## 2 Theory and Simulation
The optical path change due to the substrate thermal expansion is related to the residual arm length difference:  
\[\delta_{1}(t)=\Delta{\rm T(t)}\Delta{\rm L_{residual}}\alpha_{1} \tag{7}\]  
\(\Delta{\rm T(t)}\) is the temperature change and \(\alpha_{1}\) is the linear expansion coefficient of the optical substrate. Due to the symmetrical design of the system, the common-mode noise in the reference (PD1) and measurement (PD2) optical paths can be eliminated by subtraction. However, since the reference optical path passes through the BS one more time compared to the measurement optical path, the temperature change causes the beam splitter (BS) to expand and induce the thermo-optic effect, which introduces noise that cannot be eliminated by subtraction [18; 19; 20]. Therefore, the optical path noise introduced by this difference is described by formulas (7) and (8). The optical path noise due to the thermal expansion of the optical components is:  
\[\delta_{2}(t)=\Delta{\rm T(t)}{\rm L_{BS}}\alpha_{2}({\rm n_{2}-1}) \tag{8}\]  
where \(\alpha_{2}\) is the linear expansion coefficient of the BS and \({\rm n_{2}}\) is its refractive index. The optical path noise due to the thermo-optic effect is:  
\[\delta_{3}(t)=\Delta{\rm T(t)}{\rm L_{BS}}\frac{{\rm d_{n}}}{{\rm d_{t}}} \tag{9}\]  
where \({\rm d_{n}}/{\rm d_{t}}\) is the thermo-optic coefficient, approximately 1.2 \(\times\)\(10^{-6}\) K\({}^{-1}\). Thus, the optical path noise temperature induced is:  
\[\mathrm{L_{PD1}}=\mathrm{L1+L2+L3-L2-L4-L6}=\mathrm{L1+L3-L4-L6} \tag{2}\]0  
To mitigate the impact of the environmental temperature fluctuations on the phase noise of the interferometer, this study proposes a multi-layer thermal shielding system for the heterodyne interferometer. The system is designed to achieve a large time constant and low temperature sensitivity [25; 26; 27], thereby reducing thermal noise caused by the changes in ambient temperature. Heat transfer occurs through three primary mechanisms: heat convection, heat radiation, and heat conduction [28]. Given that the equal-arm heterodyne interferometer operates in a vacuum environment, the influence of convection can be disregarded, and thus, heat conduction and radiation were primarily considered. The thermal shielding system, illustrated in Figure 2a, comprises a vacuum chamber (P\({}_{0}\)), multiple thermal shielding layers (P\({}_{1}\), P\({}_{2}\), P\({}_{3}\)), and the substrate for the equal-arm heterodyne interferometer's measurement path (P\({}_{4}\)). The vacuum chamber, constructed from stainless steel (outer diameter 1200 mm, height 1600 mm), encloses the copper shielding layers (P\({}_{1}\), P\({}_{2}\), P\({}_{3}\)), which are separated by polyether ether ketone (PEEK) support columns to minimize conductive heat transfer. The copper layers were polished to reduce radiative heat exchange, with detailed material properties provided in \[\mathrm{L_{PD1}}=\mathrm{L1+L2+L3-L2-L4-L6}=\mathrm{L1+L3-L4-L6} \tag{2}\]1  
In the thermal shield system, heat transfer between adjacent layers occurs through both radiation and conduction. These processes act in parallel, analogous to parallel resistors in an electrical circuit, as conceptually illustrated in Figure 2a. The conductive thermal resistance rij and the radiative thermal resistance rij characterize these parallel heat transfer \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Density** & **Specific Heat Capacity** & **Thermal Conductivity** & **Radiant Emissivity** \\  & \(\rho\)(**kg\(\cdot\)m\({}^{-3}\)**) & \(\mathbf{c}\)(J\(\cdot\)kg\({}^{-1}\)\(\cdot\)K\({}^{-1}\)**) & \(\mathbf{k}\)(W\(\cdot\)m\({}^{-1}\)\(\cdot\)K\({}^{-1}\)) & \(\epsilon\) \\ \hline Copper & 8920 & 385 & 400 & 0.02 \\ Stainless steel & 2700 & 900 & 385 & 0.03 \\ ULE & 2200 & 767 & 1.31 & 0.91 \\ PEEK & 1300 & 1400 & 0.25 & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermal parameters of materials. Figure 2: Modeling and simulation of the thermal shield system.