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.

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

Elliptical polarization diagram

Example

If an electromagnetic wave is created by multiple dipole antennas with 90° phase shifts relative to one another, the electric field vector will rotate, with the direction (clockwise or counterclockwise) depending on the sign of the phase shift.

The angle between the electric field vector and the x axis is given by

${\displaystyle \Phi (\mathbf {E} )=\tan ^{-1}{E_{y} \over E_{x}},}$ 

where ${\displaystyle E_{x}}$  and ${\displaystyle E_{y}}$  are perpendicular components of the electric field vector. If the x and y components have a 90° phase shift between them, these components are given by

${\displaystyle E_{y}=E_{yo}\sin(wt+\phi ),\ \mathrm {and} \,}$ 

${\displaystyle E_{x}=E_{xo}\sin(wt+\phi +{\pi \over 2}),}$ 

where ${\displaystyle \phi \,}$  is an arbitrary phase, and ${\displaystyle E_{x}o}$  and ${\displaystyle E_{y}o}$  are the amplitudes of the x and y components of the field. It can then be shown that

${\displaystyle \Phi (\mathbf {E} )=\omega t,}$ 

which indicates that the electric field vector rotates with an angular velocity ${\displaystyle \omega }$ . Since the phase shift on ${\displaystyle E_{x}}$  is positive, the rotation is counterclockwise. If ${\displaystyle E_{yo}=E_{xo}}$ , the polarization is circular. If they are different, it is elliptical.

Mathematical description of elliptical polarization

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (cgs units)

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

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

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

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

is the Jones vector in the x-y plane. Here ${\displaystyle \theta }$  is an angle that determines the tilt of the ellipse and ${\displaystyle \alpha _{x}-\alpha _{y}}$  determines the aspect ratio of the ellipse. If ${\displaystyle \alpha _{x}}$  and ${\displaystyle \alpha _{y}}$  are equal the wave is linearly polarized. If they differ by ${\displaystyle \pi /2}$  they are circularly polarized.

See also

• Polarization of classical electromagnetic waves