## 2 Principles and Methods
### Optomechanical Coupling Active Control System of Spatial Filter  
The spatial filter is a typical system of PW laser facility, as shown in Figure 4. The spectral bandwidth of PW laser is 3 nm, and the central wavelength is 1053 nm. The spatial filter transmission system is composed of (Obj.), front lens (L\({}_{i1}\)), filter pinhole (Filter  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Units** \\ \hline m & 100 & kg \\ c & 120 & N/m s\({}^{-1}\) \\ k & 40,000 & N/m \\ K\({}_{\rm F}\) & 28 & N/A \\ L & 0.0053 & H \\ R & 8.6 & \(\Omega\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters table of Optomechanical coupling active control system.  
Figure 3: Amplitude frequency and phase frequency response curves of Optomechanical structure.  
Figure 2: Triaxial vibration response signal of spatial filter support truss.  
Pinhole), rear lens (L\({}_{i,2}\)), and image plane (Img.). The object plane reflects the incident light information, which is related to the output light of the rear lens (L\({}_{i-1,2}\)) of the former spatial filter. And the image plane reflects the output light information, which is related to the input light of the front lens (L\({}_{i+1,1}\)) of the latter spatial filter. The spatial filter support truss is simplified as a mass stiffness damping single degree of freedom translational model (m\({}_{2}\)-k\({}_{2}\)-c\({}_{2}\) and m\({}_{3}\)-k\({}_{3}\)-c\({}_{3}\)), and the active control force (\(f_{a}\) and \(f_{b}\)) is applied on the two-stage lens support truss of the spatial filter transmission system to construct the coupling relationship between the Optomechanical structure response and the active control parameters, and the coupling relationship between the Optomechanical structure response and the beam pointing, The relationship between active control parameters and beam pointing is established by two-stage coupling system to improve the output stability of beam propagation spatial filter.  
In order to study the beam pointing stability of the Optomechanical coupling system of the spatial filter, the model of the active control system is constructed with the spatial filter whose beam expansion ratio is \(m\) in the PW laser device as the object, as shown in Figure 5. The incident beam passes through the thin confocal lens group \(L_{1}\) and \(L_{2}\) in the spatial filter in turn. According to the modal theory, the \(x\) direction first-order translational mode of the support truss is simplified as a single degree of freedom stiffness-mass-damping model. The spatial filter is rigidly connected with the support truss, and the sensor is installed on the support truss of the later lens by using the active feedback control method. The measured offset response is used as the reference of the two support truss systems. By actively controlling the equivalent stiffness and equivalent damping of the two lens support trusses, the difference of the offset response of the two lenses in the \(x\) direction is controlled, thus improving the convergence accuracy of the output light angle response.  
In this paper, an active control scheme of the Optomechanical system is designed. The scheme is that the two supporting trusses are the same. The supporting trusses of the front and rear lens adopt active control, and the active control law of the two trusses is the same.  
Figure 4: The typical spatial filter system of PW laser facility.