## 2 Principles and Methods
### Dynamics Derivation of Optomechanical Coupling Active Control System  
The motion differential equation of the active control system is as follows  
\[\left\{ \begin{array}{l}m_{2}\ddot{x}_{2}+c_{2}\dot{x}_{2}+k_{2}x_{2}=c_{2} \dot{x}_{20}+k_{2}x_{20}+L^{-1}(f_{a})\\ m_{3}\ddot{x}_{3}+c_{3}\dot{x}_{3}+k_{3}x_{3}=c_{3}\dot{x}_{30}+k_{3}x_{30}+L^{- 1}(f_{d})\end{array} \right. \tag{7}\]  
\[\left\{ \begin{array}{l}f_{a}=-\frac{K_{F}}{Ls+R}(g_{I}+gps)X_{2}\\ f_{d}=-\frac{K_{F}}{Ls+R}(g_{I}+g_{PS})X_{3}\end{array} \right. \tag{8}\]  
Among them, \(m_{2}\) and \(m_{3}\) are the modal mass, \(c_{2}\) and \(c_{3}\) are the modal damping, \(k_{2}\) and \(k_{3}\) are the modal stiffness, \(f_{a}\) and \(f_{d}\) are the control forces, \(x_{20}\) and \(x_{30}\) are the foundation disturbance, \(x_{2}\) and \(x_{3}\) are the lens offset. Since the two supporting trusses are identical, it is noted that \(m=m_{2}=m_{3}\), \(c=c_{2}=c_{3}\), \(k=k_{2}=k_{3}\). The active control force is shown in formula (8), and the pull transformation is carried out by substituting formula (8) into formula (7). \[G(s)=G_{2}(s)=G_{3}(s)=\frac{cs+k}{ms^{2}+cs+k-\frac{K_{F}}{Ls+R}(g_{I}+gps)} \tag{9}\]  
Taking periodic signal as disturbance source, the excitation of two supporting trusses can be expressed as  
\[\left\{ \begin{array}{l}x_{10}=A_{1}\sin(\omega_{1}t+\phi_{1})\\ x_{20}=A_{2}\sin(\omega_{2}t+\phi_{2})\end{array} \right. \tag{10}\]  
When the excitation is applied to the Optomechanical coupling active control system, the offset of the two lenses is as follows  
\[\left\{ \begin{array}{l}a_{1}=\frac{cs+k}{ms^{2}+cs+k-\frac{K_{F}}{Ls}(g_{I} +gps)}A_{1}\\ a_{2}=\frac{cs+k}{ms^{2}+cs+k-\frac{K_{F}}{Ls}(g_{I}+gps)}A_{2}\end{array} \right. \tag{11}\]\(\left( \begin{array}{c}x_{1}\\ \theta_{1}\end{array} \right)\) is the beam incident into the spatial filter, and \(\left( \begin{array}{c}x_{2}\\ \theta_{2}\end{array} \right)\) is the beam \(\left( \begin{array}{c}x_{1}\\ \theta_{1}\end{array} \right)\) the incident light is transmitted to the next spatial filter through the spatial filter of the Optomechanical coupling active control system, as shown in Figure 7.  
According to the beam propagation principle [30], considering one of the beams, the outgoing light can be expressed as  
\[\left( \begin{array}{c}x_{2}\\ \theta_{2}\\ 1\\ 1\end{array} \right)=\left( \begin{array}{cccc}1-\frac{l_{12}}{f_{1}}&l_{12}& \frac{l_{12}a_{1}(f)}{f_{1}}&0\\ -(\frac{1}{f_{1}}+\frac{1}{f_{2}}-\frac{l_{12}}{f_{1}f_{2}})&1-\frac{l_{12}}{f_ {2}}&\frac{a_{1}(f)}{f_{1}}+\frac{a_{2}(f)}{f_{2}}-\frac{l_{12}a_{1}(f)}{f_{1} /2}&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right)\left( \begin{array}{c}x_{1}\\ \theta_{1}\\ 1\\ 1\end{array} \right) \tag{12}\]  
\[\left\{ \begin{array}{c}l_{12}=f_{1}+f_{2}\\ f_{2}=mf_{1}\end{array} \right. \tag{13}\]  
where \(a_{1}(t)\) and \(a_{2}(t)\) are the displacement responses of two lenses \(L_{1}\) and \(L_{2}\) in the \(x\) direction, respectively. \(f_{1}\) and \(f_{2}\) are the focal lengths of lenses \(L_{1}\) and \(L_{2}\), which \(l_{12}\) are the distance from \(L_{1}\) and \(L_{2}\). Under ideal conditions, the beam will shoot out parallel to the optical axis; the microvibration of different phases will make the optical axis deviate from the ideal position, and the optical system will be out of alignment; the active control parameters are introduced to modify the beam transmission matrix to reduce the misalignment degree of the optical system and improve the convergence accuracy of the beam direction. Here, considering the change of the angle of the outgoing light of the lens, the following formulas are obtained. \[\theta_{2}=-\frac{1}{m}\theta_{1}-\frac{1}{mf_{1}}a_{1}(t)+\frac{1}{mf_{1}}a_{ 2}(t)=-\frac{1}{m}\theta_{1}+\frac{1}{mf_{1}}[a_{2}(t)-a_{1}(t)] \tag{14}\]  
It can be seen from Equation (8) that the angle of light emitted from the transmission system of the spatial filter is the back-and-forth vibration of the angle of \(-\frac{1}{m}\theta_{1}\) center. The angle information of outgoing light consists of two parts.