# Optimal Configurations of Mueller Polarimeter for Gaussian-Poisson Mixed Noise
## 2 Materials and Methods
### Multi Polarization Illumination/Analysis States (More Than Four)
#### 2.3.2 A Fixed Polarizer Followed by Dual Full-Wave Variable Retarders  
When using the variable retarders to generate or analyze states, the design of the polarization state will be more intuitive. It can be considered as how to move a point on the Poincare sphere to another point. According to the regularity demonstrated in Figure 2, the new polarization state can be reached by making the initial polarization state move on the surface of the Poincare sphere along the target direction. The fast axis angle is half of the angle between the target direction and \(S_{2}\), and the retardance is equal to the radian between the initial and target polarization states. However, the angle of the fast axis of the variable retarder cannot be freely changed during the actual measurement, so we chose to use two orthogonal variable retarders to generate the target polarization state, and the angle between the fast axis of the first retarder and the polarizer is also orthogonal (45\({}^{\circ}\)). When the angle of the polarizer is 0\({}^{\circ}\), \(S_{1}\) of the Stokes vector is changed by the first retarder and \(S_{2}\) is changed by the second retarder. This makes it very easy to generate arbitrary polarization states and determine measurement configurations. However, as long as the orthogonality of the angle is satisfied, other angles are also workable, but this paper will take the angles (0\({}^{\circ}\), 45\({}^{\circ}\), 0\({}^{\circ}\)) as an example.  
For two polarization states that are orthogonal to each other, each component of their Stokes vectors is opposite (except \(S_{0}\)), and in this measurement system, \(S_{1}\) and \(S_{2}\) are determined separately by the two variable retarders, so the retardance of the first retarder should be made to differ by 180\({}^{\circ}\) between the two illumination/analysis measurements:  
\[\delta_{F1}{}^{\prime}=180^{\lx@math@degree}+\delta_{F1} \tag{10}\]  
However, limited by the fast-axis angle, the polarization states are orthogonal but do not freely generate arbitrary polarization states. Therefore, a second variable retarder is needed for further modulation. When the retardance of the second variable retarder satisfies:  
\[\delta_{F2}{}^{\prime}=\delta_{F2} \tag{11}\]  
where \(\delta_{F1}\) and \(\delta_{F2}\) are the retardance of two retarders. Then, the polarization states will be orthogonal, as shown in Figure 5.  
Then, use a similar approach as Section 2.3.1, find the specific configuration of N/2 pairs of \(\delta_{F1}\) and \(\delta_{F2}\) when EWV is smallest. Finally, obtain specific configurations for N times polarization illumination/analysis states measurement by Equations (10) and (11). The measurement configuration obtained in this way satisfies the optimal instrument matrix and is optimized for both Poisson noise and Gaussian noise.  
Figure 5: A pair of orthogonal polarization states by variable retarder. The start points of two blue curves are 180 degrees apart and the length of blue curves are same.