## 2 Methods  
The principle of MASTS is illustrated in Figure 1a. The parallel beams with a small aperture emitted from the multi-beam array system pass through the optical system to be tested. The CCD camera, positioned at the focal plane of the system, captures the formed spots when the beams are focused by the optical system, which consists of a primary mirror and a secondary mirror. The wavefront slope information of the measured sub-aperture in the optical system can be calculated from the centroid coordinates of the focal spots. The multi-beam array system integrates 3 \(\times\) 3 small-aperture collimator tubes that are maintained in the same direction to make their output beam parallel. The multi-beam array then scans through the entire aperture of the optical system under test to obtain its slopes' information. The full-aperture aberration of the optical system is subsequently reconstructed using a wavefront reconstruction algorithm [17; 18; 19]. One of the scanning paths is shown in Figure 1b, where the numbers indicate the identification of each sub-beam in the multi-beam array system, and the arrows indicate the scanning path of it. During actual measurements, the posture of the multi-beam array system is not stable as it moves along the scanning path. This instability can lead the changes in the orientation of the multi-beam array after movement. Furthermore, aligning the individual sub-beams in the multi-beam array system strictly along the same orientation is not feasible. In this paper, using beam No. 5 as a reference, the centroid coordinate of the spot formed after the focusing of a certain beam by the optical system can be expressed as  
\[ \begin{cases}C_{j,x}=D_{j,x}+\Delta_{x}+E_{j,x}\\ C_{j,y}=D_{j,y}+\Delta_{y}+E_{j,y}\end{cases} \text{, where} \begin{cases}E_{5,x}=0\\ E_{5,y}=0\end{cases}  \tag{1}\]  
where the subscripts \(j\), \(x\), and \(y\) represent the sub-beam number in the array and the corresponding coordinates of the centroid in the x and y directions, respectively. The symbol 'C', which are the actual test data, represents the centroid coordinates of the spot. The symbol 'D', which represents the sub-aperture wavefront slope information, denotes the centroid offset component of the beam caused by the sub-aperture aberration of the tested system. The symbol '\(\Delta\)' represents the centroid offset component caused by the scanning motion-induced array pointing error. The symbol '\(E\)' is the centroid offset component caused by multi-beam pointing errors. After neglecting the '\(E\)', both sides of Equation (1) are divided by the focal length \(f\) of the optical system, and the slope expression can be written as  
\[ \begin{cases}M_{x}=S_{x}+T_{x}\\ M_{y}=S_{y}+T_{y}\end{cases}  \tag{2}\]  
In Equation (2), \(M\), \(S\), and \(T\) correspond to the slopes of 'C', 'D', and '\(\Delta\)' in Equation (1), respectively. Slope \(S\) can be expressed using Zernike polynomials:  
\[ \begin{cases}S_{x}=c_{1}\frac{\partial Z_{1}(x,y)}{\partial x}+c_{2}\frac{ \partial Z_{2}(x,y)}{\partial x}+\ldots+c_{m}\frac{\partial Z_{m}(x,y)}{ \partial y}\\ S_{y}=c_{1}\frac{\partial Z_{1}(x,y)}{\partial y}+c_{2}\frac{\partial Z_{2}(x, y)}{\partial y}+\ldots+c_{m}\frac{\partial Z_{m}(x,y)}{\partial y}\end{cases}  \tag{3}\]  
Solving Equations (2) and (3) simultaneously, we obtain  
\[ \begin{cases}M_{x}=c_{1}\frac{\partial Z_{1}(x,y)}{\partial x}+c_{2}\frac{ \partial Z_{2}(x,y)}{\partial x}+\ldots+c_{m}\frac{\partial Z_{m}(x,y)}{ \partial y}+T_{x}\\ M_{y}=c_{1}\frac{\partial Z_{1}(x,y)}{\partial y}+c_{2}\frac{\partial Z_{2}(x, y)}{\partial y}+\ldots+c_{m}\frac{\partial Z_{m}(x,y)}{\partial y}+T_{y} \end{cases}  \tag{4}\]  
Figure 1: Schematic diagrams of (**a**) multi-beam array measuring optical system and (**b**) the path covering the entire aperture in a multi-beam array scanning optical system. The measurement data obtained at each sampling point on the optical system for each small beam are represented by Equation (4).