## 3 Numerical Results and Analysis
### Influence of AO System Residuals  
We used the Zernike polynomial fitting method to generate wavefront simulation data according to the statistical characteristics of atmospheric turbulence and explore the effects of AO system residual aberrations on CE. \(Ia\) is the covariance matrix of Zernike coefficient vector \(a\) for simulating the atmospheric wavefront [32; 33]:  
\[\Gamma a=\overline{a_{i}a_{j}}=\left(\frac{D}{r_{0}}\right)^{5/3}\times\frac{ 2.2698(-1)^{(n_{i}+n_{j}-2m_{i})/2}[(n_{i}+1)(n_{j}+1)]^{1/2}\Gamma((n_{i}+n_ {j}-5/3)/2)}{\Gamma((n_{i}-n_{j}+17/3)/2)\Gamma((n_{j}-n_{i}-17/3)/2)\Gamma((n_ {i}+n_{j}+23/3)/2)} \tag{16}\]  
Figure 4: (**a**) Tilt-coma aberration patterns; (**b**) CE with tilt-coma item.  
Figure 5: (**a**) Astigmatic aberration patterns; (**b**) CE with an astigmatic item.  
We generated atmospheric turbulence simulation wavefront data with normalized atmospheric turbulence strength D/\(r_{0}\) ranging from 0.5 to 15, where \(r_{0}\) is the Fried parameter. We also analyzed the CE of different correction residuals of a FSOC AO system and the correction residuals margin of the AO system. The parameters in the simulation were D = 1.2 m, \(\lambda\) = 1550 nm, \(\omega_{0}\) = 5.2 \(\upmu\)m, D/\(f\) = 0.204 and \(\varepsilon\) = 0.2 (center occlusion ratio for the Kassai Green telescope).  
The typical AO system uses a fast steering mirror (FSM) to correct the tilt aberration and a deformable mirror (DM) to correct the aberrations expect tilt separately, so we analyzed the relationship between CE and residual aberrations (tip/tilt removed) under different tilt bias conditions. For convenience, the tilt amplitude is given here in arc-seconds while the remaining aberrations are given in the form of the overall RMS. The results are shown in Figure 6.  
Figure 7 shows the processed results of all simulation data under different coupling thresholds (\(\eta_{\text{th}}\)), encompassing the range of aberrations that reach the CE threshold under certain tilt bias conditions. The margin of the AO system residuals differ across these conditions. As the tilt residual increases, the margin of residuals except the tilt decreases; that is, the residual of the FSM increases, which reduces the tolerance of the system for the DM correction residual.  
Figure 6: Relation between CE and root-mean-square (RMS) of residual wavefront with different tilt biases. (**a**) Tilt = 0.025 arc-second; (**b**) tilt = 0.05 arc-second; (**c**) tilt = 0.075 arc-second; (**d**) tilt = 0.1 arc-second; (**e**) tilt = 0.125 arc-second; (**f**) tilt = 0.15 arc-second.  
Figure 7: _Cont._  
Figure 8 shows the residual control range of the AO system under different coupling thresholds. The abscissa is the tilt magnitude, which is the FSM residual. The ordinate is the RMS of residual wavefront without tilt, i.e., the DM residual. If the coupling threshold is 30%, then the FSM and DM residuals should fall within the range enclosed by the orange curve and the coordinate axis. Our results suggest that when designing the AO system for FSOC, the correction residuals of FSM and DM should be reasonably allocated under the coupling threshold according to the system requirements.