We are now left with the third case, where the
electromagnetic radiation experiences both a different refractive index and a different extinction coefficient for its left and right circular polarizations
while propagating in the material. As we have seen, different angular velocities
means rotation of the linear polarization and different scattering/absorption
means ellipticity. Consequently, the linearly polarized light in this case would
exhibit both of these features and would follow a rotated ellipse, such as the
one in figure 4.
Fig. 4 Linearly polarized light as a combination of left (s
-) and right (s + ) circularly polarized light in the case where the latter two experience
different extinction coefficients in the material and different phase
velocities.
Ellipticity and rotation are then related by the
Kronig-Kramers relations:
(Eq.4.1)
where P denotes the Cauchy Principal Value. The
Kroning-Kramers transforms are a set of general mathematical equations that link
the real and imaginary part of a complex number. They apply here since the
optical rotation is related to the real part of the complex refractive index
while ellipticity originates in its imaginary part. As you can see, it is
possible to
obtain the ellipticity at a given wavelength from a spectrum of optical rotation
evaluations and vice-versa.
You now understand that any phenomenon that gives rise to a
difference in the light-matter interaction for left and right circularly
polarized light will result in an optical rotation and/or ellepticity. In the
case of Circular Dichroism this is done because of chirality, while in the case
of the Faraday effect, it originates in the magnetic field. We can then find
that the Faraday rotation is given by:
(Eq.4.2)
where V is the Verdet
constant, B the magnetic flux density and l is again the sample
length. Identifying with Eq. 2.9 we see that the difference in refractive indices
comes from the magnetic field. Exactly how much difference there is will depend
on the material constant V, which also
contains the wavelength information.
