## 3 Calculation of Optimal Testing Conditions  
It is necessary to determine the optimal detection conditions before performing RMMS to detect and classify various types of defects on the surface of optical elements. This study uses dirty particles on the surface of optical elements and SSBDs as an example. For ultra-smooth optical elements, most of the dirty particles of interest to the surface defect detection system have a diameter of about tens of nanometers, the size of which is much smaller than the wavelength of visible light. The Rayleigh-Rice scattering model can be used to approximate the interaction of a single particulate dust with the incident light. The BRDF is used instead of the pBRDF to calculate the optimal experimental conditions, as in Equations (11) and (13), which effectively simplifies the tedious calculation process. \[\text{BRDF}_{particle}=\frac{16\pi^{4}}{\lambda^{4}}\bigg{(}\frac{{n_{1}}^{2}- 1}{{n_{1}}^{2}+2}\bigg{)}^{2}\frac{a^{6}}{\cos\theta_{i}\cos\theta_{s}}\frac{ NF}{A}\times\left|\stackrel{{\rightarrow}}{{e}}\cdot J_{particle}\right|^{2} \tag{11}\]  
Figure 4: Subsurface bubble defect particle scattering schematic. where  
\[\left\{ \begin{array}{l}J_{ss}=\left[1+\beta r_{s}(\theta_{s})\right]\left[1+ \alpha r_{s}(\theta_{s})\right]\cos\phi_{s},\\ J_{ps}=-\left[1+\beta r_{s}(\theta_{s})\right]\left[1-\alpha r_{p}(\theta_{i}) \right]\cos\theta_{i}\sin\phi_{s},\\ J_{sp}=-\left[1-\beta r_{p}(\theta_{s})\right]\left[1+\alpha r_{s}(\theta_{i} )\right]\cos\theta_{s}\sin\phi_{s},\\ J_{pp}=\left[1+\beta r_{p}(\theta_{s})\right]\left[1+\alpha r_{p}(\theta_{i} )\right]\sin\theta_{i}\sin\theta_{s}\\ -\left[1-\beta r_{p}(\theta_{s})\right]\left[1-\alpha r_{p}(\theta_{i}) \right]\cos\theta_{i}\cos\theta_{s}\cos\phi_{s}\end{array} \right. \tag{12}\]  
\[\text{BRDF}_{SSBD}=\frac{256\pi^{4}}{\lambda^{4}}\bigg{(}\frac{{n_{1}}^{2}-{n _{2}}^{2}}{{n_{1}}^{2}+2{n_{2}}^{2}}\bigg{)}^{2}a^{6}\cos\theta_{i}\Big{(}{n _{2}}^{2}-\sin^{2}\theta_{s}\Big{)}^{1/2}|\gamma\delta|^{2}{n_{2}}^{3}\frac{ NF}{A}\times\left|\stackrel{{\sim}}{{e}}\cdot JJ_{SSBD}\right|^{2} \tag{13}\]  
where  
\[\left\{ \begin{array}{l}J_{ss}=\frac{\cos\phi_{s}}{\left[\cos\theta_{i}+\left( {n_{2}}^{2}-\sin^{2}\theta_{i}\right)^{1/2}\right]\left[\cos\theta_{s}+\left( {n_{2}}^{2}-\sin^{2}\theta_{s}\right)^{1/2}\right]}\\ J_{sp}=\frac{-\sin\phi_{s}\left({n_{2}}^{2}-\sin^{2}\theta_{s}\right)^{1/2}}{ \left[\cos\theta_{i}+\left({n_{2}}^{2}-\sin^{2}\theta_{i}\right)^{1/2}\right] \left[{n_{2}}^{2}\cos\theta_{s}+\left({n_{2}}^{2}-\sin^{2}\theta_{s}\right)^ {1/2}\right]}\\ J_{ps}=\frac{-\sin\phi_{s}\left({n_{2}}^{2}-\sin^{2}\theta_{s}\right)^{1/2}}{ \left[{n_{2}}^{2}\cos\theta_{s}+\left({n_{2}}^{2}-\sin^{2}\theta_{s}\right)^ {1/2}\right]\left[{n_{2}}^{2}\cos\theta_{s}+\left({n_{2}}^{2}-\sin^{2}\theta_{ s}\right)^{1/2}\right]}\\ J_{pp}=\frac{\sin\theta_{i}\sin\theta_{s}-\left({n_{2}}^{2}-\sin^{2}\theta_{i} \right)^{1/2}\left({n_{2}}^{2}-\sin^{2}\theta_{i}\right)^{1/2}\cos\theta_{s}} {\left[{n_{2}}^{2}\cos\theta_{i}+\left({n_{2}}^{2}-\sin^{2}\theta_{s}\right)^ {1/2}\right]\left[{n_{2}}^{2}\cos\theta_{s}+\left({n_{2}}^{2}-\sin^{2}\theta_{ s}\right)^{1/2}\right]}\end{array} \right. \tag{14}\]  
In this study, we take the dirty particles and SSBDs with a particle size of \(a\) = 5 nm on the surface of the optical element as an example, and draw the pseudo-color maps with radial coordinates \(\theta_{s}=0^{\circ}\sim 90^{\circ}\), azimuthal angle \(\phi_{s}=0^{\circ}\sim 360^{\circ}\), and incidence angle of \(0^{\circ}\) and \(70^{\circ}\), respectively, as shown in Figures 5 and 6. The different combinations of polarization states are indicated by the subscripts of BRDF, with the first one representing the polarization state of the incident light and the second one representing the polarization state of the outgoing light.