## 2 Theoretical Analysis of Phase Noise Generated by the Backscattered Light  
The off-axis four-mirror optical system used in the Taiji Telescope is shown in Figure 1a, where the primary mirror is a paraboloid, the secondary mirror is a hyperboloid, and the third and fourth mirrors are both spherical [21]. The detailed parameters are shown in Table 1. The Taiji interferometer adopts a balanced orthogonal four-quadrature detection scheme to detect phase signals, and its transmitting and receiving optical paths are shown in Figure 1b. For the transmitting optical path, the beam emitted by the frequency-stabilized laser is divided into two beams by the polarization beam splitter, one of which is used as the local reference beam (polarization state is s), and the other forms the outgoing beam (polarization state is p). For the receiving optical path, the signal light becomes s light after passing through a 1/4 wave plate twice, and then interferes with the local reference beam after multiple reflections. It should be noted that both the frequency and polarization states of the backscattered light generated by the outgoing beam are consistent with the local reference beam, which means that it can interfere with the local reference beam, resulting in an additional phase. If the additional phase was a constant, it would have very little effect on the phase measurement, but in reality, the scattered light is affected by variations in the telescope's optical length. Therefore, the additional phase caused by the scattered light is also variable, that is, the phase noise. As shown in Figure 2, we use \(E_{l}\), \(E_{b}\), and \(E_{s}\) to denote the electric field complex amplitude of the local beam, the backscattered beam, and the signal beam, respectively. All three beams of light can interfere with each other. Furthermore, the distributions of the light fields \(E_{1}\) and \(E_{2}\) are described by Equations (1) and (2). \[E_{1}=E_{L}e^{(i\omega_{1}t)}+E_{S}e^{i(\omega_{2}t+\varphi_{S}+ \frac{\pi}{2})}+E_{B}e^{i(\omega_{1}t+\varphi_{B}+\frac{\pi}{2})} \tag{1}\] \[E_{2}=E_{L}e^{(i\omega_{1}t+\frac{\pi}{2})}+E_{S}e^{i(\omega_{2} t+\varphi_{S})}+E_{B}e^{i(\omega_{1}t+\varphi_{B})} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \multicolumn{2}{c}{**Parameter**} & **Requirement** \\ \hline
1 & Optical aperture & 200 mm \\ \hline
2 & Optical efficiency & 0.853 \\ \hline
3 & Working wavelength & 1064 nm \\ \hline
4 & Field-of-View (spatial acquisition) & \(\pm\)200 \(\upmu\)rad \\ \hline
5 & Field-of-View (science) & \(\pm\)20 \(\upmu\)rad out-of-plane \\ \hline
6 & Surface Flatness & \(\lambda\)/70 RMS \\ \hline
7 & Wavefront quality over science field of view & \(\lambda\)/30 RMS \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the optical system of Taiji telescope. Figure 1: (**a**) Schematic diagram of the off-axis four-mirror optical system adopted by the Taiji Telescope. Compared with the coaxial optical system, its ability to suppress stray light is stronger. (**b**) Schematic diagram of the light field distribution in the telescope. The figure shows the outgoing laser light (red), signal light (blue), and backscattered light (wave) of the Taiji telescope. Figure 2: Schematic diagram of balanced orthogonal detection. According to the design of the Taiji interferometer, the output optical power \(P_{O}\) of the frequency-stabilized laser is 2 \(W\), the optical power of the local beam \(P_{L}=2\times 10^{-3}\)\(W\), and the optical power of the signal light \(P_{S}=a^{2}P_{L}=3\times 10^{-10}\)\(W\). For the convenience of calculation, we introduce the parameter \(a^{2}\), where \(a^{2}\) is a constant with a value of \(1.5\times 10^{-7}\). Similarly, the optical power of stray light \(P_{B}=u^{2}P_{L}\), where \(u^{2}\ll 1\).