Faraday rotation

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.

Gravitational Faraday Rotation

The contents of this section are largely based on this source.

Let us consider a beam of light passing near a rotating black hole. The impact parameter is $b$, the mass of the black hole is $M$ and its rotation parameter is $a \approx J / \frac{G M^2}{c}$ (i.e. the ratio between the actual angular momentum and the maximum possible angular momentum. For simplicity, let us consider a case where the angular momentum vector of the black hole and the direction of propagation of the photon are both pointing toward the observer. Another simplification we will be making is that all the rotation of the polarisation angle happens within a distance comparable to $b$.

From the discussion of Lense Thirring precession we know that neutral particle near a rotating black hole behave like charged particles near a rotating charged sphere. However, in our case the magnetic field would be pointing toward the observer, and since it is parallel to the trajectory of the particle, the latter would not rotate. However, due to gravitational lensing, the particle will be deflected by an angle $\alpha \approx G M / b c^2$, so the effective magnetic field the particle feels not zero, by a factor $\alpha$ than its magnitude. This means that the rotation rate of the polarisation vector is smaller by a factor $\alpha$ compared to the Lense Thirring precession frequency. Since the time the photon spends at a distance $b$ is $b / c$, the rotation angle of the polarisation vector is given by

$\theta_p \approx \alpha \omega_{lt} \frac{b}{c} \approx \left(\frac{G M}{b c^2}\right)^3$