II.2 Refraction in anisotropic media

In general the behaviour of light in crystals is determined by the atomic structure causing the light to be transmitted with different velocities in different directions. Atoms are polarized in the direction of strongest attraction. The plane of maximum polarization is usually the plane of maximum atom population density in a crystal. A light wave (an oscillating electrical field) is strongly affected by the polarization of the atoms in a crystal. The stronger the atomic polarization is, the greater the effect i.e. slower velocity. Unevenly distributed ionic or covalent forces in a crystal create different polarization environments in different directions.

Let's have a look at the refraction of a very small part of the ray velocity surface, so small that we can assume it looks like a planar surface because the light source is infinitely far away.

Fig. 2.2

Case 1: Ray perpendicular to crystal surface (Fig. 2.2). Let's assume that the optical axis makes an angle of 45° with the crystal surface and lies in the plane of the drawing. In this case the wave front will not be refracted, neither will be the front normal. This already means that Snell's law is not applicable for light rays in this situation. In the crystal two wave fronts will be formed with two different velocities. Besides, both rays are polarised and the vibrational directions are within the wave front. The polarisation direction of the extraordinary ray is in the plane of the figure, while the polarisation of the ordinary ray is perpendicular to the plane of the figure.

Fig. 2.3

Case 2: Ray on angle with crystal surface (Fig. 2.3). Let's assume that a planar wave OP reaches the crystal surface OQ that makes an angle of 80° with the optical axis. The oridnary ray is now OG. For this ray Snell's law is valid, so:

sin i/sin r = sin COE/sin DOG = sin POQ/sin GQO = (PQ/OQ)/(OG/OQ) = PQ/OG = v(air)/v(crystal)

For the extraordinary ray follows:

sin i/sin r = sin COE/sin DOR = sin POQ/sin DOR ≠ PQ/OR

for the front normal follows:

sin i/sin r = sin COE/sin DON = sin POQ/sin NQO = (PQ/OQ)/(ON/OQ) = PQ/ON = v(air)/v(crystal)

Here we see that Snell's law is again valid, i.e. the ratio sin i/sin r is the same as the ratio of the velocities of the wave fronts in air and in the crystal. This law is thus valid for both the front normals, but not for both the rays. Therefore, as a definition of refractive index we use:

refractive index = velocity wave front in air/velocity wavefront in crystal = sin i/sin r (// front normal)