# Linear polarization

Diagram of the electric field of a light wave (blue), linear-polarized along a plane (purple line), and consisting of two orthogonal, in-phase components (red and green waves)

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.

The orientation of a linearly polarized electromagnetic wave is defined by the direction of the electric field vector.[1] For example, if the electric field vector is vertical (alternately up and down as the wave travels) the radiation is said to be vertically polarized.

## 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)

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

for the magnetic field, where k is the wavenumber,

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

is the angular frequency of the wave, and ${\displaystyle c}$ is the speed of light.

Here

${\displaystyle \mid {\mathbf {E} }\mid }$

is the amplitude of the field and

${\displaystyle |\psi \rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}\psi _{x}\\\psi _{y}\end{pmatrix}}={\begin{pmatrix}\cos \theta \exp \left(i\alpha _{x}\right)\\\sin \theta \exp \left(i\alpha _{y}\right)\end{pmatrix}}}$

is the Jones vector in the x-y plane.

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

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

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

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

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

If unit vectors are defined such that

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

and

${\displaystyle |y\rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}0\\1\end{pmatrix}}}$

then the polarization state can written in the "x-y basis" as

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