## 1 Introduction  
In space-based gravitational wave detection programs, such as China's \"Taiji\" [1], \"Tianqin\" [2], and the European Space Agency's LISA [3], require picometer-precision laser interferometry between satellites to detect gravitational wave signals. Achieving this precision interferometry relies on the PAAM [4, 5], which serves to steer a laser beam to compensate for the angle generated by the relative motion of the two spacecrafts (SCs) during the approximately 10 s of flight time a laser beam will take from one SC to reach a distant SC of three million kilometers away. The TTL noise of the PAAM directly affects the detection sensitivity of gravitational wave signals [6, 7, 8]. Its noise budget is constrained to below 8 \(\mathrm{pm}/\sqrt{\mathrm{Hz}}\) in the frequency range from 1 mHz to 1 Hz, which necessitates that the interferometer's measurement noise remains under this critical threshold to ensure the precision required for the gravitational wave detection in space. Recent studies have advanced PAAM testing methodologies. The Albert Einstein Institute (Hannover) conducted tests on the PAAM for the LISA mission [4]. The OPD is measured using a triangular resonance cavity. The system achieved an OPD sensitivity of \(8\sqrt{1+(2.8\,\mathrm{mHz}/\mathrm{f})^{4}\mathrm{pm}/\sqrt{\mathrm{Hz}}}\) across the frequency range from 0.1 mHz to 1 Hz. However, specific details of their testing methodology, such as relevant parameters of the triangular resonance cavity or relevant experimental constraints, have not been fully reported in the publicly available literature. In our previous work, [PERSON] et al. [9] constructed an equal-arm heterodyne interferometer on an optical quartz substrate with UV-bounded Super-Invar (4J32) components. However, the thermal expansion of the substrate (\(\alpha_{\mathrm{4J32}}=9\ \times\ 10^{-7}\mathrm{K}^{-1}\)) and the stress relief of the UV adhesive introduced noise exceeding \(390\) pm\(/\sqrt{\mathrm{Hz}}\) at 1 mHz-1 Hz, limiting long-term stability. [PERSON] et al. [10] built a symmetric heterodyne interferometer on an aluminum baseplate (\(\alpha_{\mathrm{Al}}=2.36\ \times\ 10^{-6}\mathrm{K}^{-1}\)), reporting a \(2\) pm\(/\sqrt{\mathrm{Hz}}\) precision above 0.1 Hz. However, at lower frequencies, the noise level approximately follows a \(1/\mathrm{f}\) trend. The low-frequency performance of the interferometer is largely limited by the thermal noise of its system [11; 12]. Various methods have been employed to reduce the thermal noise of the system, including the reduction of optical substrate noise and the reduction of the mechanical loss of the coatings [13; 14; 15]. These studies demonstrate that picometer precision has been achieved in the high-frequency bands through optical substrate optimization and symmetric interferometer configurations [9; 10]. However, in the PAAM OPD measurements, the interferometer is required to have a background noise of less than 8 pm\(/\sqrt{\mathrm{Hz}}\) in the frequency range of 1 mHz to 1 Hz. In laboratory environments, the impact of mechanical vibrations on the interferometer primarily occurs within the 0.1 Hz to 100 Hz frequency range [15; 16], and its magnitude is below the requirement levels for space gravitational wave detection [17]. Therefore, this study focuses on suppressing thermal noise effects [18; 19; 20; 21] as they primarily limit the low-frequency sensitivity in the mHz band, rather than detailing the vibration noise. In previous studies, the suppression of the PAAM OPD measurement noise is mainly focused on the substrate material optimization and the design of the interferometer optical path. Despite this progress, significant challenges remain in suppressing the low-frequency noise to meet the stringent picometer-level requirements for the PAAM OPD measurements, and the strategies to reduce the noise from the temperature fluctuations and improve the low-frequency measurement sensitivity require more detailed investigation.