# Applications of Mueller Matrix Polarimetry to Biological and Agricultural Diagnostics: A Review
## 2 Principles of Mueller Matrix Analysis  
The Mueller matrix \(\mathbf{M}\) describes the transformation of the Stokes vector \(\overset{\rightarrow}{S}\), which represents the state of light polarization due to the interaction of light with an object. The components of the Stokes vector that are called the Stokes parameters can be expressed in terms of the intensities of the basic polarized components of the light wave as follows:  
\[\overset{\rightarrow}{S}=\left[ \begin{array}{c}I\\ Q\\ U\\ V\end{array} \right]=\left[ \begin{array}{c}I_{x}+I_{y}\\ I_{x}-I_{y}\\ I_{45^{\circ}}-I_{-45^{\circ}}\\ I_{R}-I_{L}\end{array} \right], \tag{1}\]where \(I,\ Q,\ U\) and \(V\) are the Stokes parameters, \(I_{x},\ I_{y}\) and \(I_{\pm 45^{\circ}}\) are the intensities of linearly polarized components oriented along the corresponding directions (\(x,\ y,\ \pm 45^{\circ}\)) and \(I_{R,L}\) are the intensities of right and left circular polarized components. \[\stackrel{{\rightarrow}}{{S}}_{out}=\mathbf{M}\stackrel{{ \rightarrow}}{{S}}_{in}, \tag{2}\]  
\[\mathbf{M}=\left[ \begin{array}{cccc}M_{11}&M_{12}&M_{13}&M_{14}\\ M_{21}&M_{22}&M_{23}&M_{24}\\ M_{31}&M_{32}&M_{33}&M_{34}\\ M_{41}&M_{42}&M_{43}&M_{44}\end{array} \right], \tag{3}\]  
For more details on the algebraic aspects of the Mueller-Stokes formalism, see [59; 60]. In the case of dispersed media, the light scattering matrix (LSM) defined as the Mueller matrix (MM) contains the most complete information on particles scattering light that is available for static scattering. The LSM elements, being functions of the scattering angle, also depend on the wavelength of the probe radiation, optical properties and size distribution of the dispersed entities [3; 4; 5; 14; 15; 16; 17; 61]. In the particular case of spherical particles, it has a block-diagonal structure:  
\[\mathbf{M}=\left[ \begin{array}{cccc}M_{11}&M_{12}&0&0\\ M_{21}&M_{22}&0&0\\ 0&0&M_{33}&M_{34}\\ 0&0&M_{43}&M_{44}\end{array} \right], \tag{4}\]  
where \(M_{11}=M_{22}\), \(M_{12}=M_{21}\), \(M_{34}=-M_{43}\) and \(M_{33}=M_{44}\). The physical meaning of the element \(M_{11}(\theta)\) is the scattering indicatrix, that is, the angular distribution of the scattered light intensity (\(\theta\) is the scattering angle). For example, the \(M_{34}\) element normalized to \(M_{11}\) or \(M_{33}\) is sensitive to changes in the morphological properties of bacterial colonies during growth [16; 62], while the normalized \(M_{14}\) element is sensitive to changes in the structure of chiral polymers, such as DNA [63; 64; 16]. To better understand how the angular variations are related to the properties of the medium, numerical modeling of the angular-resolved radiation transfer problem was considered in [65]. It was found that light is anisotropically reflected from all media encountered in practice, and that the angular changes depend on the absorption and transmission of the medium, as well as on the angular distribution of the incident light. Particle diagnostics using the MM are carried out, in fact, by the mathematical processing of its elements measured for scattering, absorption and emission of light in accordance with the selected structural model of the investigated medium or sample in order to extract the values of its microphysical parameters [4]. The simplest structural model is a collection of spherical particles. However, particles can be both solid elements and clusters of such particles, as well as have different shapes and sizes, to which the MM is also sensitive [5]. Various mathematical approaches are used to solve the inverse problem from the MM. Most often, it is decompositions of directly measured MM [11; 13; 20; 60], in particular, using group theory [66] or stochastic simulations by the Monte Carlo method [67; 68; 69; 70]. Monte Carlo simulations of MMs for multiple scattering media are very important in interpreting MMs measured for turbid media such as biological tissues [23; 71]. It is worth dwelling on some details of the MM polar decomposition method [20; 60].