## 2 Theoretical Analysis of Phase Noise Generated by the Backscattered Light
Since the optical power \(P\) is proportional to the square of the electric field, \(P\propto E^{2}\), we rewrite Equations (1) and (2) into the following form:  
\[E_{1}=\sqrt{P_{L}}e^{i\omega_{1}t}\Big{(}1+a_{1}e^{i(\Delta\omega t+\varphi_{S }+\frac{\pi}{2})}+u_{1}e^{i(\varphi_{B}+\frac{\pi}{2})}\Big{)} \tag{3}\]  
\[E_{2}=\sqrt{P_{L}}e^{i\omega_{1}t}\Big{(}e^{\frac{\pi}{2}}+a_{2}e^{i(\Delta \omega t+\varphi_{S})}+u_{2}e^{i\varphi_{B}}\Big{)} \tag{4}\]  
where \(\Delta\omega=\omega_{2}-\omega_{1},u_{1}^{2}+u_{2}^{2}=u^{2},a_{1}^{2}+a_{2}^ {2}=a^{2}\). Since the common divisor \(\sqrt{P_{L}}e^{i\omega_{1}t}\) has no effect on the calculation of phase, we omit it when calculating the current signal below. The output currents \(I_{1}\), \(I_{2}\) of the two detectors are proportional to \(E_{1}\)\({}^{2}\), \(E_{2}\)\({}^{2}\), respectively. Here we assume that the conversion efficiency of the detector is 1, and only the AC signal is output. The AC terms of \(I_{1}\), \(I_{2}\) are described by Equations (5) and (6), where \(a_{1}\approx a_{2}\). \[I_{1}(t)=-2u_{1}\sin(\varphi_{B})-2a_{1}\sin(\Delta\omega t+\varphi_{S})+2a_{1 }u_{1}\cos(\Delta\omega t+\varphi_{S}-\varphi_{B}) \tag{5}\]  
\[I_{2}(t)=-2u_{2}\sin(\varphi_{B})-2a_{2}\sin(\Delta\omega t+\varphi_{S})+2a_{2 }u_{2}\cos(\Delta\omega t+\varphi_{S}-\varphi_{B}) \tag{6}\]  
The final output signal can be obtained by \(I_{1}-I_{2}\), and the measurement equation for the phase \(\varnothing\) of the interference signal is described by Equation (7) [22], where \(\widetilde{u}=(a_{1}u_{1}-a_{2}u_{2})/(a_{1}+a_{2})\approx(u_{1}-u_{2})/2\). Let \(\varphi_{n}=\widetilde{u}\sin(\varphi_{B})+\widetilde{u}^{2}\sin(\varphi_{B}) \cos(\varphi_{B})\), then \(\varphi_{n}\) is the phase noise. \[\varnothing=\varphi_{S}+\frac{\pi}{2}+\widetilde{u}\sin(\varphi_{B})+ \widetilde{u}^{2}\sin(\varphi_{B})\cos(\varphi_{B}) \tag{7}\]  
Here, we introduce a constant \(\delta\) to represent the splitting ratio error of BS (as shown in Figure 2), then \(u_{1}=\sqrt{\frac{1+\delta}{2}}u\), \(u_{2}=\sqrt{\frac{1-\delta}{2}u}\). After Taylor expansion for \(u_{1}\) and \(u_{2}\), respectively, and keeping only the low-order terms, we get \(u_{1}\approx\frac{2+\delta}{2\sqrt{2}}u\), \(u_{2}\approx\frac{2-\delta}{2\sqrt{2}}u\). For the phase noise \(\varphi_{n}\), its second-order term is tiny and can be ignored. Thus, the simplified phase noise is described by Equation (8). \[\varphi_{n}=\frac{\delta u\sin(\varphi_{B})}{2\sqrt{2}} \tag{8}\]  
Here, we use variance \(\sigma^{2}\) to describe the variation of the phase noise \(\varphi_{n}\). For convenience, the backscattered light can be regarded as the sum of \(n\) individual rays [22], and it is assumed that the distribution of all these rays obey \(\frac{1}{\pi\sqrt{1-x_{j}^{2}}}\), then:  
\[u\sin(\varphi_{B})=\sum_{j=1}^{n}u_{j}\sin\Big{(}\varphi_{B}^{j}\Big{)} \tag{9}\]  
In order to calculate the variance, we first need to calculate the mean of the above formula. The mean calculation process is as follows:  
Figure 1:0  
According to the definition of variance and the previous definition of the optical power of stray light and local beam, \(P_{B}=u^{2}P_{L}\), we can get:Figure 1:1  
Therefore, the variance of the phase noise is described by Equation (12). Figure 1:2  
According to the requirements of Taiji Project, the phase noise caused by the backscattered light should be less than 1 pm, and the spectral error of BS should be less than 0.001. According to Equation (12), the requirement for the backscattered light energy of the telescope can be obtained. Figure 1:3  
There are various evaluation functions for stray light, the most commonly used of which is point source transmittance (PST) [23; 24; 25]. For the stray light analysis to be performed in this paper, the PST that meets the phase noise requirement is:  
Figure 1:4