## 3 Design of the Interferometer  
Based on the principle of a laser heterodyne interferometer, a high-precision interferometry system is constructed. Figure 5 shows the layout of the entire interferometer. The entire system consists of a frequency modulation bench and an optical bench. The modulation bench consists of a laser, a signal generator, acousto-optic modulators (AOMs), etc. And the modulation bench is designed to obtain two lasers with a certain frequency difference. The optical bench consists of the steering mirror (SM) to be measured, photodetectors (PDs), linear polarizers (LPs), beam splitters (BSs), a mirror (M), etc. And the optical bench comprises a reference interferometer and a measurement interferometer. Here, the function of the AOMs is to achieve frequency shifting in the laser, while the LPs serve to ensure that the laser entering the interferometer is not affected by polarization changes. The laser with a frequency of \(f_{0}\) emits a light that passes through a one-to-two optical fiber coupler and is divided into two paths. The two laser paths pass through the AOMs, resulting in two laser beams with frequency shifts \(f_{1}\) and \(f_{2}\), respectively. At this point, the laser modulation bench has obtained two laser beams (\(\overset{\rightarrow}{E}_{1}\) and \(\overset{\rightarrow}{E}_{2}\)) with frequencies (\(f_{0}+f_{1}\)) and (\(f_{0}+f_{2}\)), respectively. \[\overset{\rightarrow}{E}_{1}\left(\overset{\rightarrow}{r},\mathbf{t}\right) =\overset{\rightarrow}{A}_{1}\left(\overset{\rightarrow}{r} \right)\cdot\cos[2\pi(f_{0}+f_{1})\mathbf{t}+\varphi_{1}] \tag{9}\] \[\overset{\rightarrow}{E}_{2}\left(\overset{\rightarrow}{r}, \mathbf{t}\right) =\overset{\rightarrow}{A}_{2}\left(\overset{\rightarrow}{r} \right)\cdot\cos[2\pi(f_{0}+f_{2})\mathbf{t}+\varphi_{2}] \tag{10}\]  
where \(\overset{\rightarrow}{A}_{1}\) and \(\overset{\rightarrow}{A}_{2}\) are the amplitude vectors, respectively, and \(\varphi_{1}\) and \(\varphi_{2}\) are the initial phases, respectively. Figure 4: Numerical fitting graph of partial piston noise. The two laser beams then pass through the linear polarizers, beam splitters, the steering mirror, and the reference mirror, and finally combine at the beam splitters to form a heterodyne beat optical signal. The frequency of the heterodyne beat optical signal is the difference between the frequencies of the two lasers; that is, the heterodyne frequency is \(f_{het}=f_{1}-f_{2}\). The heterodyne signal is then converted into electrical signals \(I_{PD1}\) and \(I_{PD2}\) through the detectors. \[I_{PD1}(t)=\eta_{PD}\Big{(}{A_{1a}}^{2}+{A_{2a}}^{2}+2{A_{1a}}{A_{2a}}\times \cos(2\pi f_{het}t+\delta\varphi 1)\Big{)} \tag{11}\]  
\[I_{PD2}(t)=\eta_{PD}\Big{(}{A_{1b}}^{2}+{A_{2b}}^{2}+2{A_{1b}}{A_{2b}}\times \cos(2\pi f_{het}t+\delta\varphi 2)\Big{)} \tag{12}\]  
where \(\eta_{PD}\) is the photoelectric conversion efficiency; \(A_{1a}\), \(A_{2a}\), \(A_{1b}\), and \(A_{2b}\) are the amplitudes of the light field before interference; and \(\delta\varphi 1\) and \(\delta\varphi 2\) are the optical path phase differences of the measurement interferometer and the reference interferometer. The phase difference can be obtained with a phase meter. Then, we apply the optical path and phase conversion formula (Formula (13)):  
\[L=\frac{\varphi}{2\pi}\lambda \tag{13}\]  
The optical path difference of the interference optical path can be obtained. Due to the stationary nature of the reference mirror in the reference interferometer, the measured optical path difference, obtained as the steering mirror rotates in the measurement interferometer, can be attributed entirely to the rotation of the steering mirror.