## 4 Numerical and Discussion
Comparing Figure 3a,c, heterodyne efficiency increases with increasing \(\sigma_{lx}\) and \(\sigma_{ly}\), respectively. Heterodyne efficiency on the downlink path is higher than on the uplink path in Figure 3a,c. Additionally, the waist radius of the LO beam in the \(x\) direction  
Figure 2: Effect of waist radius of signal beam on the DoP and heterodyne efficiency of an uplink path and a downlink path in atmospheric turbulence (**a**) \(\sigma_{sx}\), (**b**) offset from the axial position \(\rho\) and (**c**) \(\sigma_{sy}\). component \(\sigma_{lx}\) has a relatively strong effect on the heterodyne efficiency. Comparing Figure 3a,b, heterodyne efficiency improves and decreases with increasing \(\sigma_{Sx}\) and \(\sigma_{Sy}\), respectively. Finally, heterodyne efficiency eventually stabilizes. When \(\sigma_{Sy}\leq 0.02\) m, heterodyne efficiency is higher for a downlink path than for an uplink path, as shown in Figure 3b,d. However, when \(\sigma_{Sy}\) is larger than 0.02 m, the heterodyne efficiency of the downlink path is lower than that of the uplink path. That is because the degree of mismatch between the signal beam and the LO beam increases as \(\sigma_{Sy}\) increases, which has a greater impact than the effect of the turbulence term on an uplink and downlink path. Comparing Figure 3c,d, heterodyne efficiency improves and decreases with increasing \(\sigma_{Sx}\) and \(\sigma_{Sy}\), respectively, which is consistent with the results presented in Figure 3a,b. Overall, heterodyne efficiency is significantly impacted by the signal beam waist radius in the Y direction component \(\sigma_{Sy}\). Meanwhile, \(\sigma_{Sy}\) plays a significant role in determining the trend of heterodyne efficiency with increasing \(\sigma_{lx}\) (or \(\sigma_{ly}\)) and this change of degree. Figure 4a-d show heterodyne efficiency and DoP as a function of \(\delta_{Sxx}\), \(\delta_{Syy}\), \(\delta_{Sxy}\) and \(\delta_{Syx}\) of an uplink and downlink path, respectively. As seen in Figure 4a, heterodyne efficiency increases and the DoP of the signal beam on-axis points decreases with an increase in \(\delta_{Sxx}\). The heterodyne efficiency and DoP of the signal beam propagation on a downlink path are higher than on an uplink path. That is because the difference between uplink and downlink affected by atmospheric turbulence is expressed in the turbulence term \(M_{3}\) in Equation (7), i.e., \(M_{3}\) of a downlink transmission is less than that of an uplink transmission. The heterodyne efficiency and DoP on a downlink and uplink path decrease with an increase in \(\delta_{Syy}\), which is consistent with the results presented in Figure 2c. The heterodyne efficiency of a downlink path is lower than that of an uplink path when \(\delta_{Syy}\) reaches a certain value. Namely, the variation of heterodyne efficiency of a downlink and uplink path is increased with an increase in \(\delta_{Syy}\). As seen in Figure 4c,d, heterodyne efficiency remains consistent as \(\delta_{Syy}\) and \(\delta_{Syx}\) increase, but the DoP of the on-axis points increases with an increase in \(\delta_{Syy}\) and \(\delta_{Syx}\). The heterodyne efficiency and DoP of a downlink path are higher than that of an uplink path. This demonstrates that the coherence length of the signal beam in the \(xy\) (or \(yx\)) direction component has a significant effect on the DoP, but is less sensitive with regard to heterodyne efficiency. Generally, \(\delta_{Syy}\) has the most significant effect on the heterodyne efficiency and DoP of the signal beam propagation on a downlink and uplink path. Figure 5a,b show the heterodyne efficiency as a function of \(\delta_{Lxx}\) for various \(\delta_{Sxx}\) and \(\delta_{Syy}\) of the uplink and downlink path, respectively. Figure 5c,d show the heterodyne efficiency as a function of \(\delta_{Lyy}\) for various \(\delta_{Sxx}\) and \(\delta_{Syy}\) of the uplink and downlink path, respectively. In Figure 5a-d, the top and bottom symbols in each box represent the maximum and minimum values, respectively.