Wave Propagation in Linear Media | re-examined

4.4 The square barrier
1
0.8
0.6
0.4
0.2
|C5|
0.5 1 1.5 2
Figure 4.18: Transmission coe cient jC 5j depending on the normalised propagation constant for
several values of the barrier thickness D (dotted line: 1, dashed line: 5, solid line: 10). Note the
relation 2 = !=!p.
As for the signs of the square roots, the same considerations as in section 4.1 apply, so
that for < ,1, the negative value of the square roots must be taken, whereas > 1
corresponds to the positive solution. Note that the wave functions inside the barrier were
de ned for the transmissive case. If the energy of the monochromatic wave is below V0, then
p 2 , 1 7! j p 1 , 2 , and hyperbolic functions take the places of the circular functions.
The transmission coe cient, jC 5j, is shown in g. 4.18 in dependence of the barrier thickness.
Not surprisingly, the thicker the barrier, the closer the frequency of an evanescent wave,
!, must be to the cuto frequency, !p, in order to allow a noticeable fraction to tunnel
through. Above the top of the barrier ( >1), the transmission coe cient exhibits oscillations
depending on as well as D, and only very high-frequency waves are transmitted entirely.
With the coe cients for the individual parts of the wave, we can formulate the wave function
in the three regions as Fourier integrals with an arbitrary spectrum of the initial wave, A( ),
1 = !p
c
Z 1
A( ) e
,1
jX ,j 2T d ,
Z 1
!p
c ,1
A( ) j sin Dp 2 , 1
N( )
95
e ,jX ,j 2 T d ;
ξ
(4.41)