Crystals and Interference

Some technical details

The basis of the program is formed by a not too difficult analytical calculation. Each point on the screen represents  a direction in space. You may think of the light beam forming a cone with an origin about 5 cm behind your screen. Along each direction the birefringence and optical activity of the crystal is calculated using an extended version of Fresnels equations (extended by the gyration tensor). The calculation for the electrical field is easily done starting from Maxwells equations, assuming a linear dielectricity and plane waves. With a plain doubly refracting material without damping and optical activity we arrive at Fresnels equations for the refraction indices, which lead to a quadratic equation for n^2. The two solutions of this equation correspond to the two orthogonally linear polarized waves that traverse the crystal. If we add optical activity, we get additional terms into Fresnels equations, but still arrive at a quadratic formula for n^2, which is solved just accordingly. The two found refraction indices now correspond to two orthogonal elliptically polarized waves (see ref. Born/Wolf). To calculate the ellipticity of these two waves is actually the more difficult part. I followed mostly Ramachandran/Ramasehan (ref. see intro). All calculations were done in the coordinate system of the crystals principal axes. These can be rotated by the reader using the angles theta and phi.

Knowing the ellipticity (Ramachandran/Ramasehan), the polarization planes of the D-field (Fresnel) and the phase difference of the two traversing waves (difference in refraction indices, thickness of the plate), we can readily deduce the brightness of a single beam passing through polarizer, crystal and analyzer with the help of Jones matrices. Again for plain doubly refracting materials without optical activity we arrive at simple, intuitively understandable formulas, adding the gyration tensor makes formulas look a little bit awkward.

The possible polarizers/analyzers that can be used in the applet are linear polarizers at all angles and a circular polarizer, resulting in three basic setups that can be investigated.
1) linear polarizer --- crystal --- linear analyzer
2) linear polarizer --- crystal --- circular analyzer
3) circular polarizer --- crystal --- circular analyzer
(exchanging polaizer and analyzer in setup 2 leads obviously to the same result).

Finally the generated interference pattern depends also on the employed wave length. The applet allows for monochromatic illumination and white light illumination. The latter leads to colored interference patterns since different wave lengths result in different path lengths. But it is rather crudely implemented and by no means as beautiful as in non-virtual life. There should always be some reason left to walk out of your room from time to time.

The main approximations used in the calculations are the following:
1) Neglecting double refraction at the entrance of the rays into the crystal, that is, the direction of the incident ray is not split up into two different ray paths only into two different rays on the same spacial path experiencing different refraction indices. This is allowed since the crystal plate is considered to be very thin. Setting the thickness parameter in the program to arbitrary large values results in interesting pixel interference figures, that probably do not relate much to our reality anymore.
2) No difference between energy and wave propagation (s-vector and k-vector, or equivalently direction of D- and E-field) was made
3) Neglecting the influence of the magnetic field
4) Simplifying the color calculations. Dispersion at the entrance of the rays into the crystal is neglected for the same reason as in (1), inside the crystal a crude empirical estimate for dispersion is used (instead of Sellmeyer equations or the like which again would need new parameters). So the main color effect is that due to the different ratio thickness/lambda for the red/green/blue wave (the interference maxima and minima of the three separate waves occur at different places).

Better versions of the applet later on may avoid some of the approximations, which is rather a matter of calculation time.  Approximation 1 is the reason why effects such as Haidinger rings cannot be made visible by the present applet.
Approximation 2 cancels off something known in nonlinear optics as beam walk-off.
Approximation 3 affects (to an unknown amount)  only the interference figures of  optically active crystals.