Definitions

# Elliptical polarization

In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

Other forms of polarization, such as circular and linear polarization, can be considered to be special cases of elliptical polarization.

## Mathematical description of elliptical polarization

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (cgs units)
$mathbf\left\{E\right\} \left(mathbf\left\{r\right\} , t \right) = mid mathbf\left\{E\right\} mid mathrm\left\{Re\right\} left \left\{ |psirangle exp left \left[i left \left(kz-omega t right \right) right \right] right \right\}$

$mathbf\left\{B\right\} \left(mathbf\left\{r\right\} , t \right) = hat \left\{ mathbf\left\{z\right\} \right\} times mathbf\left\{E\right\} \left(mathbf\left\{r\right\} , t \right)$

for the magnetic field, where k is the wavenumber,

$omega_\left\{ \right\}^\left\{ \right\} = c k$

is the angular frequency of the wave, and $c$ is the speed of light.

Here

$mid mathbf\left\{E\right\} mid$

is the amplitude of the field and

$|psirangle stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} begin\left\{pmatrix\right\} psi_x psi_y end\left\{pmatrix\right\} = begin\left\{pmatrix\right\} costheta exp left \left(i alpha_x right \right) sintheta exp left \left(i alpha_y right \right) end\left\{pmatrix\right\}$

is the Jones vector in the x-y plane. Here $theta$ is an angle that determines the tilt of the ellipse and $alpha_x - alpha_y$ determines the aspect ratio of the ellipse. If $alpha_x$ and $alpha_y$ are equal the wave is linearly polarized. If they differ by $pi/2,$ they are circularly polarized.