## 3 Performance Analysis of Direct Detection CPolSK System
### Bit Error Rate  
Atmospheric turbulence induces fluctuations in the received intensity of a GSM laser beam, leading to variations in the BER [27]. The initial circular polarization condition derived above is applied here, ensuring the input beam satisfies that condition. To evaluate the performance of a CPolSK optical communication system in a downlink channel, the  
Figure 2: CPolSK system structure diagram. intensity scintillation probability distribution is assumed to follow the gamma-gamma distribution model, expressed as  
\[f(I)=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha)\Gamma(\beta)I }\left(\frac{I}{\langle I\rangle}\right)^{\frac{\alpha+\beta}{2}}K_{\alpha- \beta}\left(2\sqrt{\alpha\beta\frac{I}{\langle I\rangle}}\right),\text{I>0}. \tag{22}\]  
Here, \(K_{\alpha\text{-}\beta}(\bullet)\) denotes the modified Bessel function of the second kind of order \(m\), and \(\Gamma(\bullet)\) represents the gamma function. \(I\) is the received intensity, and \(<\)\(I\)\(>\) denotes the mean received intensity, which is normalized to unity. The parameters \(\alpha\) and \(\beta\) correspond to the effective numbers of small- and large-scale scatters, respectively. They are expressed as follows. \[\alpha=\frac{1}{exp\sigma_{lnx}^{2}-1} \tag{23}\]  
\[\beta=\frac{1}{exp\sigma_{lny}^{2}-1} \tag{24}\]  
Here, \(\sigma_{lnx}^{2}\) and \(\sigma_{lny}^{2}\) denote the variances associated with the large- and small-scale fluctuations, respectively. For a GSM laser beam source, the flux variance \(\sigma_{I}^{2}(z,\ d)\) of the intensity fluctuations at the receiver plane can be formulated as  
\[\sigma_{I}^{2}=exp[\sigma_{lnx}^{2}+\sigma_{lny}^{2}]-1. \tag{25}\]  
For a downlink channel, the scintillation \(\sigma_{lnx}^{2}\) and \(\sigma_{lny}^{2}\) can be expressed as follows. \[\sigma_{lnx}^{2}=\frac{0.49\sigma_{\text{B}}^{2}}{\left[1+\left[0.98/\sigma_{ \text{B}}^{4/5}\gamma\right]^{6/7}\sigma_{\text{B}}^{12/5}\right]^{7/6}} \tag{26}\]  
\[\sigma_{lny}^{2}=\frac{0.51\sigma_{\text{B}}^{2}}{\left(1+0.69\sigma_{\text{B }}^{12/5}\right)^{5/6}} \tag{27}\]  
\(\sigma_{B}^{2}\) is the scintillation index of the beam propagating through the downlink channel under weak turbulence, which is given by  
\[\sigma_{\text{1,new}}^{2}=\sigma_{B}^{2}=8.702\mu_{1}k^{7/6}(H-h_{0})^{5/6} sec^{11/6}(\theta), \tag{28}\]  
where \(\mu_{1}\) can be expressed as  
\[\mu_{1}=Re\int_{h_{0}}^{H}C_{n}^{2}(h)\left\{\xi^{5/6}\left[\Lambda\xi+i\left( 1-\overline{\Theta}\xi\right)\right]^{5/6}-\Lambda^{5/6}\xi^{5/3}\right\} \text{d}h. \tag{29}\]  
\(\gamma=\sigma_{1,strong}^{2}-1\), where \(\sigma_{1,strong}^{2}\) is the scintillation index under strong fluctuation conditions, \(Re\) denotes the real-part operator. It is expressed as  
\[\sigma_{1,strong}^{2}=1+\frac{1.23L^{-1/3}sec^{-2/5}(\theta)}{\left[k^{1/3}\int _{h_{0}}^{H}C_{n}^{2}(h)dt\right]^{7/5}}\int_{h_{0}}^{H}C_{n}^{2}(h)\frac{ \left(1-\overline{\Theta}\xi\right)^{2}}{\xi^{1/3}\left[\left(1-\overline{ \Theta}\xi\right)^{5/3}-\left(1-\overline{\Theta}\right)^{8/3}\right]^{7/5}} \text{d}h. \tag{30}\]  
where \(L\) is the propagation distance. The other parameters are given by  
\[f(I)=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha)\Gamma(\beta)I }\left(\frac{I}{\langle I\rangle}\right)^{\frac{\alpha+\beta}{2}}K_{\alpha- \beta}\left(2\sqrt{\alpha\beta\frac{I}{\langle I\rangle}}\right),\text{I>0}. \tag{22}\]0where \(R(z)\) and \(w(z)\) are the wavefront curvature radius of the GSM laser beam and beam width at the receiver plane in free space, respectively.