## 4 Conclusions  
We propose a method where the characteristics of beams in a free-space channel can be calculated by the extended Huygens-Fresnel principle and the coherent Gaussian Schell-model (GSM). Based on the unified theory of coherence and polarization, a general model is proposed for determining the rotation and depolarization effects, which gives more practical results than those in the previous free-space QKD work. We also compare the key rates in both the asymptotic case and the case with finite-size effect, which show a  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Name** & **Value** \\ \hline \(\lambda\) & wavelength of the laser beam & 1550 nm \\ \(\omega_{0}\) & waist of the laser beam & 3.5 cm \\ \(N\) & total number of signals & \(10^{14}\) \\ \(\eta_{d}\) & detection efficiency & 50\% \\ \(e_{0}\) & error probability of dark counts & 0.5 \\ \(e_{d}\) & error probability of optical misalignment & 0.015 \\ \(f_{e}\) & Error correction efficiency & 1.16 \\ \(Y_{0}\) & background rate & \(3\times 10^{-6}\) \\ \(\in\) & security bound & \(10^{-7}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Lists of Necessary Parameters.  
Figure 3: (**a**) Key rate comparison against coherent length difference \(\Delta\delta\) in the asymptotic case and (**b**) key rate comparison against finite/infinite data size or fixed/dynamic probability of misalignment error \(e_{d}\) in our model.  
more critical impact on the key rate from the misalignment error than that of the finite-size effect. Furthermore, our numerical simulations show that the polarization effect caused by turbulence can be effectively mitigated when maintaining good beam coherence properties. Hence, it is suggested that the use of a laser beam in free-space QKD experiments should be considered more cautiously. We remark that our method can also be applied to different QKD protocols, including decoy-state BB84 QKD and MDI-QKD. Of course, more parameters of laser sources will be taken into consideration when our method is applied in MDI-QKD, as discussed in References [34; 35; 36].
* [3]**Author Contributions:** Conceptualization, T.W.; Data curation, T.W. and Q.P.; Formal analysis, C.L.; Funding acquisition, L.S., S.Z. and C.D.; Investigation, C.L.; Methodology, T.W.; Project administration, Y.Z. and C.D.; Resources, T.W.; Software, T.W. and C.L.; Supervision, T.W.; Validation, T.W., Q.P. and C.L.; Visualization, T.W.; Writing--original draft, X.W.; Writing--review and editing, X.W. All authors have read and agreed to the published version of the manuscript.
* [4]**Funding:** This work has been supported by the National Natural Science Foundation of China under Grant No. 61971436, in part by the National University of Defense Technology under Grant No. 19-QNCXJ, in part by the Innovative Talents Promotion Plan in Shaanxi Province under Grant No.2020 KJXX-011, in part by the Key Research and Development Program of Shaanxi under Grant No. 2019 ZDLGY09-01), in part by the Innovative Key Projects Promotion in Information and Communication College (No. YJKT-ZD-2105), and in part by the Innovative Talents Promotion in Information and Communication College (No. YJKT-RC-2113).
* [5]**Institutional Review Board Statement:** Not applicable.
* [6]**Informed Consent Statement:** Not applicable.
* [7]**Data Availability Statement:** Data are contained within the article.
* [8]**Acknowledgments:** We thank [PERSON] for helpful discussions.  
The authors declare no conflict of interest.