Consider an electromagnetic wave propagating in a magnetized plasma. For simplicity we assume the plasma is magnetized in the $ \hat{z} $direction and that the wave is travelling along the same direction. Electromagnetic waves in the plasma can have a right or left handed circular polarization. The dispersion relation for these waves is:

$ n_{R,L}=1-\frac{\omega_p^2/\omega^2}{1\pm\frac{\omega_c}{\omega}} $

where $ n=\frac{ck}{\omega} $ is the refraction index of the plasma corresponding to each polarization, $ \omega_p $ is the plasma frequency and $ \omega_c $ is the Larmour frequency. It is immediately seen that the two polarizations are traveling at different speeds. As a result, the direction of the polarization vector of a linearly polarized electromagnetic wave will rotate as the wave travels along the magnetized plasma. This is because, a linear polarization can be built from a superposition of right and left handed polarizations, and since these components travel at different speed, the superposition will result in different linear polarizations along the path of the wave. This phenomena is known as Faraday rotation. File:DOC050115-05012015091054.pdf

We now show this in a more quantitative fashion. The phase accumulated as the plasma travels a distance $ d $ is:

$ \Phi_{R,L}=\int_0^d k_{R,L} ds. $

A linearly polarized electromagnetic wave will rotate by (see figure):

$ \Delta \theta =(\Phi_R-\Phi_L)/2. $

For $ \omega \gg \omega_p, \omega \gg \omega_c $ we obtain:

$ k_{R,L}= \frac{\omega}{c} \sqrt{1-\frac{\omega_p^2/\omega^2}{1\mp\frac{\omega_c}{\omega}}} \approx \frac{\omega}{c} \sqrt{1-\frac{\omega_p^2}{\omega^2}(1\pm\frac{\omega_c} {\omega})} \approx \frac{\omega}{c} (1- \frac{\omega_p^2}{2\omega^2}(1 \pm \frac{\omega_c}{\omega})) $

From this it follows that:

$ \Delta \theta = \frac{1}{2} \int_0^d (k_R-k_L)ds=\frac{1}{2}\int_0^d \frac{\omega_p^2 \omega_c}{c\omega_2}ds=\frac{2\pi e^3}{m_e^2 c^2 \omega^2} \int_0^d n B_{||}ds $

where $ n $ is the plasma density and $ B_{||} $ is the magnetic field parallel to the direction of the wave's propagation.

Since, for a given line of sight, $ \Delta \theta \propto \omega^{-2} $, then it is possible to determine the last integral (known as the dispersion measure) by measuring $ \Delta \theta $ at different frequencies. This technique is used to provide a measure of the magnetic fields and densities in the inter stellar medium. However, if the direction of the magnetic field changes significantly along the line of sight, this will only provide a lower limit on the real magnetic field strength.