# Error Analysis and Calibration Improvement of the Imaging Section in a Mueller Matrix Microscope
## 3 Application and Discussion
### Calibration Application on a Standard Sample of Air
However, the system parameters in the polarizing section are calculated in terms of \(\alpha_{2}\), \(\alpha_{6}\), \(\alpha_{8}\), \(\alpha_{10}\), \(\beta_{2}\), \(\beta_{6}\), \(\beta_{8}\), and \(\beta_{10}\), while the system parameter \(a_{3}\) is calculated in terms of \(\alpha_{6}\). Therefore, the fourth row and column of the Mueller matrix have been regarded as redundant and ignored. This, however, results in information loss and calibration inaccuracy. From Equation (11), it can be observed that the matrix elements in the fourth row and fourth column of the error magnification coefficient matrices of the polarizing section are all zeros. To discuss whether these results were the outcome of some particular default values, we also assigned values other than the default values to the system parameters in the polarizing section. The results show that the calculated error magnification coefficients are stable, and the elements in the fourth row and column are always near zero when the five system parameters simultaneously deviate from the default; this indicates that the errors of the measurement system that propagate to these elements cannot be calibrated using the conventional method. One simulation result for the error transform coefficient matrix is shown in Figure 3, where the horizontal axis of each subfigure corresponds to the five parameters in the polarizing section, the vertical axis corresponds to the simulation times, and different colors correspond to the values of the matrix elements. As the maximum absolute value of any element of the error transform coefficient matrices in Equation (11) is 4, random values in the range of \(-0.05\) to \(0.05\) were assigned to every system parameter (this range was used because it results in a maximum absolute error of 0.2 before the calibration and is available for the visualization of the results, according to the calculated values determined using Equation (9)). Thus, the error magnification coefficient matrix of the imaging section is supplemented, thereby reflecting the error induced by the objective lens, and the calculated results are presented by Equation (12):  
\[H_{\theta_{O}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],H_{\psi_{O}}=\left[ \begin{array}{cccc}0&2&0&0\\ 2&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],H_{\Delta_{O}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&-1&0\end{array} \right]. \tag{12}\]  
The calculated results for rotation angle \(\theta_{O}\) and ellipsometric angles \(\psi_{O}\) and \(\Delta_{O}\) are shown in Equation (12). Because all the elements in \(H_{\theta_{O}}\) are zero, Equation (12) indicates that the rotation angle \(\theta_{O}\) has no effect at any point in time. To test whether the zero results were caused by particular default values, deviating values were once again assigned to the parameters, and the Mueller matrix of the objective lens was reset to \(M_{\text{obj}}(0.01,\pi/4+0.01,0.01)\), although 0.01 is the recommended value according to our experimental results. The error magnification coefficients of the polarizing section were recalculated to ensure accurate calibration. The results, as presented in Table 1(a), Table 1(b), and Figure 4a, show that the maximum error could usually be reduced to 0.01 or less. Elements \(M_{24}\), \(M_{34}\), \(M_{42}\), and \(M_{43}\), even after the polarizing section was calibrated, constitute an exception; therefore, these elements must have been mainly influenced by the imaging section, instead of the polarizing one. Figure 3: Simulation results of the error transform coefficient matrix of the polarizing section of the microscope using the conventional calibration method. In each subfigure, the horizontal axis shows the five polarizing section system parameters, and the vertical axis shows the time. The range of colors indicates the numerical values of the matrix elements.