## 2 Theory and Simulation
(**a**) Schematic diagram of the heat shielding system and its equivalent multilevel RC integral circuit. (**b**) Temperature distribution at different time points. pathways [25; 26; 27]. These combine to form the equivalent thermal impedance \(Z_{ij}\) between the layers, calculated as follows:  
\[r_{ij}=\frac{L}{\kappa_{S}A_{s}} \tag{11}\]  
\[r_{ij}=\frac{\beta_{ij}}{4\sigma A_{i}T_{0}^{3}} \tag{12}\]  
\[Z_{ij}=\frac{R_{ij}\cdot r_{ij}}{R_{ij}+r_{ij}} \tag{13}\]  
where L is the pillar height, \(\kappa_{s}\) is the thermal conductivity of the support material (PEEK), \(A_{s}\) is the total pillar cross-sectional area, \(\beta_{ij}\) is the heat radiation coefficient, \(\sigma\) is the Stefan-Boltzmann constant, \(A_{j}\) is the surface area of \(P_{j}\), and \(T_{0}\) is the initial temperature. The characteristic time constant \(\tau_{ij}=Z_{ij}\cdot C_{j}\) quantifies the thermal response time between the adjacent layers i and j. When the external temperature of the vacuum chamber changes from \(T_{0}\) to \(T_{f}\), the temperature dynamics across the shielding layers are governed by the following differential equations [29]:  
\[ \begin{cases}C_{1}\frac{dT_{1}}{dt}=\beta_{01}\left(T_{f}^{4}-T_{1}^{4}\right) -\beta_{12}\left(T_{1}^{4}-T_{2}^{4}\right)+\alpha_{01}\left(T_{f}-T_{1} \right)-\alpha_{12}\left(T_{1}-T_{2}\right)\\ C_{2}\frac{dT_{2}}{dt}=\beta_{12}\left(T_{1}^{4}-T_{2}^{4}\right)-\beta_{23} \left(T_{2}^{4}-T_{3}^{4}\right)+\alpha_{12}\left(T_{1}-T_{2}\right)-\alpha_{2 3}\left(T_{2}-T_{3}\right)\\ C_{3}\frac{dT_{3}}{dt}=\beta_{23}\left(T_{2}^{4}-T_{3}^{4}\right)-\beta_{34} \left(T_{3}^{4}-T_{4}^{4}\right)+\alpha_{23}\left(T_{2}-T_{3}\right)-\alpha_{3 4}\left(T_{3}-T_{4}\right)\\ \qquad\qquad\qquad C_{4}\frac{dT_{4}}{dt}=\beta_{34}\left(T_{3}^{4}-T_{4}^{4} \right)-\alpha_{34}\left(T_{3}-T_{4}\right)\end{cases}  \tag{14}\]  
Here, \(C_{i}=c_{i}\)\(m_{i}\) represents the heat capacity of the i-th layer, where \(c_{i}\) is the specific heat capacity and \(m_{i}\) is the mass; \(T_{i}\) is the temperature of the i-th layer; \(\alpha_{ij}\) and \(\beta_{ij}\) are t the conduction and radiation heat transfer coefficients between layers i and j, respectively. The conduction coefficient \(\alpha_{ij}\) is the inverse of the conductive thermal resistance \(r_{ij}\) (\(\alpha_{ij}=1/r_{ij}\)), primarily associated with the heat transfer through the support pillars. This model captures both radiative (via the \(T^{4}\) terms) and conductive (via the linear T terms) contributions, aligning with the physical conditions of the vacuum-based interferometer setup. Thermal simulations were performed using Finite Element Analysis (FEA) to evaluate the system's performance. A step change in the vacuum chamber's outer wall temperature from \(T_{0}=20\)\({}^{\circ}C\) (293.15 K) to \(T_{f}=21\)\({}^{\circ}C\) (294.15 K) was modeled. Figure 2b, illustrating the simulated temperature distribution at different time points, shows largely uniform temperatures within each shield layer (\(P_{1}\)-\(P_{3}\)). The profiles also visualize the progressive attenuation effect, with the temperature changes decreasing significantly layer by layer towards the substrate (\(P_{4}\)), substantiating the multi-layer structure's damping effectiveness. The temperature response of the optical substrate (\(P_{4}\)) was calculated for the single-layer (1L), double-layer (2L), and triple-layer (3L) shielding configurations, as shown in Figure 3a. The thermal time constant (\(\tau\)) was determined as the time required for the substrate temperature to reach \(T=(23-e^{-1})\)\({}^{\circ}C\approx 22.632\)\({}^{\circ}C\). The results yielded time constants of \(\tau_{1}=42\) h for the single-layer, \(\tau_{2}=72\) h for the double-layer, and \(\tau_{3}=90\) h for the triple-layer configurations. The progressive increase in \(\tau\) with additional layers--culminating in a 58% extension for the triple-layer system compared to the single-layer design--demonstrates a slower thermal response to external disturbances.