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.
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
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.
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.
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.
