## 1 Introduction
For a signal beam propagating in a slant path, the Hufnagel-Valley model is widely used to approximate the atmospheric refractive index structure constant \(C_{n}^{2}(h)\)[22]:  
\[C_{n}^{2}(h)=8.148\times 10^{-56}\upsilon_{RMS}^{2}h^{10}e^{-h/1000}+2.7\times 1 0^{-16}e^{-h/1500}+C_{0}e^{-h/100} \tag{9}\]  
where \(\upsilon_{RMS}\) = ( \( \begin{smallmatrix}2\\ g\end{smallmatrix} +30.69\upsilon_{g}+348.91\))\({}^{1/2}\) is the speed of wind in the vertical path, \(\upsilon_{g}\) is the surface wind speed and \(h\) is the distance from the ground. Generally, \(\upsilon_{g}\) = 21 m/s, and the refractive index structure constant \(C_{0}\) is 1.7 \(\times 10^{-14}\) m\({}^{-2/3}\). By substituting Equations (5), (7)-(9) by Equation (6) and a complex integral simplification [24], \(\mathbf{W}_{aij}(\mathbf{\rho}_{1},\mathbf{\rho_{2}},z)\) can be simplified as:  
\[ \begin{array}{l}\mathbf{W}_{aij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)=\frac{k^{2}A_{i}A_ {j}B_{ij}}{4z^{2}\Delta_{aij}\sqrt{(Q(z)z_{aij}^{2}+T(z)z_{aij}^{2}}}\\ \exp\left[M_{aij}(z)(\rho_{1}-\rho_{2})^{2}+I_{aij}(z)(\rho_{1}+\rho_{2})^{2}+ G_{aij}(z)\left(\mathbf{\rho}_{1}^{2}-\mathbf{\rho}_{2}^{2}\right)\right]\\ \times\exp\left[iL_{aij}(z)(\rho_{1}-\rho_{2})^{2}+i\gamma_{aij}(z)(\rho_{1}+ \rho_{2})^{2}+iN_{aij}(z)\left(\rho_{1}^{2}-\rho_{2}^{2}\right)\right]\end{array}  \tag{10}\]  
By substituting Equation (10) by Equation (3), and converting to the polar coordinates \(\rho\) and \(\varphi\), the power associated with the IF signal photocurrent can be expressed as:\[P_{IF}=\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{D/2}\int_{0}^{D/2}\mathrm{Re}\Big{\{} \mathbf{W}_{Lifj}(\rho_{1},\rho_{2},z)\cdot\mathbf{W}_{Sij}^{*}(\rho_{1},\rho_{2},z)\Big{\}} \rho_{1}\rho_{2}d\rho_{1}d\rho_{2}d\varphi_{1}d\varphi_{2} \tag{11}\]  
where \(D\) is the hard aperture diameter of the detector. To simplify the calculation, we use the Gaussian limiting aperture, or soft aperture, to replace the hard aperture. Using the relationship between the soft aperture \(w\) and the hard aperture, \(D\) is \(D^{2}\) = \(8w^{2}\)[25] and \(\rho_{1}\rho_{2}\) = \(\rho_{1}\rho_{2}\)cos(\(\varphi_{1}\)-\(\varphi_{2}\)). The complex integration is performed as shown in detail in Appendix A. Then, \(P_{Lifj}\) can be simplified as:  
\[ \begin{array}{ll}P_{IFij}=&\frac{\pi^{2}k^{4}A_{i}{}^{2}A_{j}{}^{2}B_{ij}{} ^{2}}{8z^{4}\Delta_{\mathrm{Lifj}}\Delta_{\mathrm{Sij}}\big{[}s_{ij}-s_{ij}-i \big{(}\epsilon_{ij}-m_{ij}\big{)}\big{]}\sqrt{Q(z)^{2}_{Lifj}+T(z)^{2}_{Lifj} \sqrt{Q(z)^{2}_{Sij}+T(z)^{2}_{Sij}}}}\\ &\times\frac{1}{2\Bigg{[}-\big{(}s_{ij}+s_{ij}+i\big{(}m_{ij}+s_{ij}\big{)} \big{)}-\frac{\big{(}s_{ij}+i\epsilon_{ij}\big{)}^{2}}{s_{ij}-s_{ij}-i\big{(} \epsilon_{ij}-m_{ij}\big{)}}\Bigg{]}}\end{array}  \tag{12}\]  
Based on the results obtained in Appendix A and Equation (12), it is possible to express the heterodyne efficiency of the partly polarized GSM beam in the turbulence of heterodyne detection as:  
\[\eta_{h}=\frac{\sum_{i,j=x,y}\,P_{IFij}}{\sum_{i=x,y}\,P_{LIi}\sum_{i=x,y}\,P _{Sii}} \tag{13}\]  
Mathematical Model of DoP with Partially Polarized GSM Beam Propagating along an Uplink Path and a Downlink Path in Turbulence  
When the signal beam passes through the receiver in the uplink or downlink slant path, let \(\rho_{1}\) = \(\rho_{2}\) = \(\rho\). Based on Equation (10) and the integral formulation, \(\mathbf{W}_{Sij}(\mathbf{\rho}\),\(\mathbf{\rho}\),\(z)\) can be expressed as:  
\[\mathbf{W}_{Sij}(\mathbf{\rho},\mathbf{\rho},z)=\frac{k^{2}A_{i}A_{j}B_{ij}}{4z^{2}\Delta _{\mathrm{Sij}}\sqrt{Q(z)^{2}_{Sij}+T(z)^{2}_{Sij}}}\exp\Big{[}4I_{Sij}(z)\mathbf{ \rho}^{2}\Big{]}\exp\Big{[}i4Y_{Sij}(z)\mathbf{\rho}^{2}\Big{]} \tag{14}\]  
The DoP of the received signal beam is described with the generalized Stokes parameter, which can be expressed as [15]:  
\[P(\mathbf{\rho};z)=\frac{\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}}{S_{0}} \tag{15}\]  
where \(S_{0}\), \(S_{1}\), \(S_{2}\) and \(S_{3}\) can be expressed with the cross-spectral density matrix (CSDM), namely,  
\[\left\{