## 3 Simulations
In (**d**), the green circle represents the optical system aperture (1 m), and the blue circle with red dots indicates the effective sampled points). Figure 5: _Cont_. From Figure 5, it can be seen that the RMS values of the residual aberration obtained using the iterative method are all less than one-thousandth of a wavelength after the reconstructed wavefront is subtracted from the known aberration distribution. The RMS value shown in Figure 5c approaches one-ten-thousandth of a wavelength, which demonstrates the accuracy of the iterative method in simulation results. As the array scanning step size decreases, the aberration obtained through the iterative method becomes closer to the known aberration. Stated differently, reducing the step size enhances the accuracy of the iterative method and reduces the number of iterations needed for convergence. This is attributed to the fact that a smaller scanning step size brings adjacent measurement positions of the multi-beam array closer, resulting in a denser sampling of beams. As a consequence, more data are collected, and a larger dataset implies more accurate fitting results for Equations (5) and (6). Additionally, the averaging method proposed in reference [16] assumes the wavefront in the optical system is uniformly sampled by an individual beam. It calculates the average of a large amount of multi-beam pointing error data based on statistical principles to eliminate sub-aperture aberration signals. But, a larger reconstruction error will be induced. The residuals after subtracting the reconstructed wavefront distribution using the averaging method from the preset aberrations are shown in Figure 6.  
Under the same conditions, the residuals of the iterative method are significantly smaller than those of the averaging method, as shown in Figure 6. Figure 7 illustrates the comparison among the ground truth of aberration, the result using the iterative method, and the result using the averaging method with the coefficients of Zernike polynomial. Figure 5: In the first row, the reconstructed wavefront through the iterative method when the scanning step (**a**) refers to Figure 4a, (**b**) refers to Figure 4b, and (**c**) refers to Figure 4c, respectively. In the second row, the residuals after subtracting the reconstructed wavefront distribution from the preset aberrations when the scanning step (**d**) refers to Figure 4a, (**e**) refers to Figure 4b, and (**f**) refers to Figure 4c, respectively. In the third row, the curves depicting the convergence of aberration PV values and RMS values of the difference between the preset and the reconstructed when the scanning step (**g**) refers to Figure 4a, (**h**) refers to Figure 4b, and (**i**) refers to Figure 4c, respectively. Figure 6: Distribution plots obtained by subtracting the preset aberrations from the reconstructed wavefront using the averaging method when the scanning step (**a**) refers to Figure 4a, (**b**) refers to Figure 4b, and (**c**) refers to Figure 4c, respectively. Figure 7: _Cont_. From Figure 7, it can be seen that the Zernike coefficients of the reconstructed wavefront using the two methods exhibit errors in the lower-order terms, particularly evident in the fifth and sixth terms (astigmatism). Conversely, the high-order Zernike coefficients show a similar trend. This corresponds well with the extensive simulation results we conducted. In summary, with different scanning step sizes, the results obtained through the iterative method exhibit a clear advantage over those obtained through the averaging method.