# Jones Matrix

Jones Matrix
관련코스 현대광학
소분류 물리
선행 키워드 편광
연관 키워드 Mueller Matrix

In Jones Matrix representation, the polarization state of light is represented as a 2x1 matrix, usually in normalized form.

When the light is assumed to be propagating in z-direction, the two components in the column matrix represent the electric field amplitude of light in x- and y-direction, respectively.

$\textbf{E}_0=E_{0x}\textbf{i}+E_{0y}\textbf{j}$

## Polarization State

Polarization State Jones vector
Linearly polarized in x-direction $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$
Linearly polarized in y-direction $\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]$
Linearly polarized in 45 degrees from x-direction $\frac1{\sqrt{2}}\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$
Left circularly polarized $\frac1{\sqrt{2}}\left[ \begin{array}{c} 1 \\ i \end{array} \right]$
Right circularly polarized $\frac1{\sqrt{2}}\left[ \begin{array}{c} 1 \\ -i \end{array} \right]$

## Polarizing Optical Elements

### Linear polarizer

 Horizontal polarizer $\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right]$ Vertical polarizer $\left[ \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right]$ Polarizer in 45 degree from x-axis ${1\over 2}\left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right]$ Polarizer in -45 degree from x-axis ${1\over 2}\left[ \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right]$

### Phase retarder

Note that an error has been fixed for QWP with slow axis at $\pm45$ degree cases.

 Quarter-wave plate, vertical slow axis $\left[ \begin{array}{cc} 1 & 0 \\ 0 & i \end{array} \right]$ Quarter-wave plate, horizontal slow axis $\left[ \begin{array}{cc} 1 & 0 \\ 0 & -i \end{array} \right]$ Quarter-wave plate, slow axis at -45 degrees from x-axis $\frac{1}{\sqrt{2}}\left[ \begin{array}{cc} 1 & -i \\ -i & 1 \end{array} \right]$ Quarter-wave plate, slow axis at 45 degrees from x-axis $\frac{1}{\sqrt{2}}\left[ \begin{array}{cc} 1 & i \\ i & 1 \end{array} \right]$ Half-wave plate, horizontal or vertical slow axis $\left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]$

### Reflection and Transmission

 Reflection Matrix $\left[ \begin{array}{cc} -r_\parallel & 0 \\ 0 & r_\perp \end{array} \right]$ Transmission Matrix $\left[ \begin{array}{cc} t_\parallel & 0 \\ 0 & t_\perp \end{array} \right]$

## Coordinate Transformation

Even in cases where a new x-y coordinate is used, the physics shouldn't change.

What if we use a new coordinate system which is rotated by an angle $\alpha$?

Then both Jones vector $\mathbf{J}$ and Jones matrix $\mathbb{A}$ transforms accordingly:

$\mathbf{J}'=\mathbb{R}(\alpha) \mathbf{J}$
$\mathbb{A}'=\mathbb{R}(\alpha) \mathbb{A} \mathbb{R}(-\alpha)$

where rotation matrix is defined as below:

$\mathbb{R}(\alpha)=\left[ \begin{array}{cc} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{array} \right]$

Now let's deal with some examples below.

### Waveplate with fast axis at 45 degree from x-direction

Let's see if we can derive a Jones matrix for QWP with fast axis at +45 degree, by coordinate rotation.

We'll start with the Jones matrix for QWP with fast axis vertical, and apply the rotation transformation.

$\mathbb{A}'=\mathbb{R}(\frac{\pi}{4}) \mathbb{A} \mathbb{R}(-\frac{\pi}{4}) \\ =\left[\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right] \left[\begin{array}{cc} 1 & 0 \\ 0 & -i \end{array}\right] \left[\begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array}\right] = \left[\begin{array}{cc} 1-i & -1-i \\ -1-i & 1-i \end{array}\right] = (1-i) \left[\begin{array}{cc} 1 & -i \\ -i & 1 \end{array}\right]$

where we have neglected multiplicative constant for normalization throughout.

This result is consistent with the one listed in the table, and when this matrix operates on a horizontal polarization, it generates right-handed circular polarization as expected.

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