Microwave Tunneling

1

The discrepancy between classical mechanics and quantum mechanics is often mirrored in the discrepancy between geometrical optics and physical optics. You will explore one example in this experiment. In Modern Physics, you have learned that there is a finite probability that an electron can tunnel through energy barriers. Classical mechanics does not allow for this possibility. Specifically, in modern physics class we have learned that a particle in a potential well of depth Vo can tunnel into the potential barrier formed by the difference in energy between the particle’s energy, E, and the potential Energy, Vo. Schrodinger’s equation gives us the following equation for the particle in the region just outside of the box. (Inside the box the equation is the same except Vo = 0.)

-

!2

2md2ψ(x)/dx2 + Voψ(x) = Eψ(x)

We can rearrange this equation so that

d2ψ(x)/dx2 = -

2m!2

(E – Vo) ψ(x) = -

p2

!2ψ(x)

The solution to this equation outside the well is

! ! = !!!!! where ! = 2m!2

!"#

$%& Vo ' E( ) (1)

We can see the solution outside the well is an exponentially decaying function into the potential barrier that depends on the barrier height, Vo-E. The probability of the electron tunneling through this barrier depends on the height of the barrier (Vo-E) and the width of the barrier, a. One application of quantum mechanical tunneling is the Scanning Tunneling Microscope. Electromagnetic radiation behaves in a manner analogous to the tunneling particle. In optics, geometric optics is analogous to classical mechanics, and physical optics is analogous to quantum mechanics. While geometric optics predicts a photon can’t tunnel across a barrier, physical optics says it can. With light, a barrier occurs when photon undergo total internal reflection (TIR). This is like the potential barrier described above. While geometric optics predicts that no light is transmitted through the interface, physical optics tells us that there exists an exponentially decaying electromagnetic wave on the other side of the interface where TIR occurs. This exponentially decaying light is called an Incident Light

Exponentially DecayingEvanescent Field

Vo

E

a

Microwave Tunneling

2

evanescent field. In the figure on the right, light incident from below is totally reflected at the interface. Above the interface is an exponentially decaying, evanescent field. The barrier is the medium on the opposite side of the interface from the totally internally reflected light. We can bring another object close to this interface, such as a detector or another prism. The light can then “tunnel” through the air gap to the detector or prism. This is called “frustrated total internal reflection” or FTIR.

In this experiment, we will be investigating the tunneling of electromagnetic radiation using microwaves. The evidence of this effect will be the observation of an exponentially decaying signal. One application of photon tunneling is the beam splitter. Library research: Research the following topics: Index of refraction, total internal reflection, evanescent waves, frustrated total internal reflection, and quantum mechanical tunneling. See if you can explain how a beam splitter works. Find and describe an application of quantum mechanical tunneling. I. Optical Total Internal Reflection (“Optical TIR”) Set up a diode laser and a hemi-cylindrical 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.

Direct the laser onto the curved surface of the acrylic hemi-cylinder. Using a hemi-cylinder is advantageous because there is no refraction when the laser first enters the acrylic, so the angle of incidence is easy to read. Adjust the angle of incidence until you just reach the point where no light is emitted from the flat side. The angle where this total internal

reflection first occurs is called the critical angle, θc, and it is measured from the surface normal as seen in the figure. Observe and record this angle. 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 hemi-cylindrical prism only because of some slight scattering inside the prism.

θc

Microwave Tunneling

3

Now position the laser to produce an incident angle greater than the critical 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 you can NOT see a laser beam transmitted into the air from that spot. In fact, the light is penetrating the surface, but its intensity is decreasing exponentially as it gets further from the surface so that by the time it reaches your eye, you can’t see it. You would need to position your eye a few nanometers from the interface to see this light. II. Optical Tunneling (“Frustrated TIR”) With the apparatus set so that the laser is at an angle greater than the critical angle, no light is being transmitted through the interface, and all of the light is undergoing total internal reflection or TIR. At this point, there exists an electromagnetic wave that is decaying exponentially away from the interface. We can “frustrate” the TIR and capture that light by bringing another optically transparent object close enough to this surface so that the light can essentially tunnel through the air gap. Using this angle where you have total internal reflection, 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. Hold on to the hemi-cylinder so that it doesn’t move, and press the triangular prism into the hemi-cylinder’s flat surface. You should see the laser beam exiting the hemi-cylinder. 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. Microwaves, Snell’s law and index of refraction: Now you are going to set up a microwave emitter and receiver to measure the index of refraction of a wax hemi-cylinder for microwaves. To do this experiment, you will measure the incident and refracted angles and apply Snell’s Law:

ni sin!i = nt sin!t (1) where ni is the index of refraction of the prism, θi is the angle inside the wax and θt is the angle of transmission into the air and nt is the index of refraction of the air. One way to do this experiment is to do the following:

