Propagation Through a Homogeneous Plasma

Even above the cutoff frequency there are various propagation effects that are important for a radio wave passing through a plasma. Let us start with the most straightforward ones. Consider a radio signal passing through a homogeneous slab of plasma of length L. The signal is delayed (with respect to the propagation time in the absence of the plasma) by the amount

$$\Delta T = {L\over v_g} - {L \over c} = {L\over c}(1/\mu -1) \simeq {L\over c} {1 \over 2} {\nu_p^2 \over \nu_2} .$$

The magnitude of the propagation delay can hence be written as

$$\vert\Delta T\vert = {L \over c} \times {4 \times 10^6 \over \nu_{\rm Hz}^2}n_e .$$

The propagation delay can also be considered as an ``excess path length'' $\Delta L$ = c $\Delta T$. Further since ($v_g/c-1)$ and $(v_p/c-1)$ differ only in sign<sup>16.1</sup>, the magnitude of the ``excess phase'' (viz. $2\pi\nu(L/v_p -L/c)$) is given by $\Delta \Phi = 2\pi\nu\Delta T$. Note that since the propagation delay is a function of frequency $\nu$, waves of different frequencies get delayed by different amounts. A pulse of radiation incident at the far end of the slab will hence get smeared out on propagation through the slab; this is called ``dispersion''. If the plasma also has a magnetic field running through it then it becomes birefringent - the refractive index is different for right and left circularly polarized waves. A linearly polarized wave can be considered a superposition of left and right circularly polarized waves. On propagation through a magnetized plasma the right and left circularly polarized components are phase shifted by different amounts, or equivalently the plane of polarization of the linearly polarized component is rotated. This rotation of the plane of polarization on passage through a magnetized plasma is called ``Faraday rotation''. The angle through which the plane of polarization is rotated is given by

$$\Theta = RM \lambda^2 = 0.81 \lambda^2 \int n_e B_{\vert\vert} dl.$$

and $RM$ is called the rotation measure. For the second equality $\lambda$ is in meters, $n_e$ is in cm$^{-3}$, $B_{\vert\vert}$ is in $\mu G$ and the length is in parsecs.

Footnotes

... sign^16.1

to first order for $\nu >> \nu_p$, as can be easily verified.