Elliptical polarization: Difference between revisions

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{{Short description|Polarization of electromagnetic radiation}}

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{{more footnotes|date=November 2018}}

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The [[Classical physics|classical]] [[sinusoidal]] plane wave solution of the [[electromagnetic wave equation]] for the [[Electric field|electric]] and [[Magnetic field|magnetic]] fields is ([[Gaussian units]])

:<math> \mathbf{E} ( \mathbf{r} , t ) = \midleft| \mathbf{E} \midright| \mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \} </math>

:<math> \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t ) ,</math>

where <math>k</math> is the [[wavenumber]], <math display=inline> \omega = c k</math> is the [[angular frequency]] of the wave propagating in the +z direction, and <math> c </math> is the [[speed of light]].

for the magnetic field, where k is the [[wavenumber]],

:Here <math>| \omega_mathbf{ E}^{ } = c k|</math> is the [[amplitude]] of the field and

is the [[angular frequency]] of the wave propagating in the +z direction, and <math> c </math> is the [[speed of light]].

Here <math>\mid \mathbf{E} \mid</math> is the [[amplitude]] of the field and

:<math> |\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} </math>

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==Polarization ellipse==

[[File:Polarisation ellipse.svg|250px|right]]At a fixed point in space (or for fixed z), the electric vector <math> \mathbf{E} </math> 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

:<math> A=|\mathbf{E}|\sqrt{\frac{1+\sqrt{1-\sin^2(2\theta)\sin^2\beta}}{2}}</math>

and