# Applications of Mueller Matrix Polarimetry to Biological and Agricultural Diagnostics: A Review
## 2 Principles of Mueller Matrix Analysis
[PERSON] and [PERSON] proposed a procedure for decomposing a MM into a successive product of three basis matrices [72]. \[\mathbf{M}\equiv M_{11}\left[ \begin{array}{cc}1&\stackrel{{ \rightarrow}}{{D}}\\ \stackrel{{\rightarrow}}{{P}}&\mathbf{m}\end{array} \right]= \mathbf{M}_{\Delta}\mathbf{M}_{R}\mathbf{M}_{D}, \tag{5}\]The original matrix \(\mathbf{M}\) is partitioned into the polarizance vector \(\stackrel{{\rightarrow}}{{P}}\), the diattenuation vector \(\stackrel{{\rightarrow}}{{D}}\):  
\[\stackrel{{\rightarrow}}{{P}}\equiv\frac{1}{M_{11}}\big{[} \begin{array} []{cc}M_{21}&M_{31}&M_{41}\end{array} \end{array}^{T}\!\!,\ \stackrel{{\rightarrow}}{{D}}\equiv\frac{1}{M_{11}}\big{[} \begin{array}{cc}M_{12}&M_{13}&M_{14}\end{array} \end{array}^{T} \tag{6}\]  
and \(3\times 3\) submatrix \(\mathbf{m}\):  
\[\mathbf{m}\equiv\frac{1}{M_{11}}\left[ \begin{array}{ccc}M_{22}&M_{23}&M_{24 }\\ M_{32}&M_{33}&M_{34}\\ M_{42}&M_{43}&M_{44}\end{array} \right]\!. \tag{7}\]  
The matrices \(\mathbf{M}_{A}\), \(\mathbf{M}_{R}\) and \(\mathbf{M}_{D}\) have the following form:  
\[\mathbf{M}_{D}=M_{11}\left[ \begin{array}{cc}1&\stackrel{{ \rightarrow}}{{D}}\\ \stackrel{{\rightarrow}}{{D}}&\mathbf{m}_{D}\end{array} \right]\!,\ \ \mathbf{M}_{R}=\left[ \begin{array}{cc}1&\stackrel{{ \rightarrow}}{{0}}\\ \stackrel{{\rightarrow}}{{0}}&\mathbf{m}_{R}\end{array} \right]\!,\ \ \mathbf{M}_{\Delta}=\left[ \begin{array}{cc}1&\stackrel{{ \rightarrow}}{{0}}\\ \stackrel{{\rightarrow}}{{P}}_{\Delta}&\mathbf{m}_{\Delta}\end{array}  \right]\!. \tag{8}\]  
A set of polarimetric parameters that are related to the intrinsic properties of the tissue can be derived from this representation of the MM. These parameters are considered to be as follows [20]:  
Depolarization:  
\[\Delta=1-\frac{\left|\mathrm{Tr}(\mathbf{M}_{\Delta})-1\right|}{3}, \tag{9}\]  
Diattenuation:  
\[d=\frac{1}{M(1,1)}\sqrt{M(1,2)^{2}+M(1,3)^{2}+M(1,4)^{2}}, \tag{10}\]  
Total retardance:  
\[R=\cos^{-1}\left\{\frac{\mathrm{Tr}(\mathbf{M}_{R})}{2}-1\right\}\!, \tag{11}\]  
Linear retardance:  
\[\delta=\cos^{-1}\left\{\sqrt{\left[M_{R}(2,2)+M_{R}(3,3)\right]^{2}+\left[M_{ R}(3,2)-M_{R}(2,3)\right]^{2}}-1\right\}\!, \tag{12}\]  
Optical rotation:  
\[\psi=\tan^{-1}\left\{\frac{M_{R}(3,2)-M_{R}(2,3)}{M_{R}(2,2)+M_{R}(3,3)}\right\} \tag{13}\]  
Such a representation of the MM and the parameters thus defined are especially informative in tissue diagnostics. For light transmitted through a homogeneous, anisotropic material, the polarization effects are described by a set of eight parameters with a clear physical interpretation: circular dichroism (\(CD\)), circular birefringence (\(CB\)), horizontal linear dichroism (\(LD\)), \(45^{\circ}\) linear dichroism (\(LD^{\prime}\)), horizontal linear birefringence (\(LB\)), \(45^{\circ}\) linear birefringence (\(LB^{\prime}\)), isotropic retardation (\(\eta\)) and isotropic amplitude absorption (\(\kappa\)). In terms of these parameters, the MM of a non-depolarizing sample can be written as a matrix exponent [8]:  
\[\mathbf{M}=e^{\mathbf{L}},\ \ \mathbf{L}=\left[ \begin{array}{ccc}-\kappa&-LD&- LD^{\prime}&CD\\ -LD&-\kappa&CB&LB^{\prime}\\ -LD^{\prime}&-CB&-\kappa&-LB\\ CD&-LB^{\prime}&LB&-\kappa\end{array} \right], \tag{14}\]  
If an anisotropic material is only transversely homogeneous, then the differential expansion of the MM can be applied to analyze the evolution of its polarization properties along the direction of light propagation [60]. In the context of the diagnostics of organic substances, the results of comparing three approaches to determine the parameters of optical chirality [12]--direct evaluation from the MM elements, differential decomposition of the MM and electromagnetic modeling using Tellegen's constitutive relations--may be of interest. The operation principle of the polarimetric devices measuring the MM (MM polarimeters) can be illustrated by a general block diagram (Figure 1).