# 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  
As mentioned in the introduction section, for the polarizing section, it is sufficient to use the conventional calibration method based on the discrete Fourier coefficients [13, 15, 16] illustrated in Appendix A. However, in the case of important polarization parameters such as birefringence, the imaging section should be considered for accurate measurement results. We performed computer simulations and experimental measurements on a standard sample of air to reveal the cause of the errors arising from the polarizing section and the imaging section and their propagating effects on the Mueller matrix measurement results. The robustness of the proposed calibration methods was investigated in comparison to the conventional calibration method. It can be seen from Equation (9) that, assuming an optimized combination of controllable small errors when measuring under stable system conditions, the errors of the measurement results caused by different error sources are independent of each other, i.e., \(ex_{j}\) is independent of \(ex_{i}(j\
eq i)\); therefore, we can separately calibrate the polarizing section and imaging section. When calibrated using the conventional calibration method, the maximum error for the absolute values of the Mueller matrix elements of the measurement configuration without an objective lens could be reduced to 0.01. With the same predefined values of the PSG and PSA (\(a_{2}\), \(a_{3}\), \(a_{4}\), \(b_{2}\), and \(b_{3}\) set to \(0^{\circ}\), \(0^{\circ}\), \(0^{\circ}\), \(90^{\circ}\), and \(90^{\circ}\), respectively), we first analyzed the parasitic polarization effects in the objective lenses of the imaging section using the conventional method. The Mueller matrix of the objective lens is discussed as follows. This matrix was set to \(M_{\text{obj}}(0,\pi/4,0)\), identical to the Mueller matrix of air, and this indicated that there were no effects caused by the objective lens. In this case, we obtained the error magnification coefficient matrices of the five system parameters \(a_{2}\), \(a_{3}\), \(a_{4}\), \(b_{2}\), and \(b_{3}\), as expressed in Equation (11):  
\[ \begin{split} H_{a_{2}}=\left[\begin{array}{cccc}0&0&0&0\\ 0&0&-4&0\\ -2&4&0&0\\ 0&0&0&0\end{array}\right]H_{a_{3}}=\left[\begin{array}{cccc}0&0&-2&0\\ 0&0&4&0\\ 0&-4&0&0\\ 0&0&0&0\end{array}\right]\\ H_{a_{4}}=\left[\begin{array}{cccc}0&0&2&0\\ 0&0&-2&0\\ 0&2&0&0\\ 0&0&0&0\end{array}\right],H_{b_{2}}=\left[\begin{array}{cccc}0&0&0&0\\ -1&1&0&0\\ 0&0&1&0\\ 0&0&0&0\end{array}\right]\\ H_{b_{3}}=\left[\begin{array}{cccc}0&-1&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\end{array}\right].\end{split}  \tag{11}\]  
Equation (11) can be used with Equation (9) to obtain the errors in the Mueller matrix for air, which are caused by the polarizing elements. For example, the orientation error of the quarter-wave plate in the PSG (\(a_{2}\)) influences the Mueller matrix elements \(M_{23}\), \(M_{31}\), and \(M_{32}\), and the orientation error of the quarter-wave plate in the PSA (\(a_{3}\)) influences the Mueller matrix elements \(M_{13}\), \(M_{23}\), and \(M_{32}\). However, it also follows from Equation (11) that the error magnification coefficients that may influence the Mueller matrix elements \(M_{11}\), \(M_{14}\), \(M_{24}\), \(M_{34}\), \(M_{41}\), \(M_{42}\), \(M_{43}\), and \(M_{44}\) are zero, implying that the current system parameters in the polarizing section have no effect on these matrix elements. Generally, the Mueller matrix should be analyzed in normalized form, and the influence on the matrix element \(M_{11}\) is therefore ignored. The error sources for the other matrix elements need to be investigated. For the conventional calibration method, as described in Table 11, the Fourier coefficients \(\alpha_{3}\), \(\alpha_{4}\), \(\alpha_{6}\), \(\alpha_{7}\), \(\alpha_{9}\), \(\alpha_{11}\), \(\beta_{1}\), \(\beta_{3}\), \(\beta_{5}\), \(\beta_{7}\), \(\beta_{9}\), and \(\beta_{11}\) are combinations of one or more terms from the fourth row or column of the Mueller matrix.