## 5 Conclusions  
In this paper, we created a new statistical model of the CE based on the non-Kolmogorov turbulence theory. The power law \(\beta\) was introduced as a correction factor so that the variance of the wavefront residual variance could fit the actual distribution. A numerical simulation was done to test its robustness and rationality. The results showed that the statistical model was suitable for both weak and strong turbulence because the \(\beta\) was not constant. Finally, we set up an experiment to verify the statistical model. After counting the CE data, we found that the PDF built on the non-Kolmogorov turbulence model suited the experimental distribution well, because \(\beta\) was variable, and so we could choose a proper value so that variance of the wavefront was corrected. In conclusion, the statistical model of the CE based on non-Kolmogorov turbulence could describe the distribution of the CE well in the presence of atmospheric turbulence. With the application of this model, it is beneficial to evaluate the properties of the PSO terminals so to reduce the design costs at the beginning. Further study should focus on choosing a proper \(\beta\) in different conditions. Conceptualization, L.M. ; methodology, S.G.; software, L.M. ; validation, S.G. and Y.L. ; formal analysis, L.M. ; investigation, L.M. ; resources, S.G.; data curation, L.M. and Y.L. ; writing--original draft preparation, L.M. ; writing--review and editing, L.M. and B.C. ; visualization, L.M. and B.C. ; supervision, S.G.; project administration, S.G.; funding acquisition, S.G. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The authors declare no conflict of interest. Appendix A Calculation of \(\overline{\chi_{r}}\), \(\overline{\chi_{i}}\), \(\sigma_{r}^{2}\) and \(\sigma_{i}^{2}\)  
According to the definition of \(\chi_{r}\) and \(\chi_{i}\), and replacing \(-\frac{r_{k}^{2}}{\omega_{2}^{2}}\) with \(\Lambda_{k}\) for convenience, Equation (9) could be written as Equation (A1)  
\[ \begin{cases}\chi_{r}\approx\frac{S}{N}\exp\bigl{(}\overline{\Lambda_{k}} \bigr{)}\sum\limits_{k=1}^{N}\exp\bigl{(}\Lambda-\overline{\Lambda_{k}}\bigr{)} \cos\phi_{k}\\ \chi_{i}\approx\frac{S}{N}\exp\bigl{(}\overline{\Lambda_{k}}\bigr{)}\sum \limits_{k=1}^{N}\exp\bigl{(}\Lambda-\overline{\Lambda_{k}}\bigr{)}\sin\phi_{ k}\\ N=\biggl{\{}1.09\bigl{(}\frac{r_{0}}{D}\bigr{)}^{\frac{1}{2}}\Gamma\biggl{[} \frac{\phi}{1.08\bigl{(}\frac{D}{r_{0}}\bigr{)}^{\frac{5}{5}}\biggr{]}\biggr{\}} ^{-1}\end{cases} \] (A1)  
After some algebraic calculus, the mean value of \(\chi_{r}\) and \(\chi_{i}\), which are \(\overline{\chi_{r}}\) and \(\overline{\chi_{i}}\), is given by Equation (A2), and the variance, which are \(\sigma_{r}^{2}\) and \(\sigma_{i}^{2}\), is shown in Equation (A3). \[ \begin{cases}\overline{\chi_{r}}=\frac{S}{2}\exp\bigl{(}\overline{\Lambda_{k}} \bigr{)}\overline{\exp\bigl{(}\Lambda_{k}-\overline{\Lambda_{k}}\bigr{)}} \bigl{[}M_{\phi}(1)+M_{\phi}(-1)\bigr{]}\\ \overline{\chi_{i}}=-\frac{iS}{2}\exp\bigl{(}\overline{\Lambda_{k}}\bigr{)} \overline{\exp\bigl{(}\Lambda_{k}-\overline{\Lambda_{k}}\bigr{)}}\bigl{[}M_{ \phi}(1)-M_{\phi}(-1)\bigr{]}\end{cases} \] (A2)  
\[ \begin{cases}-\frac{\sigma_{r}^{2}}{S}=\frac{\exp\bigl{(}2\overline{\Lambda_{k }}\bigr{)}\overline{\exp\bigl{(}\Lambda_{k}-\overline{\Lambda_{k}}\bigr{)}} }{4N}\bigl{[}2+M_{\phi}(2)+M_{\phi}(-2)\bigr{]}\\ \frac{\exp\bigl{(}2\overline{\Lambda_{k}}\bigr{)}\overline{\exp\bigl{(} \Lambda_{k}-\overline{\Lambda_{k}}\bigr{)}}}{4N}\bigl{[}2M_{\phi}(1)M_{\phi}( -1)+M_{\phi}^{2}(1)+M_{\phi}^{2}(-1)\bigr{]}\\ -\frac{\exp\bigl{(}2\overline{\Lambda_{k}}\bigr{)}\overline{\exp\bigl{(} \Lambda_{k}-\overline{\Lambda_{k}}\bigr{)}}}{4N}\bigl{[}2M_{\phi}(1)M_{\phi}( -1)-M_{\phi}^{2}(1)-M_{\phi}^{2}(-1)\bigr{]}\end{cases} \] (A3)  
\(M_{\phi}(\omega)\) is the Fourier transformation of the PDF of the random phase \(\phi\).