## 4 Results and Analysis  
### Simulation and Experimentation of Reflectance Mueller Matrix Spectra of Roughness and Pockmarks  
Using the pBRDF theoretical model, the RMMS of surface roughness and pockmark defects of optical elements are simulated, and the parameters to be set in advance are shown in Table 1.  
Among the 16 Muller matrix elements, \(M_{11}\) represents the intensity properties of light, and before analyzing the Muller matrix spectra, the remaining 15 Muller matrix elements are normalized to \(M_{11}\) to filter out the light intensity interference. The simulation results show that the diagonal block elements \(M_{12}\) (\(M_{21}\)), \(M_{33}\) (\(M_{44}\)), and \(M_{34}\) (\(M_{43}\)) provide effective information, and the non-diagonal block elements \(M_{13}\) = \(M_{14}\) = \(M_{23}\)= \(M_{24}\) = \(M_{31}\) = \(M_{32}\) = \(M_{41}\) = \(M_{42}\) = 0. In addition, the Muller matrix elements \(M_{12}\)(\(M_{21}\)), \(M_{33}\)(\(M_{44}\)), and \(M_{34}\)(\(M_{43}\)) effectively discriminate the surface roughness and digs on the optical component, as shown in Figure 8.  
To verify the simulation results, an optical surface defect plate conforming to MIL-PRF-13830 B and ANSI/OEOSC OP1.002 standards was selected for experimental verification. The VK-X3000 laser microscope system was used to measure the depth of the 500 \(\upmu\)m (\(\pm\)5 \(\upmu\)m) diameter dig (Area1, Figure 9a) on the optical test plate is 50 nm (\(\pm\)20 nm), as shown in Figure 9b. Moreover, the arithmetic mean height of the roughness of the non-defective area on the surface of the test plate (Area2, Figure 9a) is 0.025 \(\upmu\)m. The surface roughness and the depth of the pockmarks are much smaller than the visible wavelength. This fulfils the preconditions of the Rayleigh-Rice vector scattering theory. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Set Value** & \\ \hline \multirow{6}{*}{Detection system related} & Wavelength \(\lambda\) & 300–700 nm \\  & Angle of incidence \(\theta_{i}\) & 70\({}^{\circ}\) \\  & Scattering angle \(\theta_{s}\) & 70\({}^{\circ}\) \\  & Azimuth \(\phi_{b}\) & 0\({}^{\circ}\)/180\({}^{\circ}\) \\  & Dig depth \(d\) & 50 nm \\  & The refractive index of the test plate \(n_{2}\) & 1.5151 \\ Surface Related & Relevant length \(l\) & 1 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General parameter settings for surface defect scattering simulation. Figure 8: The simulated comparison of RMMS curves for roughness and pockmarks on optical test panels. The RMMS of the surface roughness and speckles on the optical test plate was measured using the HORIBA UVISEL PLUS ellipsometer [26; 27]. The ellipsometer uses a 75 W xenon lamp as the light source, and the spectral range covers 190-2100 nm. The experiments were performed with a spot size of 50 \(\upmu\)m and a light source wavelength of 300-700 nm. The ellipsometer was configured according to the optimal detection conditions determined in Section 3, with the incident and scattering angles set at \(\theta_{i}=\theta_{s}=70^{\circ}\) and the azimuth angle at \(\phi_{s}=\pi\), as shown in Figure 10.  
The measured RMMS of the surface roughness and digs of the optical test plate are shown in Figure 11. The difference curves between the actual measured values and the simulated values are shown in Figure 12, with the mean value of the difference being \(-0.005\) and the maximum value being 0.06. From the results of the difference between the simulated and experimentally measured Muller matrix elements, the simulated curves and the measured curves overlap almost completely, which verifies that the RMMS model of this study is accurate. In combination with Figure 8, the Mueller matrix elements \(M_{12}\) (\(M_{21}\)), \(M_{33}\) (\(M_{44}\)), and \(M_{34}\) (\(M_{43}\)) can effectively differentiate between the surface roughness and speckmarks of optical elements.