**3. Stokes vector-Mueller matrix formulism**

258 Soybean – Genetics and Novel Techniques for Yield Enhancement

A comprehensive understanding of light propagation and scattering for the most general case of highly scattering media is yet to be attained. An analysis based on the Stokes vector and Mueller matrix approach provides a theoretical framework, which can be directly related to the experimentally measurable parameters [23-24]. The Stokes vector - Mueller matrix approach for scattering has been extended to characterization of spatially varying polarization patterns for scattered light. In this approach, determining 16 components of the Mueller matrix for the studied object gives a comprehensive description of scattering properties of a sample or a medium in the spatial domain. Rather then being just one number, each of the 16 components of the Mueller matrix is, in fact, a two-dimensional (2D) array of numbers, corresponding to different spatial locations across the surface of the object

In this study, we consider experimental Mueller matrix of soybean oil (highly tissue like phantom) for their polarization and depolarization observations. The transmitted photons preserve their polarization memory and Mueller matrix represents this information in the

The scattering through soybean oil is the principle mechanism that modifies the initial polarization state of the incident light. The polarization state of light after a single scattering event depends on the direction of scattering and incident polarization state. [26-27] In many turbid media such as tissue, scattering structures have a large variance in size and are distributed or oriented in a complex and sometimes apparently random manner. Because each scattering event can modify the incident polarization state differently, until finally the polarization state is completely randomized. An important exception is when the media consists of organized linear structures, such as birefrengent soybean oil, and then the phase retardation between orthogonal polarization components is proportional to the distance traveled through the birefrengent medium. The phase retardation of the scattering medium

<sup>2</sup>*nx*

The phase retardation measurement through turbid media is aimed at retrieving useful information from such multiply scattered light. The behavior of light in random media is well-known from the extensive study of wave propagation. Light traveling in a random medium can be classified into three categories, the ballistic, the snake and the diffuse light. The ballistic light either remains unscattered, or undergoes coherent forward scattering in the medium. This light travels undeviated and has the shortest path length in the medium. The snake light is that which undergoes near-forward scattering, and follows path that undulate about the ballistic path.[28] The diffuse photons largely exceed the other two categories in number and undergoes multiple scattering. We considered all three kinds of

A large number of different experiments are possible if one wants to study the polarization dependent scattering properties of turbid media. The probing light may be linearly polarized at various angles, right and left-hand circularly, or elliptically polarized. Light coming from the scattering medium can be analyzed in the same numerous ways. However, only a few measurements are needed to completely characterize the optical properties of

(1)

**2. Optical properties of the scatterer that influence polarization**

or medium.[25]

is given as.

photons in our study.

form of matrix array and intensity patterns.

The research of polarized scattered light deals with the entire scattering process in the context of Stokes-Mueller matrices and polarizations. [29] An introduction to optical polarization often starts with a description of the optical elements which physically act as polarizers and retarders. The Stokes vector-Mueller matrix- calculus is then used to show mathematically how these optical elements affect a light beam.

A Stokes vector, a 4 x 1 vector, is a mathematical representation of the polarization state of light. [30] It can be represented as a set of six intensity measurements recorded through a set of various polarizing filters. The Stokes vector is composed of four elements, *I, Q, U*, and *V* and provides a complete description of the light polarization state. If the total irradiant intensity *<sup>t</sup> <sup>I</sup>* incident on the sample and <sup>0</sup><sup>0</sup> *<sup>I</sup>* , <sup>900</sup> *<sup>I</sup>* , <sup>450</sup> *<sup>I</sup>* , <sup>450</sup> *<sup>I</sup>* , *rc I* , and *lc I* the irradiances transmitted by a polarizer-retarders are focused to the detector, then, the Stokes parameters are defined by:[30]

$$\begin{aligned} S = \begin{bmatrix} I \\ Q \\ U \\ V \end{bmatrix} = \begin{bmatrix} S\_o \\ S\_1 \\ S\_2 \\ S\_3 \end{bmatrix} = \begin{bmatrix} I\_r \\ I\_{0r} - I\_{so0} \\ I\_{\ast 49} - I\_{\ast 69} \\ I\_{\ast 49} - I\_{\ast 69} \\ I\_{\kappa} - I\_k \end{bmatrix} = \begin{bmatrix} \left< E\_{0x}^2 \right> + \left< E\_{0y}^2 \right> \\ \left< E\_{0x}^2 \right> - \left< E\_{0y}^2 \right> \\ 2 \left< E\_{0x} E\_{0y} \cos \delta \right> \\ 2 \left< E\_{0x} E\_{0y} \sin \delta \right> \end{bmatrix} \end{aligned} \tag{2}$$


Table 1. A matrix array showing the polarization measurements, necessary to measure each particular matrix element of the different configurations (polarizer and analyzer) setup.

Polarization Sensitive Optical Imaging

**4. Determining the Mueller matrix**

and Characterization of Soybean Using Stokes-Mueller Matrix Model 261

If the Mueller matrix is not known, all the elements can be determined experimentally. It can be shown that 49 intensity measurements with various orientations of polarizers and

> ( )( ) ( )( ) ( )( )

*hh vv vh hv hv hv lh rv rh lv*

( )( ) ( )( ) ( )( )

*hv vh*

 

(7)

*lr rl*

( )( ) ( )( ) ( )( )

*hl vr vl hr lr lr ll rr rl lr*

analyzers are necessary to obtain the 16 elements of the Mueller matrix. [25]

 

*m I mII mII mII mII*

*h v*

*l ro h v*

*m II II m II II m II II*

 

*m II II m II II m II II*

 

22 23 24

32 33 34

42 43 44

the medium is completely described in terms of its optical properties.

