# Applications of Mueller Matrix Polarimetry to Biological and Agricultural Diagnostics: A Review
## 2 Principles of Mueller Matrix Analysis
In general, an MM polarimeter consists of a light source (LS), a polarization state generator (PSG), a test sample cuvette (S), a polarization state analyzer (PSA) and a photodetector (PD). Light sources can vary depending on the purpose of the polarimetric instrument and the accuracy of the measurements, from broadband lamps and super luminescent diodes [12; 73; 74; 75] to lasers [14; 61], including tunable lasers [76]. A polarization state generator (PSG) is a transmissive optical system that allows the generation of any arbitrary polarization state (generally elliptical) and is used to convert the polarization of source radiation which is not initially polarized or has constant polarization into a set of polarized light states that is required for measuring the Mueller matrix of an arbitrary object. They can have different designs; the most common PSG in polarimetric instruments consists of a combination of phase modulators, phase plates and a linear polarizer [14; 15; 74; 75; 76]. Note that PSGs based on liquid crystals are very promising [74; 77], since they are compact and have no moving parts. On the basis of full Poincare beams, which represent all possible completely polarized states of polarization in their cross-section and can be generated by focusing a laser beam on the front face of a uniaxial crystal, it is possible to implement a parallel-type PSG [78]. After the PSG, the polarization state of the radiation is changed (time-modulated). Two ways are possible here:  
* Polarization is modulated continuously or pulse-modulated (periodically, as a rule);
* Polarization sequentially passes a fixed, discrete set of time-constant states. In the first case, the Stokes vector of radiation, after passing through the PSG to be incident on the object, has the form:  
\[S_{i}=y_{i}(t),\;\;i=1\ldots 4 \tag{15}\]  
where \(y_{i}(t)\) are known time functions determined by the modulation method. The Stokes vector at the output of the object is expressed as follows:  
\[S_{i}^{*}(t)=\sum_{k=1}^{4}M_{ik}S_{k}=\sum_{k=1}^{4}M_{ik}y_{k}(t),\;\;i=1 \ldots 4 \tag{16}\]  
where \(\mathbf{M}\) is the MM of the object. If the functions \(y_{i}(t)\) are linearly independent, then the matrix elements \(M_{ik}\) can be obtained as the corresponding coefficients of the expansions of the measured Stokes vector components \(S_{i}^{*}\) in these functions. In the second case, it is necessary to set at least four discrete polarization states of the input radiation, which are described by linearly independent Stokes vectors \(\overset{\rightarrow}{S}^{(j)}(j=1\ldots 4)\)  
Figure 1: General block diagram of MM polarimeters: light source (LS), polarization state generator PSG, sample (S), polarization state analyzer (PSA), photodetector (PD). Having measured the four corresponding output Stokes vectors \(\overset{\rightarrow}{S}^{*(j)}\) (\(j=1\ldots 4\)), we obtain a system of 16 linear algebraic equations:  
\[S_{i}^{*(j)}=\sum_{k=1}^{4}M_{ik}S_{k}^{(j)},\;\;i,j=1\ldots 4 \tag{17}\]  
The solution of this system gives the values of all 16 elements of the Muller matrix of the object. Similar to a PSG, a PSA is also a combination of phase optical elements (phase modulators, phase plates) and a linear polarizer, which serves to measure the components of the Stokes vector of radiation coming out of an object. For instance, a PSA using a certain set of relative azimuthal orientations of a linear polarizer together with a quarter-wave plate enables complete Stokes vector measurements. Actually, this PSA successively implements particular cases of a linear polarizer oriented at \(0^{\circ}\), \(90^{\circ}\) and \(\pm 45^{\circ}\) and a circular polarizer. For spectroscopic measurements of the MM, it is attractive to use polarimeters with four photoelastic modulators, two each in the PSG and PSA, since modulators of this type are easily adjusted to different wavelengths [79]. Having no moving parts, they perform relatively fast MM measurements, which, additionally, can be made with spatial resolution.