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Ellipticity

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

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 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:

(Eq.3.4)

After expanding in Taylor series

(Eq.3.5)

we find that, in radians:

(Eq.3.6)


V. K. Valev

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