# Optimal Configurations of Mueller Polarimeter for Gaussian-Poisson Mixed Noise
## 2 Materials and Methods
### Four Polarization Illumination/Analysis States
#### 2.4.3 A Fixed Polarizer Followed by Dual Half-Wave Variable Retarders  
Whether the rotating quart-wave plates followed a rotating polarizer or two full-wave variable retarders followed a fixed polarizer measurement system can generate every polarization state to make it possible to generate arbitrary instrument matrices. If using two half-wave variable retarders followed a fixed polarizer as the measurement system, when the angles of the polarizer and retarders are orthogonal as 2.3.2, it can only generate half the polarization states on the Poincare sphere at most. All the polarization states that can be generated will form a hemisphere on the surface of the Poincare sphere. It will not be able to make the polarization states uniformly distributed on the Poincare sphere and obtain the optimal instrument matrix.  
It can be noted that the top view of the regular tetrahedron formed by the optimal instrument matrix in Equation (6) is a square whose diagonals are just past the origin and along 45\({}^{\circ}\) and 135\({}^{\circ}\). If the angles of the polarizer and the two delays are \(-22.5^{\circ}\), \(22.5^{\circ}\), and 45\({}^{\circ}\) respectively, the possible generated polarization states will include the arcs where the two diagonals are located as shown in Figure 8a.  
The four polarization states are divided into two groups, one for stokes1 and stokes3 and one for stokes2 and stokes4. When changing the polarization states by the first retarder, make the two retardance of each group equal. It will make the two points coincide on the Poincare sphere. In addition, the two retardance of the two groups should complement each other. Thus, the constraints of the first retarder are:  
\[\delta_{H1} =\delta_{H1}{}^{\prime\prime} \tag{21}\] \[\delta_{H1}{}^{\prime} =\delta_{H1}{}^{\prime\prime}\] (22) \[\delta_{H1} +\delta_{H1}{}^{\prime} =180^{\circ} \tag{23}\]  
Figure 8: (**a**) shows the optimal instrument matrix and all possible generated polarization states. (**b**) is the polarization states geometric relationships by the first retarder and (**c**) shows the relationships by two retarders. All the four states are at the edge of the surface.  
Then, the retardance of the second retarder is made to be 0\({}^{\circ}\) and 180\({}^{\circ}\) for two measurements in each group, respectively:  
\[\delta_{H2}=\delta_{H2}{}^{\prime}=0^{\circ} \tag{24}\]  
\[\delta_{H2}{}^{\prime\prime}=\delta_{H2}{}^{\prime\prime}=180^{\circ} \tag{25}\]  
where \(\delta_{H1}\) and \(\delta_{H2}\) are the retardance of the two half-wave variable retarders.  
After all the constraints, three orthographic projections of the tetrahedron formed by the instrument matrix on the Poincare sphere will be rectangle and the top view will be square. Then, the value of \(\delta_{H1}\) can be calculated either by genetic algorithm or directly to make the instrument matrix reach the optimum. This configuration makes the retardance of the second half-wave variable retarder to be 0\({}^{\circ}\) and 180\({}^{\circ}\), making maximum use of its retardance range.  
However, since there are two optimal instrument matrices for four polarization illumination/analysis states measurements, only one of them can be achieved with this configuration. However, when both the polarizer and the retarders are rotated 45\({}^{\circ}\) in the same direction, another optimal instrument matrix can be achieved.