# 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
#### 3.2.2 The Relationships Between Correlation Length and Mueller Matrix Elements  
The lateral dimension of randomly rough surfaces is characterized by the correlation length, \(\tau\), while the ratio of rms roughness to correlation length, \(\sigma/\tau\), is typically used in practice. Figure 9 shows that the Mueller matrix elements of rough surfaces change as \(\sigma/\tau\) increases from 0.125 to 0.75. Here, \(\sigma\) is fixed at 0.05 \(\upmu\)m, with only \(\tau\) varying. An incident angle \(\theta=65^{\circ}\), an incident wavelength \(\lambda=4\)\(\upmu\)m, and an extinction coefficient \(k=0\) are selected, and the refractive index, \(n\), is the same as that of silicon. For \(m_{22}\), \(m_{33}\), \(m_{34}\), \(m_{43}\), and \(m_{44}\), Figure 9 exhibits the same trend as Figure 7. It is clear that the values of \(m_{12}\) and \(m_{21}\) are the same, and both of them remain unchanged as \(\sigma/\tau\) increases. Combining this with Equations (27) and (28), we conclude that the effect of the correlation length on amplitude ratios is also negligible.  
Extinction coefficients of \(k=0\), \(k=2\) and \(k=4\) are selected, with the other parameters remaining unchanged, as shown in Figure 9. Figure 10 shows the variation in the Mueller matrix elements of three material types with \(\sigma/\tau\). From Figure 10a, we can conclude that the effect of rough surfaces' morphologies on amplitude ratios is minimal and material-independent. As shown in Figure 10b, the \(m_{33}\) element of rough surfaces increases linearly with \(\sigma/\tau\) for \(k=2\) and \(k=4\), and the slopes of these curves are identical but differ from those of surfaces with \(k=0\). Therefore, the presence or absence of absorptivity in the material will affect the relationship between \(m_{33}\) and \(\sigma/\tau\). Figure 10c shows that \(m_{34}\) increases linearly with \(\sigma/\tau\) for randomly rough surfaces with \(k=0\), \(2\), and \(4\), exhibiting identical slopes. This demonstrates that the relationship between m34 and \(\sigma/\tau\) is independent of \(k\). Combining this with Figure 6c, we observe that the effects of \(\sigma/\tau\) on \(m_{34}\) are independent of \(k\) within an extinction coefficient range of \(0\leq k\leq\)10. This can be used to deduce the correlation  
Figure 9: The Mueller matrix of randomly rough surfaces as a function of the ratio of rms roughness to correlation length \(\sigma/\tau\). The rms roughness remains unchanged at \(\sigma=0.05\)\(\upmu\)m, and the correlation length, \(\tau\), is varied from 0.067 \(\upmu\)m to 0.4 \(\upmu\)m. An extinction coefficient \(k=0\) is selected, and the refractive index, \(n\), is the same as that of silicon. The incident wavelength and angle are fixed at \(\lambda=4\)\(\upmu\)m and \(\theta=65^{\circ}\), respectively.  
length, \(\tau\), of randomly rough surfaces. By combining this with Equation (4), we derive the following expression:  
\[(S+\beta_{2})\propto\sigma/\lambda \tag{30}\]  
Therefore, the \(\tau\) of rough surfaces may be deduced by \(m_{34}\), which can be measured by an ellipsometer.