Jones Vectors and Matrices

I think it is important to have an introduction into Jones vectors and matrices. A Jones vector is when you represent the Electric field as such

(1)   \begin{equation*} \vec{E}}=\begin{bmatrix} E_{x}\\E_{y} \end{bmatrix} \end{equation*}

A Jones matrix is a 2 by 2 matrix that does some kind of operation on the Jones vector thereby transforming it. These Jones matrices vary depending on the optical element that your electromagnetic wave is traveling through. Some examples of Jones matrices

Horizontal Polarizer:

\begin{vmatrix} 1 & 0\\ 0& 0 \end{vmatrix}\rightarrow \begin{bmatrix} 1 & 0\\ 0& 0 \end{bmatrix}\begin{bmatrix} E_{x}\\ E_{y}  \end{bmatrix}=\begin{bmatrix} E_{x}\\ 0  \end{bmatrix}

Vertical Polorizer:

\begin{vmatrix} 0 & 0\\ 0& 1 \end{vmatrix}\rightarrow \begin{bmatrix} 0 & 0\\ 0& 1 \end{bmatrix}\begin{bmatrix} E_{x}\\ E_{y}  \end{bmatrix}=\begin{bmatrix} 0\\E_{y} \end{bmatrix}

Half wave plate:

\begin{vmatrix} cos2\theta & sin2\theta\\ sin2\theta& -cos2\theta \end{vmatrix}\rightarrow \begin{bmatrix} cos2\theta & sin2\theta\\ sin2\theta& -cos2\theta \end{bmatrix}\begin{bmatrix} E_{x}\\ E_{y}  \end{bmatrix}=\begin{bmatrix} E_{x}cos2\theta+E_{y}sin2\theta\\E_{x}sin2\theta-E_{y}cos2\theta  \end{bmatrix}

\theta is the angle that the half wave plate is oriented at. These Jones vectors will be useful in describing how the electromagnetic wave is polarized in the following posts