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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* and *plane of 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,

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

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 be written in the "x-y basis" as

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

## See also

## References

- Jackson, John D. (1998).
*Classical Electrodynamics (3rd ed.)*. Wiley. ISBN 0-471-30932-X.

**^** Shapira, Joseph; Shmuel Y. Miller (2007). *CDMA radio with repeaters*. Springer. p. 73. ISBN 0-387-26329-2.

## External links

This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".

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