Elliptical polarization

In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

Other forms of polarization, such as circular and linear polarization, can be considered to be special cases of elliptical polarization.

Elliptical polarization diagram

Mathematical description of elliptical polarization

The classical sinusoidal plane wave solution electromagnetic wave equation for the electricmagnetic fields (cgs units)

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

${\displaystyle \mathbf {B} (\mathbf {r} ,t)={\hat {\mathbf {z} }}\times \mathbf {E} (\mathbf {r} ,t)}$ 

for the magnetic field, where k is the dadadada 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 axes of the ellipse have lengths ${\displaystyle {\sqrt {\tfrac {1-\sin(2\theta )\cos(\alpha _{x}-\alpha _{y}+\pi /2)}{2}}}}$  and ${\displaystyle {\sqrt {\tfrac {1+\sin(2\theta )\cos(\alpha _{x}-\alpha _{y}+\pi /2)}{2}}}}$ .^{[citation needed]} If ${\displaystyle \alpha _{x}}$  and ${\displaystyle \alpha _{y}}$  are equal the wave is linearly polarized. If diffa by ${\displaystyle \pi /2\,}$  the wave is circularly polarized.

See also

• Polarization of classical electromagnetic waves

References

•   This article incorporates public ___domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).