Crystals and Interference


This site intends to give some insight into the propagation of optical waves in non-isotropic crystals. The main part consists of a virtual polarizing microscope which, according to the parameters set by the reader, displays interference figures of coherent light passing through a transparent crystal that exhibits double refraction and/or optical activity. In the past these interference figures played an important role in the field of classical optics, being used for the characterization of optical materials, determination of crystal symmetry, or specifying complementary colors. They can be found in (or even on) most standard textbooks on optics (references see below). For more general informations about waves, interference, refraction and so on you may want to have a look at one of the introductory pages 1 - 3 of the link list given below.

When entering a 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. 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. The image of a polarization microscope does not contain informations about the spatial properties of the specific piece of crystal (at least if it is used in the particular way described here), but about the crystal structure itself.

Apart from showing the beauty of interference patterns, the applet may help to get acquainted with the notion of crystal axes, polarization planes, different forms of polarization, and optical activity. 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 1-7 provide you with a short introduction to the basics of the optics the applet is about. 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. Informations concerning directly this applet are given by the following links.

Some technical details
How to use the program

Java virtual polarizing microscope, interference figures

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, ;

Related Web sites:

Gemology Project: Polariscope
Mineralogy Tutorial, Tulane Univ. Home Page. Especially the chapter Interference of Light
Electrical field and polarization, basics, Andras Szilagyi
Optical mineralogy introduction, Universidad Granada, spanish
Optical Mineralogy, Brock Univ., CA
Minerals under the Microscope, Univ. Bristol
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 Java animation. An experiment is outlined which makes use of the properties of optical nonlinearity and birefringence of crystals.

Curves of planetary motion in geocentric perspective: Epitrochoids. This site gives an introduction to planets orbits from a geocentric point of view.

Vibrating Strings, musical Intervals and Lissajous Curves. This site allows you to visualize two simultaneous oscillations via the motion of one or two vibrating strings or the Lissajous curves. By changing the wavenumbers of the two strings and the phase between the two oscillations graphs of different musical intervals, beats and interferences are generated.

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