# Optimal Configurations of Mueller Polarimeter for Gaussian-Poisson Mixed Noise
## 2 Materials and Methods
### Four Polarization Illumination/Analysis States
#### 2.4.1 A Rotating Polarizer Followed by a Rotating Quarter-Wave Plate  
Likely, we optimize the optimal configuration of the RPRQ measurement system by geometric constraint.  
To make the final instrument matrix optimal, we first consider making three orthographic projections of the tetrahedron formed by the instrument matrix on the Poincare sphere rectangular. The three polarization states are constrained to satisfy certain geometric relations with an arbitrary polarization state on the Poincare sphere. The four polarization states are denoted as stokes1, stokes2, stokes3, and stokes4, as shown in Figure 6.  
Figure 6: The geometric relationship between the polarization states after constraint. Respectively, the geometric relationships between (**a**) stokes2 and stokes1, (**b**) stokes3 and stokes1, and (**c**) stokes4 and stokes1. (**d**) is the tetrahedron formed by the four polarization states on the Poincaré sphere.  
The \(S_{3}\) of stokes1 and stokes2 should be equal and the \(S_{1}\), \(S_{2}\) of stokes1 and stokes2 should be opposite. This means stokes2 is the point on the Poincare sphere where stokes1 has rotated \(180^{\circ}\) along the latitude. Therefore, if the position of the polarization state on the 8-shaped curve generated by the rotating waveplate is fixed during two measurements, and the two 8-shaped curves are orthogonal, two polarization states will satisfy this geometric relationship. Combined with the effect of the waveplate on the polarization state, the configuration of these two measurements should satisfy:  
\[\left\{\begin{array}{l}\theta_{P}{{}^{\prime}}=\theta_{P}+90^{\circ}\\ \theta_{Q}{{}^{\prime}}=\theta_{Q}{{}^{\circ}}+90^{\circ}\end{array}\right. \tag{12}\]  
Since the polarizer is rotated by \(90^{\circ}\) to make the 8-shaped curve orthogonality, the fast axis angle of the quartz-wave plate needs to be rotated by \(90^{\circ}\) to keep the relative angle between the quartz-wave plate and the polarizer constant.  
Then, the \(S_{1}\) of stokes1 and stokes3 should be equal and the \(S_{2}\), \(S_{3}\) of stokes1 and stokes3 should be opposite. The two 8-shaped curves are symmetrical about the \(S_{1}S_{3}\) plane, and the relative positions on the 8-shaped curve should complement each other. Thus, the geometric constraints are:  
\[\left\{\begin{array}{l}\theta_{P}{{}^{\prime\prime}}=-\theta_{P}\\ \theta_{Q}{{}^{\prime\prime}}=-\theta_{Q}\end{array}\right. \tag{13}\]  
The geometric relationship between stokes4 and stokes1 is like the relationship between stokes3 and stokes1. The only difference is that the two 8-shaped curves are symmetrical about the \(S_{2}S_{3}\) plane. Therefore, similarly, their geometric constraints are:  
\[\left\{\begin{array}{l}\theta_{P}{{}^{\prime\prime}}=90^{\circ}-\theta_{P} \\ \theta_{Q}{{}^{\prime\prime}}=90^{\circ}-\theta_{Q}\end{array}\right. \tag{14}\]  
where \(\theta_{P}\), \(\theta_{P}\)\({}^{\prime}\), \(\theta_{P}\)\({}^{\prime\prime}\), \(\theta_{P}\)\({}^{\prime\prime\prime}\) are the angles of polarizer and \(\theta_{R}\), \(\theta_{R}\)\({}^{\prime}\), \(\theta_{R}\)\({}^{\prime\prime}\), \(\theta_{R}\)\({}^{\prime\prime\prime}\) are the angles of quartz-wave plate. With those geometric constraints, the three orthographic projections of the tetrahedron that formed by the four polarization states on the Poincare sphere are all rectangular. Then, find \(\theta_{P}\) and \(\theta_{R}\) of the first illumination/analysis state to make the EWV of the final instrument matrix minimal by genetic algorithm, and the optimal configuration can be determined.  
In addition, the rotating polarizer can also be replaced by a rotating half-wave plate followed a fixed polarizer. Then, the geometric constraints of the polarizer angle will change to the half-wave plate angle. However, the angle is half of the polarizer angle.