## 3 Numerical Results and Analysis  
### Effects of Spatial Frequency Characteristics  
Equations (6)-(8) show that the CE of SMF depends on the matching between the incident beam wavefront and the SMF mode-field. Equation (9) is used to represent the incident beam wavefront. Figure 2a shows the pattern of first three-order Zernike aberrations and Figure 2b shows the relationship between these aberrations and the CE as analyzed by the proposed method, the SR approximation method, and the PIB method. The approximate formula for SR is [23]:  
\[\text{CE}_{\text{SRmethod}}\approx\exp(-\phi^{2})\times\text{CE}_{\text{max}} \tag{14}\]  
where \(\text{CE}_{\text{max}}\) is the max CE without wavefront aberration, which for the SMF is 81.45% usually; and the PIB method formula is [27]: \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**n** & **m** & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
0 & \(\mathbf{Z}_{1}\) & & & & & & & & & & circularly \\
1 & \(\mathbf{Z}_{4}\) & & & & & & & & & & & \\
2 & \(\mathbf{Z}_{11}\) & & & & & & & & & & & \\
3 & \(\mathbf{Z}_{21}\) & & & & & & & & & & & \\
4 & \(\mathbf{Z}_{11}\) & & & & & & & & & & & \\
5 & & \(\mathbf{Z}_{22}\) & & & & & & & & & & \\
6 & \(\mathbf{Z}_{22}\) & & & & & & & & & & & \\
7 & & & & & & & & & & & & \\
8 & \(\mathbf{Z}_{37}\) & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Noll–Zernike Polynomial Sequence. \(\varphi\) is the root-mean-square (RMS) of wavefront phase, D is the beam diameter, \(f\) is the focal length, \(\Phi\) is the phase structure function of incident wavefront, and \(l\) is the distance between any two points. Based on the theory discussed above, we next analyzed the influence of aberrations of different spatial frequencies on the CE under the condition that \(\lambda=1550\) nm, \(w_{0}=5.2\) um, beam diameter D = 1.2 m and relative aperture D/\(f=0.211\). Both SR and PIB methods can only calculate the relationship between the RMS value of the whole wavefront phase and the CE, so they do not reflect the influence of different aberration spatial frequencies on CE. By contrast, the proposed method reflects differences in different wavefront on CE even though the RMS of wavefront is the same; the differences are particularly obvious when the aberration amplitude is larger than 2 rad. The SR method has a high similarity with Z4 terms in the small aberration part. However, large errors occurred under the PIB method as it is mainly based on the calculation of geometrical optical information and generally used for the coupling calculation of few-mode and multi-mode optical fibers. We next further classified the aberrations, beginning with the circularly symmetric aberration (\(m=0\)) as shown in Figure 3. The overall trend of the curve for this item's aberration on CE is consistent. Before 1.5 rad, the values are almost identical; above 1.5 rad, various oscillations emerged. Figure 3: (**a**) Circularly symmetric aberration patterns; (**b**) CE with the circularly symmetric item. Figure 2: (**a**) Pattern of the first three-order Zernike aberrations; (**b**) coupling efficiency (CE) with the first three-order Zernike aberrations. Figure 4 shows the patterns and the relationship between the tilt-coma item (\(m=1\)) and CE. These aberrations have a similar effect on CE in that they all decay rapidly within 2 rad. When the aberration is greater than 2 rad, the CE is almost 0. The effects of \(m=\max\), the astigmatic aberration item, on CE are shown in Figure 5b. There is rapid CE attenuation in the early stage (within 0.5 rad) and gentle attenuation with continued increase in the RMS of aberration. In addition, for this kind of aberration, when the RMS value of aberration is the same, the higher the spatial frequency of aberration, the larger the corresponding CE.