## 3 Simulations  
A Gregorian optical system model with an optical system aperture of 1 m and a focal length of 8 m is established in optical design software, as depicted in Figure 2(a). The optical system aberration is illustrated in Figure 2(b), and its Zernike coefficients are presented in Table 1. This paper utilizes the image simulation function in optical design software to generate images of all small-aperture beams focused by the optical system. Through collaborative simulation between programming software and optical design software, interaction between the two is achieved using the Dynamic Link Library (DLL) pyzdde [20], programming software that plans the sampling positions of the beams and sends commands. Optical design software receives the commands, executes them, and outputs spot images corresponding to the sampled beams to a specified folder. Subsequently, programming software reads these spot images and calculates their centroids. In order to simulate the  
Figure 2: Iterative process flowchart. array pointing errors caused by scanning motion, different field-of-view angles are set for nine beams at various measurement positions of the array system, as illustrated in Figure 4. Additionally, multi-beam pointing errors are simulated by assigning different field-of-view angles to the beams with different numbers. Measured using an auto-collimating way in our lab, the array pointing error angle caused by scanning motion ranges from 6 to 9 arcseconds, and after a rough calibration, the aligned pointing error angle between the nine beams ranges from 8 to 12 arcseconds, which are random. The magnitudes of the array pointing error and multi-beam pointing error are plugged into the simulation. Additionally, the scanning way of the multi-beam array system is designed with three kinds of steps, as shown in Figure 4a-c, the scanning path of it for the full-aperture sampling using the multi-beam array system is illustrated in Figure 1b, and the distribution of all sampled points is shown in Figure 4d. Points sampled at the edge of the optical system aperture are considered effective sampled points when the energy of their beams entering the optical system reaches 80% of the total beam energy. When a higher wavefront measurement resolution is required, smaller step sizes can be considered to increase sampling density. However, as the step size decreases, the sampled data increase, which will cause a rapid decline in measurement efficiency. Therefore, it is essential to choose an appropriate step size based on the requirements of accuracy and efficiency. Due to the use of a significant number of random values in the simulation, the process is repeated 1000 times. The average of all results is taken as the final simulation result. When the array scans with different step sizes, the iterative method yields the following results in Figure 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Standard Zernike Terms** & **Coefficients** \\ \hline Z4 & \(-0.003\,\lambda\) \\ Z5 & \(0.0155\,\lambda\) \\ Z6 & \(-0.0152\,\lambda\) \\ Z7 & \(-0.0124\,\lambda\) \\ Z8 & \(0.0124\,\lambda\) \\ Z9 & \(0.00\,\lambda\) \\ Z10 & \(0.0122\,\lambda\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The standard Zernike coefficients for the aberration illustrated in Figure 3b. Figure 3: Schematic diagrams of (**a**) the simulation model of the optical system and (**b**) the aberrations in the simulation model of the optical system. Figure 4: Schematic diagrams of array scanning way with (**a**) non-overlapping, (**b**) partially overlapping (one column), (**c**) extensively overlapping (two columns), and (**d**) schematic diagrams of sampled points distribution. (The dashed line in figure represents the position of a multi-beam array system before the scanning movement, while the solid line represents the position after the movement. The red circles indicate beams undergoing repeated sampling.