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.
the following topics: Index of refraction, total internal reflection,
evanescent waves, frustrated total internal reflection, and quantum mechanical
I. Optical Total Internal Reflection (“Optical TIR”):
Set up a diode laser and a hemicylindrical acrylic prism to observe the phenomenon of total internal reflection. 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.
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.
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.