Polarization

A pure right-handed circularly polarized wave propagating along the $z$-axis takes the form

$$E_x = A\,\cos(k\,z-\omega \,t),$$ (504)

$$E_y = -A\,\sin (k\,z-\omega \,t).$$ (505)

In terms of complex amplitudes, this becomes

$$\frac{{\rm i}\,E_x}{E_y} = 1.$$ (506)

Similarly, a left-handed circularly polarized wave is characterized by

$$\frac{{\rm i}\,E_x}{E_y} = -1.$$ (507)

The polarization of the transverse electric field is obtained from the middle line of Eq. (490):

$$\frac{{\rm i}\,E_x}{E_y} = \frac{n^2 -S}{D} = \frac{2n^2 - (R+L)}{R-L}.$$ (508)

For the case of parallel propagation, with $n^2 = R$, the above formula yields ${\rm i}\,E_x/E_y = 1$. Similarly, for the case of parallel propagation, with $n^2 = L$, we obtain ${\rm i}\,E_x/E_y = -1$. Thus, it is clear that the roots $n^2 = R$ and $n^2 = L$ in Eqs. (499)-(501) correspond to right- and left-handed circularly polarized waves, respectively.