## 3 Calculation Method of the Backscattered Light of the Taiji Telescope
### Analysis of the Backscattering Caused by Particulate Contamination  
For the optical system with a wavelength of 1064 nm, the impact of particle contamination on the BSDF of the optical surface cannot be ignored, but it is very difficult to simulate stray light on a non-ideal optical surface contaminated by particles. Since the distribution of particles and the undulation of the optical surface are random, it is quite challenging to obtain a BSDF equation that takes into account the surface roughness and particle contamination factors. Therefore, this paper only considers the situation of particle contamination on ideal optical surfaces and obtains the control level of environmental cleanliness by analyzing the stray light generated by particle contamination. In other words, at this cleanliness, we can believe that the PST of the Taiji telescope is mainly affected by the surface roughness, while the scattered energy contributed by particle contamination can be neglected. [PERSON] and [PERSON] elaborated in their paper that MIE scattering can be used to simulate the surface scattering of mirrors polluted by particles [34], which has been widely used. This paper focuses on the relationship between scattering and the distribution of particle size, since both the incident wavelength (1064 nm) and the incident medium (air or vacuum (\(n=1.0+0i\))) are determined, while the only unknown parameter in the calculation of this paper is the refractive index of the particle. According to the paper by [PERSON] et al., the refractive index of the particle is not an essential factor in the MIE scattering model [35], but can be replaced by an approximation. In this paper, the refractive index of particulate contamination used is \(n=1.53+0.001i\). The BSDF of an optical surface polluted by particles is closely related to the actual distribution of particles on its surface. The distribution of standard particles on the clean surface is defined in CC1246 [36]. The number (\(N\)) of particles with a diameter greater than d per square foot is described by Equation (19), where \(CL\) represents cleanliness, generally taking \(h\) as 0.926 for a clean surface and 0.383 for an uncleaned surface. By special transformation of Equation (19), the number of particles \(N(d)\) with a particle size of \(d\) per square foot can be obtained, and its expression is shown in Equation (20). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**M1** & **M2** & **M3** & **M4** & **PST** \\ \hline
25 & 4 & 1 & 1 & \(4.05\times 10^{-10}\) \\
20 & 5 & 1 & 1 & \(4.23\times 10^{-10}\) \\
20 & 4 & 1 & 1 & \(3.89\times 10^{-10}\) \\
10 & 4 & 1 & 1 & \(3.70\times 10^{-10}\) \\
10 & 4 & 2 & 1 & \(7.06\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: PST analysis results of optical surfaces under different surface roughness. Figure 5: The light intensity distribution diagram of the detector at the most lenient surface roughness distribution. \[\log_{10}N=h\left(\log_{10}^{2}CL-\log_{10}^{2}d\right) \tag{19}\]  
\[N(d)=10^{h(\log_{10}^{2}CL1-\log_{10}^{2}d)}-10^{h[\log_{10}^{2}CL1-\log_{10}^{2} (d+1)]} \tag{20}\]  
In addition, the percentage space coverage \(f\) is required for analysis using ASAP, which is related to cleanliness and is described by Equation (21). When \(h\) is taken as 0.926, \(K=-7.245\), while when \(h\) is taken as 0.383, \(K=-5.683\). \[f=10^{K+|h|log_{10}^{2}(CL)-2} \tag{21}\]  
The percentage space coverage \(f\) used in this simulation is shown in Table 5. Figure 6 shows the BSDF for an ideal optical surface with particle diameter from 1 to 200 \(\upmu\)m at 500 cleanliness level. It has become a standard operation to clean the optical surface with high-speed flowing carbon dioxide gas [37].