Let us first examine the case where left and right circularly
polarized light are affected differently by the refractive index but not by the
extinction coefficient. The real part of the complex refractive index (i.e. the
refractive index) is defined as
n = c / v_{P}where c is the speed of light in empty
space. If while traveling within a material, left and right circularly polarized
light propagate with a different phase velocity, then after emerging from the
medium, the arrows and would complete different parts of the circle. However, since there is
no difference in absorption, both arrows would exhibit equal amplitudes. The
resulting polarization is schematized in figure 2, where, as you can see, at
time t = 0, ^{
}has not completed its circular motion and the resulting linear
polarization vector (blue arrow) appears to be rotated from its original
position by an angle
. This is called
optical rotation or circular birefringence.

Fig. 2 Linearly polarized light as a combination of left () and right () circularly polarized light in the case where the latter two propagate with
different phase velocities in the material, (i.e. the medium has a different
refractive index forand) but experience similar extinction coefficients.

It is very straightforward to evaluate
in terms of n^{-}
and n^{+} the respective refractive indices for and . Circularly polarized light is described by a set of two linearly polarized
electromagnetic waves. Thus for left and right circular polarizations:

(Eq.2.1)

which can be rewritten as: (since)

(Eq.2.2)

and we can then further rewrite:

(Eq.2.3)

If we then define:

(Eq.2.4)

equations 2.3 become:

(Eq.2.5)

.
Recognizing the opportunity to apply the formula for sin and cos sum and
difference:

(Eq.2.6)

Now we can combine the x-components and the y-components of these circular
polarizations in order to evaluate the resulting linear polarization:

(Eq.2.7)

Equations 2.7 represent a linearly polarized wave having an
angle of polarization
with respect to the x-axis such as:

(Eq.2.8)

where you can recognize:

(Eq.2.9)

with l being the length of
the material. Note that this angle is in radians and must be multiplied by
in order to obtain the value in degrees.