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# 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.

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) $\mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \}$ $\mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t )$

for the magnetic field, where k is the wavenumber, $\omega_{ }^{ } = c k$

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

Here $\mid \mathbf{E} \mid$

is the amplitude of the field and $|\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix}$

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