## 4 Experimental Results and Discussion  
The entire system was tested in a vacuum chamber with a pressure of \(10^{-3}\) Pa to reduce the effects of refractive index fluctuations and air turbulence. The laser source is coupled into the system through the vacuum feed-throughs, and the optical signal is detected after being transmitted through the vacuum feed-throughs, too. Figure 7 shows the operating environment of the preliminary test. A small vacuum chamber is placed on  
Figure 6: Actual picture of the interferometer (M: A reference mirror. BS: A beam splitter. SM: A steering mirror to be measured. PD1: A photodetector in the measurement interferometer. PD2: A photodetector in the reference interferometer. The red and green lines represent two laser beams with a certain frequency difference, while the yellow line represents the interfered light obtained after combining these two laser beams). an air-floating optical platform (LBTEK, AFT1510) to isolate vibrations above its resonance (1 Hz). The modulation bench is also placed on a commercial isolator (Herz, TS-300/LP) to isolate environmental vibrations. Paste heating film around the vacuum chamber and wrap it with thermal insulation cotton to achieve approximately 10 mK temperature control. Place the optical substrate into the vacuum chamber and use three heat-insulating brackets (PEEK) to raise it so that the optical substrate is not in direct contact with the bottom of the vacuum chamber to reduce heat conduction. To evaluate the background noise of the interferometer, we do not apply voltage to the steering mirror, so that the steering mirror is in a static state, equivalent to a reference mirror. Figure 8 shows the measurement results of the reference interferometer, the measurement interferometer, and their differences, which represent the sensitivity level of the entire interferometer. Measurements were performed for 8 h in the operating environment described above. Figure 8a shows the time series of the measured OPD for individual interferometers and the differential results, with a measured displacement change of 0.6 nm over a duration of 1081 s (40,000 sampling points). Figure 8b is their Linear Amplitude Spectral Density (LASD) plot, showing that the reference and measurement interferometers achieve a sensitivity level of 3.5 \(\upmu\)m/\(\sqrt{\mathrm{Hz}}\) in the frequency range of 1 mHz to 1 Hz. Since the two interferometers share similar paths, the traces of the reference and measurement interferometers are highly overlapping. They contain some common noise, including laser frequency noise, laser Relative Intensity Noise (RIN), temperature fluctuations, etc. Through the common-mode suppression scheme, the sensitivity is increased to a level of 390 pm/\(\sqrt{\mathrm{Hz}}\) in the 1 mHz to 1 Hz frequency band, which is about four orders of magnitude higher than that of a single interferometer. Such differential experimental results highlight the advantages of the equal-arm interferometer. The reference interferometer and measurement interferometer follow similar paths, enabling the effective elimination of common-mode noise as much as possible. This achieves measurement accuracy spanning from micrometers to hundreds of picometers. Figure 7: Preliminary test system diagram: (**a**) is the modulation bench, and the purpose is to obtain two laser beams with a certain slight frequency difference; (**b**) is the control system for the steering mirror, designed to control the rotation of the steering mirror during experimentation; (**c**) is the optical bench, placed within a vacuum chamber to generate interferometric measurement signals and reference signals; (**d**) is the optical signal acquisition and processing part, designed to record and monitor experimental data. However, this interferometer cannot directly measure the OPD of the steering mirror in the frequency range of 1 mHz to 1 Hz because the OPD of the interferometer is higher than 8 \(\mathrm{pm}/\sqrt{\mathrm{Hz}}\) in the 1 mHz to 1 Hz frequency band.