I'm looking to find an expression for the Faraday rotation of a wave in a magnetized plasma, propagating parallel to a magnetic field.

It's more math help than physics help I need here though. I know that I'll have to start with the dispersion relations for a right and left circularly polarized waves:

Note that capital omegas $$\Omega$$ represent cyclotron frequencies, small omegas $$\omega$$ represent plasma frequencies, and the i/e subscripts refer to electrons or ions. The $$\omega$$ with no subscript is the frequency of the propagating wave.

$$\frac{k_R^2 c^2}{\omega^2}=1-\frac{\omega_{pe}^2}{\omega (\omega - \Omega_e)}-\frac{\omega_{pi}^2}{\omega (\omega + \Omega_i)}$$

$$\frac{k_L^2 c^2}{\omega^2}=1-\frac{\omega_{pe}^2}{\omega (\omega + \Omega_e)}-\frac{\omega_{pi}^2}{\omega (\omega - \Omega_i)}$$

Note that I'm dealing with an Alfven wave, so when manipulating the above formulae, the following assumption can be made: $$\omega << \Omega_i << \Omega_e$$

The angle that the wave's plane of polarization rotates through is
$$\Theta = \frac{(k_R-k_L)\Delta z}{2}$$

The problem lies in finding the $$k_R-k_L$$ term from the first two equations listed.

Using the relationship $$\frac{\omega_{pi}^2}{\Omega_i} = \frac{\omega_{pe}^2}{\Omega_e}$$ I can simplify the term for the R-wave to get

$$\frac{k_R^2 c^2}{\omega^2} \approx 1-\frac{\omega}{\Omega_i}+\frac{\Omega_i^2}{\omega_{pi}^2}$$

I know this is correct, but repeating the same process for the L-wave doesn't yield a term that can easily be combined with this one to get the $$k_R-k_L$$ expression I'm looking for.

If anyone could guide me through this, it would be much appreciated!