## 3 Results  
In this section, we put the above models into a simulation program for decoy-state BB84 QKD in a turbulent channel where the refraction structure constant \(C_{n}^{2}\) is fixed at \(10^{-13}\). Here, we first estimate the spatially-dependent polarization effects caused by the use of curved optics in an off-axis configuration. The 45-degree polarized state under a coherent beam with the different coherence length (i.e., the difference \(\Delta\delta\) between the coherence length \(\delta_{xx}\) in the \(x\) direction and \(\delta_{yy}\) in the \(y\) direction) is taken as an example, and the results of the misalignment errors of the beam in the vacuum channel are shown in Figure 2b. It can be observed that one factor that causes the rotation of the polarization angle is the coherence characteristic of the laser beam itself. Once the difference \(\Delta\delta\) is close to zero (i.e., the ideal coherent beam), the rotation of the polarization angle is negligible. This is because, if the beam has the same coherence length in x/y directions, the variation in cross-spectral density function is equal according to Equation (2), thus, there are almost no rotations. In this case, the expected misalignment error is consistent with the reported value of 1.5%, which means the polarization misalignment is due to the measurement setting.  
To perform the influence of polarization analysis on the secret key generation rate under different beam qualities, we plot the comparison results both in the asymptotic case and finite data size. Here, we only fix the signal and decoy intensities to \(\mu=0.3\) and \(\
u=0.05\). The probabilities of sending them are equal and the probability of sending X basis is 0.5. Other experimental parameters are listed in Table 1. As shown in Figure 3a,  
Figure 2: The evolution of the polarization angle (**a**) and misalignment error (**b**) of the 45-degree polarization state versus transmission distance in a free-space channel. Since the misalignment error only depends on the rotation of polarization angle \(\Delta\theta\), the result is same with respect to the \(-45\)-degree, and the vertical polarized and horizontal polarized states in the theoretical simulations.  
the key rate under a coherent beam with \(\Delta\delta=0.0001\) m is very close to the ideal case. As the coherent length difference \(\Delta\delta\) of the laser beam increases from 0.0001 m to 0.005 m, the maximum tolerable transmission distance is reduced from over 120 km to approximately 60 km. Moreover, the valid secret key rate is hard to realize over 20 km when a coherent beam with \(\Delta\delta=0.0075\) m or \(\Delta\delta=0.01\) m is applied.  
The practical key rates in finite data size are also shown in Figure 3b. We compared the key rate of QKD in the dynamic misalignment errors calculated by our model (blue line) and the fixed misalignment errors (1.5%) reported in the previous works (pink line), respectively, and we took a moderate value \(\Delta\delta=0.005\) m in the range of \(\Delta\delta=0.0001\)\(\sim\) 0.01 m, as shown in Figure 3a. It can be seen that the maximum tolerable transmission distances are both around \(\sim\)125 km under infinite data size (\(N=10^{99}\)) or finite data size (\(N=10^{12}\)). Compared to the finite length effect, the maximum tolerable transmission distance is reduced to 60 km with a dynamic probability of misalignment error \(e_{d}\) in our model, which shows that the probability of misalignment error \(e_{d}\) has a severe impact on the key rate. These results highlight the importance of improving the beam quality in the practical implementation of free-space QKD.