## 3 Experimental Results and Discussion
To further evaluate the system's ability to suppress the low-frequency thermal noise, an amplitude spectral density (ASD) analysis was conducted, as shown in Figure 7b. At a frequency of 1 mHz, the temperature noise levels were measured as follows: 286 mK\(/\sqrt{\mathrm{Hz}}\) for P\({}_{0}\), 86 mK\(/\sqrt{\mathrm{Hz}}\) for P\({}_{1}\), 55 mK\(/\sqrt{\mathrm{Hz}}\) for P\({}_{2}\), 49 mK\(/\sqrt{\mathrm{Hz}}\) for P\({}_{3}\), and 37 mK\(/\sqrt{\mathrm{Hz}}\) for P\({}_{4}\)). Relative to P\({}_{0}\), the thermal shielding layers reduced the temperature noise by approximately 69.9% for P\({}_{1}\), 80.8% for P\({}_{2}\), 82.9% for P\({}_{3}\), and 87.1% for P\({}_{4}\). These results highlight the multi-layer shielding system's capability to significantly attenuate the low-frequency thermal noise, with each successive layer enhancing the thermal stability of the optical substrate. Figure 6: Test system structure. To quantify the improvement effect of the thermal shielding system on the measurement precision of the interferometer, this study measured the system's local noise under the conditions of without thermal shielding (WTS), single-layer thermal shielding (1L), double-layer thermal shielding (2L), and triple-layer thermal shielding (3L). The correlation between the number of thermal shielding layers and the characteristics of thermal noise was analyzed using the data of displacement noise amplitude spectral density (as shown in Figure 8). The experimental data show that at the characteristic frequency of 1 mHz, the noise for single-layer, double-layer, and triple-layer thermal shielding is 380 pm/\(\sqrt{\rm Hz}\), 140 pm/\(\sqrt{\rm Hz}\), and 60 pm/\(\sqrt{\rm Hz}\), respectively, achieving noise suppression rates of 81.1%, 93.9%, and 97.1% compared to the no thermal shielding state (2070 pm\(\sqrt{\rm Hz}\)). Meanwhile, in the high-frequency segment, the system's noise shows a decreasing trend as the number of layers increases. This phenomenon may be attributed to the  
Figure 8: Interference measurement results. Figure 7: Temperature test. (**a**) Temperature fluctuation of optical substrate under different layers of heat shielding system. (**b**) Frequency domain analysis of temperature fluctuation of optical substrate under different layers of heat shielding system. reduction in electronic noise coupling resulting from improved thermal stability; when the temperature fluctuations of the interferometer system are effectively suppressed, such noise sources as dark current fluctuations and thermoelectric drift in the photoelectric detectors and circuits are reduced [32]. This study, through the synergistic optimization design of multi-layer thermal shielding and low-expansion materials, achieved a background noise of 60 pm\(/\sqrt{\mathrm{Hz}}\) at 1 mHz, which is a reduction of 84.6% compared to previous research [9] of 390 pm\(/\sqrt{\mathrm{Hz}}\). The three-layer thermal shielding system extended the thermal time constant to 78.5 h (a 58% increase compared to the single-layer system) and effectively suppressed thermal-induced phase noise by reducing temperature sensitivity. However, the current system's background noise in the 1 mHz to 1 Hz frequency range is still higher than 8 pm\(/\sqrt{\mathrm{Hz}}\) and does not meet the requirement for direct measurement of the OPD of the PAAM. The main limitations are from the vacuum environment, mechanical installation errors, and detector thermal noise. In this study, the experimental system's fixed vacuum degree was only 10\({}^{-2}\) Pa, and the residual gas refractive index fluctuations introduced additional noise. Through the IFOCAD simulation, when the lens installation position deviation is \(\pm\)1 mm and \(\pm\)5 mm, under temperature fluctuations, the noise introduced by Formula (12) is 1.40 pm and 3.55 pm (with \(\Delta\)T = 20 mK), respectively.