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THE
LIGHT FANTASTIC
Donald H.
Lyons,Department of Physics,University of Massachusetts/Boston
March 1987
The title of
my talk is "The Light Fantastic." It might have been something
like "Fascinating Physics," or "Why I Love Physics," or
maybe, more deeply, something like "The Explanatory Power
of Physics," or, dressing it up, "Modern Paradigms for Light."
But really, it's mainly going to be a kind of magic show.
Let me tell
you a bit of how I got into this. Lexington, where I live,
has an art gallery called "Gallery on the Green." [Editor's
note: No longer in business. See Home Page for up-to-date
list of galleries representing Austine] One day I wandered
into the gallery and saw these things called Polages. I
have one here on exhibit1. If you look at it carefully,
you'll see that the colors in it continually shift and change.
The first question for me was, how does it work? By the
way, the artist who makes these Polages is in the audience.
Her name is Austine Wood-Comarow, and she is here with her
husband to supervise the mounting of a huge 25 foot by 27
foot permanent exhibit of Polages in the Museum of Science.
When I shine
light through this Polage, you don't see much of anything,
just some nearly colorless plastic2. Now, if I put this
other piece of colorless plastic in front of the Polage,
suddenly I get intense colors which shift and change as
I rotate the plastic. They also change if I tilt the plastic
or you view it at a different angle. If I take away the
plastic and you look at the Polage in reflection, using
an ordinary mirror, again you don't see much of anything,
nothing unexpected. But if I substitute a shiny piece of
black plastic for the mirror, you'll see colors again. The
image formed by reflection is colored, but the object which
is reflected is not! There are a lot of little mysteries
here!
To help explain
what we have seen, I'll run through a brief history of optics.
(Fig.1)The first thing we note is the stellar galaxy of
people who contributed to the science of optics of thought
about light. We can go back to Euclid, who contributed the
idea that there are rays of something which travel in straight
lines. According to Euclid, however, that something traveled
outward from the eye - a common theory in earlier times.
One didn't receive light or anything else, rather the eye
actively sought out objects by emitting something. After
Euclid, we come to another famous figure, Claudius Ptolemy,
who lived in Alexandria. Ptolemy made a table of angles
of incidence vs. angles of refraction for a light beam passing
from air to water. This data is perhaps the oldest recorded
table of physical measurements. Ptolemy, of course, is much
better known for his theories of how the sun and planets
move around the Earth, He was also the leading astrologer
of the day. Kepler, known chiefly for his immortal three
laws of planetary motion - for example, that planets move
in elliptical orbits - in addition laid the foundation for
geometrical optics. Descartes was the first to give a proper
explanation for the formation of rainbows; here we again
have colorless material, water droplets, making colors.
He also said, supposedly, "I think, therefore I am." By
thinking about rainbows, he proved his own existence. Newton's
experiments with prisms were famous. Newton not only showed
that light could be dispersed by a prism into a spectrum
of colors, he showed that a second prism could recombine
the light of different colors into white light. That demonstrated
that the colors were not manufactured in the prism itself
but were actually present in the white light.
Bartholinus,
a Danish physicist not now generally known, discovered double
refraction. Double refraction, also called birefringence,
is at the heart of our story, the heart of how a Polage
works3.
Here I show
an arrow, and now I put a calcite crystal over it and you
see two arrows instead of one. When I rotate the crystal,
you notice that one arrow stays fixed while the other one
rotates. We'll get back to this after the next bit of history,
which is Huygens discovery of polarization. Huygens, known
today mostly for his principle of secondary wavlets which
contributed a lot to the development of the wave theory
of light, also discovered the phenomenon of polarization.
Before explaining polarization, I want to show you what
effect it has. I have here something called a Polaroid filter,
which polarizes light. If I lay it on top of the calcite
crystal and orient the polaroid one way, we see only one
arrow . If I rotate it by 90 degrees, we see only the other
arrow. This proves that the light in one arrow is polarized
one way and the light in the other arrow is polarized along
the perpendicular direction. Birefringence and polarization
are the two phenomena required to explain how a Polage works.
