The Ultimate Jones calculus - American History Information Guide and Reference

In optics one can describe polarisation using the Jones calculus, invented by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarisation of the emerging light is simply the Jones matrix of the optical element multiplied by the Jones vector of the incident light.

The following table gives examples of Jones vectors. ( $i$ is the imaginary unit, i.e., $\sqrt{-1}$ .)

Polarisation	Corresponding Jones vector
Linear polarized in the x-direction	$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$
Linear polarized in the y-direction	$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$
Linear polarized at 45 degrees from the x-axis	$\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$
Left circular polarized	$\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ i \end{pmatrix}$
Right circular polarized	$\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ -i \end{pmatrix}$

The following table gives examples of Jones matrices.

Optical element	Corresponding Jones matrix
Linear polarizer with axis of transmission horizontal	$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
Linear polarizer with axis of transmission vertical	$\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$
Linear polarizer with axis of transmission at 45 degrees	$\frac12 \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$
Linear polarizer with axis of transmission at -45 degrees	$\frac12 \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$
Linear polarizer with axis of transmission at $\varphi$ radian	$\begin{pmatrix} \cos^2\varphi & \cos\varphi\sin\varphi \\ \sin\varphi\cos\varphi & \sin^2\varphi \end{pmatrix}$
Left circular polarizer	$\frac12 \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}$
Right circular polarizer	$\frac12 \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}$
Half-wave plate with fast axis pointing along x-direction	$\begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}$
Quarter-wave plate with fast axis pointing along x-direction	$\begin{pmatrix} \frac12 - \frac i2 & 0 \\ 0 & \frac12 + \frac i2 \end{pmatrix}$