We can now consider what happens when left and right
circularly polarized light are affected differently by the extinction
coefficient but not by the refractive index. From what we have just seen, we can
deduce that s^{
-} and s^{ +
} would have the same phase velocity and that therefore they would
complete the same circular paths. Consequently, there would be no rotation of
the linear polarization vector (f = 0).
However, since the two circular polarizations do not experience the same amount
of scattering and absorption in the material, the relative amplitude of their
vectors changes. In terms of our picture, the arrow s^{
-} would have a smaller amplitude and will thereby complete a
smaller circle as it is shown in figure 3. As you can see, the resulting linear
polarization follows an ellipse, and hence it is said to exhibit ellipticity
denoted by q.

Fig. 3 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 but similar phase
velocities.

Quantitatively, the ellipticity is defined as:

(Eq.3.1)

where |s+| and
|s-| designates the amplitude of the
vectors. In practice, we can measure the intensities of both circularly
polarized light waves I^{+} and I^{-}. and since
the ellipticity is usually very small, we can assume that tan(q)
~ q. It then follows that

(Eq.3.2)

We can then make use of Beer's law. This is an empirical law
that gives the relationship between absorbance in a material and the intensity
of transmitted light which reads:

(Eq.3.3)

where A is again the amplitude
of the electromagnetic wave from Eq.1.1 and then Eq. 3.2 becomes: