## 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{align} \Delta \theta & = \frac{2\pi e^3}{m_ e^2c^2\omega ^2}\int _0^ d n_ eB_{||}\, ds \\ & =\lambda ^2\mathcal{RM} \end{align}\,\!

where

\begin{align} \mathcal{RM}=\frac{e^3}{2\pi m_ e^2 c^4}\int _0^ d n_ eB_{||}\, ds \end{align}\,\!

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

## 2 Derivation

Suppose we have a circularly polarized wave

\begin{align} \mathbf{E}(t)=E_0 e^{-i\omega t}(\hat{\mathbf{x}}\mp i\hat{\mathbf{y}}) \end{align}\,\!

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

It can be shown that the velocity of the electron is

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

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

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{align} 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{align}\,\!

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

\begin{align} \Delta \theta = \frac{2\pi e^3}{m_ e^2c^2\omega ^2}\int _0^ d n_ e B_{||}\, ds \end{align}\,\!

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{align} \langle B_{||}\rangle _{n_ e} = \frac{\int _0^ d n_ e B_{||}\, ds}{\int _0^ d n_ e\, ds} \end{align}\,\!

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.