## 3 Results and Analysis of Multi-Physics Coupling  
### Displacement Distribution of FP Cavity under the Fixing Force  
Taking into account the thermal effect of the laser beam in optical and thermal coupling, the magnitude of the beam power \(P_{0}\) in Equation (3) was set to \(10^{-4}\) Watt. In thermal and mechanical coupling, the heat radiation to the environment (default 20 \({}^{\circ}\)C) and heat conductivity were taken into consideration. In addition, the temperature fluctuation of the vacuum chamber was assumed as \(10^{-3}\) Kelvin. Then, the finite element model was assigned materials, meshed, and imposed constraints in a solid mechanical analysis. A fixed force of 200 N was applied to each of the four truncated symmetrical vertices. Finally, a virtual prototyping multi-physics coupling model was established and the results were calculated by FEM.  
The displacement and Von Mises stress distribution of the FP cavity under the fixing force is shown in Figure 2, with the color map scaled logarithmically for better visualization. It is evident that the displacement and stress amplitude are higher at the four supporting vertices, which have symmetrical distributions as a result of the given fixing strategy. The performance index that should receive the most emphasis is the cavity length change, that is, the displacement of the cavity along the \(Z\) axis. Therefore, the \(Z\) component displacement of the plane mirror, the concave mirror, and the cavity length change, i.e., the sum of the above two, are shown in Figure 3. It can be seen that the displacement of the two mirrors is symmetrical about the line of \(45^{\circ}\) and \(-45^{\circ}\), respectively. In order to further study the variation characteristics of the cavity length change \(\Delta L\), two stripe regions of data in Figure 3c with a width of 2 mm along the \(45^{\circ}\) and \(-45^{\circ}\) were extracted separately. Each consisted of a total of 40,000 data points and was converted to a one-dimensional signal.  
Next, a noise-assisted data analysis method called Ensemble Empirical Mode Decomposition (EEMD) is introduced, which consists of an ensemble of workers, each performing Empirical Mode Decomposition (EMD) on a copy of the input signal with added noise. The mean of these EMD results is considered the final result [32]. EMD recursively decomposes a non-stationary signal into data-dependent basis functions termed Intrinsic Mode Functions (IMFs) [33]. In the EEMD process, white noise with a standard deviation of 0.1 is added for the calculation and the number of realizations is set to 100. The EEMD results for the cavity length change along \(45^{\circ}\) and \(-45^{\circ}\) are shown in Figure 4. It can be concluded that the data comprise a fundamental half-sinusoidal component with a low frequency (IMF6) along with other higher-frequency components. The spectral kurtosis (SK) is a common dimensionless time series statistic for detecting and characterizing non-stationarities in a signal that can reflect the random distribution of time series data. A high SK level corresponds to a high level of nonstationary or non-Gaussian behavior [34,35]. The spectral kurtosis of the cavity length change \(\Delta L\) along \(45^{\circ}\) and \(-45^{\circ}\) are computed and visualized by kurtosis as shown in Figure 5, for respectively detecting and characterizing the non-stationarities in a signal. The kurtogram uses the normalized frequency, i.e., the sample rate is set to 40,000 Hz for time normalization to 1 s. The kurtosis reveal that the maximum K value is higher in the \(-45^{\circ}\) direction compared to the \(45^{\circ}\) direction, with both directions exhibiting low levels at \(K_{max}\), which indicates that the impact of random signal fluctuations with high frequencies is negligible.  
Figure 3: Z component displacement of mirrors under fixing force.  
Figure 2: Displacement and Von Mises stress distributions.