MICROWAVE TUNNELING
The discrepancy between classical mechanics and quantum mechanics is often
mirrored in the discrepancy between geometrical optics and physical optics. For
example, Heisenberg’s Uncertainty Principle is analogous to single-slit
diffraction of a wave.
Classical
mechanics says that objects cannot cross regions where their kinetic energy would
have to be negative. Geometrical optics says that electromagnetic waves
approaching a dielectric/air boundary at an angle greater than the critical
angle for total internal reflection are totally reflected inside the material:
there is no refracted wave. This week we will prove geometrical optics in the
same way that quantum mechanical tunneling proves classical mechanics wrong.
Library
research:
Research
the following topics: Index of refraction, total internal refraction,
evanescent waves, frustrated total internal refraction, and quantum mechanical
tunneling.
I. Optical Total Internal Reflection
(“Optical TIR”):
Set up a
diode laser and a hemicylindrical acrylic prism to
observe the phenomenon of total internal refraction. Your instructor will instruct
you in the basic rules of laser safety. You must follow these rules.
Observe and
record the angle which produces total internal reflection. Turn the lights off
and observe the paths of the light rays both inside and outside the prism at
angles both less than and greater than the critical angle. Sketch these in your
notes. Notice that you can see the light rays inside the hemicylindrical
prism only because of some slight scattering of the beam inside the prism.
Now
position the laser to produce an incident angle greater than this angle. You
will see the spot where the laser beam totally reflects from the flat surface
of the prism (but only because of scattering). Confirm to your own satisfaction
that there is no laser beam transmitted into the air from that spot.
II.
Optical Tunneling (“Frustrated TIR”):
For this angle, press a triangular acrylic prism into the spot where the laser
beam hits the flat surface. Hold it so that, if a beam were to be transmitted
at an angle close to the incident angle, it would shine on the palm of the hand
holding the triangular prism. Now, pressing down on the hemicylinder
so that it doesn’t move, press the triangular prism into the hemicylinder’s flat surface. You should see the
results of a laser beam exiting the hemicylinder.
Describe it in your notes. By pressing the two imperfect surfaces together, you
are actually just decreasing the width of the air gap between them.
III. Microwave TIR:
Set up a microwave emitter and receiver to measure the incident and refracted
angles for a hemicylindrical prism of polystyrene
pellets held in a cardboard form. Measure incident and
refracted angles for at least five angles for which there is a refracted angle
and then calculate the index of refraction of the prism. Verify that the
microwaves obey Snell’s Law: nsinθi = sinθt,
where n is the index of refraction of the prism, θi
is the incident angle inside the acrylic and θt
is the angle of transmission into the air, both angles measured from the normal
to the surface.
Microwave tunneling apparatus seen
from above (left) and Lock-in amplifier and oscilloscope (right). The scope is displaying both the
reference signal and the amplified signal that
comes from the diode.
IV.
Microwave Tunneling (Microwave “Frustrated TIR”):
Position the emitter to produce an incident angle halfway between the critical
angle, qc, and 90º. Using a diode detector
mounted on a micrometer stage, and using an AC modulated emitter and sending
the diode signal through an amplifier and then a lock-in amplifier, measure the
signal as a function of distance from the “microwave prism”. Make
sure that the emitter is mounted as close to the prism as possible in order to
maximize the signal. To analyze the results, we will assume that the intensity
is proportional to the signal measured. In fact, it only has to be proportional
to a power law of the signal.
Signal = S0 exp (-x/X)
The
distance X is of the same order of magnitude as the wavelength of the radiation
used, in this case 2.9cm microwaves, but varies with the incident angle and the
index of refraction. [F. Albiol, et
al., Am. J. Phys. 61(2), 165 (1993).] You will probably want to collect
data for the entire 1” travel of the micrometer. Time permitting, you can
use Malus’ Law to calibrate the detector and
convert signal, S, into intensity, I.
V. Are we really seeing TIR?
Repeat Step
IV for an incident angle of 0º. In this case, there is no evanescent wave. In
fact, the prism may even act as a lens, increasing the signal as your
probe’s distance from the prism increases. In any event, collect some
data, try to fit it to an exponential function, and compare it to the genuine
Frustrated TIR data. Remember, the point is to convince the reader of your report
that you actually observed exponential behavior in the TIR regime, but
non-exponential behavior in the non-TIR regime.