**4. Determining the Mueller matrix**

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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

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

> 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44

*S MS out system in* (4)

(3)

*mmmm mmmm*

*mmmm mmmm*

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

where [*Sout*] the output Stokes vector, [*Sin*] the Stokes input vector and [*Msystem*] is the Mueller

[ ] [ ][ ][ ][ ][ ] *Msystem QW A M QW P <sup>M</sup> <sup>M</sup>* (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

*QUV S S S DOP*

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

222 2 2 2

*I S*

123 2 0

(6)

of right (*V*= +1) or left (*V*= -1) circularly polarized light.

*M*

matrix representing the entire experimental optical system given as

the calculation of the degree of polarization (DOP), defined as

parameters for the original input beam. [34]

as [30]

[32]

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

$$\begin{aligned} m\_{11} &= I\_{0o} \\ m\_{12} &= I\_{0o} - I\_{-o} \\ m\_{13} &= I\_{-o} - I\_{-o} \\ m\_{14} &= I\_{0o} - I\_{-o} \\ m\_{21} &= I\_{0o} - I\_{0o} \\ m\_{22} &= (I\_{+h} + I\_{-v}) - (I\_{-h} + I\_{sc}) \\ m\_{23} &= (I\_{+h} + I\_{-v}) - (I\_{-h} + I\_{+v}) \\ m\_{24} &= I\_{1o} + I\_{-v} \\ m\_{31} &= I\_{0o} + I\_{-v} \\ m\_{31} &= I\_{1o} + I\_{-v} \\ m\_{33} &= (I\_{+h} + I\_{-v}) - (I\_{-v} + I\_{-v}) \\ m\_{41} &= I\_{0i} - I\_{-v} \\ m\_{41} &= I\_{1o} - I\_{-v} \\ m\_{41} &= I\_{1o} - I\_{0o} \\ m\_{42} &= (I\_{l} + I\_{-v}) - (I\_{-l} + I\_{+v}) \\ m\_{50} &= (I\_{l} + I\_{-v}) - (I\_{-l} + I\_{+v}) \\ m\_{63} &= (I\_{l} + I\_{-v}) - (I\_{-l} + I\_{+v}) \\ \end{aligned} \tag{7}$$

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