## 2 Theoretical Analysis  
The Kolmogorov atmosphere model is widely used to describe turbulence in free-space laser communication links, but it is not accurate because the boundary conditions are not always satisfied [26; 27]. The statistical model of the wavefront based on non-Kolmogorov turbulence is essential to evaluate the PDF of the CE for SMF. Figure 1 shows the optical structure of a free-space optical communication terminal and how special signal light couples into SMF. Signal and beacon light propagate though the atmosphere and are affected by atmospheric turbulence. Both of them are received by the optical antenna, which consist of a telescope. It is designed to minify the radius of the signal light at the optical pupil and beneficial to minimize the optical structure. The telescope is set on the coarse tracking element to follow the beacon light. Reflected by the fast steering mirror (FSM), which is the fine tracking element, the beacon and signal light are separated by the beam splitter 1 (BS1). The beacon light is received by the coarse tracking camera, which provides the feedback signal for the coarse tracking element. The signal light transmits though the beam splitter 2 (BS2), which splits the receiving and launching signal, and splits them with the beam splitter 3 (BS3). Most of the energy, e.g., 95%, is coupled into the SMF by the coupling optical system for the next processing, while the rest transmits into the fine tracking camera, providing the feedback signal for the FSM. Therefore, the CE is essential to evaluate the performance of the receiving terminal. Firstly, the wavefront residual variance based on the non-Kolmogorov model is calculated. Zernike polynomials are widely used to expend the random wavefront. Considering a plane where a random wavefront is expended, a polar coordinate system is needed. Here, \(r\) represents the polar radius and \(\theta\) is for the polar angle. Zernike polynomials are defined by Equation (1) [28]  
\[ \begin{array}{ll}Z_{even~{}j}=\sqrt{(n+1)}R_{n}^{m}(r)\sqrt{2}\cos m\theta,&m \
eq 0\\ Z_{odd~{}j}=\sqrt{(n+1)}R_{m}^{m}(r)\sqrt{2}\sin m\theta,&m\
eq 0\\ Z_{j}=\sqrt{(n+1)}R_{n}^{0}(r),&m=0\end{array}  \tag{1}\]  
where,  
\[R_{n}^{m}(r)=\sum_{s}^{(n-m)/2}\frac{(-1)^{s}(n-s)!}{s!\left[\frac{n+m}{2}-s \right]!\left[\frac{n-m}{2}-s\right]!}r^{n-2s} \tag{2}\]  
Figure 1: Free-space signal light couples into a single-mode fiber. where \(n\) and m are integral and satisfy \(m\leq n\), \(n-|m|=\) even and \(j\) is a mode-ordering number which is the function of \(n\) and \(m\). Table 2 shows the first 10 polynomials. It is convenient to use Zernike polynomials expending a random wavefront \(\phi(r,\ \theta)\), given by Equations (3)-(5). \[\phi(r,\theta)=\sum_{j}a_{j}Z_{j}(r,\theta) \tag{3}\]  
\[a_{j}=\frac{1}{R^{2}}\int W\Big{(}\frac{r}{R}\Big{)}\phi(r,\theta)Z_{j}\Big{(} \frac{r}{R},\theta\Big{)}d^{2}r \tag{4}\]  
\[W(r)= \begin{cases}\frac{1}{\pi}&r\leq 1\\ 0&r>1\end{cases}  \tag{5}\]  
The more terms we use in Equation (3), the more accurately will the wavefront be described. However, we can only expand the wavefront with finite Zernike terms, so the mean square residual error \(\Delta_{J}\) should be well discussed, which is shown in Equation (6). \[\Delta_{J}=\langle\varphi^{2}\rangle-\sum_{j=1}^{J}\left\langle|a_{j}|\right\rangle ^{2} \tag{6}\]  
\(\left\langle|a_{j}|^{2}\right\rangle\) represents the Zernike coefficient variances and \(\langle\varphi^{2}\rangle\) is total wavefront error shown by Equation (7). \[\langle\varphi^{2}\rangle=\sum_{j=1}^{\infty}\left\langle|a_{j}|^{2}\right\rangle \tag{7}\]  
Consider the distorted wavefront effected by non-Kolmogorov turbulence. The three-dimensional power spectral density of phase fluctuations is  
\[\Phi_{\varphi}(k)=\frac{A_{\beta}k^{-\beta}}{r_{0}^{\beta-2}} \tag{8}\]  
\(\beta\) is the power law at the range from 2 to 4. When it equals to \(11/3\), \(\Delta_{J}\) has the same value of the residual variance based on the Kolmogorov model [29].