## 2 Theoretical Model of Polarization Characteristics
\[P(\mathbf{\rho},z=0;\omega)=\sqrt{\frac{\left(A_{x}^{2}-A_{y}^{2}\right)^{2}+4A_{ x}^{2}A_{y}^{2}\left|B_{xy}\right|^{2}}{\left(A_{x}^{2}+A_{y}^{2}\right)^{2}}} \tag{6}\]  
\[\varepsilon=\frac{A_{minor}}{A_{major}}=\sqrt{\frac{\left[\left(A_{x}^{2}-A_{ y}^{2}\right)^{2}+4A_{x}^{2}A_{y}^{2}\left|B_{xy}\right|^{2}\right]^{\frac{1}{2}}- \left[\left(A_{x}^{2}-A_{y}^{2}\right)^{2}+4A_{x}^{2}A_{y}^{2}Re\left|B_{xy} \right|^{2}\right]^{\frac{1}{2}}}{\left[\left(A_{x}^{2}-A_{y}^{2}\right)^{2}+4 A_{x}^{2}A_{y}^{2}Re\left|B_{xy}\right|^{2}\right]^{\frac{1}{2}}}}=1 \tag{7}\]  
The parameters of GSM beam satisfy the following conditions. \[A_{x}=A_{y},\;\;\;Re\big{|}B_{xy}\big{|}=0 \tag{8}\]The elements of the matrix in the downlink are expressed as [22] as follows. \[ \begin{array}{l}W_{ij}(\mathbf{r}_{1},\mathbf{r}_{2},z;\omega)=\left(\frac{k}{2\pi z} \right)^{2}\iint d^{2}\mathbf{\rho}_{1}\iint d^{2}\mathbf{\rho}_{2}W_{ij}(\mathbf{\rho}_{1}, \mathbf{\rho}_{2},0;\omega)\\ \times exp\left[-ik\frac{(\mathbf{r}_{1}-\mathbf{\rho}_{1})^{2}-(\mathbf{r}_{2}-\mathbf{\rho}_{ 2})^{2}}{2z}\right]\\ \times\left(exp[\psi^{*}(\mathbf{r}_{1},\mathbf{\rho}_{1},z)+\psi(\mathbf{r}_{2},\mathbf{\rho}_ {2},z)]\right)_{m}\end{array}  \tag{9}\]  
Here, \(\mathbf{\rho}_{1}\) and \(\mathbf{\rho}_{2}\) represent the transverse coordinates at the transmitter plane, whereas \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) denote those at the receiver plane. The wave number is given by \(k=2\pi/\lambda\). The final term, \(\langle\ldots\rangle_{m}\), represents the ensemble average over statistical realizations in the turbulent downlink and can be expressed as [18] follows. \[ \begin{array}{l}\langle exp[\psi^{*}(\mathbf{r}_{1},\mathbf{\rho}_{1},z)+\psi(\mathbf{r} _{2},\mathbf{\rho}_{2},z)]\rangle_{m}\\ \approx exp\left\{-\frac{1}{2}B_{0}\Big{[}B_{1}(\mathbf{\rho}_{1}-\mathbf{\rho}_{2})^{ 2}+B_{2}(\mathbf{\rho}_{1}-\mathbf{\rho}_{2})(\mathbf{r}_{1}-\mathbf{r}_{2})+B_{3}(\mathbf{r}_{1}- \mathbf{r}_{2})^{2}\Big{]}\right\}\end{array}  \tag{10}\]  
with  
\[\left\{ \begin{array}{l}B_{0}=2\pi^{2}k^{2}sec\theta\int_{0}^{\infty}x^{3} \Phi_{n}(\kappa)dx\\ B_{1}=\int_{h_{0}}^{H}C_{n}^{2}(h)(1-\xi)^{2}dh\\ B_{2}=2\int_{h_{0}}^{H}C_{n}^{2}(h)(1-\xi)\xi dh\\ B_{3}=\int_{h_{0}}^{H}C_{n}^{2}(h)\xi^{2}dh\end{array} \right. \tag{11}\]  
In the downlink channel, \(H\) and \(l_{0}\) represent the altitudes of the transmitter and receiver relative to the horizontal plane, respectively. The zenith angle is denoted as \(\theta\) (as illustrated in Figure 1). The power spectrum of the refractive index fluctuations follows the [PERSON]'s power law [23]. \[\Phi_{n}(\kappa)=0.033C_{n}^{2}\kappa^{-11/3}, \tag{12}\]  
where \(1/l_{0}\leq\kappa\leq 1/L_{0}\); \(L_{0}\) and \(l_{0}\) represent the outer and inner turbulence scales, respectively. \({C_{n}}^{2}(h)\) is the refractive-index structure parameter at altitude \(h\) and quantifies the turbulence intensity downlink. A widely adopted model for \({C_{n}}^{2}(h)\) is expressed as follows [24]. \[ \begin{array}{l}C_{n}^{2}(h)=0.00594\big{(}\frac{v}{27}\big{)}^{2}\big{(}10^ {-5}h\big{)}^{10}exp\Big{(}-\frac{h}{1000}\Big{)}\\ +2.7\times 10^{-16}exp\Big{(}\frac{-h}{1500}\Big{)}+C_{0}exp\Big{(}-\frac{h}{10 0}\Big{)}\end{array}  \tag{13}\]  
Here, \(C_{0}\) denotes the refractive index structure parameter and \(v\) denotes the wind velocity [25]. Substituting Equations (10)-(13) into Equation (9) and performing the necessary calculations, the following equations are obtained.