## 4 Compensator Design and Simulation Verification
Figure 10b shows the variation of the radius of a circle with an encircled energy fraction higher than 0.9 with different wedge angles. The smaller the wedge angle during the adjustment process, the smaller the aberration generated, and the higher the energy concentration. This is because the OAR angle, which is included in the aberration coefficient, is the main factor for all plane symmetric aberrations. Reducing  
Figure 8: Schematic diagram of field of view stitching. Figure 7: Schematic diagram of mask template deformation. the surface tilt angle can effectively reduce the contribution of surface aberration. To avoid collisions during the rotation of the prism, a wedge angle of 4\({}^{\circ}\) is selected for the prism. The parameters of the compensation device are set according to the doubly telecentric projection optical system. Increasing the wedge angle will reduce the step size, but increases the system's aberration, and it is limited by resolution, which reduces the compensation effect. To meet the 5\({}^{\prime\prime}\) angle resolution requirement, the rotation angle is processed, and a curve is drawn. The schematic diagram in Figure 11 shows the adjustment of prism rotation angle with different positions of image plane tilt. Its maximum travel is within 0.6\({}^{\circ}\). Additionally, the telecentricity is 0.05 mrad, which meets the design specifications. Figure 11: Schematic diagram of adjusting the rotation angle of prism 2 with the tilt of the image plane at different positions. Figure 10: (**a**) Dual-Prism RMS WFE and Maximum RMS WFE in Initial State; (**b**) the radius of a circle with an encircled energy fraction higher than 0.9 varies with different wedge angles. Figure 9: Schematic diagram of optical system and compensation device. To meet the displacement resolution of 2 \(\upmu\)m, the thickness of the prism in the optical path is processed, and the curve is plotted. Figure 12 shows the schematic diagram of adjusting the distance of prism thickness variation in the optical path with different positions of image plane defocus. Its maximum travel is within 0.2 \(\upmu\)m. The sensitivity of adjusting the image plane is 28.125 nm/\(\upmu\)m, meaning that moving the prism diagonally by 1 \(\upmu\)m will change the focal plane by 28.125 nm. This low sensitivity and minimal machinery and control requirements make it very easy to achieve precise control of the focusing surface. The deformation after compensation using the dual-prism designed with the above parameters is shown in Figure 13. Table 2 shows the results of compensating for image plane deformation using different compensation methods. Among them, only defocusing compensation still results in a significant amount of deformation, with a maximum deformation of 20.087 \(\upmu\)m, which exceeds the depth of focus range of the doubly telecentric projection optical system. After applying both defocusing and tilt compensation, the maximum deformation is 3.011 \(\upmu\)m, which is within the depth of focus range of the doubly telecentric projection optical system, proving the necessity of tilt compensation. Under the condition of acceptable image quality, effective compensation for image plane deformation has been achieved. Due to the advanced and specialized technical nature of this research area and the limited quantitative data on the available compensation methods published for confidentiality reasons, relevant publicly available comparative data could not be found. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode of** & **Peak** & **Valley** & **RMS** \\
**Compensation** & & & \\ \hline None & 0 & \(-\)123 \(\upmu\)m & 75.264 \(\upmu\)m \\ Defocus & 20.087 \(\upmu\)m & \(-\)19.028 \(\upmu\)m & 6.700 \(\upmu\)m \\ Defocus and tilt & 3.011 \(\upmu\)m & 2.738 \(\upmu\)m & 0.668 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of compensation results. Figure 12: Schematic diagram of prism thickness transformation and image plane defocus adjustment at different positions.