# Linear polarization

In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. See polarization for more information.

Historically, the orientation of a polarized electromagnetic wave has been defined in the optical regime by the orientation of the electric vector, and in the radio regime, by the orientation of the magnetic vector.

## Mathematical description of linear 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,

$\omega _{}^{}=ck$ 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.

The wave is linearly polarized when the phase angles $\alpha _{x}^{},\alpha _{y}$ are equal,

$\alpha _{x}=\alpha _{y}\ {\stackrel {\mathrm {def} }{=}}\ \alpha$ .

This represents a wave polarized at an angle $\theta$ with respect to the x axis. In that case the Jones vector can be written

$|\psi \rangle ={\begin{pmatrix}\cos \theta \\\sin \theta \end{pmatrix}}\exp \left(i\alpha \right)$ .

The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that

$|x\rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}1\\0\end{pmatrix}}$ and

$|y\rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}0\\1\end{pmatrix}}$ then the polarization state can written in the "x-y basis" as

$|\psi \rangle =\cos \theta \exp \left(i\alpha \right)|x\rangle +\sin \theta \exp \left(i\alpha \right)|y\rangle =\psi _{x}|x\rangle +\psi _{y}|y\rangle$ . 