# Optimal Configurations of Mueller Polarimeter for Gaussian-Poisson Mixed Noise
## 2 Materials and Methods
### Multi Polarization Illumination/Analysis States (More Than Four)  
#### 2.3.1 A Rotating Polarizer Followed by a Rotating Quarter-Wave Plate  
When we use the rotating waveplates to generate or analyze a polarization state, all the polarization states of the instrument matrix can only form an 8-shaped curve on the Poincare sphere. However, if the polarization direction of the incident polarized light can be rotated, the 8-shaped curve will rotate along the equator of the Poincare sphere so we can get any polarization state we need to optimize the instrument matrix. In our previous study, we used a PSG based on the rotating polarizer and rotating quarter-wave plate (RPRQ) to generate any polarization state and demonstrate the optimal configurations. Thus, we will first demonstrate the optimization method by the RPRQ measurement system. It is known that when the waveplate is rotating, the \(S_{3}\) of its Stokes vector is only related to the angular difference between the waveplate fast axis angle and the polarizer transmission axis angle. In addition, the angle of the polarizer transmission axis determines the starting point of the 8-shaped curve on the Poincare sphere; therefore, to generate a  
Figure 3: The trajectories on the Poincaré sphere of polarized light passing through a rotating waveplate with (**a**) different incident polarized light generated by a polarizer with different angles, (**b**) different retardance, and (**c**) different fast axis angles. Notice that all the trajectories are 8-shaped curves, and with the incident polarized light changes, the curve’s position also changed. pair of orthogonal polarization states, the angles of the polarizer of two polarization states must be orthogonal to make the 8-shaped curve on the Poincare sphere symmetrical:  
\[\theta_{p}{}^{\prime}=\theta_{p}+90^{\lx@math@degree} \tag{8}\]  
After that, one polarization state is taken on each of the two 8-shaped curves, and makes sure that the \(S_{3}\) of the two polarization states are opposite to each other. The position of the polarization state on the 8-shaped curve depends on the relative angle between the polarizer transmission axis angle and the waveplate fast axis angle. Therefore, the two relative angles should meet the relationship of 90 degrees difference:  
\[\theta_{Q}{}^{\prime}=\theta_{p}{}^{\prime}+\theta_{Q}+90^{\lx@math@degree} =\theta_{Q} \tag{9}\]  
where \(\theta_{p}\) is the angle of polarizer transmission axis, \(\theta_{Q}\) is the angle of quarter-wave plate fast axis. When two polarization states are generated, if the configurations of \(\theta_{p}\) and \(\theta_{Q}\) satisfy Equations (8) and (9), the geometric relationship of two polarization states on the Poincare sphere is shown in Figure 4 and becomes orthogonal. Thus, to optimize an instrument matrix with a number of N polarization illumination/analysis states, the first step is obtaining N/2 pairs of orthogonal polarization states by Equations (8) and (9). Then, use the angles of the two rotating devices in the RPRQ as the optimized parameters to minimize the EWV of the final instrument matrix by means of genetic algorithms. This gives the actual configuration of \(\theta_{p}\) and \(\theta_{Q}\) for 2/N pairs of orthogonal polarization states; then, the optimal configuration for N times polarization illumination/analysis states measurement is obtained based on the angular relationship between each pair of orthogonal polarization states. However, the rotating polarizer requires the polarization state of the light source to be a unpolarized light as much as possible. Otherwise, it may lead to different light intensities in different polarization directions and make measurement errors. To avoid this problem, consider using an FPRHRQ measurement system to generate linearly polarized light in different directions. Then, to keep the pair of polarizer states orthogonal, the polarization directions of the initial linearly polarized light should remain unchanged, which means the half-wave plate angle should be the half of the polarizer angle.