 | What kind of properties need a material
have to have, so it linearly polarizes light? |
|  | 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? |
| | 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. |
| 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. |
| 1. Special geometries. |
| | 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. |
| | 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: |
| |
|
| | 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: |
| 2. Polarization filters or foils |
| | 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.
|
| | 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. |
| The first principle is easy to understand: |
| | 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: |
| |
|
| Surprise! The polarization direction is not parallel to
the rods as one would naively imagine but perpendicular to it. |
| | 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. |
| | 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. |
| 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?
|
| | 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. |
|
| 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. |
| So back to square 1: Any ideas of how to align conducting
nanorods on a 100 nm or so scale? |
| | 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 |
| |
 | | Si nanorods (actually rather microrods but one could make them
smaller if required) |
|
| | 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. |
| | 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. |
| | 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. |
| 3. Polarizing materials |
| | 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. |
| | 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. |
| | 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. |
| 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". |
| | 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. |
| | |
| 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 two-colored). |
| | 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. |
| 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
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. |
|
| 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
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. |
| | 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. |
| | 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 |
| |
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. |
| 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. |
| | 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).
|
| | 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. |
| | 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 |
| 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 |
| 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 |
|
| 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). |
| | Here are some
pictures taken in my office: |
| |
|
© H. Föll (Advanced Materials B, part 1 - script)
