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Linear polarization

## Summary

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. The term linear polarization (French: polarisation rectiligne) was coined by Augustin-Jean Fresnel in 1822.[1] See polarization and plane of polarization for more information.

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)

The orientation of a linearly polarized electromagnetic wave is defined by the direction of the electric field vector.[2] 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

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 be written in the "x-y basis" as

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

## References

• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.
1. ^ A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe", read 9 December 1822; printed in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), Oeuvres complètes d'Augustin Fresnel, vol. 1 (1866), pp. 731–51; translated as "Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis", Zenodo4745976, 2021 (open access); §9.
2. ^ Shapira, Joseph; Shmuel Y. Miller (2007). CDMA radio with repeaters. Springer. p. 73. ISBN 978-0-387-26329-8.