# Correlation of Optical Constants and Morphologies with Mueller Matrix for Micro-Rough Surfaces
## 1 Introduction  
Rough solid surfaces refer to solid samples whose surfaces are imperfect no matter how well they are polished [1]. A \(4\times 4\) Mueller matrix comprising 16 elements contains information about anisotropy and depolarization. Therefore, the Mueller matrix is usually utilized to investigate rough surfaces. [PERSON] et al. [2] used the Mueller matrix to obtain the polarization characteristics in the specular reflection direction of a rough surface. [PERSON] et al. [3] obtained the Mueller matrix of a randomly rough surface under specific morphological parameters through the Kirchhoff approximation method. [PERSON] et al. [4] used six polarization states of incident light, calculated their scattering polarization states, and obtained a rough surface Mueller matrix through combination. [PERSON] et al. [5] found that \(m_{34}\) and \(m_{43}\) are key elements in the Mueller matrix that can be used to identify materials with rough surfaces. [PERSON] et al. [6] investigated the interplay between incident wavelength, angle of incidence, roughness parameters, and the contributions of diffuse and specular reflections utilizing the Mueller matrix of rough surfaces. The roughness of surfaces can significantly influence the precision of optical constants, which are measured by spectroscopic ellipsometry (SE) based on the Fresnel equation [7, 8, 9, 10]. In order to reduce this influence, the effective medium approximation (EMA) is usually applied to model the rough surfaces of solid materials [11; 12; 13]. However, the EMA model may cause a large deviation in the SE data analysis since it only accounts for the height irregularities of rough surfaces but neglects the effect of lateral irregularities on electromagnetic scattering from rough surfaces [14]. [PERSON] et al. [15] proposed a novel inversion method without using the EMA to extract optical constants from SE parameters for solid materials with micro-rough surfaces based on the first-principles calculations of Maxwell's equations and the Levenberg-Marquardt optimization algorithm. This method does indeed have wider applicability and higher precision than the EMA model, but it also has a defect in that non-unique inversion may occur. The Mueller matrix with 4\(\times\)4 elements of rough surfaces of solid materials can be used to develop more relations with optical constants to avoid this non-uniqueness. The morphologies of randomly rough surfaces are closely connected with remote sensing [16; 17], target identification [18; 19], and the Synthetic Aperture Radar imaging technique [20; 21]. Normally, the root mean square (rms) roughness, \(\sigma\), and correlation length, \(\tau\), representing height and lateral irregularities, respectively, are utilized to characterize the morphologies of rough surfaces. Currently available metrology tools, such as scanning electron microscopy (SEM), atomic force microscopy (AFM), and transmission electron microscopy (TEM), are mostly suitable for the measurement of the morphologies of randomly rough surfaces [22]. These instruments have the advantages of being reliable and having high resolution, but they also have defects like working slowly or destructively. A 4 \(\times\) 4 Mueller matrix associating the Stokes vectors of incident electromagnetic waves with emergent ones can fully reflect the morphological information of rough surfaces [23; 24; 25]. Compared with scanning tools, the method of directly deducing the morphologies of rough surfaces utilizing a Mueller matrix has attractive merits, such as being low-cost, time-saving, and non-destructive and allowing for in situ measurements. In this work, we utilized the finite-difference time-domain (FDTD) method, which can simulate electromagnetic responses to obtain the reflection coefficients of randomly micro-rough surfaces for incident light in the p and s directions, respectively. Then, we calculated the 4 \(\times\) 4 Mueller matrix of rough surfaces according to the formula used for the Jones-to-Mueller matrix conversion.