## 2 Theoretical Analysis
\(A_{\beta}\) is a coefficient related to \(\beta\), which has the value that \(\Delta_{1}\) is normalized to \(1\) rad\({}^{2}\) when the diameter of receiving aperture equals to \(r_{0}\), given by Equation (9) [29]. \[A_{\beta}=\frac{2^{\beta-2}\Big{[}\Gamma\Big{(}\frac{\beta+2}{2}\Big{)}\Big{]} ^{2}\Gamma\Big{(}\frac{\beta+4}{2}\Big{)}\Gamma\Big{(}\frac{\beta}{2}\Big{)} sin\Big{(}\tau\frac{\beta-2}{2}\Big{)}}{\pi^{\beta}\Gamma(\beta+1)} \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(n\) & \(m\) & \(0\) & \(1\) & \(2\) & \(3\) \\ \hline \(0\) & \(Z_{1}=0\) & & & \\ \(1\) & & \(Z_{2}=2 rcos\ \theta\ Z_{3}=2 rsin\ \theta\) & & \\ \(2\) & \(Z_{3}=\sqrt{3}\big{(}2r^{2}-1\big{)}\) & & \(Z_{5}=\sqrt{6}r^{2}sin\ 2\theta\) & \\  & & \(Z_{6}=\sqrt{6}r^{2}cos\ 2\theta\) & \\ \(3\) & & \(Z_{7}=2\sqrt{2}\big{(}3r^{3}-2r\big{)}sin\ \theta\) & & \(Z_{9}=2\sqrt{2}r^{3}sin\ 3\theta\) \\  & & \(Z_{8}=2\sqrt{2}\big{(}3r^{3}-2r\big{)}cos\ \theta\) & & \(Z_{9}=2\sqrt{2}r^{3}cos\ 3\theta\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: First 10 items of Zernike polynomials. According to the definition of \(a_{j}\) and the \(A_{\beta}\), the Zernike coefficient variances \(\left\langle\left|a_{j}\right|^{2}\right\rangle\) are given by Equation (10). \[\left\langle\left|a_{j}\right|^{2}\right\rangle=\left(\frac{D}{r_{0}}\right)^{ \beta-2}\frac{(n+1)}{\pi}.\frac{\Gamma\left(\frac{2n+2-\beta}{2}\right)\Gamma \left(\frac{\beta+4}{2}\right)\Gamma\left(\frac{\beta}{2}\right)\sin\left(\pi \cdot\frac{\beta-2}{2}\right)}{\Gamma\left(\frac{2n+4+\beta}{2}\right)} \tag{10}\]  
Substitute Equations (7) and (10) for Equation (6), and \(\Delta_{I}\) is calculated. We assume that the field of SMF could transfer backwards to the pupil plane of the receiving telescope, at which the coupling is processed and in which the condition of the CE is defined by Equations (11) and (12). \(E_{A}(r)\) represents the optical field of the signal light at the pupil plane, which is a plane-wave function. \(F_{A}(r)\) is the model field of the SMF transferring backwards to the pupil plane, and it is a Gaussian function, approximately. \(\chi\) is an intermediate variable, and \(\chi_{r}\) and \(\chi_{i}\) represent the real and imaginary parts of \(\chi\). Considering atmospheric turbulence, \(\phi\) turns into a random variable and is related to \(r\), so the integrals in Equations (11) and (12) have no analytic solution. \[\eta=\frac{\left|\int\int E_{A}^{*}(r)F_{A}(r)\mathrm{d}r\mathrm{d}\phi\right| ^{2}}{\left|\int\int E_{A}(r)\mathrm{d}r\mathrm{d}\phi\right|^{2}\left|\int \int F_{A}(r)\mathrm{d}r\mathrm{d}\phi\right|^{2}}=\frac{2}{\pi^{2}R^{2}\omega _{a}^{2}}\Big{(}\chi_{r}^{2}+\chi_{i}^{2}\Big{)} \tag{11}\]  
\[ \begin{cases}\chi_{r}=\int\int\exp\left(-\frac{r^{2}}{\omega_{a}^{2}}\right) \cos\phi\mathrm{d}r\mathrm{d}\phi=\sqrt{\chi^{2}}cos\theta\\ \chi_{i}=\int\int\exp\left(-\frac{r}{\omega_{a}^{2}}\right)\sin\phi\mathrm{d}r \mathrm{d}\phi=\sqrt{\chi^{2}}sin\theta\\ \chi^{2}=\chi_{r}^{2}+\chi_{i}^{2}\end{cases}  \tag{12}\]  
If we get the PDF of \(\chi^{2}\), which is \(p_{X^{2}}\big{(}\chi^{2}\big{)}\), the PDF of the CE could be described as Equation (13), based on Equation (11). \[p_{\mathrm{H}}(\eta)=\frac{\pi^{2}R^{2}\omega_{a}^{2}}{2}p_{X^{2}}\bigg{(} \frac{\pi^{2}R^{2}\omega_{a}^{2}}{2}\chi^{2}\bigg{)} \tag{13}\]  
When the signal light reaches the pupil of the receiving telescope after transmitting though the atmospheric turbulence, several speckles could be seen, and \(N\) is the number of these speckles. We assume that these speckles fulfill two conditions [30]:  
(1) The speckles are independent of each other. (2) The phase distribution function is a Gaussian function related to \(\Delta_{I}\). Considering strong turbulence, \(N\) is larger \(1\times 10^{4}\). Thus, the integrals in Equation (12) are replaced by the summation in Equation (14). \[