# Correlation of Optical Constants and Morphologies with Mueller Matrix for Micro-Rough Surfaces
## 3 Results and Discussion
### The Relationships Between Morphological Parameters and Mueller Matrix Elements
#### 3.2.1 The Relationships Between Relative Roughness and Mueller Matrix Elements  
Relative roughness, \(\sigma/\lambda\), is usually used to describe the degree of roughness of solid surfaces. Figure 7 shows the Mueller matrix elements changing with \(\sigma/\lambda\). The morphological parameters and optical constants of randomly rough solid surfaces are as follows: \(\sigma\) = 0.06 \(\upmu\)m, \(\tau\) = 0.08 \(\upmu\)m, \(n\) = 3.42, and \(k\) = 2. An incident angle, \(\theta\), of 65\({}^{\circ}\) is selected, and the incident wavelength range is 4 \(\upmu\)m\(\leq\lambda\leq\) 20 \(\upmu\)m. Studying all the Mueller matrix elements, it is obvious that the curve of each element varying with \(\sigma/\lambda\) is approximately linear. The values of \(m_{12}\) and \(m_{21}\) are the same, and both of them remain unchanged as \(\sigma/\lambda\) increases. The specific expression of \(m_{21}\) in the NSC representation in Equation (4) is as follows:  
\[m_{21}=-N-\alpha_{sp}=(\gamma^{2}-\gamma_{sp}^{2}+\gamma_{ps}^{2}-1)/(\gamma^ {2}+\gamma_{sp}^{2}+\gamma_{ps}^{2}+1) \tag{28}\]  
By combining this with Equation (27), we see that \(m_{12}\) and \(m_{21}\) are only influenced by the amplitude ratios of polarized light. That is to say, the effect of relative roughness on the amplitude ratios is negligible. In addition, the values of \(m_{33}\) and \(m_{44}\) are also identical, and both of them increase linearly as \(\sigma/\lambda\) increases. \(m_{34}\) and \(m_{43}\) are the negative of each other, and \(m_{34}\) increases linearly with \(\sigma/\lambda\), while \(m_{43}\) decreases correspondingly. \(m_{22}\) remains unchanged at 1.  
The extinction coefficient is set to \(k\) = 0, while the other parameters remain unchanged, as shown in Figure 7. Figure 8 shows the variation in the Mueller matrix elements with \(\sigma/\lambda\) for two material types. An extinction coefficient, \(k\), of 0 is selected, and the other  
Figure 7: The Mueller matrix of randomly rough surfaces as a function of the relative roughness, \(\sigma/\lambda\). The rms roughness remains unchanged at \(\sigma\) = 0.06 \(\upmu\)m, and the wavelengths are varied from 4 \(\upmu\)m to 20 \(\upmu\)m. The optical constants are as follows: \(n\) = 3.42 and \(k\) = 2. The correlation length and incident angle are fixed at \(\tau\) = 0.08 \(\upmu\)m and \(\theta\)= 65\({}^{\circ}\), respectively.  
parameters remain unchanged, as shown in Figure 7. A comparison of two kinds of materials' Mueller matrix elements changing with \(\sigma/\lambda\) is shown in Figure 8. It can be seen from Figure 8a that the \(m_{12}\) element of both rough surfaces remains constant as \(\sigma/\lambda\) increases. This indicates that the effect of relative roughness on amplitude ratios is unrelated to the material composition of the surfaces. As shown in Figure 8b, the \(m_{33}\) element of surfaces with different extinction coefficients increases linearly with \(\sigma/\lambda\), and the slopes of the curves depend on \(k\). Hence, the dependence of \(m_{33}\) on \(\sigma/\lambda\) is related to the intrinsic properties of the materials. Figure 8c shows that the \(m_{34}\) element of randomly rough surfaces with a \(k\) value of 0 and 2 increases linearly with \(\sigma/\lambda\), and the slopes of the two curves are identical. This implies that the effect of \(\sigma/\lambda\) on \(m_{34}\) is independent of the extinction coefficient, \(k\), of rough surfaces, which can be used to deduce the root mean square roughness (rms), \(\sigma\), of randomly rough surfaces. Combining this with Equation (4), we can obtain the following expression:  
\[(S+\beta_{2})\propto\sigma/\lambda \tag{29}\]  
Therefore, the \(\sigma\) of rough surfaces may be deduced by \(m_{34}\), which can be measured by an ellipsometer.  
Figure 8: The independent Mueller matrix element (**a**) \(m_{12}\); (**b**) \(m_{33}\); (**c**) \(m_{34}\) of randomly rough surfaces as a function of the relative roughness, \(\sigma/\lambda\), at different extinction coefficients, \(k\).