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What kind of properties need a
material have to have, so it linearly
polarizes light? |
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For starters: if you run the light through the material, it needs to be transparent.
But who says that you can only polarize light by running it through a material? How about bouncing it off some
material or, in other words, reflecting it
at some mirror-like surface? |
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This should trigger a flash-back. As you already
know, it is perfectly possible to polarize light by utilizing reflection; just
look once more at the Fresnel
equations. |
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All things considered, there are
several ways to polarizes light, each one with its own requirements,
advantages, and disadvantages. Here we'll just take a first glimpse at some
major polarizing methods. |
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1. Special geometries. |
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As we have learned under the heading "Fresnel
equations", a non-polarized light beam impinging on
any material under some special angle
("Brewster angle") will
produce a fully polarized reflected beam.
Of course, some materials are more suitable than others (the index of
refraction should have a decent value, not too close to n = 1 but
not too large either) but no special
properties of the materials are required. |
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Using the "Brewster angle" approach is
indeed a major way to produce polarized light. It is more or less limited to
"advanced" application, however, because there are some problems.
Time for a quick exercise: |
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For everyday applications like your
"Polaroid" sun glasses or the cheap glasses used for watching
3D movies in the cinema, you need something less special and far, far
cheaper. You need: |
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2.
Polarization filters or foils |
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We have a typically thin and
bendable transparent foil (or a thin unbendable glass-like sheet). Unpolarized
light goes in from one side, linearly
polarized light comes out on the other side. There are two basic ways or principles for that:
- The material contains "nanorod" conductors arranged in a grid
with dimensions in the wavelength region.
- The material is "birefringent" or in neutral terms, it is
optically anisotropic, meaning that
e and therefore also n are
tensors.
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The bad news
are: The second principle is the more important one!
The good news are: We are not going to look at that in great detail, since
there is not enough time. Tensor optics also may cause the bulk intake of large
quantities of alcohol to soothe your brain, and you are too young for
this. |
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The first principle is easy to understand: |
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Just imagine for a moment that you need to
polarize radio waves with a wave length in
the cm region and not light. All you need to do is to use a bunch of
aligned conducting rods as shown
below: |
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Surprise! The polarization direction is not parallel to the rods as one would naively
imagine but perpendicular
to it. |
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It's clear what happens, though. The field
components parallel to the conducting rods
will simply be short circuited, causing currents j =
sE in the rod, and thus heating.
The components perpendicular to the rods
cannot cause much current and pass mostly through. The figure above, by the
way, is identical to a figure we had before except that the symbolic polarizer
in this picture has now materialized into a defined device or product. |
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If you wonder about possible diffraction effects at the grid: There aren't any if
the distance between the rods is smaller than a wavelength. Just do an
Ewald
sphere construction to see this. |
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The priciple, of course, also works for the
electromagnetic waves we call light. All that remains to do, is to produce
aligned conductive rods on a 100 nm or so scale; nanorods in other words. Any ideas? |
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Well, a guy named Edwin Herbert Land (1909
- 1991) had an idea - in 1932. Take a polymer foil that can easily be stretched
to a large extent (look at
this
link to get an idea), get some proper stuff inside (microscopic
herapathite crystals in
Lang's case; funny things in their own right involving
dog piss), and then stretch the
whole foil, aligning the little crystals. Use
the link to get the full
story. |
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I don't know what Lang thought when he did this.
Some people (and text books) believe that he did align
conducting rods, indeed. Chances are that he did not, however. The
herapathite crystals in Lang's case are actually very complex crystals; their
structure was found out only a few years ago. Herapathite (also known as
Chininbisulfatpolyiodid) forms intrinsically polarizing crystals, and all the
stretching of the foil just insured that their polarization axes were
aligned. |
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So back to square 1: Any
ideas of how to align conducting nanorods on a 100 nm or so scale? |
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Of course you have an idea: Use the basic
micro (or by now nano) technologies of
semiconductor technology and produce
something as shown in the picture below |
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| Si nanorods (actually rather microrods but one could make
them smaller if required) |
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The rods shown consist of Si and they
would certainly polarizes (IR) light besides doing a few more things
(due to their 3-dim.arrangement). It's however not a practical way for cheap
mass-production stuff. Moreover, while light coming out on the
"other" side would certainly be linearly polarized, one can just feel
that it would not be a lot of light that makes it through the structure. More
about this structure (which was not made for optics but for something utterly
different) in the link. |
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If we look a bit beyond "normal" optics
into the far UV and IR, polarizing a beam is not all that easy
and structures as shown above might be the "future" in this case.
