Crystals and Interference

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..

• If you press START, using the parameters set by the program, you will see the interference image produced by a uniaxial non-optically active crystal whose optical axis lies perpendicular to the screen. You can observe two different kinds of black regions in the figure, isogyres and isochromes. The large triangular areas  forming the shape of a windmill sail  result from the setting of the linear polarizer and analyzer. They are called isogyres (lines of the same inclination of the D-field's polarization plane). Since xi, the angle between the polarizer and the analyzer, is set to 0.5 pi, the black regions where the light is not transmitted by the analyzer correspond to directions along which the light's polarization plane is not altered by the crystal. This is the case, if the polarization of the incident ray of light coincides with the polarization plane defined by the crystal symmetry, thus the ray is not split up into two rays with paths of different refraction indices but keeps its polarization throughout its whole passage through the crystal. The intersection of the isogyres marks the direction of the optical axis. The second kind of black regions, the isochromes, are caused by the interference of the two beams passing through the crystal. With the setting of the default parameters they are just circles around the center indicating the ray which passes exactly along the optical axis.
• You can check the remarks above by changing the relative analyzer angle xi  to 0.0 pi. Here the exact inverse (photographic negative) of the original image should appear. What happens to the isogyres if you change the polarizer angle eta?
• To make the shape of the isogyres more clear choose the very last setup "draw D-field vector" via the choice menu and press START. This will generate the vector field of D. Think now about the polarization of the incident ray and the setting of the analyzer. Along which paths will the light beam be extinguished?
• You can investigate the isochromes without having to be disturbed by the pattern of the isogyres by simply switching the setup to "circular polarizer, circular analyzer".
• Return to the setup "linear polarizer, linear analyzer". The image produced by a biaxial crystal is generated by changing the refraction index, making ny different from nx. For realistic images the change should be in the order 0.001 - 0.01. Start by giving ny a value between nx and nz..
• The angles theta and phi define the orientation of the crystal´s optical axis with respect to the screen coordinates. Change theta to see what a rotation of the crystal does to the interference image. At theta= 0.5 pi hyperbolic lines appear, indicating that the crystals optical axis is now lying parallel to the plane of the screen.  What happens if you  interchange the values of nx, ny and nz?
• The type of curves (isochromes)  in the  image produced by a biaxial crystal (nx != ny, both < nz) at the angle theta=0 are called lemniscates. The two center points of a lemniscate indicate the two directions of the two optical axes of the crystal. The angle between these axes can be calculated to be  beta = ARCTAN(nz/ny*SQRT((ey - ex)/(ez - ey))) where ex,ey,ez are the dielectricities given by the squares of the refraction indices, that is ex=SQR(nx), ey=SQR(ny), ez=SQR(nz). You can check this formula, by calculating beta for your values of nx,ny,nz and then rotating the crystal in such a way that once the first center and then the second center of the lemniscate lies in the middle of the applets frame. How big was the angle by which you had to rotate?
• To study optical activity you have to give one of the gyroscopic tensor elements gij a value different from zero. Optical activity is a phenomenon much smaller than birefringence, so it is best observed in a region, where the refraction indices coincide. This is the case along the optical axes. If we set nx=ny, the optical axis will be the z-axis, so the element of the gyration tensor to consider will be gzz.
  The most impressive phenomenon due to optical activity are the Airy spirals. Choose the setup "linear polarizer, circular analyzer" with nx=ny=1.544, nz=1.553 and theta=0.0, thickness=3.0. Now set gzz=0.001 and press START. What happens if you set ny different from nx?
• Note that the optical activity does not influence at all the graph of the D-vector field, since the polarization planes follow directly from Fresnels (not extended) equations. The eigen polarizations of an optically active crystal are elliptical, so what is drawn in the vector field are now not the polarization planes but the principal axes of the elliptical eigen polarizations. You may think, that since an optically active medium does rotate the polarization of a traversing beam, it also should affect the crystal's polarization planes. This is not the case, since the rotation of the polarization is again an effect of splitting and recombining a beam at entrance and exit of the crystal.
• All points listed above can of course be also checked in the colored version. Here the name isochromes becomes more evident, they are the curves formed by light of the same wavelength (chroma gr.=color).
• The applet does not have any zoom function, but you can inspect interesting regions more closely by reducing the crystals thickness and rotating the crystal to the desired area.

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