g\beta+\Omega^{\prime}\right)^{1-k/2}}{(k-1 )! }\left(\frac{\Omega^{\prime}}{g}\right)^{k-1}\left(\frac{\alpha}{g}\right) ^{k/2}\), \(A=\frac{2\alpha^{\prime 2}}{g^{1+\alpha+2}\Gamma(\alpha)}\left(\frac{g\beta}{g \beta+\Omega}\right)^{\alpha/2+\beta}\), and \(B=\frac{g\beta+\Omega^{\prime}}{\alpha\beta}\). \(\alpha\) is a positive parameter for the effective number of large-scale cells in the scattering process, and \(\beta\) is a natural number indicating the number of the fading parameters. \(g=2b_{0}(1-\rho)\), where \(2b_{0}\) is the average power of the total scattered component, and \(\rho\) is the ratio of the scattered power of the line-of-sight (LOS) coupling to the power of all scattered components. \(\Omega^{{}^{\prime}}=\Omega+2b_{0}+2\sqrt{2b\Omega\rho}\cos(\phi_{A}-\phi_{B})\), where \(\Omega\) is the average power of the LOS term, \(\phi_{A}\) and \(\phi_{B}\) are the determined phases of the LOS and the coupled-to-LOS scattering terms, respectively, and \(K_{v}(\bullet)\) is the Bessel correction function of the second kind of order \(v\). Under the M-distributed channel model, the PDF of the communication system with EGC at the receiver has been given in [16]. It can be written as  
\[f_{s-p}(I)=C\sum_{k=0}^{N_{E}(\beta-1)}c_{k}I^{\frac{N_{E}+N_{E}+k}{2}-1}K_{N _{E}\alpha-N_{E}-k}\left(2\sqrt{\frac{N_{E}I}{B}}\right) \tag{3}\]  
Figure 1: Schematic diagram of the RIS-SHTLC system. where \(c_{k}= \begin{pmatrix}N_{E}\beta-1\\ N_{E}(\beta-1)\end{pmatrix} \frac{1}{k! }\bigg{(}\frac{\Omega^{\prime}}{(g\beta+ \Omega^{\prime})_{g}}\bigg{)}^{k}\bigg{(}\frac{\beta}{(g\beta+\Omega^{\prime}) N_{E}\alpha}\bigg{)}^{(N_{E}\alpha-N_{E}-k)/2},\) and \(C=\frac{2(N_{E}\alpha)^{N_{E}\alpha}}{g^{N_{E}}(N_{E}\beta)_{N_{E}-1}\Gamma(N_ {E}\alpha)}\bigg{(}\frac{\beta\beta}{\beta+\Omega^{\prime}}\bigg{)}^{N_{E} \beta}.\)  
Using \(K_{\varphi}(x)=\frac{1}{2}G_{0,2}^{2,0}\bigg{[}\frac{x^{2}}{4}\bigg{|}_{ \begin{subarray}{c}v/2,-v/2\end{subarray} }^{2}-\bigg{]}\) and \(\gamma=\overline{\gamma}I^{2}\), \(G_{p,4}^{m,n}[t|_{b_{1}\ldots b_{m},b_{m+1}\ldots b_{q}}^{4\ldots a_{n}a_{n+1} \ldots a_{p}}]\) is the Meijer G-function. Equation (3) can be rewritten as  
\[f_{s-p}(\gamma)=\frac{C}{4\sqrt{\gamma\overline{\gamma}}}\sum_{k=0}^{N_{E}( \beta-1)}c_{k}\Big{(}\sqrt{\gamma/\overline{\gamma}}\Big{)}^{\frac{N_{E} \alpha+N_{E}+k}{2}-1}G_{0,2}^{2,0}\Bigg{[}\frac{N_{E}\sqrt{\gamma/\overline{ \gamma}}}{B}\Bigg{|}_{ \begin{subarray}{c}(N_{E}\alpha-N_{E}-k)\\ \frac{(N_{E}\alpha-N_{E}-k)}{2}\end{subarray} }-\frac{(N_{E}\alpha-N_{E}-k)}{2} \Bigg{]} \tag{4}\]  
The CDF of \(\gamma\) can be calculated using \(F_{s-p}(\gamma)=\int_{0}^{\infty}f_{s-p}(\gamma)d\gamma\).