# Error Analysis and Calibration Improvement of the Imaging Section in a Mueller Matrix Microscope
## 2 Materials and Methods
### Mathematical Model and Calibration Algorithm
To make the Mueller matrices of the objective lens and air similar, i.e., similar to the identity matrix, the default values of \(\theta_{O}\), \(\psi_{O}\), and \(\Delta_{O}\) are set to 0, \(\pi/4\), and 0, respectively. Furthermore, if the values of \(\theta_{O}\), \(\psi_{O}\), and \(\Delta_{O}\) deviate from the default values, the objective lens will cause a change in the polarization characteristics of the incident light. Therefore, the instrument matrix depends not only on the parameters of the polarizing elements but also on those of the objective lens in the imaging section, which can change the polarization state of the incident light. The conventional calibration method involves derivation of the equations using the Fourier coefficients of the measured intensities, and the calibration performance was reported in Reference [10]. However, the robustness of this method needs to be improved, and the polarization influence, i.e., the birefringence effect of the imaging section must be discussed. In the present study, we re-establish the relationship between the errors caused by the optical elements, including the objective lens, and the Mueller matrix measurement result of the sample. First, it is important to note that random errors of light intensity are also a potential source of measurement error, and as such, Equation (1) should be modified to  
\[\overset{\rightarrow}{P}+\varepsilon\overset{\rightarrow}{P}=(W+ \varepsilon W)\overset{\rightarrow}{(\overset{\rightarrow}{M}+ \varepsilon\overset{\rightarrow}{M})} \tag{4}\]  
where \(\varepsilon\overset{\rightarrow}{P}\) is the error of the light intensity measured by the CCD, \(\varepsilon\overset{\rightarrow}{M}\) is the error of the Mueller matrix of the sample, and \(\varepsilon W\) is the error caused by misalignments and imperfections of the polarizing elements and objective lens introduced in the process of constructing the instrument matrix. Then, according to Equations (1) and (4),  
\[\varepsilon\overset{\rightarrow}{M}=pinv(W+\varepsilon W)\varepsilon \overset{\rightarrow}{P}-pinv(W+\varepsilon W)\varepsilon W\overset{ \rightarrow}{M} \tag{5}\]  
where the symbol \(pinv\) indicates pseudoinverse. In the present dual-rotating-retarder configuration, two fixed linear polarizers are placed in front of the light source and the CCD; therefore, the sensitivity of the CCD and the error caused by the partially polarized source are not considered. Thus, the most crucial factor determining \(\varepsilon\overset{\rightarrow}{P}\) is the light intensity stability of the light source, and this random error could be reduced by performing repeated measurements. As the light intensity stability of the light source depends on the specific light source used, we will not describe it further in this paper and will reduce Equation (5) to  
\[\varepsilon\overset{\rightarrow}{M}=-pinv(W+\varepsilon W)\varepsilon \overset{\rightarrow}{W}. \tag{6}\]The instrument matrix used in the experiment is defined by \(W_{\text{exp}}=W+\varepsilon W\). It depends on the various sources of error: \(W_{\text{exp}}(x_{1},x_{2},\ldots,x_{n})\), where the \(j\)th error source \(x_{j}\) has the default value \(x_{j}^{(0)}\). Then, \(W_{\text{exp}}\) can be expanded in a Taylor series as  
\[ \begin{array}{l}W_{\text{exp}}(x_{1},x_{2},\ldots,x_{n})=W(x_{1}^{(0)},x_{2} ^{(0)},\ldots,x_{n}^{(0)})+\sum\limits_{j=1}^{n}\frac{\partial W}{\partial x_{ j}}(x_{j}-x_{j}^{(0)})\\ +\frac{1}{2}\sum\limits_{j=1}^{n}\frac{\partial^{2}W}{\partial x_{j}\partial x _{k}}(x_{j}-x_{j}^{(0)})(x_{k}-x_{k}^{(0)})+\ldots\end{array}  \tag{7}\]  
Furthermore, when the higher-order terms in Equation (7) are small, they can be ignored. We first substitute \(W+\varepsilon W\) with \(W_{\text{exp}}\), where \(\varepsilon W\) is the first-order term in Equation (7) and \(x_{j}-x_{j}^{(0)}\) is defined as \(\varepsilon x_{j}\).