Elliptical polarization

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (Gaussian 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 propagating in the +z direction, 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 normalized Jones vector. This is the most complete representation of polarized electromagnetic radiation and corresponds in general to elliptical polarization.

At a fixed point in space (or for fixed z), the electric vector ${\displaystyle \mathbf {E} }$ traces out an ellipse in the x-y plane. The semi-major and semi-minor axes of the ellipse have lengths A and B, respectively, that are given by

${\displaystyle A=|\mathbf {E} |{\sqrt {\frac {1+{\sqrt {1-\sin ^{2}(2\theta )\sin ^{2}\beta }}}{2}}}}$

and

${\displaystyle B=|\mathbf {E} |{\sqrt {\frac {1-{\sqrt {1-\sin ^{2}(2\theta )\sin ^{2}\beta }}}{2}}}}$,

where ${\displaystyle \beta =\alpha _{y}-\alpha _{x}}$ with the phases ${\displaystyle \alpha _{x}}$ and ${\displaystyle \alpha _{y}}$. The orientation of the ellipse is given by the angle ${\displaystyle \phi }$ the semi-major axis makes with the x-axis. This angle can be calculated from

${\displaystyle \tan 2\phi =\tan 2\theta \cos \beta }$.

If ${\displaystyle \beta =0}$, the wave is linearly polarized. The ellipse collapses to a straight line ${\displaystyle (A=|\mathbf {E} |,B=0}$) oriented at an angle ${\displaystyle \phi =\theta }$. This is the case of superposition of two simple harmonic motions (in phase), one in the x direction with an amplitude ${\displaystyle |\mathbf {E} |\cos \theta }$, and the other in the y direction with an amplitude ${\displaystyle |\mathbf {E} |\sin \theta }$. When ${\displaystyle \beta }$ increases from zero, i.e., assumes positive values, the line evolves into an ellipse that is being traced out in the counterclockwise direction (looking in the direction of the propagating wave); this then corresponds to left-handed elliptical polarization; the semi-major axis is now oriented at an angle ${\displaystyle \phi \neq \theta }$. Similarly, if ${\displaystyle \beta }$ becomes negative from zero, the line evolves into an ellipse that is being traced out in the clockwise direction; this corresponds to right-handed elliptical polarization.

If ${\displaystyle \beta =\pm \pi /2}$ and ${\displaystyle \theta =\pi /4}$, ${\displaystyle A=B=|\mathbf {E} |/{\sqrt {2}}}$, i.e., the wave is circularly polarized. When ${\displaystyle \beta =\pi /2}$, the wave is left-circularly polarized, and when ${\displaystyle \beta =-\pi /2}$, the wave is right-circularly polarized.

The reflected light from some beetles (e.g. Cetonia aurata) is elliptical polarized.[2]

• Ellipsometry
• Fresnel rhomb
• Photon polarization
• Sinusoidal plane-wave solutions of the electromagnetic wave equation

•  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).

• Henri Poincaré (1889) Théorie Mathématique de la Lumière, volume 1 and Volume 2 (1892) via Internet Archive.
• H. Poincaré (1901) Électricité et Optique : La Lumière et les Théories Électrodynamiques, via Internet Archive

• Animation of Elliptical Polarization (on YouTube)
• Comparison of Elliptical Polarization with Linear and Circular Polarizations (YouTube Animation)