## 3 Image-Motion Compensation Technique of the Secondary Mirror Motion
### Aberration Field Constant Technique
#### 3.2.2 Secondary Mirror Motion Induces Third-Order Aberrations Effects  
Since the third-order spherical aberration is not affected by the size of the FOV, when the motion of the secondary mirror does not change the spacing of the primary mirror and secondary mirror, it does not affect the balance of the third-order spherical aberration of the optical system. The influence of the secondary mirror off-axis on the optical system is mainly manifested in the third-order coma and the third-order astigmatism. The third-order coma is linearly related to the size of the FOV and has only one coma node, and the third-order astigmatism is squared with the size of the FOV and has two astigmatic nodes [16; 17; 18].  
According to the vector aberration theory, the third-order coma of the misaligned R-C system can be calculated according to Equations (11) and (12).  
\[W_{COMA3}=\begin{bmatrix}\overset{\rightarrow}{-A_{131,x}}\\ \overset{\rightarrow}{-A_{131,y}}\\ \overset{\rightarrow}{A_{131,x}}\\ \overset{\rightarrow}{A_{131,y}}\\ \end{bmatrix}=\begin{bmatrix}\overset{\rightarrow}{\sigma}_{SM,x}^{( sph)}&\overset{\rightarrow}{\sigma}_{SM,x}^{(soph)}\\ \overset{\rightarrow}{\sigma}_{SM,y}^{(sph)}&\overset{\rightarrow}{\sigma}_{ SM,y}^{(sph)}\\ \end{bmatrix}\begin{bmatrix}W_{131,SM}^{sph}\\ W_{131,SM}^{sph}\\ W_{131,SM}^{asph}\\ \end{bmatrix}, \tag{12}\]  
where \(W_{COMA3}\) is the third-order coma, \(\overset{\rightarrow}{A_{131,x}}\overset{\rightarrow}{A_{131,y}}\) is the constant coma of the field of view, \(\overset{\rightarrow}{\rho}\) is the pupil vector, \(W_{131}\) is the coefficient of the third-order coma, \(\overset{\rightarrow}{\sigma}_{SM,x}^{(sph)}\) and \(\overset{\rightarrow}{\sigma}_{SM,y}^{(sph)}\) is the normalized spherical aberration eccentricity vector, and \(\overset{\rightarrow}{\sigma}_{SM,x}^{(asph)}\)\(\overset{\rightarrow}{\sigma}_{SM,y}^{(asph)}\) is the normalized aspheric aberration eccentricity vector.  
Applying the same method, the third-order astigmatism of the misaligned R-C system is given by the following equation:  
\[W_{AT53}=\frac{1}{2}\begin{bmatrix}W_{222}\overset{\rightarrow}{H}^{2}-2 \overset{\rightarrow}{H}\overset{\rightarrow}{A_{222}}+\overset{ \rightarrow}{B_{222}}\\ \end{bmatrix}(\overset{\rightarrow}{\rho}^{2}), \tag{13}\]  
where \(W_{AT53}\) is the third-order astigmatism, \(\overset{\rightarrow}{H}\) is the FOV vector, and \(W_{222}\) is the coefficient of the third-order astigmatism.  
Define \(\overset{\rightarrow}{A_{222}}\) and \(\overset{\rightarrow}{B_{222}}\) as follows:  
\[\begin{bmatrix}\overset{\rightarrow}{A_{222,x}}\\ \overset{\rightarrow}{A_{222,y}}\\ \overset{\rightarrow}{A_{222,x}}\\ \overset{\rightarrow}{B_{222,x}}\\ \overset{\rightarrow}{\rho}_{SM,x}^{(sph)}\\ \overset{\rightarrow}{\sigma}_{SM,y}^{(sph)2}-\overset{\rightarrow}{\sigma} _{SM,y}^{(asph)2}-\overset{\rightarrow}{\sigma}_{SM,x}^{(asph)2}\\ \overset{\rightarrow}{2\sigma}_{SM,x}^{(sph)}\overset{\rightarrow}{\sigma} _{SM,y}^{(asph)}\\ \end{bmatrix}\begin{bmatrix}W_{222,SM}^{sph}\\ W_{222,SM}^{asph}\\ W_{222,SM}^{asph}\\ \end{bmatrix}, \tag{14}\]  
The third-order astigmatism induced by the secondary mirror off-axis exhibits a two-nodes phenomenon [19], and the position of the astigmatic nodes \(\overset{\rightarrow}{H}_{ATS3}\) in the FOV can be determined using Equation (15).  
\[\overset{\rightarrow}{H}_{ATS3}=\frac{\overset{\rightarrow}{A_{222}}}{W_{2 22}}\pm i\sqrt{\frac{\overset{\rightarrow}{B_{222}}}{W_{222}}-(\frac{ \overset{\rightarrow}{A_{222}}}{W_{222}})}^{2}, \tag{15}\]where \(\frac{\overset{\rightarrow}{A}_{222}}{W_{222}}\) is a vector from the intersection of the optical axis and the image plane to the midpoint of the line between the two astigmatic nodes, and \(\pm i\sqrt{\frac{\overset{\rightarrow}{B}_{222}}{W_{222}}-\left(\frac{ \overset{\rightarrow}{A}_{222}}{W_{222}}\right)^{2}}\) are two vectors that point from the end point of the \(\frac{\overset{\rightarrow}{A}_{222}}{W_{222}}\) vector to the two astigmatic nodes.