## 4 Performance Evaluation of the Active Support System for the 4m SiC Primary Mirror
### Relationship between the Orders of Bending Modes and the Correction Effect  
For the 4m SiC primary mirror and the designed support system, there were 51 available bending modes. If only the first few orders of the bending modes were used  
Figure 14: Correction of gravity deformation of the primary mirror using the first 40 orders of bending modes at an elevation angle of 45\({}^{\circ}\). (**a**) Surface shape correction result (force on hardpoint not compensated); (**b**) Surface shape before correction (force on hardpoint compensated); (**c**) Surface shape correction result (force on hardpoint compensated).  
Figure 13: Schematic diagram of the deformation of the 4m SiC primary mirror surface due to gravity. (**a**) Elevation angle of 90\({}^{\circ}\); (**b**) Elevation angle of 45\({}^{\circ}\); (**c**) Elevation angle of 0\({}^{\circ}\).  
to calculate the correction force, the amplitude of the force could be reduced. To ensure a good accuracy was attained for the corrected primary mirror shape, the order of the bending modes used in the correction had to be determined. Therefore, we conducted a series of simulations at elevation angles of 90\({}^{\circ}\), 45\({}^{\circ}\), and 0\({}^{\circ}\), and the gravity deformation of the primary mirror was corrected using different orders of bending modes. The corrected primary mirror shape and the amplitude of the correction force are shown in Figure 15, where the horizontal axis represents the order of the bending modes used for the surface correction, the vertical axis on the left represents the rms of the corrected primary mirror shape, and the vertical axis on the right represents the amplitude of the correction force. Figure 15 shows that as the orders of the bending modes used increased, the rms of the primary mirror shape gradually decreased, and the amplitude of the correction force gradually increased. Therefore, if too many orders of bending modes was used, the amplitude of the correction force could not be reduced. If too few orders were used, the corrected surface shape accuracy would be poor. After comprehensive considerations, we believe that using the first 30 orders of bending modes for the surface shape correction was the most reasonable. This not only effectively reduced the amplitude of the correction force, but also ensured a good accuracy was achieved for the corrected primary mirror surface shape.  
The corrected surface shapes using the first 30 orders of bending modes are shown in Figure 16, and the relevant data before and after the surface shape correction are listed in Table 3. A comparison of Figures 13 and 16 revealed that the corrected primary mirror surface was significantly improved. According to Table 3, the rms values of the corrected surface shape under the three elevation angles were less than 16 nm, which met the requirement of being less than \(\lambda/30\) (\(\lambda\) = 632.8 nm). The above results indicated that using the first 30 orders of bending modes was sufficient to effectively correct the gravity deformation of a 4m SiC primary mirror.  
The Zernike polynomial was used to fit the surface shape before and after the correction, as shown in Figure 17. Due to space limitations, only the results at the elevation angle of 90\({}^{\circ}\) were given. The main components of the surface shape error before the correction were defocus and spherical aberration. The corrected surface shape significantly improved.