## 1 Introduction
The signal beam and the LO beam are partially polarized and partially coherent beams at the source field, of which the BCP matrix can be expressed as [20]:  
\[\mathbf{W}_{aij}^{(0)}(\mathbf{r}_{1},\mathbf{r}_{2})=A_{ai}A_{aj}B_{aij}\exp\left[-\left( \frac{\mathbf{r}_{1}^{2}}{4\sigma_{ai}^{2}}+\frac{\mathbf{r}_{2}^{2}}{4\sigma_{aj}^{2} }\right)\right]\times\exp\left(-\frac{|\mathbf{r}_{1}-\mathbf{r}_{2}|^{2}}{2\delta_{ aij}^{2}}\right) \tag{5}\]  
where \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) denote any two-point position vector at plane \(z=0\) (\(z\) is the propagation distance of the signal beam). The subscript is \(\alpha=L\) (LO beam) or \(S\) (signal beam) and (\(i\),\(j\)) denote (\(x\),\(y\)), respectively. The equation \(B_{ij}=1\), \(i=j\), \(\mid B_{ij}\mid\leq 1\), \(I\
eq j\); \(B_{j}=B_{ji}^{*}\) is satisfied by the parameter \(B_{ij}\). \(A_{ai}\) and \(A_{aj}\) are the amplitude of the light field, and \(\sigma_{ai}\) and \(\sigma_{aj}\) are the beam waist radius in the \(x\) and \(y\) axes, respectively. The transverse coherence length of the beam is \(\delta_{aij}\) (\(z=0\)), which meets \(\delta_{ij}=\delta_{ji}\). Assuming that a signal beam propagates along the \(z\) axis, based on the generalized Huygens-Fresnel diffraction principle, each element of the BCP matrix of the partially polarized coherent beam propagated through the atmospheric turbulence can be easily obtained, which can be expressed as [22]:  
\[ \begin{array}{l}\mathbf{W}_{aij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)=\frac{1}{\chi^{2}z }\iint\mathbf{W}_{aij}(r_{1},\mathbf{r}_{2},0)\times\exp\Bigl{\{}-\frac{ik}{2z}\Bigl{[} (\mathbf{\rho}_{1}-\mathbf{r}_{1})^{2}-(\mathbf{\rho}_{2}-\mathbf{r}_{2})^{2}\Bigr{]}\Bigr{\}} \\ \times\langle\exp[\varphi^{*}(\mathbf{\rho}_{1},\mathbf{r}_{1},z)+\varphi(\mathbf{\rho}_{2 },\mathbf{r}_{2},z)]\rangle d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2}\end{array}  \tag{6}\]  
where \(\mathbf{\rho_{1}}\) and \(\mathbf{\rho_{2}}\) denote the position vectors of any two points at the receiver plane, \(\lambda\) is the wavelength, \(k\) = 2\(\pi/\lambda\) and \(\varphi\) is the cross-correlation function of the complex phase of the spherical wave, which can be expressed as [23]:  
\[\langle\exp[\varphi^{*}(\mathbf{\rho}_{1},\mathbf{r}_{1},z)+\varphi(\mathbf{\rho}_{2},\mathbf{ r}_{2},z)]\rangle\approx\exp\left[-\frac{1}{2}\Bigl{(}M_{1}Q^{2}+M_{2}P\cdot Q +M_{3}P^{2}\Bigr{)}\right] \tag{7}\]  
where \(P\) = \(\mathbf{r_{1}}\) - \(\mathbf{r_{2}}\), \(Q\) = \(\mathbf{\rho_{1}}\) - \(\mathbf{\rho_{2}}\), \(M_{1}\), \(M_{2}\) and \(M_{3}\) are, respectively:  
\[\left\{ \begin{array}{l}M_{1}=2\pi^{2}k^{2}\sec\theta\int_{I_{0}}^{H}\int_{0} ^{\infty}\Phi_{n}(\kappa,h)\times(1-\xi)^{2}\kappa^{3}dxdh\\ M_{2}=4\pi^{2}k^{2}\sec\theta\int_{I_{0}}^{H}\int_{0}^{\infty}\[PERSON]_{n}(\kappa,h )\times(1-\xi)\xi\kappa^{3}dxdh\\ M_{3}=2\pi^{2}k^{2}\sec\theta\int_{I_{0}}^{H}\int_{0}^{\infty}\Phi_{n}(\kappa,h )\xi^{2}\kappa^{3}dxdh\end{array} \right.\]  
where \(\xi\) denotes \((h\)-\(h_{0})/(H\)-\(h_{0})\) in the downlink path and \(\xi\) denotes \(1-(h-h_{0})/(H-h_{0})\) in the uplink path. \(\Phi_{n}(\kappa)\) is the spectrum of the refractive index fluctuations. Here, to overcome the application limitations of the Kolmogorov turbulence spectrum and considering both inner scale and outer scale effects at the same time, the modified Hill spectrum model \(\Phi_{n}(\kappa)\) was used in this paper [22]. It is expressed as:  
\[\Phi_{n}(\kappa,h)=0.033C_{n}^{2}(h)\left[1+1.802\frac{\kappa}{\kappa_{I}}-0. 254\biggl{(}\frac{\kappa}{\kappa_{I}}\biggr{)}^{7/6}\right]\frac{\exp\bigl{(} -\kappa^{2}/\kappa_{I}^{2}\bigr{)}}{\bigl{(}\kappa^{2}+\kappa_{0}^{2}\bigr{)} ^{11/6}} \tag{8}\]  
where \(\kappa_{I}\) = 3.3/\(l_{0}\); \(\kappa_{0}\) = 2\(\pi\)/L\({}_{0}\), \(l_{0}\) is the inner scale and \(L_{0}\) is the outer scale.