# Optimal Configurations of Mueller Polarimeter for Gaussian-Poisson Mixed Noise
## 2 Materials and Methods  
### Instrument Matrix Optimization Theory for Gaussian-Poisson Mixed Noise  
EWV is the correct metric for polarimeters. Therefore, we used covariance analysis to quantify the effect of Gaussian or Poisson noise on the estimation variance of the Mueller matrix, and more details can be found in [9]. When additive Gaussian noise with the variance \(\sigma^{2}\) is present in the system, it follows that the estimation variance of the Mueller matrix that is caused by Gaussian noise \(\mathrm{C}_{\textit{Gaussian}}\) is:  
\[ \begin{array}{ll}\mathrm{C}_{\textit{Gaussian}}&=\sigma^{2}trace\Big{(} \big{[}A^{-1}\otimes W^{-1}\big{]}\big{[}A^{-1}\otimes W^{-1}\big{]}^{T} \Big{)}\\ &=\sigma^{2}trace(Q_{A})trace(Q_{W})\\ &=\sigma^{2}EWV_{A}EWV_{W}\end{array}  \tag{1}\]  
where \(Q_{A}=\big{(}A^{T}A\big{)}^{-1}Q_{W}=\big{(}W^{T}W\big{)}^{-1}\), \(\otimes\) denotes the Kronecker product, \(A\) and \(W\) are the instrument matrix of PSA and PSG, respectively and \(EWV_{A}\) (\(EWV_{W}\)) is the \(EWV\)of \(A(W)\). Each column of \(W\) and \(A\) represents the Stokes vector corresponding to an illumination/analysis polarization state. It is clear that when the EWV of PSG and PSA are minimal, the estimation variance of the Gaussian noise is minimal as well. For the Poisson noise in the system, using covariance analysis, the estimation variance that is caused by Poisson \(C_{Poisson}\) noise in the system is [9]:  
\[ \begin{array}{ll}C_{Poisson}&=\sum\limits_{k=1}^{16}[V_{M}]_{k}\left[\sum \limits_{i=1}^{8\times N}[Q_{A}\otimes Q_{W}]_{i,j}[A\otimes W]_{i,k}^{T} \right]\\ &=\frac{1}{4}[V_{M}]_{1}\sum\limits_{i=1}^{8\times N}[Q_{A}\otimes Q_{W}]_{i,j }+\sum\limits_{k=2}^{16}[V_{M}]_{k}\left[\sum\limits_{i=1}^{8\times N}[Q_{A} \otimes Q_{W}]_{i,i}[A\otimes W]_{i,k}^{T}\right]\end{array}  \tag{2}\]  
where \(V_{M}\) is a 16-dimensional vector that consists of 16 elements of the Mueller matrix and \([V_{M}]_{k}\) is the \(k\)th element of \(V_{M}\). For the normalized Mueller matrix, its first term \([V_{M}]_{1}\) is always equal to 1. From the Equation (2), we can see that the first term is independent of the sample, and the second term is related to the last 15 Mueller matrix elements of the sample, which will lead to a consequent change in the estimation variance that is caused by Poisson noise as the sample changes. Fortunately, \([Q_{A}\otimes Q_{W}]_{i,j}\) is a constant when the EWV of the instrument matrix of PSG and PSA is optimal. Therefore, the second term of \(C_{Poisson}\) is set to zero when the following equation is satisfied, thereby making the estimation variance caused by Poisson noise independent of the sample:  
\[\forall k\in[2,16],\sum\limits_{i=1}^{8\times N}[A\otimes W]_{i,k}^{T}=0 \tag{3}\]  
The equivalence condition of Equation (3) is: the sum of each row of W and A is zero [19]. When this condition is satisfied and the EWV of the instrument matrix of PSG and PSA is optimal, the estimation variance caused by Poisson noise is independent of the sample, and the estimation variance reaches the minimum value. At this time, the optimized \(C_{Poisson}\) can be expressed as:  
\[C_{Poisson\_Opt}=\frac{1}{4}\sum\limits_{i=1}^{8\times N}[Q_{A}\otimes Q_{W} ]_{i,j}=\frac{1}{4}EWV_{A}EWV_{W} \tag{4}\]  
To sum up, when the sum of each row of instrument matrices \(W\) and \(A\) is 0, and their EWV are optimal, the estimation variance caused by Gaussian-Poisson mixed noise is minimal and independent of the sample, and its minimum value is:  
\[C_{Optimal}=\left(\frac{1}{4}+\sigma^{2}\right)EWV_{A}EWV_{W} \tag{5}\]  
It is obvious that the sum of the last three rows in the measurement matrix is naturally zero when we divide the polarization states corresponding to the instrument matrix with N polarization illumination/analysis states into N/2 pairs of orthogonal polarization states [19]. The two polarization states symmetric about the center of the sphere are a pair of orthogonal polarization states on the Poincare sphere.