θi

θt R

T

ni

nt

Microwave Tunneling

4

1) Find the center of the flat side of the hemi-cylinder and mark it on the bottom in some way that you can see. Position that mark over the screw on the goniometer that holds the microwave transmitter and receiver. The transmitter will be on the fixed arm of the goniometer at an angle of zero degrees.

2) Line up the flat side of the hemi-cylinder so that it is aligned between the 90° and 270° marks on the protractor provided on the goniometer. Now your angle of incidence is zero degrees.

3) The receiver should have a maximum signal when the movable arm is at 180 degrees which corresponds to a transmitted angle of zero degrees. Check to be sure that this is the case.

4) You will continue to make measurements in ten degree increments by re-aligning the flat side of the hemi-cylinder with the protractor. To change the angle of incidence to 10°, line up the flat side with 100° and 280°.

5) Now adjust the receiver until you find the maximum signal. Read the angle off the protractor and figure out the angle of transmittance. Be careful, it isn’t what you read off the protractor.

6) Repeat this process as far as you can. You should be able to get at least 5 measurements (including zero)

7) Repeat the process again using the other side of the surface normal (lining up the flat side with the 80° and 260° mark for a new 10° angle of incidence.

8) Average your two values for each angle of incidence and make a plot of sinθI vs sinθt to find the index of refraction of the hemi-cylinder?

IV. Microwave Tunneling (Microwave “Frustrated TIR”) Does photon tunneling really exist? Is there an exponentially decaying signal on the opposite side of an interface undergoing TIR? If we use microwaves with large wavelengths (cm), distances that are the order of 10’s of nm with visible light become on the order of mm and we can measure this phenomenum.

180 0

100 280

Microwave Tunneling

5

Position the emitter to produce an incident angle larger than the critical angle, θc, and less than 90º so that you have TIR. You should measure this angle with a protractor. Make sure that the emitter is mounted as close to the prism as possible in order to maximize the signal. To detect the signal, we will use a receiver where the horn has been removed. It is mounted at the center of the flat side of the hemi-cylinder and it is attached to a stage that can be moved in very small increments using a micrometer. Mount the receiver as close as possible to the hemi-cylinder. Adjust the receiver to get the highest possible signal using the multiplier, and make measurements of the signal as you move the receiver away from the hemi-cylinder. Be sure to get a zero reading with the transmitter off. Graph the measured signal minus the measured zero offset as a function of distance from the flat surface. To analyze the results, we will assume that the intensity of the microwaves, I, is proportional to the signal measured, S. In fact, it only has to be proportional to a power law of the signal, I=Sm. I= I0 e(-bx) so S=So e (-bx/m) (2) Here x is the distance from the prism to the point where it is detected. So is the signal at x=0. The constant b depends on the wavelength of the microwaves, λ, the index of refraction of the acrylic, n, the angle of incidence and the critical angle where

b = 2!"n sin#isin#c

$

% &

'

( ) 2

*1$

% & &

'

( ) )

1/ 2

(3)

Notice that equation (2) looks very much like equation (1). 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 data as you move the detector away from the surface of the sample. Adjust the receiver so that the signal strength that is the same order of magnitude as what you started with in experiment IV. Plot this data as a function of distance from the sample and compare it to the Frustrated TIR data from experiment IV. What can you conclude about the dependence on distance of the signal strength for the two sets of data? Remember, the point is to convince yourself that you observed exponential behavior in the TIR regime, but non-exponential behavior in the non-TIR regime. If time permits, repeat experiment IV for another angle of incidence that provides TIR. Plot this new data set with the previous one and decide whether or not it is consistent with what you would expect based on equation (3).

Microwave Tunneling

6

This experiment is adapted from the following reference: F. Albiol, et al., Am. J. Phys. 61(2), 165 (1993).