# Correlation of Optical Constants and Morphologies with Mueller Matrix for Micro-Rough Surfaces
## 3 Results and Discussion
### The Relationships Between Optical Constants and Mueller Matrix Elements
#### 3.1.1 The Relationships Between the Refractive Index and Mueller Matrix Elements  
Figure 3 shows the variation in Mueller matrix elements with the refractive index, \(n\), increasing from 2 to 10, while the extinction coefficient, \(k\), remains unchanged at 0. The morphological parameters of randomly rough solid surfaces are as follows: \(\sigma\) = 0.05 \(\upmu\)m and \(\tau\) = 0.1 \(\upmu\)m. The wavelength, \(\lambda\), of the incident light is 6 \(\upmu\)m, and the incident angle, \(\theta\), is set to 65\({}^{\circ}\). Compared with the main diagonal elements, the values of the off-diagonal elements are so small that they can be neglected. The surfaces that we simulated are randomly micro-rough surfaces (relative roughness \(\sigma/\lambda\) = 0.0083), and the root mean square (rms) roughness, \(\sigma\), and correlation length, \(\tau\), are much smaller than the wavelength, \(\lambda\), of the incident light. Therefore, randomly micro-rough surfaces can be considered isotropic, and we can only discuss the main diagonal elements. It can be seen that the values of the Mueller matrix elements \(m_{12}\) and \(m_{21}\) are the same, and both of them increase as \(n\) increases from 2 to 10. The values of \(m_{34}\) and \(m_{43}\) are mutually opposite, resulting in inversely related curves. In addition, \(m_{33}\) and \(m_{44}\) are identical. It is worth noting that the values of \(m_{22}\) remain unchanged at 1.  
Through the above analysis, we find that not all Mueller matrix elements are independent. Here, we only need to discuss the sensitivities of three elements that are independent of rough surfaces.  
Figure 2: Comparison of the measured and simulated Mueller matrices of the sample. The wavelengths vary from 200 nm to 1000 nm. The incident and azimuthal angles are fixed at \(\theta\) = 55\({}^{\circ}\) and \(\varphi\) = 30\({}^{\circ}\), respectively.  
Figure 4 illustrates the comparisons of the \(m_{12}\), \(m_{33}\), and \(m_{34}\) elements of rough surfaces and those of smooth plates, whose optical constants are the same as those of rough surfaces. As shown in Figure 4a, the curve correlating with \(m_{12}\) of rough surfaces is coincident with that of smooth plates when \(n\leq 4\), while the difference between them increases as \(n\) increases from 4 to 10. This indicates the Mueller matrix element \(m_{12}\) is not sensitive to rough surfaces until \(n\) increases to 4. The specific expression of \(m_{12}\) in the NSC representation in Equation (4) is as follows [24]:  
\[m_{12}=-N-\alpha_{ps}=(\gamma^{2}+\gamma_{sp}^{2}-\gamma_{ps}^{2}-1)/(\gamma^{ 2}+\gamma_{sp}^{2}+\gamma_{ps}^{2}+1) \tag{27}\]  
We find that the value of \(m_{12}\) only depends on the amplitude ratios of polarized light. Therefore, the amplitude ratios of polarized light induced by rough surfaces change little until \(n\) exceeds 4. Figure 4b shows that the \(m_{33}\) curves of rough surfaces and smooth plates changing with n follow the same trend, and \(m_{33}\) is the least sensitive to rough surfaces within the refractive index range of \(3\leq n\leq 4\). From Figure 4c, it can be observed that the differences in \(m_{34}\) between the two types of surfaces increase as \(n\) increases.  
Figure 3: The Mueller matrix of randomly rough surfaces as a function of the refractive index. The extinction coefficient remains unchanged at \(k=0\) and the morphological parameters are as follows: \(\sigma=0.05\)\(\upmu\)m and \(\tau=0.1\)\(\upmu\)m. The incident wavelength and angle are fixed at \(\lambda=6\)\(\upmu\)m and \(\theta=65^{\circ}\), respectively.