A complete Polage is a sandwich of two polaroids with birefringent
material in the middle (Fig. 2). So we have three elements:
polaroid. birefringent material, another polaroid. When
we understand how these work together, we'll understand
Polages.
Goethe (Fig.
I), not a famous name in science (although he wrote something
like fourteen volumes on science) wrote a huge tome of some
fifteen hundred pages on the theory of colors, part of which
is still useful; not an equation in it, no mathematics at
all. The first part is an immense compendium of all the
phenomena that have to do with color, the next part is a
polemical attack on Newton's theory of light, and then there
is a final historical part. Up to the time of the early
nineteenth century there was a dispute as to whether light
consisted of waves or particles. The great authority Newton
maintained that light consisted of particles despite the
fact that many of his own experiments could be easily explained
with the wave theory. The wave theory triumphed in the nineteenth
century as a result of the experiments of Young and Fresnel,
who demonstrated that light exhibits the properties of a
particular kind of wave, namely transverse waves. The only
kind of wave that can be polarized is a transverse wave.
The next name,
Brewster, is not very well known but Brewster comes into
our story because he studied how reflected light is polarized.
When light is reflected by a non-metallic surface, the reflected
light is partially or completely polarized even if the incident
light is unpolarized. That is why we saw a colored image
of the Polage when we saw it as a reflection in the black
plastic sheet. We didn't see colors in the image formed
by the mirror because the reflecting surface, the film of
silver, was metallic. In the course of his studies of reflection,
Brewster invented the kaleidoscope, which became a rage
throughout Europe and a principal form of home entertainment.
This should have made Brewster rich but, unfortunately for
him, the market quickly became captured by a flood of cheap,
inferior kaleidoscopes. I have something here which looks
like a kaleidoscope. called a "Karascope". Designed by Judith
Karelitz and sold through the Museum of Modern Art, it works
on the same principles as Polages; if you look through it
and rotate it, you see various shifting colors made by polarizers
and birefringent plastic.
The next, and
very important, figure is James Maxwell, who added to and
unified the existing theories of electricity and magnetism.
Here is a T-shirt on which are printed the celebrated Maxwell
equations of electromagnetism. Such shirts are commonly
owned by physics students who study these equations at length
and learn that most of the properties of light can be predicted
from them. From his equations, Maxwell inferred the existence
of electromagnetic waves which travel with the known speed
of light, and concluded that light consisted of such waves.
Light waves, because they are electromagnetic waves, exert
forces on the charged particles of matter.
Einstein also
had a little bit to do with light. According to Einstein,
light travels at the same speed, with respect to an observer,
no matter how fast the observer is moving. That'screwy!
But not really - that idea led to the special theory of
relativity! Still, Einstein didn't get a Nobel prize for
that or for his general theory of relativity; he got it
for explaining certain features of the photoelectric effect.
To do that, he invented the photon concept, in which light
is modeled as a stream of particles. So, is light a particle
or a wave? I could say a lot about that, but I won't. For
us, it's a wave: it acts like a wave. The other, final bit
of history which made Polages possible was the invention
of a cheap, simple and convenient polarizer. In 1928, when
he was an undergraduate at Harvard University, Edwin Land
invented something called J-sheet. In 1938 he invented H-sheet,
which is what we're using today. He called the material
"polaroid" and went on to found the Polaroid Corporation.
As I've said, two polaroid sheets make up the front and
back of a complete Polage.
I want to turn
now to what polarized light is and how polaroids work, how
they polarize light. When light is incident on or passing
through a medium, the charged particles (nuclei, electrons)
experience a force. I want to focus on the force. (Fig.