Here is the link for some
more information about this. |
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Time for the famous last word: I could not find
out in April 2011 if there are actually (cheap) polarizing foils out there that
actually work on the "conducting rod" principle. So let's go for a
third way. |
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3. Polarizing materials |
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A good polarizing material, e.g. some crystal and not a "foil", transmits an
incoming beam without too much absorption and emits one or even two beams that are more or less polarized.
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We can already make one statement about the
properties such a material must have: All directions must not be equal since the polarization direction is per
definition special. Isotropic materials like most amorphous stuff and all cubic crystals, thus cannot be polarizing materials. |
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A stretched polymer foil is no longer isotropic
since the polymer chains now are somewhat aligned in the stretching direction.
However, it will not necessarily polarize light. Try
it!. Stretch some suitable candy wrap and see if it now polarizes
light. Changes are it will not. So just being structurally anisotropic is not a good enough condition for being
a polarizer. |
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That means we must now deal with not-so-simple
Bravais lattices, crystals, or other structures. It also means that properties
like the dielectric constant er
and thus the index of refraction n, the conductivity sr, possible magnetic properties, the
modulus of elasticity (Young's modulus) and so on are no longer simple scalar number but
tensors
of second, third or even fourth rank! We are now discussing crystalline
"optical tensor materials". |
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Talk about opening a can of worms! Nevertheless,
understanding and exploiting the tensor properties of materials (mostly
crystals) is where the action is right now (2011), and where it will be for the
foreseeable future.
At this point I will only enumerate some of the important effects of tensor
materials on polarization. |
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Optical Anisotropy and
Tensor Materials |
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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): |
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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). |
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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 two-colored). |
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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" ("more-colored") is
occasionally used as a more general term, containing all of the above. |
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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 |
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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. |
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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. |
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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. |
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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. |
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We then can always define three principal refractive indices
n1, n2,
n3 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. |
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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? |
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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. |
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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. |
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The two figures above illustrate the basic effects
that will occur. It is best to discuss birefringences for the two special cases
shown |
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1. case: The incoming (unpolarized) beam is
not parallel to a principal axis of the
crystal but its angle of incidence is 90o, 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. |
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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 =
co/n. |
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Exactly the same thing happens for the
ordinary wave or o-wave in our tensor
material here. However, the o-wave now is fully
polarized as shown |
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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 e-wave is
also generated at the interface. This e-wave travels under some angle
with a velocity that is different from that
of the o-wave. The e-wave is also fully
polarized, but with a direction orthogonal to that of the
o-wave. |
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2. case: The incoming (unpolarized) beam is
parallel to an optical axis of the crystal, and its angle of
incidence is still 90o, 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. |
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The so-called 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). |
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The o-wave and the e-wave travel in the same direction = optic axis but with different velocities. Both waves are still fully
polarized in orthogonal directions. |
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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. |
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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. |
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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. |
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| Dielectric tensor in principal axes |
| Anisotropic Materials |
Isotropic Materials |
| er |
= |
æ
ç
è |
e1 0 0
0 e2 0
0 0 e3 |
ö
÷
ø |
|
| er |
= |
æ
ç
è |
e1 0 0
0 e1 0
0 0 e3 |
ö
÷
ø |
|
| er |
= |
æ
ç
è |
e1 0 0
0 e1 0
0 0 e1 |
ö
÷
ø |
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General case
Two optical axes |
Practical case
One optical axis |
Simple case
All directions / axes are equal |
| Mica |
e3 >
e1
Positive uniaxial |
e3 <
e1
Negative uniaxial |
Glass
Diamond
Cubic crystal
(NaCl, CaF2, Si, GaAs, ..) |
Ice
quartz
TiO2 (rutile) |
Calcite (CaCO3)
Tourmaline
LiNbO3 |
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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 (CaCO3 or
calcite; easy to get), put it on
some paper with writing on it and you see everything twice (see below). |
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Here are some pictures taken in my office: |
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© H. Föll (Advanced Materials B, part 1 - script)
