## 4 Numerical and Discussion  
Based on Equations (13) and (17), the effects of the source parameters, turbulence parameters and propagation distance on the DoP and the performance of the partially coherent GSM beam heterodyne detection system are investigated. For convenience and with no special instructions, the simulation parameters were set as Table 1, the correlation length \(B_{ij}\) satisfies \(\max\{\delta_{xxx},\,\delta_{ayy}\}\leq\delta_{axy}\leq\min\{\delta_{xxx},\, \mid\,B_{xy}\,\mid\,0.5,\,\delta_{ayy}\,/\mid\,B_{xy}\,\mid\,0.5\}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Light Source Parameters**} & \multicolumn{2}{c}{**Transmission Medium and Other Parameters**} \\ \hline Wavelength & \(\lambda=1550\) nm & Detector diameter & \(D=2\) mm \\ \hline \multirow{4}{*}{Coherence length of the signal beam (\(z=0\))} & \(\delta_{sxx}=2\) cm & \multirow{4}{*}{Atmospheric refractive index structure constant} & \multirow{2}{*}{\(C_{0}=10^{-14}\) m\({}^{-2/3}\)} \\ \cline{2-2} \cline{4-4}  & \(\delta_{sxy}=1\) cm & & \\ \cline{2-2} \cline{4-4}  & \(\delta_{sxy}=2.5\) cm & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & \(\mathit{inf}\) & & \\ \cline{2-2} \cline{4-4}  & Waist radius of signal beam (\(z=0\)) & & \\ \cline{2-2} \cline{4-4}  & \(\sigma_{xy}=1\) cm & & \\ \cline{2-2} \cline{4-4}  & Waist radius of LO beam (\(z=0\)) & & \\ \cline{2-2} \cline{4-4}  & \(\sigma_{lx}=2\) cm & & \\ \cline{2-2} \cline{4-4}  & \(\sigma_{ly}=2\) cm & & \\ \hline Correlation length & \(B_{ij}=0.5\mathrm{exp}(i\pi/4)\) & Vertical height & \(H=500\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameter settings. Figure 2a illustrates the DoP and heterodyne efficiency of on-axis points (\(\rho=0\)) as a function of \(\sigma_{Sx}\) for uplink and downlink paths. As seen in Figure 2a, the DoP of the signal beam increases and eventually stabilizes as \(\sigma_{Sx}\) increases, which is similar to the heterodyne efficiency as a function of \(\sigma_{Sx}\). Overall, signal beam propagation on a downlink path has a greater Dop and heterodyne efficiency than on an uplink path, which is consistent with Ref [15]. Figure 2b illustrates the DoP of deviation from the axial position for various \(\sigma_{sy}\) on the uplink and downlink paths. The heterodyne efficiency as a function of \(\sigma_{sy}\) for the uplink and downlink paths is shown in Figure 2c. As shown in Figure 2b, the variation of DoP with the deviation from the axial position on the downlink and uplink paths was decreased with an increase in \(\sigma_{sy}\). As seen in Figure 2c, heterodyne efficiency decreases with increasing \(\sigma_{sy}\); however, there seems to be a key point at \(\sigma_{sy}=0.02\) m. When \(\sigma_{sy}\leq 0.02\) m, the heterodyne efficiency of the signal beam propagation on a downlink path is higher than on an uplink path; however, when \(\sigma_{sy}>0.02\) m, the result of heterodyne efficiency on a downlink vs. an uplink path is precisely the opposite. This is attributed to the fact that \(\sigma_{sy}\) has a major effect on the DoP of the downlink path, which has a significant impact on heterodyne efficiency. Figure 3a,b show the heterodyne efficiency as a function of \(\sigma_{lx}\) for various \(\sigma_{Sx}\) and \(\sigma_{sy}\) of the uplink and downlink path, respectively. Figure 3c,d show the heterodyne efficiency as a function of \(\sigma_{ly}\) for various \(\sigma_{sx}\) and \(\sigma_{sy}\) of the uplink and downlink path, respectively. In Figure 3a-d, the top and bottom symbols in each box represent the maximum and minimum values, respectively. The mean value is indicated by the number at the top or bottom of each box.