2) Let the little black dot in the center of the screen
represent a charged particle, and suppose there is a light
wave traveling from the screen toward you. There will be
a force on the particle. The arrow represents the force,
the length of the arrow being proportional to the size of
force and the direction of the arrow being the direction
of the force, which lies parallel to the screen. I can resolve
the force into components, one horizontal component and
one vertical component. Together, the two components make
up a force that is at an angle to the horizontal. The picture
shows the force at one instant of time. As time passes,
the two components rapidly change their size, and go back
and forth from positive to negative. For visible light,
these fluctuations are extremely fast - on the order of
one million, billion times each second. By definition, when
the light is Iinearly polarized, the two components will
fluctuate together and the resultant force will always be
parallel (or, anti-parallel) to the direction shown on the
screen. The two components, in this case, are precisely
correlated and we say they are in phase. On the other hand,
for unpolarized light, the two components are uncorrelated
and the tip of the arrow will move all around the center
of the diagram, and at any Instant might be pointing in
any direction. If unpolarized light passes through a polarizer,
it emerges linearly polarized.
How does a polarizer
change unpolarized light to polarized light? The polarizers
I'm going to talk about are called dichroic polarizers.
They absorb or reflect the energy from one component and
let the other component through. Now imagine a lot of wires
like this, vertical, and light incident on the wires (Fig.
4). The force on the electrons in the wire at one instant
of time is represented by the arrow. One can break the arrow
into two components, one that fluctuates vertically, parallel
to the wires, and the other component that fluctuates horizontally.
The vertical component causes the electrons to move up and
down in the wire, which results in the energy in that component
of the light being partly reflected and partly turned into
heat in the wire. The other component of the light exerts
a horizontal force on the electrons, but they are confined
to the wire and cannot move horizontally. As a result, the
horizontal component passes through the wire grid and the
light that emerges is horizontally polarized - the force
that it exerts on a charged particle is horizontal. That's
how a wire grid polarizer works. To make one, the wires
must be much closer than the wavelength of the light, which
means there has to be a lot of wires very close together.
Someone succeeded in making one with fifty thousand wires
per inch, so that in one one-thousandth of an inch, there
are fifty wires. How do you make something like that? You
rule a diffraction grating. That is, you have a machine
which scratches a straight line on a metal plate with a
diamond point. The point is attached to a very fine screw
which can be advanced 1/50,000 of an inch between scratches.
Then, you use the plate to form a transparent plastic sheet
imprinted with the scatched lines. Having done all that,
you evaporate gold atoms so that they are incident on the
plastic at a very shallow angle. The gold gets plated on
the edges of the ridges as shown by the black dots in the
diagram. You end up with a wire grid,50,000 wires per inch,
that polarizes light. This little trick was accomplished
in 1960.
Edwin Land had
a better idea in 1938, when he invented H-sheet (Fig.5).
It consists of polyvinyl alcohol (PVA) plastic sheet which
has been stretched and mounted on cellulose acetate. Before
you stretch it, the molecules are all twisted every which
way, but when you stretch it out, the long molecules line
up in the direction of the stretch. Iodine atoms are attached
to the PVA molecules which make the molecules conducting.
The PVA sheet is attached to the cellulose acetate to keep
it from shrinking back to its original size. So now you
have microscopic wires all lined up and the system acts
just like a wire grid polarizer.
Now let's see
polaroid in action. I have two large pieces here. If you
want some, what you do is call up the Polaroid company and
ask them to send you a price list and some samples. They
send you these big sheets. The second time you do it, they
send you these little tiny sheets. Unfortunately, the process
converges rapidly and you only get a finite amount of polaroid
out of them. Now, I put a polaroid on the projector and
a second polaroid on top of it. You don't see anything much,
just two ordinary pieces of gray plastic. But if I turn
the top polaroid 90 degrees, it blocks out almost all the
light. The first polaroid passes only the horizontal component
of the unpolarized light, and the second polaroid blocks
it now because it allows only a vertical component to pass.
But now watch this. I insert a piece of glass between the
polaroids and you see that nothing changes. Now, I put a
piece of tape on the glass - Tuck tape which is cellulose
tape - and insert it between the polaroids. Now light passes
through where the tape is If I rotate the top polaroid,
light passes everywhere except where the tape is. Now cross
the first tape with a second piece and you see that the
second piece cancels out the effect of the first where they
cross, so that the center is light, but the rest of the
tape is dark. Rotating the top polaroid again reverses that.
