# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 4 Simulation Verification
### Segmented Mirror Aberration Correction  
In Section 4.2, the co-phase maintenance accuracy of the two segmented mirror control modes was verified and compared. In this section, the WH technique is presented to correct the residual error of the mirror after the segmented mirror is in the co-phase state as per Scheme 1 presented in Section 4.2. WH technique is also used to correct the mirror error under the T condition. As in Section 4.2, Scheme 1 is set as an aberration correction mode with the segmented mirror aberration as the correction target. While Scheme 2 is the correction of the segment's aberration followed by the co-phase maintenance of the  
Figure 5: Zerodur segment segmented mirror surface after correction by two schemes, (**a**) Scheme 1, deformation of mirror surface in Y condition, (**b**) Scheme 1, deformation of mirror surface in Z condition, (**c**) Scheme 2, deformation of mirror surface in Y condition, (**d**) Scheme 2, deformation of mirror surface in Z condition. segmented mirror by using the conventional co-phase retention technique as mentioned in the literature [11; 12; 13; 14]. In Scheme 1, the WH stiffness matrix \(E\) is obtained by applying a unit force to the end of the WH spring leaf in the segment system and fitting it according to Equation (1). For the residual aberrations of the segmented mirror surface after the correction done by Scheme 1 presented in Section 4.2, \(Y\) and \(Z\) conditions are set as \(U_{DY}^{(0)}\) and \(U_{DY}^{(0)}\). Moreover, the aberrations of the segmented mirror surface in \(T\) condition are set as \(U_{DT}^{(0)}\). Similar to the correction process discussed in Section 4.2, the aberrations of the segmented mirror surface after the first correction are calculated according to Equation (27) and the algorithm in Section 3. It is done with the solution vector dimension \(M=12\times 7+1=85\), the number of differential evolutionary populations of 850, the evolutionary algebra \(G=50\), the crossover probability \(CR=0.8\), and the actuator correction forces \(f_{Y}{}^{(0)}\), \(f_{Z}{}^{(0)}\) and \(f_{T}{}^{(0)}\). Finally, these values are substituted into the model. The residuals of \(U_{DY}^{(1)}\), \(U_{DY}^{(1)}\) and \(U_{DT}^{(1)}\) are calculated. It is assumed that the change in the correction forces \(\Delta f=f^{(t)}-f^{(t-1)}\) is smaller than \(\xi=10^{-6}\)N after \(t\) cycles, and the correction is cut off. In Scheme 2, firstly, the correction force and segment surface aberration stiffness matrix of the segment system are established. Then, each segment surface shape error is analyzed under Y, Z and T working conditions. The sub-aperture aberration coefficient matrix is fitted according to the Fringe Zernike polynomial \(Z_{Y_{1}}^{(0)},Z_{Z_{1}}^{(0)},Z_{T_{1}}^{(0)}\). Where Y, Z and T in the subscript represent the Y, Z and T conditions. \(i\) represents the segment code as shown in Figure 1. According to Equation (27) and the algorithm in Section 3, this is done with the solution vector dimension \(M=13\), the number of differential evolutionary populations of 130, the evolutionary algebra \(G=50\), and the crossover probability \(CR=0.8\), when \(\Delta f=f^{(t)}-f^{(t-1)}\) is smaller than the accuracy of the finite element software solution \(\xi=10^{-6}\)N, the segment aberration correction is finished. Then the segmented mirror co-phase maintenance is carried out according to Scheme 1 in Section 4.2. The comparison of residual deformation of Scheme 1 and Scheme 2 in this section is shown in the following figure. It can be seen that the two schemes adopted in this section are from Figure 6, which have improved the face shape accuracy to different degrees after mirror correction. Among these, the correction effect of Scheme 1 is the most obvious, with the RMS value decreasing from 149.3 nm to 2.2 nm in the Z condition. This indicates that Scheme 1 can effectively correct the segmented mirror error introduced by the rigid body displacement along the X and Y directions of the segment and the residual aberration of the segment together.