## 2 The Methods of Coupling Efficiency Calculation  
The scheme of spatial light coupling into a SMF is shown in Figure 1. The laser beam in the FSOC link generally travels a long distance, so we consider the beam entering the receiving terminal as a plane wave. \(E_{A}\) is the light field at the pupil plane \(A\), while \(E_{O}\) is the light field on the focal plane \(O\) after the plane wave passes through the lens (usually an Airy spot). The mode-field radius of the SMF is small and only the fundamental mode transmission is allowed in the SMF, so the process of the plane wave coupling into the SMF through the focusing lens can be considered the filtering effect of the spatial filter with a radius of \(\omega_{0}\) placed at the focus on the optical field \(E_{O}\). In line with many other researchers [22; 23; 24], we use the fiber mode field to be coupled Gaussian, not J/K Bessel functions, for convenience. The transfer function of the spatial filter is:  
\[F_{O}(r,\theta)=\sqrt{\frac{2}{\pi\omega_{0}^{2}}}\exp(-\frac{r^{2}}{\omega_{ 0}^{2}}) \tag{1}\]  
where \(\omega_{0}\) is the mode-field radius of SMF and (\(r\),\(\theta\)) are the polar coordinates of a certain point in the focal plane \(O\). Polar coordinates are adopted in this paper to analyze wavefront aberrations. We calculated the influence of different wavefront aberrations on the CE in plane A for convenience, because this can avoid Fourier transform of Zernike polynomial, simplify the mathematical process and achieve fast calculation. The backpropagated mode-field of the SMF can be expressed as follows [31]:\[F_{A}(r,\theta)=F^{-1}[F_{O}(r,\theta)]=\sqrt{\frac{2}{\pi\omega_{a}^{2}}}\exp(- \frac{r^{2}}{\omega_{a}{}^{2}}) \tag{2}\]  
\[\omega_{a}=\frac{\lambda f}{\pi\omega_{0}} \tag{3}\]  
where \(\omega_{a}\) is the radius of backpropagated SMF mode-field radius, \(f\) is the focal length of the coupling lens, and \(\lambda\) is the wavelength of laser. Here, we consider the intensity of the incident beam to be uniform and only the phase of the wavefront to change as we observe the influence of spatial frequency characteristics of aberrations on the CE. The laser wave field can be expressed as:  
\[E_{A}(r)=E_{s}\exp[-j\phi(r,\theta)] \tag{4}\]  
where \(E_{s}\) is the amplitude of incident light transmission and \(\phi\)(\(r\),\(\theta\)) is the phase of incident light wave field. The CE can be expressed as [23; 24]:  
\[\eta=\frac{\left|\int E_{A}{}^{*}(r)F_{A}(r)ds\right|^{2}}{\int\left|E_{A}(r) \right|^{2}ds\int\left|F_{A}(r)\right|^{2}ds} \tag{5}\]  
The integral range \(S\) is the effective full aperture of the entrance pupil plane. Substituting Equations (1)-(4) into Formula (5) yields the following CE expression:  
\[ \begin{array}{ll}\eta&=\frac{\left|\int E_{A}{}^{*}(r,\theta)F_{A}(r,\theta )n\text{d}n\theta\right|^{2}}{\int\left|E_{A}(r,\theta)\right|^{2}n\text{d}n \theta\cdot\iint\left|F_{A}(r,\theta)\right|^{2}n\text{d}n\theta}\\ &=\left|\iint\exp[-j\phi(r,\theta)]\cdot\sqrt{\frac{2}{\pi\omega_{a}{}^{2}}} \exp(-\frac{r^{2}}{\omega_{a}{}^{2}})r\text{d}r\theta\right|^{2}\\ &=\frac{2}{\pi\omega_{a}{}^{2}}(\alpha_{r}^{2}+\alpha_{\beta}^{2})\end{array}  \tag{6}\]  
where  
\[\alpha_{r}=\iint\exp(-\frac{r^{2}}{\omega_{a}{}^{2}})\cos[\phi(r,\theta)]r \text{d}r\text{d}\theta \tag{7}\]  
\[\alpha_{j}=\iint\exp(-\frac{r^{2}}{\omega_{a}{}^{2}})\sin[\phi(r,\theta)]r \text{d}r\text{d}\theta \tag{8}\]  
The Zernike polynomial is generally adopted to decompose the wavefront phase with distortion to a sum of weighted orthogonal polynomials, which represent various types of aberrations. Wavefront phase \(\phi\)(\(r\),\(\theta\)) can be expressed as follows:  
\[\phi(r,\theta)=\alpha_{0}+\sum_{i=1}^{\infty}\alpha_{i}Z_{i}(r,\theta) \tag{9}\]  
Figure 1: Scheme of spatial light coupling into a single-mode fiber (SMF). where \(Z_{i}(r,\theta)\) indicates the \(i\)th Zernike polynomial and \(a_{i}\) is the coefficient of polynomials. There are at least six different schemes available for Zernike polynomials.