## 3 Active Optics Correction of the 4m SiC Primary Mirror
### Surface Shape Correction of the 4m SiC Primary Mirror
#### 3.1.2 Influence Function and Bending Mode of the 4m SiC Primary Mirror  
The purpose of this section was to obtain the influence function, interaction matrix, and bending mode of the 4m SiC primary mirror based on the theory described in Section 3.1.1. To obtain the influence function \(\mathbf{A}_{i}\) of the primary mirror, we used the equilibrium force \(\mathbf{F}_{Ai}\) approximated using the unit force. The characteristic of \(\mathbf{F}_{Ai}\) was that the force at the \(i\)-th support point was large, whereas that at the other support points was very small. The effect of \(\mathbf{F}_{Ai}\) was similar to that of the unit force, except that the resultant force and moment of \(\mathbf{F}_{Ai}\) on the primary mirror were zero, avoiding the trend of a rigid displacement of the primary mirror. The 54 actuators were distributed across four concentric circles. Applying \(\mathbf{F}_{Ai}\) sequentially to the primary mirror resulted in 54 influence functions \(\mathbf{A}_{i}\). The influence functions and equilibrium forces are shown in Figure 4 (because of space limitations, only the influence functions corresponding to two actuators on each concentric circle were displayed; the surface shapes that were not provided were similar, owing to the symmetry of the structure). Based on the 54 influence functions \(\mathbf{A}_{i}\), we obtained the 4m SiC primary mirror interaction matrix \(\mathbf{A}\). Then, SVD was performed on \(\mathbf{A}\) to obtain the 54 bending modes \(\mathbf{B}_{i}\) of the primary mirror (only the first 51 of the 54 modes were valid). As mentioned in Section 3.1.1, the rms values of all \(\mathbf{B}_{i}\) were equal, and we further normalized them to 500 nm. Each \(\mathbf{B}_{i}\) corresponded to a bending-mode force \(\mathbf{F}_{Bi}\), as shown in Figure 5. The \(\mathbf{B}_{i}\) in Figure 5 eliminated the tilt, and owing to space limitations, only the odd-order bending modes and forces in the first 20 orders were displayed. The bending modes in Figure 5 were derived from SVD, so they were the theoretical values of \(\mathbf{B}_{i}\). Figure 4: Influence function \(\mathbf{A}_{i}\) and equilibrium force \(\mathbf{F}_{Ai}\) of some pneumatic actuators. (**a**) Influence function \(\mathbf{A}_{1}\) and equilibrium force \(\mathbf{F}_{A1}\); (**b**) Influence function \(\mathbf{A}_{5}\) and equilibrium force \(\mathbf{F}_{A5}\); (**c**) Influence function \(\mathbf{A}_{7}\) and equilibrium force \(\mathbf{F}_{A7}\); (**d**) Influence function \(\mathbf{A}_{16}\) and equilibrium force \(\mathbf{F}_{A16}\); (**e**) Influence function \(\mathbf{A}_{19}\) and equilibrium force \(\mathbf{F}_{A19}\); (**f**) Influence function \(\mathbf{A}_{28}\) and equilibrium force \(\mathbf{F}_{A28}\); (**g**) Influence function \(\mathbf{A}_{31}\) and equilibrium force \(\mathbf{F}_{A31}\); (**h**) Influence function \(\mathbf{A}_{50}\) and equilibrium force \(\mathbf{F}_{A50}\). Figure 5: Theoretical values of partial bending mode \(\mathbf{B}_{i}\) of the 4m SiC primary mirror and corresponding bending-mode forces \(\mathbf{F}_{Bi}\). (**a**) Bending mode \(\mathbf{B}_{1}\) and bending-mode force \(\mathbf{F}_{B1}\); (**b**) Bending mode \(\mathbf{B}_{3}\) and bending-mode force \(\mathbf{F}_{B3}\); (**c**) Bending mode \(\mathbf{B}_{5}\) and bending-mode force \(\mathbf{F}_{B5}\); (**d**) Bending mode \(\mathbf{B}_{7}\) and bending-mode force \(\mathbf{F}_{B7}\); (**e**) Bending mode \(\mathbf{B}_{9}\) and bending-mode force \(\mathbf{F}_{B9}\); (**f**) Bending mode \(\mathbf{B}_{11}\) and bending-mode force \(\mathbf{F}_{B11}\); (**g**) Bending mode \(\mathbf{B}_{13}\) and bending-mode force \(\mathbf{F}_{B13}\); (**h**) Bending mode \(\mathbf{B}_{15}\) and bending-mode force \(\mathbf{F}_{B15}\); (**i**) Bending mode \(\mathbf{B}_{17}\) and bending-mode force \(\mathbf{F}_{B17}\); (**j**) Bending mode \(\mathbf{B}_{19}\) and bending-mode force \(\mathbf{F}_{B19}\). Figure 5 shows that the higher the order, the higher the spatial frequency of the bending mode.