## 3 Active Optics Correction of the 4m SiC Primary Mirror
### Surface Shape Correction of the 4m SiC Primary Mirror  
#### 3.1.1 Principle of Surface Shape Correction Based on the Bending Mode  
A surface shape correction requires force actuators to output axial forces to the primary mirror. Therefore, the key to surface correction is the calculation of the correction force, which is based on measuring the surface error of the primary mirror. The designed active support system for the 4m SiC primary mirror used the bending-mode method to calculate the correction force. Figure 3: Finite element model of the active support system for the 4m SiC primary mirror. According to [PERSON]'s law, the relationship between the deformation of the primary mirror and the force is linear [20]. Therefore, if we could determine the surface deformation \(\mathbf{A}_{i}\) after applying the axial unit force to each support point and obtain the interaction matrix \(\mathbf{A}=[\mathbf{A}_{1},\mathbf{A}_{2},\ldots,\mathbf{A}_{54}]\), we could obtain the relationship between the force and deformation of the primary mirror, where \(i\) represents the serial number of the support point. \(\mathbf{A}_{i}\) is called the influence function of the unit force and is a column vector of length \(m\), where \(m\) is the number of sampling points for the primary mirror shape, requiring \(m>54\). For a set of forces \(\mathbf{F}\) applied to the support points, the resulting deformation of the primary mirror surface \(\mathbf{W}\) satisfied  
\[\mathbf{W}=\mathbf{A}\mathbf{F}=\sum_{i=1}^{54}\mathbf{A}_{i}F_{i} \tag{1}\]  
where \(F_{i}\) is the \(i\)-th element of vector \(\mathbf{F}\). The mapping relationship between the force acting on the primary mirror and the deformation of the mirror surface was established based on the interaction matrix \(\mathbf{A}\). The correction force could be determined using Equation (1). The above method is the direct least squares method. However, the correction force obtained using this method is often extremely high, because the method attempts to correct all low- and high-order components in the primary mirror shape error, and correcting higher-order aberrations would result in a large correction force, which may exceed the output force threshold of the force actuator and may also damage the primary mirror. In addition, the shape error of the primary mirror is often dominated by low-order aberrations, and correcting high-frequency aberrations results in a very small improvement in the shape. Therefore, the benefits of this method are limited. The fundamental flaw of this method is that the influence function \(\mathbf{A}_{i}\) cannot reflect the difficulty of the surface shape correction. Therefore, excluding high-frequency components in surface shape errors is challenging. The bending-mode method can improve this situation. According to the principle of the method, SVD was performed on the interaction matrix \(\mathbf{A}\):  
\[\mathbf{A}=\mathbf{U}\mathbf{S}\mathbf{V}^{T} \tag{2}\]  
Then, the bending mode \(\mathbf{B}_{i}\) and the bending-mode matrix \(\mathbf{B}\) were obtained as follows:  
\[\mathbf{B}= \begin{bmatrix}\mathbf{B}_{1}&\mathbf{B}_{2}&\cdots&\mathbf{B}_{54}\end{bmatrix} = \mathbf{U}=\mathbf{A}\mathbf{V}\mathbf{S}^{-1} \tag{3}\]  
where  
\[\mathbf{V}= \begin{bmatrix}\mathbf{V}_{1}&\mathbf{V}_{2}&\cdots&\mathbf{V}_{54}\end{bmatrix}  \tag{4}\]  
\[\mathbf{B}_{i}=\mathbf{A}\mathbf{V}_{i}\frac{1}{\mu_{i}} \tag{5}\]  
where \(\mu_{i}\) is the element in matrix \(\mathbf{S}\) and the singular value that affects matrix \(\mathbf{A}\). By introducing the bending mode \(\mathbf{B}_{i}\) into Equation (1), the required force \(\mathbf{F}_{Bi}\) for \(\mathbf{B}_{i}\) could be calculated as follows:  
\[\mathbf{W}=\mathbf{B}_{i}=\mathbf{A}\mathbf{V}_{i}\frac{1}{\mu_{i}}=\mathbf{A}\mathbf{F}_{Bi} \tag{6}\]  
\[\mathbf{F}_{Bi}=\mathbf{V}_{i}\frac{1}{\mu_{i}} \tag{7}\]  
where \(\mathbf{F}_{Bi}\), known as the bending-mode force, is the force required to generate the surface \(\mathbf{B}_{i}\).