## 4 Design Case  
Based on the research and analysis in this paper, a catadioptric optical system was designed with imaging and a motion-compensation function of the secondary mirror. The focal length of the catadioptric optical system was 450 mm, the spectral range was 435 nm\(\sim\)900 nm, the F number was 4.2, and the FOV of the optical system was 4.17\({}^{\circ}\times 3.13^{\circ}\). The specific parameters were shown in Table 1.  
The optical system was designed with a catadioptric structural form, which allowed for the off-axis complication of the secondary mirror. The two-mirror telescope system of the optical system was constructed in the form of an R-C structure [20; 21; 22; 23; 24; 25]. A correction lens was positioned between the secondary mirror and the focal plane, thereby increasing the field of view of the optical system. The TF3 special glass was employed for the purpose of correcting the secondary spectrum [26; 27; 28; 29]. Through the matching of optical materials and structural materials, the athermalization design of the optical imaging system was realized [30; 31; 32]. The optical system employed spectroscopic prisms to segregate panchromatic light into four distinct spectral bands, namely R, G, B and NIR, and then imaged them individually. The optical system adopted the multi-dimensional movement of the secondary mirror to achieve comprehensive image-motion compensation of the four spectral bands cameras. The final design results are shown in Figures 5 and 6.  
In this design case, the image-motion compensation range of \(\pm\)0.05\({}^{\circ}\) was selected based on the deflection angle allowed by the secondary mirror and the angular motion range of the secondary mirror driving mechanism. When the integration time of the optical imaging system was set to 1 ms, the secondary mirror was capable of compensating for the image motion caused by the scanning angular velocity of the aerial camera at 65\({}^{\circ}\)/s. The compensation accuracy and compensation speed of the secondary mirror were  
Figure 5: Schematic diagram of the optical system and spectroscopic prisms for the aerial camera.  
Figure 6: Schematic diagram of aerial camera structure.  
contingent upon the control accuracy and compensation bandwidth of the actuator. As shown in Figure 7, when the secondary mirror was deflected to its maximum angle, the coma of the optical system increased by 0.018\(\lambda\), the astigmatism increased by 0.06\(\lambda\), and the modulation transfer function decreased by 0.05. However, the modulation transfer function of the optical imaging system was better than 0.28. Although asymmetric astigmatism was introduced into the optical system, the MTF field distribution of the optical system remained symmetrical, which ensured a good aberration field distribution and enabled the achievement of excellent imaging.  
Figure 7: Effect of secondary mirror deflection angle on the aberration field: (**a**) coma aberration field with 0\({}^{\circ}\) deflection angle; (**b**) coma aberration field with \(\pm\)0.03\({}^{\circ}\) deflection angle; (**c**) coma aberration field with \(\pm\)0.05\({}^{\circ}\) deflection angle; (**d**) astigmatic aberration field with 0\({}^{\circ}\) deflection angle; (**e**) astigmatic aberration field with \(\pm\)0.03\({}^{\circ}\) deflection angle; (**f**) astigmatic aberration field with \(\pm\)0.05\({}^{\circ}\) deflection angle; (**g**) modulation transfer function with 0\({}^{\circ}\) deflection angle; (**h**) modulation transfer function with \(\pm\)0.03\({}^{\circ}\) deflection angle; and (**i**) modulation transfer function with \(\pm\)0.05\({}^{\circ}\) deflection angle.