On the other hand, the tape does nothing if it is lined
up with the edges of the polaroids. Tuck tape can do all
this because, like calcite, it is birefringent. You can
see that now we have all kinds of possibilities. Before
explaining how polaroids combined with birefringent material
can exhibit colors, let me show you some more examples of
the phenomenon. Brewster, who made the first study of polarization,
also invented two new applied fields. One is called optical
mineralogy, the study of minerals, generally under a microscope,
while being sandwiched between polarizers. Here's an example,
a large piece of mica, and you see I can make a "Polage"
out of it . If I place the mica between polaroids, it becomes
iridescent because it's birefringent. If I replace the mica
with a folded and crumpled sheet of cellophane, it appears
very similar to the mica. We can learn a lot about minerals
by using polarized light. The second field created by Brewster
is called photoelasticity. This piece of Plexiglass plastic
is only weakly birefringent, but if I stress it, the birefringence
increases and you can see intense and varied colors appear
in the regions of maximum stress. If you're clever enough,
before you build a bridge, you build a model of the bridge
out of plastic, illuminate the model with polarized light
and examine it through a polaroid. You can deduce a lot,
even quantitatively, about the stresses. When you have a
complex problem in stress analysis, that's a way to do it.
Now let's talk
about how light passing through calcite produces two images
and how the polarization of light is changed when it passes
through cellophane; both effects are consequences of birefringence.
When a ray of unpolarized light is incident on the calcite
from the direction shown, it divides into two rays, each
one linearly polarized (Fig. 6). The ray which goes straight
through is the ordinary ray, so-called because it obeys
Snell's law of refraction. The X's and 0's represent arrows
into and out of the plane of the figure, respectively, and
give the direction of the force field at their locations.
Thus, the ordinary ray is linearly polarized perpendicular
to the figure. The lower ray, the extraordinary ray, so-called
because instead of going straight through, it goes through
in a crazy direction, creates a second image. The arrows
indicate that this ray is polarized in the plane of the
figure. The two rays with different polarizations - and
this fact is crucially important for the following - go
through in different directions because they travel at different
speeds.
Cellophane is
a little different from calcite, but the principle involved
is very similar(Fig. 7). There are two perpendicular axes
in the plane of the cellophane, which I'll call the fast
axis and the slow axis. Light which is polarized along the
fast axis travels faster through the cellophane than light
which is polarized along the slow axis. The diagram shows
incident light which is linearly polarized in a direction
between the fast and slow axes. This is equivalent to a
combination of light polarized along the fast axis plus
light polarized along the slow axis, whose force fields
are in phase, as illustrated by placing the arrows directly
under the X's and 0's. The two components have been displaced
vertically in the diagram in order to more clearly illustrate
what happens. In the cellophane, the light travels more
slowly, as can be inferred by the shorter wavelengths, given
by the distance between vertical arrows, for example. (Incidently,
this slower travel is in contradiction to Newton's theory
of light particles which he thought must travel faster in
material.) Note that the wavelength of light polarized in
the plane is slightly shorter than the wavelength of light
polarized perpendicular to the figure, when the light is
in the cellophane. Hence, when the light emerges from the
cellophane. the two components are no longer in phase. Here's
a second diagram (Fig. 8) which is slightly different. Instead
of representing the force field by arrows, it shows graphs
of the two components of the field. As before, the two components
have been displaced vertically in the diagram in order to
more clearly illustrate what happens. The height of the
peaks of both components are equal, implying that the components
are equal in magnitude. This means that the resultant force
field makes equal angles, 45 degrees, with the fast and
slow axes. You see that two and a quarter slow waves fit
in the material, whereas only two waves of the fast wave
fit inside. Consequently, when the peaks of the two waves
emerge, they are separated in space by a quarter of a wavelength.
