# Correlation of Optical Constants and Morphologies with Mueller Matrix for Micro-Rough Surfaces
## 4 Conclusions  
In this work, the electromagnetic responses of randomly micro-rough solid surfaces changing with optical constants and morphological parameters are simulated by the FDTD method, and the reflection coefficients of the surfaces with incident light in the p and s directions are obtained. Then, the Mueller matrix is calculated in the form of a Jones-Mueller matrix conversion. The simulation method is validated with the experimental results, which are measured by MMSE. The Mueller matrix of micro-rough surfaces has three independent elements, and the effects of optical constants and morphological parameters on them are specifically investigated. We find that the Mueller matrix element \(m_{12}\) is insensitive to rough surface morphologies when the optical constants lie within the ranges \(0\leq n\leq 4\) and \(0\leq k\leq 10\). There may exist a functional relationship between the optical constants of such surfaces and \(m_{12}\), suggesting \(m_{12}\)'s potential value for inverting optical constants. The morphological parameters \(\sigma/\lambda\) and \(\sigma/\tau\) are proportional  
Figure 10: The independent Mueller matrix elements (**a**) \(m_{12}\); (**b**) \(m_{33}\); (**c**) \(m_{34}\) of randomly rough surfaces as a function of the ratio of rms roughness to correlation length, \(\sigma/\tau\), at different extinction coefficients, \(k\).  
to the Mueller matrix element \(m_{34}\), and the expressions \((S+\beta_{2})\propto\sigma/\lambda\) and \((S+\beta_{2})\propto\sigma/\tau\) can be obtained within morphological parameter ranges of \(0.003\leq\sigma/\lambda\leq 0.015\) and \(0.125\leq\sigma/\tau\leq 0.75\). And the ratio (sensitiveness to rough surfaces) does not change with \(k\), so \(m_{34}\) has the potential to predict the morphologies of rough solid surfaces. The Mueller matrix element \(m_{33}\) can be regarded as a function of several variables, including \(n\), \(k\), \(\sigma\), and \(\tau\), which is more complicated, requiring further in-depth research.  
In summary, exploring the correlation of optical constants and morphological parameters with Mueller matrix elements for randomly rough surfaces is a complex process. This study lays the foundation for developing a new method to invert the optical constants and morphologies of rough surfaces with a Mueller matrix.  
Conceptualization, M.H.; methodology, M.H.; software, M.H.; validation, F.J.; formal analysis, M.H.; investigation, M.H.; resources, F.J.; data curation, M.H.; writing--original draft preparation, M.H.; writing--review and editing, F.J.; visualization, M.H.; supervision, F.J.; funding acquisition, F.J. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the National Natural Science Foundation of China under grant number 62205334.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
We would like to thank [PERSON] from the Harbin Institute of Technology for guidance on simulation and experimental methods.  
The authors declare no conflicts of interest.