## 2 Theory
### Scattering Model for Dig Defects  
Digs refer to pits and blemishes on the surface of an optical element, as shown in Figure 2, which gives a composite scattering schematic of digs. For tiny-sized dig defects with depths smaller than the wavelength of visible light, often between tens to hundreds of nanometers, the Rayleigh-Rice theory effectively describes the scattering characteristics of such surface features.  
Figure 1: Schematic of BRDF scattering geometry.  
According to the electromagnetic theory of physical optics, the relationship between the corresponding \(s\)- and \(p\)-components \(E_{inc}^{s}\) and \(E_{inc}^{p}\) of the incident field and the \(s\)- and \(p\)-components \(E_{scat}^{s}\) and \(E_{scat}^{p}\) of the scattered field can be established through the Jones matrix [3];  
\[\begin{bmatrix}E_{scat}^{s}\\ E_{scat}^{p}\end{bmatrix}=\frac{e^{ikr}}{R}\begin{bmatrix}\begin{array}{cc}J_{ss }&J_{ps}\\ J_{sp}&J_{pp}\end{array}\begin{bmatrix}E_{inc}^{s}\\ E_{inc}^{p}\end{bmatrix} \tag{3}\]  
The following transformation relationship exists between the Mueller and Jones matrices [22]:  
\[\begin{array}{l}M_{11}=\left(\left|[PERSON]_{pp}\right|^{2}+\left|[PERSON]_{ss}\right|^{2}+ \left|[PERSON]_{ps}\right|^{2}+\left|[PERSON]_{sp}\right|^{2}\right)/2,\quad M_{12}=\left( \left|[PERSON]_{ps}\right|^{2}+\left|[PERSON]_{ss}\right|^{2}-\left|[PERSON]_{pp}\right|^{2}-\left| [PERSON]_{sp}\right|^{2}\right)/2,\\ M_{13}=\operatorname{Re}\left([PERSON]_{ss}[PERSON]_{ps}+J_{pp}[PERSON]_{ps}\right),\quad M_{14}= \operatorname{Im}\left([PERSON]_{ss}[PERSON]_{ps}\right.^{*}-J_{pp}[PERSON]_{ps}\right),\\ M_{21}=\left(\left|[PERSON]_{ss}\right|^{2}-\left|[PERSON]_{pp}\right|^{2}-\left|[PERSON]_{ps} \right|^{2}+\left|[PERSON]_{ps}\right|^{2}\right)/2,\quad M_{22}=\left(\left|[PERSON]_{ss} \right|^{2}+\left|[PERSON]_{pp}\right|^{2}-\left|[PERSON]_{ps}\right|^{2}-\left|[PERSON]_{sp} \right|^{2}\right)/2,\\ M_{23}=\operatorname{Re}\left(J_{ss}J_{ps}\right.^{*}-J_{pp}J_{ps}\left.{}^{* }\right),\quad M_{24}=\operatorname{Im}\left(J_{ss}J_{ps}\right.^{*}+J_{pp}J_{ ps}\left.{}^{*}\right),\\ M_{31}=\operatorname{Re}\left(J_{ss}J_{ps}+J_{pp}J_{ss}\left.{}^{*}\right), \quad M_{32}=\operatorname{Re}\left(J_{ss}J_{ps}-J_{pp}J_{ps}\left.{}^{*} \right),\\ M_{33}=\operatorname{Re}\left(J_{pp}J_{ss}\right.^{*}+J_{sp}J_{ps}\left.{}^{* }\right),\quad M_{34}=\operatorname{Im}\left(J_{ss}J_{pp}+J_{ps}J_{ps}\left.{} ^{*}\right),\\ M_{41}=\operatorname{Im}\left(J_{pp}J_{ps}\left.{}^{*}+J_{ps}J_{ss}\left.{}^{* }\right),\quad M_{42}=\operatorname{Im}\left(J_{pp}J_{ps}\left.{}^{*}-J_{ps}J_{ ps}\left.{}^{*}\right),\\ M_{43}=\operatorname{Im}\left(J_{pp}J_{ss}\left.{}^{*}+J_{sp}J_{ps}\left.{}^{* }\right),\quad M_{44}=\operatorname{Im}\left(J_{pp}J_{ss}\left.{}^{*}-J_{sp}J_{ ps}\left.{}^{*}\right).\right.\end{array} \tag{4}\]  
The expression for the scattering characteristics of the Mueller matrix (pBRDF) due to the modulation of the incident optical field by the surface digs of the optical element is given by the following equation:  
\[\text{pBRDF}_{dig}=\frac{16\pi^{2}}{\lambda^{4}}\cos\theta_{i}\cos\theta_{i}S _{PSD}\left(f_{x},f_{y}\right)\times M_{ij} \tag{5}\]  
the pBRDF comprises 16 different elements, each with a distinct meaning. The value of \(M_{11}\) reflects the target's ability to transmit, scatter, and reflect incident light. \(M_{12}\), \(M_{13}\), and \(M_{14}\) reflect the target's ability to attenuate incident light horizontally, vertically, and in a circular bidirectional manner, respectively. \(M_{21}\), \(M_{31}\), and \(M_{41}\) reflect the target's ability to polarize incident unpolarized light, and the remaining nine elements reflect the target's ability for incident light depolarization and phase delay [15].