## 2 Theoretical Model of Polarization Characteristics  
To determine the performance of a CPolSK laser communication system, it is crucial to understand the polarization characteristics of the propagated signal in a turbulent atmosphere. This section discusses the polarization transmission characteristic model of a GSM beam for downlink and the sufficient conditions required to maintain the polarization transmission characteristics. In isotropic turbulence, polarization states are preserved on average, so a circularly polarized beam remains circular during propagation. Circular polarization is chosen here for convenience of modulation and detection; in the isotropic turbulence considered, orthogonal linear polarization (horizontal/vertical) would yield a similar performance. The cross-spectral density matrix components at the transmitter can be obtained as follows [17]:  
\[W_{ij}(\rho_{1},\rho_{2},0;\omega)=A_{i}A_{j}B_{ij}exp\left[-\frac{\rho_{1}^{2 }+\rho_{2}^{2}}{4w_{0}^{2}}\right]exp\left[-\frac{(\rho_{2}-\rho_{1})^{2}}{2 \delta_{ij}^{2}}\right] \tag{1}\]Here, \(\rho_{1}\) and \(\rho_{2}\) denote the position vector on the source plane. \(i\) and \(j\) index the Cartesian polarization components; \(\delta_{ij}\) and \(w_{0}\) represent the coherence lengths and half-width, respectively. The terms \(A_{i}\) and \(A_{j}\) denote two orthogonal electric field amplitude components, and \(B_{ij}\) corresponds to the correlation coefficient between them. Notably, the coefficients \(A_{i}\), \(B_{ij}\), \(w_{0}\), and \(\delta_{ij}\) are positive, frequency-dependent, and position-independent. In particular, \(B_{ij}\) and \(\delta_{ij}\) must satisfy the following expressions [18]:  
\[ \begin{array}{l}B_{ij}=1,\;when\;i=j,\\ |B_{ij}|\leq 1,\;when\;i\
eq j\\ B_{ij}=B_{il}^{*},\;\delta_{ij}=\delta_{ji}.\end{array}  \tag{2}\]  
The degree of polarization (DOP) is defined as the ratio of the intensity of the polarized portion to the total intensity [19], which is expressed as  
\[P(\mathbf{r},z;\omega)=\sqrt{1-\frac{4 det\left[\overset{\leftrightarrow}{W}(\bm {r},z;\omega)\right]}{\left[\overset{\leftrightarrow}{TrW}(\mathbf{r},z;\omega) \right]^{2}}}, \tag{3}\]  
where det represents the determinant of the matrix and Tr represents its trace. The state of polarization (SOP) of the polarized portion of the beam is characterized by the polarization ellipse, which, in turn, is characterized by the degree of ellipticity and the orientation angle [20]. \[\alpha(\mathbf{r},z;\omega)=\frac{1}{2}arctan\left(\frac{2 Re\left[W_{xy}(\mathbf{r},z; \omega)\right]}{W_{xx}(\mathbf{r},z;\omega)-W_{yy}(\mathbf{r},z;\omega)}\right),-\frac {\pi}{2}\leq\alpha\leq\frac{\pi}{2} \tag{4}\]  
\[\varepsilon=\frac{A_{minor}}{A_{major}}=\sqrt{\frac{\left[\left(W_{xx}-W_{yy} \right)^{2}+4\left|W_{xy}\right|^{2}\right]^{\frac{1}{2}}-\left[\left(W_{xx} -W_{yy}\right)^{2}+4\left[ReW_{xy}\right]^{2}\right]^{\frac{1}{2}}}{\left[ \left(W_{xx}-W_{yy}\right)^{2}+4\left|W_{xy}\right|^{2}\right]^{\frac{1}{2}}+ \left[\left(W_{xx}-W_{yy}\right)^{2}+4\left[ReW_{xy}\right]^{2}\right]^{\frac {1}{2}}}} \tag{5}\]  
The terms \(A_{major}\) and \(A_{minor}\) correspond to the major and minor semiaxes, respectively. For a circularly polarized GSM laser beam, the term \(\alpha(\mathbf{r},z;\omega)\) varies within the range of \(-\pi/2\)-\(\pi/2\), whereas the degree of ellipticity \(\varepsilon\) is unity [21]. Consequently, the DOP and SOP at the source plane can be expressed as follows.