IMAGINARY FREQUENCY OR WAVE
NUMBER, TUNNELING AND DAMPING
Ken Cheney
4/23/2006
Part II
TUNNELING IN A WAVE GUIDE
Sending electromagnetic waves (microwaves in this case) down a channel
made of parallel conducting plates (a wave guide) gives another example of
the real, observable use of an imaginary wave number. For a simple
geometry we can show that a traveling wave is a solution of the wave
equation for a “wide” wave-guide and that the solution changes to a
decaying amplitude with an imaginary wave number when the wave-guide is
“narrow”.
Electromagnetic energy or signals are transmitted by wires, coax cables,
wage guides, and in free space. In the microwave region over relatively
short distances wave-guides work very well. In the visible region fiber
optics work as wave guides for intercontinental distances, with a few
repeaters.
A wave-guide for microwaves is a hollow conducting tube, it looks like a
piece of plumbing. We can analyze a simple version consisting of just two
parallel conducting plates.
Let the wave travel in the x direction. The electric field will be in the y
direction and the plates making up the wave guide will be in an xy plane at
z=0 and z=a. Sorry, we seem to have a left handed coordinate system, no
matter here.
Since there can be no tangential electric fields at conducting surfaces the
electric field must go to zero at the plates.
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A traveling wave that will do this is:
 n 
E  ˆjE y sin 
z  cos(kx  t )
 a 
(1.1)
With
n=1, 2, 3, . . .
There are different modes for different values of n. We don’t care here but
for optimum focusing (width) or data transmission (different velocities) you
want n=1.
Conducting plates
y
z
E
a
velocity
Let’s see what restrictions the wave equation places on this wave:
2
Figure 6 2 Wave Guide
E 1 Geometry
2E
 E  0 0
t 2

c 2 t 2
x
(1.2)
Since there are no y terms in E:
2 
2
2

x 2 y 2
(1.3)
Combining Eq. (1.1). (1.2), and (1.3) we find that almost everything cancels
out and we are left with:
 n  
k 
  2
c
 a 
2
2
Then
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2
2
(1.4)
k
2
 n 


2
c
 a 
2
(1.5)
This is what we are interested in here but if you solve for velocity as omega
over k you find it is greater than the velocity of light! If you can’t stand that
investigate wave and phase velocity.
Is there some physical reason why we can’t set it up to get an imaginary k?
No, all we need is a small separation of the plates a, easily done!
To see what happens with an imaginary wave number let’s (for cases where
k is imaginary) define:
(1.6)
k  ik 
Then, as before, k  is real when k is imaginary.
Looking at the part of Eq. (1.1) involving k:
(1.7)
cos(kx  t )  Real(ei ( kx t ) )  Real(eikx eit )  Real(e-kx eit )
cos(kx  t )  e k x Real(eit )  e k x cos(t )
(1.8)
Inserting Eq. (1.8) into Eq. (1.1) we get:
 n
E  ˆjE y sin 
 a

z  e  k x cos(t )

(1.9)
which is not a traveling wave but an electric field decaying exponentially in
the x direction.
But if the wave-guide becomes wider than the critical a, cut off width, then
a traveling wave will appear again, with a smaller amplitude Ey.- This
critical a is the minimum a for real k. This is smallest for n =1. Setting k=0
in Eq. (1.5) gives this a.
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Wave Guide Experiment
microwave
generator
I
II
III
metal
plates
x
a
V
I and III: a greater than the “cut off width”
II: a less than the “cut off width”
Figure 7
Wave Guide Experiment
The experiment will add three parts in succession:
Part I has just the source, a section of wave guide wider than the cut off
width, and the detector.
Part II adds a section of wave guide narrower than the cut off width between
the wave guide of section I and the detector.
Part III adds a section of wave guide wider than the cut off width between
the narrow section and the detector.
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To start:
Calculate the wave length of the 10GHz microwave.
Calculate the critical, cut off width for a. k=0, n=1in Eq. (1.5)
Set the voltmeter to DC, the detector is a diode and will change the signal to
DC.
Be sure the microwave source and detector are oriented to give you the
maximum signal. Put them face to face and try rotating the detector.
Be sure the source is emitting waves polarized correctly for the wave guide.
With a short section of wide wave guide try the source and detector rotated
(together) both ways.
To check:
1. Is there a traveling wave in section I?
Measure the strength of the signal at the detector as you make section
I longer and longer. Plot Voltage verses the length of the wave guide.
2. Does the intensity decrease suddenly when a reaches the cut off
width?
Push the sides of I closer and closer, plot the intensity versus width.
3. Does the intensity decrease as part II becomes longer?
Restore part I to its original width (wider than the cut off width), add
part II (a wave guide narrower than the cut off width).
Measure the intensity as the length of part II increases, add to your
plot from 1.
4. Do you get a traveling wave if the wave guide becomes wider than the
cut off width?
Add a wider wave guide (part III) to the end of part II.
Measure the intensity as part III is made longer. Add to your plot.
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Tunneling and Total Internal Reflection
If waves go from a medium (lucite block) to a medium (air) where the wave
goes faster the angle of refraction is larger than the angle of incidence.
Clearly for some angle of incidence the angle of refraction is ninety degrees.
Instead of skittering along the surface the waves are totally internally
reflected, for this angle and all larger angles of incidence.
It is easy to predict the critical angle using Snell’s law:
sin(i ) nr c / vr vi
 

sin( r ) ni c / vi vr
(1.10)
If you know the indexes of refraction just set r to ninety degrees and solve
for i. If you don’t know the ns you can use Snell’s law to find them.
Lucite
Block
i
r
Microwaves
Figure 8
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Scheme to Explore Total Internal
Reflection
6
What is interesting here is whether there is any electric field outside the
block when the beam is undergoing total internal reflection, and whether a
wave can tunnel through the gap if another lucite block is placed across a
small gap from the first block.
This experiment can be done with semicircular Lucite blocks as sketched
first or with right triangular lucite blocks as shown second.
Lucite
Blocks
microwave
generator
Detector
V
Microwaves
Gap
V
Figure 9
Tunneling and Total Internal Reflection
Experiment for Tunneling and Total Internal Reflection
1. Convince yourself that total internal Reflection does occur. Explain
clearly what you did!
2. Plot, on the same graph, the reflected and transmitted intensities as a
function of the gap width. Start with zero width.
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We might expect an exponential decay in amplitude but life is not that
simple in general. Most of us (Like Feynman) will be satisfied to
observe tunneling at all, and a semi exponential decay but for those
determined to see the theoretical result the transmissivity is:
(1.11)
Tp  1/( Ap sinh2 ( z)  1)
with:
z  2 (n 2 sin 2 (i )  1)1/ 2 Z / 
(1.12)
where I denotes the angle of incidence, n the refraction index of the medium,
Z the gap thickness, and lambda the vacuum wavelength.
Ap  ATE  (n2  1)2 / 4n2 cos2 (i )(n2 sin2 (i )  1)
(1.13)
for the case where E has a component transverse to z. I can’t tell what is
divided by what either, I just copied this literally! Or:
(1.14)
Ap  ATM  ATE [(n2  1)sin2 (i )  1]
A. Kodre and J. Strnad, American Journal of Physics , Vol. 44, No. 2,
February 1976, p 181
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