## 2 Theoretical Model of Polarization Characteristics
\[\begin{array}{l}W_{ij}(r_{1},r_{2},z)=\frac{A_{i}A_{j}B_{ij}}{\Delta_{ij}^{2 }(z)}exp\left[-\frac{(r_{1}+r_{2})^{2}}{8w_{0}^{2}\Delta_{ij}^{2}(z)}\right] exp\left[\frac{ik\big{(}\frac{r_{2}^{2}-r_{1}^{2}}{2R_{ij}(z)}\big{)}}{2R_{ ij}(z)}\right]\\ \times exp\left\{  
where \(\Delta_{ij}^{2}(z)\) refers to the spatial expansion coefficient, and \(R_{ij}(z)\) represents the radius of curvature of the wavefront [26]. When considering a single point \(Q(r,z)\) at the receiver plane and assuming \(\mathbf{r}_{1}=\mathbf{r}_{2}=\mathbf{r}\), Equation (14) can be simplified as follows:  
\[W_{ij}(\mathbf{r},z;\omega)=\frac{A_{i}A_{j}B_{ij}}{\Delta_{ij}^{2}(z)}exp\left[- \frac{\mathbf{r}^{2}}{2w_{0}^{2}\Delta_{ij}^{2}(z)}\right] \tag{16}\]  
The polarization characteristics at the receiver plane can be expressed as follows. \[P(\mathbf{r},z;\omega)=\sqrt{\frac{\left[\frac{A_{i}^{2}}{\Delta_{xx}^{2}}exp\left( -\frac{r^{2}}{2\sigma^{2}\Delta_{xx}^{2}}\right)-\frac{A_{y}^{2}}{\Delta_{yy}^ {2}}exp\left(-\frac{r^{2}}{2\sigma^{2}\Delta_{yy}^{2}}\right)\right]^{2}+4 \frac{A_{x}^{2}A_{y}^{2}\left|B_{yy}\right|^{2}}{\Delta_{xy}^{2}}exp\left(- \frac{r^{2}}{\sigma^{2}\Delta_{xy}^{2}}\right)}}{\left[\frac{A_{x}^{2}}{ \Delta_{xx}^{2}}exp\left(-\frac{r^{2}}{2\sigma^{2}\Delta_{xx}^{2}}\right)+ \frac{A_{y}^{2}}{\Delta_{yy}^{2}}exp\left(-\frac{r^{2}}{2\sigma^{2}\Delta_{yy} ^{2}}\right)\right]^{2}}} \tag{17}\]  
\[\varepsilon(\mathbf{r},z;\omega)=\sqrt{\frac{\left[\left[\frac{A_{y}^{2}}{\Delta_{ xx}^{2}}exp\left(-\frac{r^{2}}{2\sigma^{2}\Delta_{xx}^{2}}\right)-\frac{A_{y}^{2}}{ \Delta_{yy}^{2}}exp\left(-\frac{r^{2}}{2\sigma^{2}\Delta_{yy}^{2}}\right) \right]^{2}+4\frac{A_{x}^{2}A_{y}^{2}Re\left[B_{yy}\right]}{\Delta_{xy}^{2}} exp\left(-\frac{r^{2}}{\sigma^{2}\Delta_{yy}^{2}}\right)\right]^{ \frac{1}{2}}}{\left[\left[\frac{A_{x}^{2}}{\Delta_{xx}^{2}}exp\left(-\frac{r^ {2}}{2\sigma^{2}\Delta_{xx}^{2}}\right)-\frac{A_{y}^{2}}{\Delta_{yy}^{2}}exp \left(-\frac{r^{2}}{2\sigma^{2}\Delta_{yy}^{2}}\right)\right]^{2}+4\frac{A_{x} ^{2}A_{y}^{2}|B_{yy}|^{2}}{\Delta_{xy}^{2}}exp\left(-\frac{r^{2}}{\sigma^{2} \Delta_{yy}^{2}}\right)\right]^{\frac{1}{2}}}}{\left[\left[\frac{A_{x}^{2}}{ \Delta_{xx}^{2}}exp\left(-\frac{r^{2}}{2\sigma^{2}\Delta_{xx}^{2}}\right)- \frac{A_{y}^{2}}{\Delta_{yy}^{2}}exp\left(-\frac{r^{2}}{2\sigma^{2}\Delta_{yy} ^{2}}\right)\right]^{2}+4\frac{A_{x}^{2}A_{y}^{2}|B_{yy}|^{2}}{\Delta_{xy}^{2}} exp\left(-\frac{r^{2}}{\sigma^{2}\Delta_{yy}^{2}}\right)\right]^{\frac{1}{2}}}} \tag{18}\]  
The SOP and DOP remain invariant during downlink propagation if the following conditions hold. \[P(\mathbf{r},z=0;\omega)=P(\mathbf{r},z;\omega),\,\varepsilon(\mathbf{r},z=0;\omega)= \varepsilon(\mathbf{r},z;\omega)=1 \tag{19}\]  
By solving the equations above, the following results are obtained. \[\delta_{xx}=\delta_{xy}=\delta_{yy},Re\left|B_{xy}\right|=0,A_{x}=A_{y} \tag{20}\]  
Therefore, conditions sufficient to maintain the polarization characteristics of a circularly polarized GSM laser beam in the downlink were derived. The conditions depend on the source parameters and are unrelated to the turbulence parameters, modeled by  
Figure 1: Schematic of the zenith angle. the extended Huygens-Fresnel integral. This finding is crucial for establishing a design foundation for CPolSK optical communication systems.