## 4 Results and Analysis
### Simulation of Reflected Mueller Matrix Spectra  
For the RMMS of dirty particles and digs and SSBDs above the surface of the optical test plate, the corresponding measurement scheme and solution method are the same as for the component surface roughness and digs. The corresponding spectral data were obtained via forward modeling employing MATLAB R2022b simulation. To investigate the difference in the RMMS of dirty particles and digs on the surface of the test plate and  
Figure 11: The comparison of the RMMS for surface roughness and digs on the optical test plate. Figure 12: The simulated (red) and experimental (blue) RMMS of the surface roughness (**a**) and pockmarks (**b**) of the test plate as well as the difference curves between simulation and experiment (black). SSBDs, the test plate simulation model is established. When the test board is exposed to an unclean environment, the surface can often be covered with dust that is mainly composed of SiO\({}_{2}\) particles. Therefore, the surface dirty particles are approximated as SiO\({}_{2}\) particles for analysis. SSBDs are formed by gases that are not discharged in time during the production or processing of optical components, and the shape of the bubbles is generally rounded due to the uniform distribution of the pressure of the gas in all directions. The RMMS simulation results of dirty particles and digs and SSBDs above the surface of the test plate are shown in Figure 13.  
The non-diagonal block elements of the Mueller matrix are zero, and the diagonal block elements provide valid information (consistent with Figure 8). In addition to this, the pBRDFs of digs and dirty particles given by Equation (5) and Equation (8), respectively, are significantly different, indicating that the two types of defects, dirty particles, and pockmarks, which are the same tiny point-like defects, have different polarization characteristics. These differences provide theoretical support for the detection and classification of dirty particles and pockmarks. Similarly, the different polarization characteristics of dirty particles and SSBDs are obtained from Equations (8) and (9). The above is reflected in the RMMS of the surface defects of the optical elements, which shows that the \(M_{12}\) (\(M_{21}\)), \(M_{33}\) (\(M_{44}\)), and \(M_{34}\) (\(M_{43}\)) elements of each type of defect are significantly different, to effectively differentiate between dirty particles and digs and SSBDs. Under optimal detection conditions, combining Figures 8 and 13 showed that \(M_{12}=M_{21}\), \(M_{33}=M_{44}\), and \(M_{34}=-M_{43}\), with all non-diagonal block elements equaling 0. To further validate the conclusion, the Mueller matrix elements of the defects were plotted in polar coordinates using a radius of 5 nm dirty particles immediately adjacent to the surface of the element (at a distance of 5 nm from the surface) as an example [28]. With the wavelength  
Figure 13: (**a**) The RMMS of dirty particles and digs above the surface of optical test panels; (**b**) the RMMS of dirty particles and subsurface bubble defects above the surface of optical test panels. as the radial coordinate and the relative azimuth as the angular coordinate, the polar coordinate plot of the 16 Mueller matrix elements of the dirty particles is obtained under the measurement scheme of \(\theta_{i}=\theta_{s}=70^{\circ},\phi_{s}=0^{\circ}\sim 360^{\circ}\), as shown in Figure 14, with different colors representing the magnitude of different Mueller matrix values. It is evident that \(M_{12}=M_{21}\), \(M_{13}=M_{31}\), \(M_{14}=-M_{41}\), \(M_{23}=M_{32}\), \(M_{24}=-M_{42}\), \(M_{34}=-M_{43}\), and \(M_{33}=-M_{44}\) when \(\phi_{s}=0^{\circ}/180^{\circ}\). Additionally, the non-diagonal block elements are negatively symmetric in the direction of \(\phi_{s}=0^{\circ}/180^{\circ}\) and equal to 0. The diagonal block elements are symmetric in the direction of \(\phi_{s}=0^{\circ}/180^{\circ}\). The above conclusions can be generalized to all types of defects on the surface of optical elements.