# Error Analysis and Calibration Improvement of the Imaging Section in a Mueller Matrix Microscope
## 3 Application and Discussion
### General Applicability  
Measured samples often have highly variable content and quality in consideration of the polarization properties. To examine the general applicability of the proposed calibration method, the measurement results of samples of air were first examined; then, a quarter-wave plate with a distinct birefringence effect was selected; and finally, Daphnia, a kind of plankton that demonstrates hierarchical structure details in its dying process, was investigated. The polarization parameter \(t3\), which characterizes the amplitude value of birefringence and is derived from elements \(M_{24}\), \(M_{34}\), \(M_{42}\), and \(M_{43}\), was considered to discuss the calibration results of the measurement. In addition, the values of the system parameters \(a_{2}\) and \(a_{3}\) were set to \(-15^{\circ}\), \(-10^{\circ}\), \(-5^{\circ}\), \(0^{\circ}\), \(5^{\circ}\), \(10^{\circ}\), and \(15^{\circ}\) in the measurement of the air and the quarter-wave plate samples, while \(a_{2}\) and \(a_{3}\) were both set to \(0^{\circ}\) for the daphnia sample to identify the improvement on its hierarchical structure details. For the air sample, the measurement results of \(t3\) (with a theoretical value of 0) shown in Figure 7a are stable when \(a_{2}\) or \(a_{3}\) deviates from the default and when the average values are 0.0461 and 0.0455, respectively, meaning that \(t3\) is independent of \(a_{2}\) and \(a_{3}\) and that the errors caused by the imaging section are constant. Thus, after calibrating the imaging section, the order of magnitude of the error of \(t3\) is reduced to 0.001, as depicted in Figure 7b. Figure 7 depicts not only the difference in the  
Figure 5: Calibration results of the polarizing section using the conventional method when (**a**) \(a_{2}\) or (**b**) \(a_{3}\) deviates from the default value. Figure 6: Calibration results of the polarizing section using the proposed method when (**a**) \(a_{2}\) or (**b**) \(a_{3}\) deviates from the default value. measurement results of \(t3\) for the conventional and proposed calibration methods but also the stability of the errors caused by the imaging section. In addition, the measurement results for the quarter-wave plate sample (Figure 8b) are closer to the theoretical value of \(t3\) (with an average error less than 0.001) and have a stable trend, indicating more stable and accurate test results than those illustrated in Figure 8a. The calibration result of parameter \(b_{3}\) in the polarizing section fluctuates considerably (shown in Figure 5b), while the results in Figure 7a are constantly stable, which suggests that the birefringence of air is mainly influenced by the imaging section. For a measurement sample that itself exhibits birefringence, such as the quarter-wave plate (with a theoretical value of 0.5 for \(t3\)), the measurement tendencies of \(t3\) as shown in Figure 8a,b are similar to those of \(b_{3}\) in Figures 5 and 6, respectively. In other words, the calibration results of parameter \(b_{3}\) affect the calculation of the birefringence parameter, \(t3\), of materials with strong birefringence. Finally, we compared the calculation accuracy and the clarity of the hierarchical details of the conventional method and the proposed method when imaging plankton samples for the \(t3\) parameter. A micrograph of the Daphnia sample is shown in Figure 9, and the measurement results of this sample are shown in Figure 10.  
Figure 8: Measurement results of \(t3\) for the quarter-wave plate sample (**a**) after calibrating the polarizing section with the conventional method and (**b**) after calibrating both the polarizing and imaging sections with the proposed method. Figure 7: Measurement results of \(t3\) for the air sample (**a**) after calibrating the polarizing section with the conventional method and (**b**) after calibrating both the polarizing and imaging sections with the proposed method. The Daphnia occupies the middle area, so the pixels at the edge correspond to the air.