# 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)

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

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.

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

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