## 3 Calculation Method of the Backscattered Light of the Taiji Telescope
### Analysis of the Backscattering Caused by Surface Roughness  
It is impossible for the optical surface to be perfectly smooth after processing. When light hits the reflector, the reflected light exhibits a scattering phenomenon centered on the mirror. The degree of scattering is usually characterized by the bidirectional scattering distribution function (BSDF). Since the measurement of the BSDF is complex and expensive [29; 30], while the measurement of surface roughness \(\sigma_{\lambda}\) is very convenient, the relationship between \(\sigma_{\lambda}\) and the BSDF needs to be established before the stray light analysis. The BSDF is defined as the ratio of scattering brightness to incident illuminance. In this paper, the Harvey-Shack model is used to describe the BSDF, and its specific expression is shown in Equation (15) [31], where \(b_{0},s\) and \(l\) are all constants. \[\text{BSDF}\bigg{(}\overset{\rightharpoonup}{\beta},\overset{\rightharpoonup}{ \beta_{0}}\bigg{)}=b_{0}[1+\left(\frac{\left|\overset{\rightharpoonup}{ \beta}-\overset{\rightharpoonup}{\beta_{0}}\right|}{l}\right)^{2}]s/2 \tag{15}\]  
Total integrated scattering (TIS) is the ratio of scattered energy to total reflected energy. For a polished optical surface, the surface roughness \(\sigma_{\lambda}\ll\lambda\). The relationship between TIS and \(\sigma_{\lambda}\) can be described by Equation (16) [32]. \[\text{TIS}\approx\bigg{[}\frac{4\pi\sigma_{\lambda}\cos(\theta_{i})}{\lambda} \bigg{]}^{2} \tag{16}\]  
According to the definitions of TIS and BSDF, TIS is equal to the integral of BSDF over the solid angle in hemispherical space. Meanwhile, it can be seen from the calculation formula of TIS that its value is maximum when the incident angle is zero. Therefore, if the worst case is considered, combined with Equation (15) yields:  
\[\text{TIS}=\left\{ \begin{array}{c}\frac{2\pi b_{0}}{P(s+2)}\bigg{[}(1+l^{2} )^{\frac{s+2}{2}}-l^{s+2}\bigg{]},s\
eq-2,\\ \pi b_{0}l^{2}ln\Big{(}1+\frac{1}{2}\Big{)},s=-2.\end{array} \right. \tag{17}\]  
Figure 3: Main scattering paths of the Taiji telescope. (**a**) In scattering path 1, the scattering energy comes from M1, accounting for 0.03%. (**b**) In scattering path 2, the scattered energy comes from M2, accounting for 1.72%. (**c**) In scattering path 3, the scattered energy comes from M3, accounting for 31.02%. (**d**) In scattering path 4, the scattered energy comes from M4, accounting for 67.22%. Combining Equations (16) and (17), the relationship between \(\sigma_{\lambda}\) and the Harvey-Shack model can be obtained. \[\sigma_{\lambda}^{2}=\left\{ \begin{array}{c}\frac{b_{0}l^{2}}{8\pi l^{2}(s+2 )}\bigg{[}(1+l^{2})^{\frac{s+2}{2}}-l^{s+2}\bigg{]},s\
eq-2,\\ \frac{b_{0}l^{2}\Lambda^{2}}{16\pi}ln\Big{(}1+\frac{1}{l^{2}}\Big{)},s=-2. \end{array} \right. \tag{18}\]  
The values of the parameters of the Harvey-Shack model for different surface roughness are given in Table 3. The parameter s is closely related to the optical surface polishing process and usually fluctuates in the range of \(-2.5\) to \(-0.5\)[33]; here, we take the typical value of \(s=-1.5\). In addition, the power spectral density of the polished optical surface generally has fractal characteristics, satisfying \(l\ll\left|\stackrel{{\rightarrow}}{{\beta}}-\stackrel{{ \rightarrow}}{{\beta_{0}}}\right|\). Therefore, the value of \(l\) must be small enough to ignore its influence on TIS; here, we take \(l=0.01\). Figure 4 shows the BSDF distribution when the value of \(b_{0},l,\) and \(s\) are, respectively, \(0.013\), \(0.01\), and \(-1.5\), and the scattering angle is \(30^{\circ}\). According to the contribution of each mirror to PST, a different surface roughness can be assigned to each optical surface to obtain the most lenient surface roughness distribution that can meet the requirements.