Faraday rotation

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Faraday Rotation

1 Overview

When a polarized electromagnetic wave propagates through a magnetized plasma, the group velocity depends on whether it is a left- or right-hand circularly polarized wave. Since a plane polarized wave is a linear superposition of a right- and left-hand circular wave, this manifests itself as a rotation of the plane of polarization. The polarization angle rotates by an amount

${\begin{aligned}\Delta \theta &={\frac {2\pi e^{3}}{m_{e}^{2}c^{2}\omega ^{2}}}\int _{0}^{d}n_{e}B_{||}\,ds\\&=\lambda ^{2}{\mathcal {RM}}\end{aligned}}\,\!$

where

${\begin{aligned}{\mathcal {RM}}={\frac {e^{3}}{2\pi m_{e}^{2}c^{4}}}\int _{0}^{d}n_{e}B_{||}\,ds\end{aligned}}\,\!$

is the rotation measure. This process is called Faraday rotation.

2 Derivation

Suppose we have a circularly polarized wave

${\begin{aligned}\mathbf {E} (t)=E_{0}e^{-i\omega t}({\hat {\mathbf {x} }}\mp i{\hat {\mathbf {y} }})\end{aligned}}\,\!$

where $-$ and $+$ correspond to right and left circular polarization, respectively. Additionally, the wave propagates through a magnetic field $\mathbf {B} =B_{0}{\hat {\mathbf {z} }}$. The equation of motion for an electron is thus

${\begin{aligned}m{\frac {d\mathbf {v} }{dt}}=-e\mathbf {E} -{\frac {e}{c}}\mathbf {v} \times \mathbf {B} \end{aligned}}\,\!$

It can be shown that the velocity of the electron is

${\begin{aligned}\mathbf {v} (t)={\frac {-ie}{m(\omega \pm \omega _{B})}}\mathbf {E} (t)\end{aligned}}\,\!$

where $\omega _{B}={\frac {eB_{0}}{mc}}$ is the cyclotron frequency. From this, it is apparent that right and left circular polarized waves will propagate through the plasma at different velocities. For a linearly polarized wave (a superposition of a right- and left- circular wave), the plane of polarization rotates by an amount

${\begin{aligned}\Delta \theta ={\frac {\phi _{R}-\phi _{L}}{2}}\end{aligned}}\,\!$

where $\phi _{R,L}=\int _{0}^{d}k_{R,L}\,ds$ is the phase angle, and (assuming $\omega \gg \omega _{p}$ and $\omega \gg \omega _{B}$)

${\begin{aligned}k_{R,L}\approx {\frac {\omega }{c}}\left[1-{\frac {\omega _{p}^{2}}{2\omega ^{2}}}\left(1\mp {\frac {\omega _{B}}{\omega }}\right)\right]\end{aligned}}\,\!$

where $\omega _{p}={\sqrt {\frac {4\pi n_{e}e^{2}}{m_{e}}}}$ is the plasma frequency. After substituting in $k_{R,L}$ for the phase angles and substituting $\omega _{p}$ and $\omega _{B}$, we find

${\begin{aligned}\Delta \theta ={\frac {2\pi e^{3}}{m_{e}^{2}c^{2}\omega ^{2}}}\int _{0}^{d}n_{e}B_{||}\,ds\end{aligned}}\,\!$

Note that the magnetic field that appears in the expression is the component along the line of sight.

3 Measuring magnetic fields

We are able to derive lower limits on the mean magnetic field, since

${\begin{aligned}\langle B_{||}\rangle _{n_{e}}={\frac {\int _{0}^{d}n_{e}B_{||}\,ds}{\int _{0}^{d}n_{e}\,ds}}\end{aligned}}\,\!$

The numerator is found by measuring the rotation measure, and the denominator is simply the dispersion measure. Note that this is a lower limit since it measures only the line-of-sight component of the magnetic field.