## 4 Numerical and Discussion
The mean value is indicated by the number at the top or bottom of each box. With increasing \(\delta_{Sxx}\) and \(\delta_{Syy}\), heterodyne efficiency improves and decreases, respectively, as seen in Figure 5a,b. Finally, the heterodyne efficiency eventually stabilizes. The heterodyne efficiency on the downlink path is also higher than that on the uplink path in Figure 5a. As demonstrated in Figure 5b, the heterodyne efficiency is higher for a downlink path than for an uplink path when \(\delta_{Syy}<0.03\) m, and when \(\delta_{Syy}\geq 0.03\) m, the heterodyne efficiency for a downlink path is lower than that for an uplink path. Additionally, this change of heterodyne efficiency of the downlink and uplink paths is increased with an increase in \(\delta_{Syy}\). Comparing Figure 5a,c, the heterodyne efficiency decreases and increases with increasing \(\delta_{Lyy}\) and \(\delta_{Lxx}\), respectively. The heterodyne efficiency on the downlink path is also lower than that on the uplink path in Figure 5c, which is an opposite trend to the results presented in Figure 5a. The reason has been explained by analyzing the results of Figure 3. Therefore, compared with the effect of \(\delta_{Sxx}\) on the heterodyne efficiency, the coherence length of the signal beam in the \(yy\) direction component \(\delta_{Syy}\) has an opposite trend to the heterodyne efficiency for a downlink path and an uplink path. The comparison of the results of Figure 5b,d can also demonstrate the correctness of this conclusion. Figure 3: Effect of waist radius of LO beam on the heterodyne efficiency on an uplink path and a downlink path (**a**) \(\sigma_{lx}\) and \(\sigma_{sx}\), (**b**) \(\sigma_{lx}\) and \(\sigma_{sy}\), (**c**) \(\sigma_{sy}\) and \(\sigma_{sx}\), (**d**) \(\sigma_{sy}\) and \(\sigma_{sy}\). The heterodyne efficiency and DoP are displayed as functions of the inner scale and outer scale, respectively, in Figure 6a,b. The heterodyne efficiency and DoP are enhanced with the increasing inner scale \(l_{0}\) and decrease with the raising outer scale \(L_{0}\), as shown in Figure 6a,b. The heterodyne efficiency and DoP on the downlink path are also higher than that on the uplink path. The DoP and heterodyne efficiency are significantly affected by the inner scale carried on by atmospheric turbulence, as seen in Figure 6a,b. Heterodyne efficiency and DoP are shown in Figure 7a-c as a function of transmission distance z for various refractive index structure parameters. As shown in Figure 7a-c, the heterodyne efficiency and DoP of an uplink path and a downlink path are extremely similar within a transmission distance of 100 m. This is because, when the transmission distance is relatively small, the random fluctuation carried upon by the refractive index may be omitted. The maximum DoP and heterodyne efficiency of a downlink propagation path correspond to a transmission distance that is greater than an uplink propagation path, and it decreases with an increasing atmospheric refractive index structure constant \(C_{n}^{2}\)\(10^{-16}\) m\({}^{-2/3}\) to \(10^{-12}\) m\({}^{-2/3}\). As the transmission distance increases, the DoP and heterodyne efficiency of a downlink path are significantly impacted by strong atmospheric turbulence, as compared to weak and medium-strong atmospheric turbulence intensities. This is due to the fact that when the number of small-scale gyres increases in the area of considerable atmospheric turbulence, the diffraction of the signal beam is strengthened, which is consistent with the findings in Figure 6.  
Figure 4: Effect of coherence length of signal beam on the DoP and heterodyne efficiency of an uplink path and a downlink path (**a**) \(\delta_{Syx}\), (**b**) \(\delta_{Syy}\), (**c**) \(\delta_{Sxy}\), (**d**) \(\delta_{Syx}\). Figure 5: Effect of coherence length of LO beam on the heterodyne efficiency on an uplink path and a downlink path.