## 5 Conclusions  
In this paper, based on the distinct reflection characteristics of pBRDF caused by the modulation of the incident light field by different kinds of nanoscale defects on the surface of the optical element, we proposed a method for detecting and categorizing nanoscale roughness, digs, dirty particles, and SSBDs on the surface of the optical elements via RMMS. Firstly, a numerical method was used to simulate and analyze the BRDFpp in order to obtain the optimal detection conditions \(\theta_{i}=\theta_{s}=70^{\circ},\phi_{s}=\pi\) for detecting and classifying various types of defects accurately on the surface of the optical element being tested. In addition, the pBRDF scattering models of surface roughness, digs, dirty particles, and SSBDs of optical elements were established, and the experimental results showed that the Mueller matrix elements \(M_{12}=M_{21}\), \(M_{33}=M_{44}\), and \(M_{34}=-M_{43}\) can significantly distinguish the various types of defects. The dimensions of the various types of defects on the surface can be calculated using the Mueller matrix elements. This study provides a theoretical and experimental foundation for the detection and classification of surface defects on ultra-precision optical components. The Mueller matrix contains rich polarization information, and in future work the polarization parameters can be separated and extracted by decomposing the transmission Mueller matrix to further analyze the internal structure of the optical element under test. Conceptualization, R.L., F.W. and J.H. ; methodology, R.L., F.W. and J.H. ; software, R.L. ; validation, R.L., F.W. and J.H. ; formal analysis, R.L. ; investigation, R.L. ; resources, R.L. ; data curation, R.L. ; writing--original draft preparation, R.L. ; writing--review and editing, R.L., F.W., H.H., W.C., Y.Z. and J.H. ; visualization, R.L. ; supervision, R.L., F.W. and J.H. ; project administration, R.L. and J.H. ; funding acquisition, J.H. All authors have read and agreed to the published version of the manuscript. This research was funded by Quanzhou Provincial Science and Technology Project Fund, grant number 2022C009R; Fujian Provincial Science and Technology Project Fund, grant number 2021T3060; Fujian Science & Technology Innovation Laboratory for Optoelectronic Information, grant  
Figure 16: The relationship between the particle size of dirty particles and subsurface bubble defects above the optical test plate surface and M33. Figure 15: (**a**) The relationship between \(\sigma\) and M33; (**b**) the relationship between dig depth and M33. number 2021 ZR107; and the External Cooperation Program of Chinese Academy of Sciences, grant number 121835 KYSB20180062. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study. **Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request. **Acknowledgments:** The authors would like to thank [PERSON] of Xiamen Rare Earth Materials Research Center. **Conflicts of Interest:** The authors declare no conflict of interest. +++ ==WARNING: Truncated because of repetitions==  
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* [PERSON] et al. (1990) [PERSON]; [PERSON]; [PERSON] _The Prediction of BRDFs from Surface Profile Measurements_; [PERSON]; SPIE: San Diego, CA, USA, 1990; p. 136. * [PERSON] et al. (2023) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Particulate Mueller Matrix Polarimetry. _Opt. Laser Technol._**2023**, _158_, 108780. [CrossRef]
* [PERSON] and [PERSON] (1987) [PERSON]; [PERSON] Light-Scattering Models for Spheres on a Conducting Plane: Comparison with Experiment. _Appl. Opt._**1987**, _26_, 2995-2999. [CrossRef]
* [PERSON] et al. (2020) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Double-Sided and Single-Sided Polished 6H-SiC Wafers with Subsurface Damage Layer Studied by Mueller Matrix Ellipsometry. _J. Appl. Phys._**2020**, _128_, 235304. [CrossRef]
* [PERSON] et al.