The Polarization Puzzle
(The Triple Phasor Paradox - A neat little trick
with three polarized filters that is against
your classical intuition)
by Roger A. Wendell
March 21, 2004
(This was an Email Roger sent on March 21st, in
response to his father and uncle's questions concerning
the Triple Phasor Paradox. Specifically, it was asked why so
much light could pass through three polarized lenses yet be
completely attenuated by just two of the same lenses...)
"The name Triple Phasor Paradox doesn't seem to be common
(cant find it on Google) and I have only seen it once in a book,
probably because the result is a more general statement about Quantum
Mechanics. This is a shame since there are not too many examples of
Quantum Physics that the average joe can pick up and see. So be happy,
YOU have seen quantum mechanics! As a AAA certified Quantum Mechanic,
I'll let you in on the secret of the paradox:
First you should know that photons have two polarization
states, which are always perpendicular to each other and to the direction
the photon is traveling so long as the photon is not in a metal box (i.e.
it’s free). A polarization state is just what you would expect it to be,
if the photon is polarized "up-down" and you put your sunglasses on with
their slats going "sideways" you wont see that photon. Actually the
polarization of a photon is labeled by the oscillation of its electric
field if you want to think of the photon as a wave instead of a
particle. For this discussion it doesn't really matter so long as you call
"polarization" the property light has which prevents it from
passing through a perpendicularly oriented lens.
(* As an aside you may notice that your sunglasses best reduce the
light coming into your eyes from things like puddles, or windows. That’s
because light which reflects off surfaces is always polarized in the plane
of the surface it reflects from. So for a puddle that gives
horizontally polarized light, but your sunglasses are usually a filter for
vertical. Give it a try sometime! Oh and this is not a quantum effect per se
*)
Anyway, the light of the ordinary world comes in a variety
of polarizations. From a photon's perspective its always in a horizontal
or vertical polarization. From your perspective however, what you call
vertical may be horizontal for the photon, or somewhere in between.
That is to say, if you plotted all the photon polarizations from your
perspective on a graph of "up-down" versus "sideways" it would look
like a circle. This is really the hardest part of the paradox to understand.
The axes of your graph form what's called a basis, and that your graph
looks like a circle is just a statement of the fact that your basis is not
the same basis as each photon you observe - and if you think about it
carefully the bases are just rotated from one another.
(* you may remember the term basis from geometry, vector calculus,
or linear algebra. As the above exhaustive description might imply, its
a fancy word for saying every point in the plane of your graph can be
written as A*x + B*y, where x, y are the axes, and A and B are
numbers and the axes are perpendicular. *)
Now that we have taken care of that, the whole paradox can
be summarized by saying that the lenses force photons into a certain
basis. Of course you are probably screaming right now saying: "But
that's all stuff I always knew and it still doesn't make sense!" And if you
are saying that its probably because I forgot to let you think quantum
mechanically!!
So the truth is that you have no idea what the polarization
of a given photon is, until you measure it. Until you make that
measurement it could be "up" it could be "sideways" you just don't know. One of
the fundamental ideas in quantum mechanics is that a measurement
"collapses" the particle into a definite state - in our case polarization. This
is an example of something made famous by Schrodinger's cat, which I can
explain also if there is any interest.
If you believe that the state of the photon is undetermined
until measurement then you have solved the paradox!
Orient the first lens in any fashion. As you expect, light
will pass and then be stopped by a second filter placed 90 degrees
relative to the first. So all the photons entering the first have random
polarizations, and only those with "up-down" say pass. You have
forced the photons into a state defined by the natural basis of the lens. The
second lens, at 90 degrees is the other half of the same basis! Once the
photons pass the first lens, they have been measured as "up-down". The
second lens tries to measure them again as "sideways" but as all of the photons
are now "up-down," none of them can pass.
Now imagine instead that the second lens was at 45 degrees
instead of 90 degrees relative to the first. From the perspective of the
first lens all the passing photons are "up-down." But from the perspective
of the second all the photons are combinations of "up-down" and
"sideways," but it does not know which since until measurement it could be
either. This is the same way the first lens viewed light coming into it,
just there is less light! So like the first lens, the second one forces
its basis on the photons and some of them collapse into states of
"up-down" with respect to the second lens. Finally, the same argument can be
applied to a third lens oriented 90degrees to the first, since this
is again 45 degrees relative to the second lens. Again you enforce a
polarization basis which is different from the most recent, and
light will go through again. Darn that tricky tricky light!
Note that there is nothing magical about the lenses being
45 degrees with respect to one another, this just makes the probability
for light surviving the first filter to pass the second, 50%. You will
see if you do the experiment that varying amounts of light will be passed
through the three filters when the second lens is at angles 0