**5. Error analysis of Mueller matrix polarimeter**

thus eliminating the systematic error.

31 0 0

41 0 0

*mII*

*l r*

*m II II m II II m II II*

 

Where the first term represents the input polarization state while the second the output polarization state of light. The states are defined as: h = horizontal, v = vertical, + = +45°, - = -45°, r = right circular and l = left circular. Once all 16 elements of the matrix are obtained,

For the retardations close to 00 or 900 the background noise on the detectors introduces a significant and systematic error of 150 at a signal to noise ratio of 10 dB. [35] The coherent detection scheme which calculates the Stokes parameters has better immunity to the system. in the calculation of the *Q* parameter the spectral density in one polarization channel is subtracted from the spectral density in the orthogonal polarization channel, thus eliminating constant background noise terms, and the *U* and *V* parameters are calculated from the cross correlation between the orthogonally polarized channels, eliminating autocorrelation noise. Noise will decrease the degree of polarization, since it will be present as autocorrelation noise in the Stokes parameter I. In the incoherent detection scheme only *V* is measured and the error in the phase retardation is introduced by the decrease of the amplitude of oscillations with increasing depth. In the coherent detection scheme, the Stokes parameters *Q, U*, and V can be renormalized on DOP, restoring the amplitude of the oscillations, and

We have analyzed system errors introduced by the extinction ratio of polarizing optics and chromatic dependence of wave retarders, and errors due to dichroism, i.e., the differences in

*mII*

Where *Ex* and *Ey* are the electric field vectors along x and y direction and is angle. After normalizing the Stokes parameters by the irradiance *I, Q* describes the amount of light polarized along the horizontal (*Q*= +1) or vertical (*Q*= -1) axes, U describes the amount of light polarized along the +45° (*U*= +1) or -45° (*U*= -1) directions, and V describes the amount of right (*V*= +1) or left (*V*= -1) circularly polarized light.

A Mueller matrix, a 4 x 4 matrix, is a mathematical description of how an optical sample interacts or transforms the polarization state of an incident light beam and given as [30]

$$\begin{aligned} \begin{bmatrix} \mathcal{M} & \end{bmatrix} = \begin{bmatrix} m\_{\text{i1}} & m\_{\text{i2}} & m\_{\text{i3}} & m\_{\text{i4}} \\ m\_{\text{i1}} & m\_{\text{i2}} & m\_{\text{i3}} & m\_{\text{i4}} \\ m\_{\text{i3}} & m\_{\text{i2}} & m\_{\text{i3}} & m\_{\text{i4}} \\ m\_{\text{i4}} & m\_{\text{i2}} & m\_{\text{i3}} & m\_{\text{i4}} \end{bmatrix} \end{aligned} \tag{3}$$

where, M is the 4 x 4 Mueller matrix of the media or sample and can be experimentally measured through the application of various incident polarization states and then by analyzing the state of polarization of the light leaving the sample. Since a Mueller matrix contains 16 elements (*mij*) of the matrix M and reconstruction requires 49 independent polarization measurements according to different polarizer and wave plate orientation as shown in table.1 and Fig.1. [31] The Mueller matrix can be thought of as the "optical fingerprint" of a sample. This matrix operates directly on an input or incident Stokes vector, thus resulting in an output 4 x 1 Stokes vector that describes the polarization state of the light leaving the sample. This is described mathematically by the equation given as [32]

$$\begin{bmatrix} \mathbf{S}\_{out} \end{bmatrix} = \begin{bmatrix} \mathbf{M}\_{s\_{system}} \end{bmatrix} \begin{bmatrix} \mathbf{S}\_{in} \end{bmatrix} \tag{4}$$

where [*Sout*] the output Stokes vector, [*Sin*] the Stokes input vector and [*Msystem*] is the Mueller matrix representing the entire experimental optical system given as

$$[M\_{\
u\_{\rm system}}] = [Q\mathcal{W}][A\_{\
u}][M][Q\mathcal{W}][P\_{\
u}] \tag{5}$$

The output stokes vector [*Sout*] can be calculated by relation in Eq. 4, putting the values of the Mueller matrix for optics and the Mueller matrix of the system. [33] The complete characterization of the polarization state of light by means of the Stokes parameters permits the calculation of the degree of polarization (DOP), defined as

$$DOP = \sqrt{\frac{Q^2 + 4I^2 + V^2}{I}} = \sqrt{\frac{S\_{\text{1}}^2 + S\_{\text{2}}^2 + S\_{\text{3}}^2}{S\_{\text{0}}^2}}\tag{6}$$

For purely polarized light, the degree of polarization is unity i.e. 1, and the Stokes parameters obey the equality I2=Q2+U2+V2, while for partially polarized light, the degree of polarization is smaller than unity, leading to I2Q2+U2+V2. An input beam can be decomposed into purely polarized beams. After propagation through an optical system, the Stokes parameters of the purely polarized beam components are added to give the Stokes parameters for the original input beam. [34]