Now a quarter of a wavelength in space translates to a quarter
of a cycle in time. A full cycle is 360 degrees, one-quarter
is 90 degrees, so the two emergent waves are ~g0 out of
phase, whereas they were in phase before entering . As a
result, the birefringent material has altered the polarization;
what went in as linearlv polarized light comes out circularly
polarized.
"Circular (I)
polarization- you never mentioned that before," I hear you
cry. Well, there's not only linear and circular polarization.
there's elliptical polarization. When two perpendicular
vectors oscillate with the same frequency, the tip of the
resultant vector traces out an ellipse (or line segment
or circle, special cases of ellipses). Figure 9 shows some
of the possibilities. The shape and size of the ellipse
depends on the amplitudes and phases of the oscillations
of the component vectors. When the phase difference of the
oscillations is zero, the ellipse flattens to a line segment
and the tip of the resultant vector moves back and forth
along a straight line. For a phase difference of 60~, the
tip moves along a skinny ellipse. If the phase difference
is 900, and the amplitudes are equal, the tip moves in a
circle. I can demonstrate this concept with the aid of a
ball on a string. As I swing the ball in a vertical circle,
you see it moving up and down, while at the same time it
moves left and right. Note that the maximum vertical displacement
from the center occurs when the horizontal displacement
is zero; the horizontal displacement is one quarter of a
cycle, or 90 degrees , out of phase with the vertical displacement
Now imagine a strong light illuminating this ball from some
arbitrary direction, with a shadow of the ball cast upon
some screen. The shadow would trace out an ellipse on the
screen. The motion of the shadow could be resolved into
two simultaneous oscillations of generally different amplitudes
and phases, the oscillations being along different axes
in the plane of the screen. Returning now to Figure 8, you
can see that the phase difference, and hence the ellipse
of polarization, clearly depends on both the wavelength
of the light and on the thickness of the material; the thicker
the material, the greater the phase difference. The two
amplitudes and the phase difference depend also on the orientation
of the fast and slow axes relative to the direction of polarization
of the incident light.
Now we can put
it all together Let's return to Figure 2 which shows light
traversing the three elements of a complete Polage. The
amplitude of light of a given wavelength which emerges from
the Polage depends on the polarization ellipse, which in
turn depends on the wavelength. Hence, the Polage alters
the proportions of the various colors which compose the
incident white light and the eye will generally perceive
color. In addition, the ellipse depends on the orientation
and thickness of the cellophane, both of which differ in
different regions of the Polage which then will exhibit
different colors. Finally, rotating either polaroid also
alters the spectrum of the light which emerges so that the
colors change. Figure 10 illustrates how the position of
the second polaroid affects the spectrum . When the pass
direction is vertical as shown, the emergent light is mainly
blue light; when the pass direction is horizontal, the emergent
light is mainly red light. The artist, Austine Wood-Comarow,
takes full advantage of all these possibilities to create
beautiful and exciting effects in her work.
The formal talk
ended here. Figure 11 illustrates a demonstration that was
exhibited informally to some of the members of the audience.]
Notes:
1. The Polage
exhibited pictured Leda and the Swan. It was a complete
Polage, i.e., could be viewed without adding another polaroid.
Mounted in a light box, its rear polaroid rotated slowly.
2. A Polage
of hummingbirds was illuminated from behind. When a square
of polaroid was held in front, the area behind the polaroid
became iridescent.
3. A plate on
which there was a transparent arrow surrounded by black
was placed on the projector.
Bibliocraphy:
Sabra, Theories
of Light from Descartes to Newton - Cambridge University
Press 1981.
Goethe, Farbenlehre,
"Didactic Part," trans. Charles Eastlake in 1820. Reprint
Van Nostrand Reinhold, 1971.
Shurcliff and
Ballard, Polarized Light . D. Van Nostrand 1964.
You
may request a hard copy of this lecture including full graphics send $20 to:
Donald H. Lyons, Ph.D.
8 Gould Road
Lexington, MA 02173 |
