## 1 Introduction  
The detection and identification of surface defects is a key aspect of the quality evaluation of optical elements. With the wide application of precision optical instruments such as ultra-smooth optical elements [1], there is an increasing demand for the detection and identification of surface defects at the nanometer scale [2]. For example, in the inertial confinement fusion system (ICF), the aperture of optical elements can reach about 1 m, while the measurement accuracy of surface defects is required to reach the micro-nanometer level. Surface defect detection includes not only the characterization of surface morphology, surface shape errors, and fluctuations in roughness but also the identification of defects such as surface pockmarks and subsurface damage. Surface defects can be categorized as pockmarks, SSBDs, and dust particles, among others. The different types of damage and the corresponding repair techniques vary greatly; so, it is important to detect and classify various types of component defects [3]. It has been common to use manual visual inspection to evaluate the quality of ultra-precision optical components [4]. However, this method relies on manual subjective judgment, and the evaluation results are unstable and inefficient. Therefore, more and more academics are devoted to the research of inspection methods that can realize the quality evaluation of optical components, such as photothermography [5], optical interferometry [6], optical coherence tomography [7], and microscopic dark-field scattering imaging [8]. The microscopic dark-field scattering imaging method is currently the main surface defect detection method and uses a specific device to collect and image the scattered light induced by surface defects. However, the method only makes use of the radiation intensity and geometric properties of electromagnetic waves and is based on limited information. It is therefore difficult to distinguish dirty particles and pockmarks that appear as dots on dark-field intensity images [1; 9]. Since different types of microscopic defects on the surface of optical elements affect the polarization state of light to different degrees, [PERSON] et al. [10; 11] added polarization information to the original scattered light intensity information to analyze the defects existing on the surface of a sample. [PERSON] and [PERSON] [12] used ray tracing to find the scattered field distribution based on the theory of polarization and put forward the idea of the Mueller's matrix. [PERSON] et al. [13] used p- and s-polarized light to establish a model of the polarization effect and used all the elements in the Mueller matrix to characterize the polarization properties of the scattered light from a damaged metal surface with roughness in the order of micrometers. However, the accurate and fast detection and classification of various types of nanoscale defects on the surface of optical elements is still a great challenge. The BRDF [10] is a commonly used function for modeling and investigating the surface scattering properties of optical elements and is defined as the ratio of the irradiance exiting through the target surface to the irradiance incident on the target surface. The polarized bidirectional reflection distribution function (pBRDF) is generalized to the polarization case by means of the Mueller matrix based on the BRDF [14]. In the field of defect detection, pBRDF can consider the spatial scattering distribution of various defects on the surface of an optical element more comprehensively than the BRDF. It is able not only to completely describe the light scattering distribution of defects on the surface of the element, but also to characterize the light scattering polarization properties of the target defects. Therefore, the pBRDF has become an important tool for obtaining polarization detection information of target defects [15].