Crystals and Interference

This virtual polarizing microscope, which is an apparatus to carry out conoscopic observations, shows interference figures, generated by thin birefringent, optically active, crystal plates. in convergent light between two polarizers, (linear or circular). The light source can be chosen to be monochrome or white ( colored interference figures ).
Click the upper left corner to choose a setup.
START - generates the interference pattern

Index of refraction

Experimental setup

Gyration tensor

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In the non-virtual world a polarization microscope consists of a thin crystal plate between two lenses and two polarizers. The crucial point is to position the two lenses in a conoscopic arrangement such that the crystal plate is illuminated by convergent light. This means that rays of the same direction that pass the crystal plate get mapped to the same point in the eye of the observer  (i.e. the same point on the screen). In contrast to this a usual microscope maps rays of the same origin of the object to the same point of the screen, resulting in an image of the object. Thus the image of a polarization microscope in conoscopic arrangement does not contain informations about the spatial properties of the specific piece of crystal but about the crystal structure itself.
When entering the nonisotropic crystal, an incident ray of arbitrary polarization splits into two separate rays, each propagating within the crystal at its own velocity and with a polarization defined by the crystals symmetry. This leads to different optical path lengths for each one of these two rays. If an analyzer behind the crystal detects the resulting phase differences between the two rays, interference will occur. The interference will be different for different directions of the incident ray, since rays of different directions passing through a nonisotropic crystal, experience different refraction indices and a different gyration (rotation of the polarization plane). Thus if we map the interference results of all the directions of incident rays to points in the xy-plane, we obtain a pattern that will yield information about the position of the crystal's symmetry axes, the relative difference between the refraction indices, the optical activity tensor and so on. This is the pattern obtained by a polarization microscope.
Since the crystal can be rotated into any position, you can gain an intuitive 3D picture of the "optical geometry" of birefringent crystals.
The reference links number provide you with a short introduction to the basics of the optics. Most of the material is described comprehensively in the excellent textbooks by Born/Wolf or Ramachandran/Ramasehan. So for more details the reader may look there or in one of the references given below.


Quartz: nx=ny=1.544   nz=1.553   gxx=0.00007   gzz = -0.0002

For further reading see:

Born M., Wolf E.,Principles of Optics, Pergamon;
Ramachandran, G.N., Ramasehan, S., Crystal optics, Handbuch der Physik Bd. XXV;
Bergmann/Schäfer, Optik, de Gruyter;
Yariv, A., Yeh,P. Optical waves in crystal, Wiley, 1984;
Czaja, A.T., Einführung in die praktische Polarisationsmikroskopie, G. Fischer 1974;
Föppl, L., Mönch, E., Praktische Spannungsoptik, Springer, Berlin 1972;
Nesse, W.D., Introduction to Optical Mineralogy, Oxford University Press;
Rinne/Berek,  Anleitung zur ... Polarisations-Mikroskopie der Festkörper im Durchlicht, Schweizerbartsche Verlagsbuchh.
Wahlstrom, E., Optical Crystallography, Wiley 1979;
Ehlers, Optical Mineralogy, Blackwell Scientific 1987;
Brossein, Polarized Light, ;
Arteaga et al., Determination of the components of the gyration tensor of quartz

Related Web sites:

Gemology Project: Polariscope
Microscopy and mineralogy by J.M.Derochette
Mineralogy Tutorial, Tulane Univ. Home Page. Especially the chapter Interference of Light
Optical mineralogy introduction, Universidad Granada, spanish
Mineral data base

A Gallery of Quantum States. This site gives an introduction to the quantum mechanical treatment of the light field. The notion of wave packets, photons etc are explained using a Javascript animation. An experiment is outlined which makes use of the properties of optical nonlinearity and birefringence of crystals. Actual experimental data of coherent states, squeezed light and single photons are presented.

to my homepage


Some details

How to use the program

The applet generates the interference figures of non-isotropic media as seen through a polarization microscope. The following suggestions may give you a starting point..

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.

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 script 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.

Last modified: Jan. 2019