# Optimal Configurations of Mueller Polarimeter for Gaussian-Poisson Mixed Noise
## 1 Introduction  
The polarization state of a beam of light can be represented by a Stokes vector, which is a four-dimensional vector, and the \(4\times 4\) transition matrix describing the change between different polarization states is called the Mueller matrix. The Mueller matrix can provide complete polarization properties of samples, which are related to their microstructural features [1]. Benefitting from this, a series of Mueller matrix imaging techniques have been developed for different applications, such as material characterization [2; 3] and the microstructure of biological tissue measurement [4; 5; 6; 7; 8]. Among the available applications, measurement accuracy is always a top concern because precise Mueller matrix measurement can provide more detailed polarization features of samples. The bedrock of Mueller matrix measurement is constructing a system of linear equations for the Mueller matrix elements by rationally setting the illumination and analysis polarization states. Therefore, the solution of the Mueller matrix depends on the matrix calculation process, in which the intensity error caused by Gaussian-Poisson mixed noise will be accumulated and amplificated [9]. In the design of the Mueller polarimeter, minimizing the estimation variance by choosing specific illumination and analysis polarization states, i.e., the instrument matrix of polarization state generator (PSG) and polarization state analyzer (PSA), has been extensively studied. For the purposes of instrument matrix optimization, the determinant [10], condition number (CN) [10], and equally weighted variance (EWV) [11] indicators were proposed to find the optimal instrumentation matrices. These methods were successfully used to obtain the optimal configurations of polarimeters, including a rotatable retarder and a fixed polarizer-based Stokes polarimeter [11] and liquid-crystal variable retarders (LCVR)-based Mueller polarimeter [12] to reduce the estimation variance caused by additive Gaussian noise. However, the Poisson shot noise in the system cannot be ignored. The estimation variance due to Poisson noise should be not only minimal but also sample-independent. In this case, the optimal configuration of the system can only be optimized by EWV [13]. From another perspective, the optimal configuration of polarimeter can be given in terms of spherical t design. [PERSON] and [PERSON] proposed that when the system is in the presence of Gaussian noise and Poisson noise, the polyhedron formed by the polarization state corresponding to the instrument matrix on the Poincare sphere should satisfy spherical 2 design and spherical 3 design, respectively [14; 15]. In addition, appropriately expanding the dimensionality of the instrument matrices of PSG and PSA helps to further reduce the estimation variance [15]. Based on these methods, [PERSON] et al. proposed a set of configurations of the Stokes polarimeter that can suppress both Gaussian and Poisson noise and reported the optimal measurement configurations for PSA, which consists of four analysis polarization states. The proposed configurations are by using a 142.1\({}^{\circ}\) wave plate and a 102.2\({}^{\circ}\) wave plate to produce two analysis polarization states, respectively, or using two quarter-wave plates [16]. However, waveplates with non-traditional retardance are not commonly used, and the PSG or PSA based on the rotating polarization elements will limit the application of polarimeters in some cases with high requirements on acquisition time. In our previous study, we used a geometric optimization method based on Coulombic energy to optimize a rotating polarizer and rotating quarter-wave plate (RPRQ)-based PSG with a different number of illumination polarization states. This is combined with the dual division of a focal plane (DoFP) polarimeter [17; 18] to build a Mueller matrix microscope capable of suppressing Gaussian-Poisson mixed noise [19].