
Optical Anisotropy and Tensor Materials 



The basic ideas are easy to state. First, if there is an optical anisotropy, it
can take two basic forms (and then mixtures of the two, of course): 

1. The index of refraction depends on the crystal direction. The optical
effects resulting from this are called "birefringence" or double
refraction (Latin: bi=two, twice, refringere=to break up, to refract). 

2. The absorption depends on the polarization (and the crystal direction).
In some directions far less light of some polarization will come out after travelling through a crystal of given thickness
than light polarized in the other direction. This effect is called dichroism (Greek for twocolored).



It is an unfortunate word because historically it was used first for a different
effect: White light is split into into distinct beams of different wavelengths because of the crystal anisotropy and this
is completely different from regular dispersion! Since the absorption effect may strongly depend on the wavelength, too,
everything can be mixed up wonderfully. The word "pleochroism" ("morecolored")
is occasionally used as a more general term, containing all of the above. 

You know, of course (?) that the two effects must be related because they must both be contained
in the complex index of refraction, which simply (haha) happens to be a tensor
now. All tensor components are functions of the wavelength because we still have dispersion
and, since we also have the Fresnel equations,
they must be functions of the polarization, too 

All we need to do now is to run through the derivation of the Fresnel equations once more.
Just consider now a complex index of refraction that is also a tensor of of second rank with a symmetry that is somehow
related to the crystal symmetry instead of a scale for boundary conditions and propagation. That would be a program that
could entertain (some of) us for many weeks  so we are not going to do it here. 
 
It is important to realize, however, that with our modern computers this is actually an easy task. As soon
as a basic program has be written, and the functions describing the tensor components of the complex index are inputted,
the rest is child's play. That's why for you, the young Materials Science and Engineer, the complex index is so important.
For old guys like me it was not important, since we simply couldn't do the necessary math with a slide ruler as the only
math tool. We had to resort to approximations and special case studies, and that's where all these fance names and distinctions
comes from. 

We are going to give the whole topic just a very superficial look, focussing on the practical
side. As a first simplification we simply ignore di or pleochroism for the time being and only give birefringence
a slightly closer look. We also ignore absorption and thus the imaginary part of the complex index of refraction. 


Recalling the little bit what we
know about tensors, we are certain that there is always a coordinate system or "axes", where only the diagonal
elements of the refractive index tensor are different from zero. These axes we call the principal
axes of the crystal under consideration. 


We then can always define three
principal refractive indices
n_{1}, n_{2}, n_{3} as the tensor components in the principal
axes system. Note that those n's will depend on the polarization and the wavelength. Of course, the principle
axes will be related to the symmetry of the crystal in some way. 


The rest is math. Not too difficult if you are used to tenor algebra but a bit mind boggling
for normal people like us. For example, the dielectric displacement D or the Poynting vector S
may no longer be parallel to the electrical field E or the wave vector k, respectively.
Now what would that mean? 

Be
that as it may, the fact of importance here is that in general, an unpolarized light
beam entering an anisotropic crystals splits into two orthogonally polarized light beams
inside the birefringent material (typically a single crystal, a collection of small aligned single crystals, or an amorphous
foil "somehow" made anisotropic and birefringent). Two different light beams
thus will run through the crystal and eventually exit on the other side as shown below. 
 


Incoming beam not along optic axis
(see below), but at right angle to some surface.
Þ No change of direction under "ordinary" circumstances and for the ordinary beam. Pronounced change of direction for extraordinary
beam. 
Incoming beam along optic axis (see beleow) and at right
angle to surface.
Þ No change of direction under "ordinary" circumstances and
for both beams in birefringent material. Both beams run parallel but with different
velocities. 


The two figures above illustrate the basic effects that will occur. It is best to discuss birefringences
for the two special cases shown 

1. case: The incoming (unpolarized) beam is not
parallel to a principal axis of the crystal but its angle of incidence is 90^{o}, i.e. it is exactly perpendicular
to the surface of the crystal. This is a special condition, but not so very special because it is easy to do in an experiment,
All you need are some flat surfaces. 


For an isotropic material with a scalar
index of refraction n we know that there will be no refraction
or bending of the beam. A little bit will be reflected (see the Fresnel equation),
and the polarization does not matter. The transmitted beam will travel through
the material at a velocity c=c_{o}/n. 


Exactly the same thing happens for the ordinary wave or
owave in our tensor material here. However, the owave now is fully polarized as shown



But in major contrast to isotropic materials with a scalar n, an extraordinary
thing happens in our birefringent material with a tensor
n. An extraordinary wave or ewave is also generated at the
interface. This ewave travels under some angle with a velocity that is different
from that of the owave. The ewave is also fully polarized, but with
a direction orthogonal to that of the owave. 

2. case:
The incoming (unpolarized) beam is parallel to an optical
axis of the crystal, and its angle of incidence is still 90^{o}, i.e. it is exactly perpendicular
to the surface of the crystal. This is a rather special condition now because it means that our crystal must have a planar
surface perpendicular to an special crystal direciion. 


The socalled optic axis of the birefringent crystal is coupled
to our principal axes from above. We have something new now (or just an extreme
case of the general situation). 


The owave and the ewave travel in the
same direction=optic axis but with different velocities.
Both waves are still fully polarized in orthogonal directions. 


The two waves emerging form the crystal then have different
phases. The exact phase difference depends on the distance covered inside the material, i.e. on the thickness
of the material. 

That's just a description of what you will observe, of course. Calculating the relative intensities
of the two beams, the intensities of reflected beams (??? (there is only one)), the two angles
of refraction, the polarization directions, and the propagation velocities from the index of refraction tensor is possible,
of course—but well beyond our scope here. 


One question remains: How many optic axes are there? The answer is: Two
for "fully" anisotropic material and one for somewhat more symmetric materials.
That tells you that the optic axis and the principal axis are not the same thing because we always have
three pricipal axes. It's all in the table below. 


Dielectric tensor in principal axes 
Anisotropic Materials 
Isotropic Materials 
e_{r} 
=_{ } 
æ ç è

e_{1} 0_{ } 0_{ }
0_{ } e_{2} 0_{ }
_{ }0_{ } 0_{ } e_{3} 
ö ÷ ø 

e_{r} 
=_{ } 
æ ç è

e_{1} 0_{ } 0_{ }
0_{ } e_{1} 0_{ }
0_{ } 0_{ } e_{3} 
ö ÷ ø 

e_{r} 
=_{ } 
æ ç è

e_{1} 0_{ } 0_{ }
0_{ } e_{1} 0_{ }
0_{ } 0_{ } e_{1} 
ö ÷ ø 
 General case Two optical axes 
Practical case One optical axis 
Simple case All directions / axes are equal 
Mica 
e_{3} > e_{1} Positive
uniaxial 
e_{3} < e_{1} Negative uniaxial

Glass Diamond Cubic crystal (NaCl, CaF_{2}, Si, GaAs, ..) 
Ice quartz TiO_{2} (rutile) 
Calcite (CaCO_{3}) Tourmaline LiNbO_{3} 


I can't end this chapter without a quick reference to the ubiquitous direct effect of birefringent
crystals discovered long ago. There is deceptively simple effect already known
to Huygens and Co: Take an "Iceland spar" crystal
(CaCO_{3} or calcite; easy to get), put it on some paper with
writing on it and you see everything twice (see below). 
 
Here are some pictures taken in my office: 


 Birefringence with a calcite crystal 
