# Stokes-Mueller formalism

Stokes-Mueller formalism
관련코스 현대광학
소분류 물리
선행 키워드 Jones Matrix
연관 키워드

Although Jones calculus is very useful in describing polarizing optical components, it has clear limitations that it only applies to completely polarized light. Now we learn about more general formalism in which general polarized state of light is included.

## Stokes parameter

Stokes vector S consists of four Stokes parameters given below:

$S_0 = \lt {E_x}^2\gt +\lt {E_y}^2\gt$
$S_1 = \lt {E_x}^2\gt -\lt {E_y}^2\gt$
$S_2 = \lt 2{E_x}{E_y}\cos\delta\gt$
$S_3 = \lt 2{E_x}{E_y}\sin\delta\gt$

Or, it can be put in simpler form as below:

$S = ( I, \lt I_0 - I_{90}\gt , \lt I_{45}-I_{-45}\gt ,\lt I_L - I_R\gt )$

where <.> refers to time average.

 Polarization state Stokes vector Linear (h) Linear (v) Linear (+45˚) Linear (-45˚) RCP LCP Unpolarized $\begin{pmatrix} 1 \\ 1 \\ 0 \\ 0\end{pmatrix}$ $\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0\end{pmatrix}$ $\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0\end{pmatrix}$ $\begin{pmatrix} 1 \\ 0 \\ -1 \\ 0\end{pmatrix}$ $\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1\end{pmatrix}$ $\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1\end{pmatrix}$ $\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix}$

## Mueller matrix

A linear optical component that converts $S_{in}$ into $S_{out}$ can be represented as a 4x4 matrix.

Mueller matrix Optical component
${1 \over 2} \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (horizontal)
${1 \over 2} \begin{pmatrix} 1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (vertical)
${1 \over 2} \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (+45°)
${1 \over 2} \begin{pmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad$ Linear polarizer (-45°)
$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad$ Quarter waveplate (slow vertical)
$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix} \quad$ Quarter waveplate (slow horizontal)
$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \quad$ Half waveplate (slow horizontal)
${1 \over 4} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \quad$ Attenuation filter (25% )

## From Jones to Mueller calculus

Any Jones vector can be represented as Stokes vector, and any Jones matrix can be represented as a Mueller matrix. The inverse is not true, though, because Jones calculus cannot deal with partially polarized state.

Then, how do we convert a Jones vector into a Stokes vector?

### From Jones vector to Stokes vector

Let us first define a quantity called 'coherency vector', as shown below:

$\mathrm{C = J \otimes J^* = \left[\begin{array}{c} E_x \\ E_y \end{array}\right] \otimes \left[\begin{array}{c} E_x^* \\ E_y^* \end{array}\right] = \left[ \begin{array}{c} E_x E_x^* \\ E_x E_y^* \\ E_y E_x^* \\ E_y E_y^* \end{array} \right]}$

Now, we want to make a Stokes vector (defined above) out of these coherency vector components, and it's done by multiplying the specially desined matrix A:

$\mathrm{S = AC = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & i & -i & 0 \end{array}\right]\left[\begin{array}{c} E_x E_x^* \\ E_x E_y^* \\ E_y E_x^* \\ E_y E_y^* \end{array}\right] = \left[ \begin{array}{c} E_x E_x^* + E_y E_y^* \\ E_x E_x^* - E_y E_y^* \\ E_x E_y^* + E_y E_x^* \\ i(E_x E_y^* - E_y E_x^*) \end{array}\right] }$

### From Jones matrix to Mueller matrix

Let's say we have an optical component which changes the polarization state from $J_{in}$ to $J_{out}$:

$\mathrm{J_{out} = M J_{in}}$

Then, let us try to convert this field equation into an intensity equation, by performing Kronecker product on both hand sides.

$\mathrm{J_{out}\otimes J_{out}^* = (MJ_{in})\otimes(MJ_{in})^* = (M\otimes M^*) (J_{in}\otimes J_{in}^*) }$

which reduces into

$\mathrm{C_{out} = (M \otimes M^*) C_{in} }$

By plugging in $\mathrm{C = A^{-1}S }$, we have

$\mathrm{ A^{-1} S_{out} = (M \otimes M^*) A^{-1} S_{in} }$

and finally,

$\mathrm{ S_{out} = \left[ A (M \otimes M^*) A^{-1}\right] S_{in} }$

## Example

### Linearly polarized light in 45 degree from x-axis passing through QWP with horizontal slow axis

In Jones calculus,

$\left(\begin{array}{cc} 1 & 0 \\ 0 & -i \end{array}\right) \left(\begin{array}{c} 1 \\ 1 \end{array}\right) = \left(\begin{array}{c} 1 \\ -i \end{array}\right)$

In Stokes-Mueller formalism,

$\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right) \left(\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right) = \left(\begin{array}{c} 1 \\ 0 \\ 0 \\ -1 \end{array} \right)$