This article describes Faraday Rotation, which is an important effect in the propagation of radio waves through the interstellar medium of both our galaxy and through external galaxies, pulsar magnetospheres, and through the Earth’s ionosphere. Credit: http://www.enzim.hu/~szia/cddemo/edemo15.htm

### The Math

A linearly-polarized wave, $A \cos{(\omega t)}\hat{x}$ (polarized in the x-direction)
can be decomposed into a sum of left- and right-circularly polarized waves at the same frequency: $A_{\rm left} = \frac{A}{2} \cos{(\omega t)}\hat{x} + \frac{A}{2} \sin{(\omega t)}\hat{y}$ $A_{\rm right} = \frac{A}{2} \cos{(\omega t)}\hat{x} - \frac{A}{2} \sin{(\omega t)}\hat{y}$
(here the left and right designations are arbitrary)

These left and right circularly-polarized components travel at different speeds through a plasma rendered anisotropic due to a magnetic field. $v_{R,L} =\frac{c}{\epsilon_{R,L}}$, where $\epsilon_{R,L} = 1 - \frac{\omega_p^2}{\omega(\omega\pm\omega_B)}$ are the effective dielectric constants for the two circular polarizations, $\omega_p$ is the plasma frequency, and $\omega_B = \frac{e B}{m_e} = 1.76\times 10^{11} {\rm radians / s} \left(\frac{B}{\rm Tesla}\right)$ is the cyclotron frequency

Upon exiting the plasma, the left- and right-circular polarization modes have picked up a net phase difference, say $2\phi$, which we can split evenly between the two modes, $A_{\rm left} = \frac{A}{2} \cos{(\omega t + \phi)}\hat{x} + \frac{A}{2} \sin{(\omega t + \phi)}\hat{y}$ $A_{\rm right} = \frac{A}{2} \cos{(\omega t - \phi)}\hat{x} - \frac{A}{2} \sin{(\omega t - \phi)}\hat{y}$

So that the net electric field, $A_{\rm left} + A_{\rm right} = \frac{A}{2} [\cos{(\omega t + \phi)} + \cos{(\omega t - \phi)}]\hat{x} + \frac{A}{2}[\sin{(\omega t + \phi)} - \frac{A}{2} \sin{(\omega t - \phi)}]\hat{y}$ $= A\cos{(\omega t)}\cos{\phi}\hat{x} + A\cos{(\omega t)}\sin{\phi}\hat{y}$ $= A\cos{(\omega t)}[\cos{\phi}\hat{x} + \sin{\phi}\hat{y}]$, which is linearly-polarized in the direction $\cos{\phi}\hat{x} + \sin{\phi}\hat{y}$

Thus a magnetized plasma rotates the plane of polarization of a linearly-polarized electromagnetic wave. Credit: http://www.enzim.hu/~szia/cddemo/edemo15.htm