# Error Analysis and Calibration Improvement of the Imaging Section in a Mueller Matrix Microscope
## 2 Materials and Methods
### Mathematical Model and Calibration Algorithm  
As discussed above, system models of a dual-rotating-retarder polarimeter were developed. The Mueller matrix measurement generally involves the establishment and solving of the linear equations described in Equation (1),  
\[\overset{\rightarrow}{P}=W\overset{\rightarrow}{\mathrm{M}} \tag{1}\]  
where \(\overset{\rightarrow}{\mathrm{M}}\) is a 16-element column vector composed of all the Mueller matrix elements of the measured sample, \(\overset{\rightarrow}{P}\) is an \(N\)-element column vector (where \(N\) is 30 times the intensity measurement in the system, as discussed above) composed of the light intensities detected, and \(W\) is an \(N\times 16\) polarimetric measurement matrix. In the conventional calibration method, however, the instrument matrix \(W\) is defined by the parameters of the polarizing elements in the PSG and PSA. Orientation errors of the polarizing elements and retardance errors of the retarders are the primary sources of error, and the evaluation of system parameters \(a_{2}\), \(a_{3}\), \(a_{4}\), \(b_{2}\), and \(b_{3}\) is used to characterize the errors [13]. The system parameters are defined in Figure 2, where \(a_{2}\) is the angle between the fast axis of the quarter-wave plate and the principal axis of the polarizer in the PSG at the initial time, \(a_{3}\) is the angle between the fast axis of the quarter-wave plate in the PSA and the principal axis of the polarizer in the PSG at the initial time, \(a_{4}\) is the angle between the principal axes of the polarizers in the PSG and PSA, and \(b_{2}\) and \(b_{3}\) are measures of the actual retardation of the wave plates in the PSG and PSA, respectively; using default values of \(0^{\circ}\), \(0^{\circ}\), \(0^{\circ}\), \(90^{\circ}\), and \(90^{\circ}\), \(a_{2}\), \(a_{3}\), and \(a_{4}\) were evaluated to indicate the orientation errors of the quarter-wave plate in the PSG, the quarter-wave plate in the PSA, and the polarizer in the PSA, respectively, while \(b_{2}\) and \(b_{3}\) were evaluated to indicate the retardance errors of the quarter-wave plates in the PSG and PSA, respectively. The aforementioned system parameters are used in the polarizing section. In the proposed calibration method, a mathematical model of the imaging section is implemented. To unify the representation of optical elements, we describe the Mueller matrices for each optical element as a function of the system parameters \(\theta\), \(\psi\), and \(\Delta\). The partially polarizing effect (\(\psi\)) and the phase-shifting effect (\(\Delta\)) are given by Equation (2):  
\[M(\theta,\psi,\Delta)=R(-\theta)\left[ \begin{array}{cccc}1&-\cos(2\psi)&0&0 \\ -\cos(2\psi)&1&0&0\\ 0&0&\sin(2\psi)\cos(\Delta)&\sin(2\psi)\sin(\Delta)\\ 0&0&-\sin(2\psi)\sin(\Delta)&\sin(2\psi)\cos(\Delta)\\ \end{array} \right]R(\theta) \tag{2}\]  
Figure 2: Five system parameters in the polarizing section. where \(\psi\) and \(\Delta\) correspond to the usual ellipsometric angles related to the Fresnel coefficients; \(\theta\) denotes the principle axis orientation angle of different optical elements; and \(R(\theta)\) is the rotation matrix expressed as follows:  
\[R(\theta)=\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos(2\theta)&-\sin(2\theta)&0\\ 0&\sin(2\theta)&\cos(2\theta)&0\\ 0&0&0&1\end{array} \right]. \tag{3}\]  
The system parameters \(\theta\), \(\psi\), and \(\Delta\) in Equation (2) and those in Figure 2 show significant similarities, i.e., the system parameters \(\Delta\), \(b_{2}\), and \(b_{3}\) have the polarization meaning of retardance, and \(\theta\), \(a_{2}\), \(a_{3}\), and \(a_{4}\) have the physical meaning of angles between the principle axes of different elements. Furthermore, we usually set the principle axis of the first polarizing element to \(0^{\circ}\). To model the imaging section in the Mueller matrix microscope, we recast the Mueller matrix of the objective lens using Equations (2) and (3) with the polarization parameters \(\theta_{O}\), \(\psi_{O}\), and \(\Delta_{O}\).