What is the mathematical difference between their refraction indices?
The conventions on the signs of the direction of magnetic field vectors and of charges.
The tutorial specifies several things that can have a sign;
- direction of the light propagation
- direction of the magnetic field
- sign of the particle's charge (electrons have negative charge)
Let's look at WIkipedia's Faraday effect; Interstellar medium where an equation for $RM$ or rotation measure is given:
$$RM = \frac{e^3}{2 \pi m^2 c^4} \int_0^d n_e(s) B_{\parallel}(s) ds$$
This expression includes $e^3$ and $B_{\parallel}$. Notice that the power of $e$ is odd. The assumption is that the interstellar medium is negative electrons, so they've replaced $q^3$ with $e^3$ but if it were positrons they'd have to write
$$q^3=-e^3$$.
Also remember that $B_{\parallel}$ can be positive or negative since the field could point either way relative to the direction of propagation.
The tutorial shows some of the intermediate steps where we get the idea that both the sign of the charge and the directions of the field will be important.
Why right handed circular polarization gets lagged when going through ionized plasma?
It could be either the RCP or LCP that is delayed more than the other, depending on the direction of the magnetic field relative to the direction of propagation, and on the sign of the lightest charged particles in the plasma that can respond most quickly to the electromagnetic wave.
In this example, the field just happened to be drawn parallel to the propagation, and normal plasma with negative electrons. In this case RCP is delated more than LHP. Had either one of those been opposite in the example, the result would have been oposite.
