
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: 


