## 2 Simulation Analysis of the Cubic FP Cavity with Reinforced Support
### Vibration Sensitivity Analysis  
Constraint settings for calculating vibration sensitivity of the FP cavity are shown in Figure 2. Two different constraints were taken to analyze vibration sensitivity of the FP cavity. Figure 2a shows the FP cavity constraint setup based on the analytical method reported in the literature [26] for four point-supported FP cavities. The bottom four polyimide shims were set to be fixed when the acceleration was in the -Z direction. Since the FP cavity needed to apply a preload force during assembly, the top four polyimide shims were set to applied force. Figure 2b shows another constraint method. Unlike the previous method, the planes of the Polyimide shims in contact with the FP cavity were set to applied force, and on the other sides of the Polyimide shims were set to fixed. Frictional contacts were set between the Polyimide shims and the FP cavity in the second method. The amount of deformation at the center of the FP cavity mirrors was used to calculate the vibration sensitivity. Both constraint methods were designed to simulate a realistic situation. For example, considering that the bottom four shims mainly play the role of fixed support when the acceleration is in the -Z direction, the first constraint method is adopted for the analysis. The second method takes into account the fixation and preload applied to the shims during assembly. Based on the first constraint setting method, vibration sensitivity of the FP cavity was calculated for different applied forces (50 N, 70 N, 100 N, and 150 N) in the X, Y, and Z directions. The results of the analysis are shown in Figure 3a. Vibration sensitivity of the FP cavity in the three directions is consistent under different applied forces. This demonstrates that the FP cavity is force insensitivity with the reinforced support. Meanwhile, the FP cavity presents vibration insensitivity in the Z direction. Vibration sensitivity of the FP cavity is consistent in the X, Y direction. With an applied force of 70 N, the vibration sensitivities in the Z, X, and Y directions are 0, \(-9.69\times 10^{-11}\)/g, and \(-9.57\times 10^{-11}\)/g, respectively. Figure 3b shows the results of vibration sensitivity analysis based on the second constraint setting method. With an applied force of 70 N, vibration sensitivities of the FP cavity in the Z, X, and Y directions are \(1.2\times 10^{-10}\)/g, \(-1.6\times 10^{-12}\)/g, and \(-1\times 10^{-12}\)/g, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Density** & **Young’s Modulus** & **Poisson’s Ratio** \\ \hline ULE & 2.21 g/cm\({}^{3}\) & 67.6 GPa & 0.17 \\ Fused silica & 2.20 g/cm\({}^{3}\) & 73 GPa & 0.16 \\ Polyimide & 1.45 g/cm\({}^{3}\) & 4 GPa & 0.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of materials in vibration sensitivity simulation. Figure 1: Models of the FP cavity support structure: (**a**) FP cavity structure with reinforced-support; (**b**) Removed structural bracket. In the X, Y direction, the FP cavity also presents force insensitivity. In the Z-direction, the second constraint method presents force insensitivity at 70 N, 100 N, and 150 N. Vibration sensitivity of the FP cavity changes by \(-2\times 10^{-11}\)/g when the applied force is 50 N.  
Regarding the two different constraint methods, FP cavity vibration sensitivity in both X and Y directions is below \(-1\times 10^{-10}\)/g and remains force insensitive in the range of 50-150 N. This indicates the consistency of the vibration sensitivity of the FP cavity in X, Y directions for both constraints setting methods. The two methods have different results in the Z direction. With the first constraint setting method, the FP cavity is vibration insensitive in the Z direction. Whereas, with the second constraint setting method, vibration sensitivity of the FP cavity is about \(10^{-10}\)/g magnitude. In addition, vibration sensitivity of the FP cavity is below \(1.5\times 10^{-10}\)/g in all three directions for both constraints setting methods.