## 2 Theory and Simulation
This enhancement, rooted in the multi-layer dynamics described by the above equations, effectively suppresses the low-frequency noise fluctuations in the interferometer. Building on the thermal response simulations discussed earlier, the periodic fluctuations in the external temperature induce corresponding periodic variations in the temperature of the optical substrate along the interference measurement path. The temperature sensitivity of the substrate, defined as:  
\[\text{S}_{\text{T}}=\text{A}_{\text{substrate}}/\text{A}_{\text{v}} \tag{15}\]  
where \(\text{A}_{\text{substrate}}\) represents the amplitude of the substrate's temperature variation and \(\text{A}_{\text{v}}\) denotes the amplitude of the external temperature oscillation, quantifies this relationship. For vacuum-based systems, a lower \(\text{S}_{\text{T}}\) is critical, as it directly correlates with the reduced low-frequency noise in the interferometer, thereby enhancing the measurement precision. To further analyze this phenomenon, we simulated the vacuum system's temperature oscillating with an amplitude \(\text{A}_{\text{v}}\) and period T. The frequency-dependent response of \(\text{S}_{\text{T}}\), illustrated in Figure 3b, reveals that the thermal shielding layers function as a low-pass filter, effectively attenuating the high-frequency components of external temperature fluctuations. For instance, when the external temperature varied by 1 K over a period of 10,000 s, the substrate temperature changes were approximately \(10^{-3}\) K, \(10^{-5}\) K, and \(10^{-7}\) K under single-layer, double-layer, and triple-layer shielding, respectively. These results underscore the efficacy of the multi-layer thermal shielding system in suppressing the impact of external temperature perturbations on the interferometer. As the number of shielding layers increases, the system's temperature sensitivity is significantly reduced, bolstering its resilience against thermal disturbances and laying a foundation for improved interferometric stability in subsequent analyses. To quantify the synergistic effect of thermal shielding optimization, this study used IFOCAD to simulate the interferometer and analyzed the optical path noise under temperature disturbances for interferometers with substrates made of aluminum alloy (6061-T6), quartz glass, and ULE glass, with material parameters as shown in Table 2 The results, as shown in Figure 4, illustrate the noise contributions arising from substrate expansion, optical elements, and the total system under different thermal shielding configurations. Figure 3: Simulation of the thermal shield system. (**a**) Temperature of optical substrate as a function of time with different multiple layers shield. (**b**) Temperature sensitivity curves of optical substrate with different multiple layers shield. The thermal expansion coefficient \(\alpha_{1}\) of the substrate directly influences the expansion noise \(\delta_{1}(t)\), as described by Equation (7). Figure 4a shows that, without thermal shielding, the aluminum alloy substrate exhibits the highest expansion noise (39.22 pm), contributing 66.7% of the total system noise. This is consistent with its large \(\alpha_{1}=23.6\times 10^{-6}\) K\({}^{-1}\). Quartz, with \(\alpha_{1}=3.2\times 10^{-6}\) K\({}^{-1}\), reduces this noise to approximately 30.9% of the aluminum alloy's value, while ULE's near-zero \(\alpha_{1}=0.01\times 10^{-6}\) K\({}^{-1}\) renders its expansion noise negligible (0.085 pm). Consistent with Equation (7), our findings show that the optical path change due to the substrate's thermal expansion (\(\delta_{1}(t)\)) scales linearly with the substrate's thermal expansion coefficient (\(\alpha_{1}\)) under constant temperature change and residual arm length difference. This highlights the importance of low-expansion substrates in minimizing thermal noise.