Wave Propagation in Linear Media | re-examined

0.75
0.5
0.25
-0.25
-0.5
-0.75
U/U0
1
50 100 150 200 T
3Wave propagation in electromagnetic transmission lines
|A|
4
3
2
1
-2 -1 1 2 η
Figure 3.24: Initial pulse and its spectrum jA( )j for the parameter values T0 = 100 and = 17. The
carrier frequency is = 0:5 in these particular graphs.
Unfortunately, a pole turns up at ! =0,preventing straightforward integration. Taking the
Cauchy principal value of the integral would solve the problem, but as we do not need the
current to gain some general insight into the propagation of a Gaussian wave, we leave it
aside.
The expression for the voltage (3.103) is valid in both the stop and pass bands. We can thus
compute the voltage inside the plasma for a given pulse with varying carrier frequencies. The
initial shape and its spectrum are shown in g. 3.24. If the cuto frequency of the plasma
is assumed to be !p = 10 10 9 s ,1 (a value roughly corresponding to Nimtz' microwave
experiments), then the pulse width is about 6 ns, which is fairly short. Accordingly, the
spectrum is quite wide, and so we must expect a small portion of the pulse to be classically
transmitted even though the major part of the eld originates from the evanescent region.
The rst case we examine is the base-band signal with =0. The result given in g. 3.25
is not very spectacular since the spectrum is practically con ned to the range below cuto .
Thus there are no propagating waves, and all we can see is the exponential decay along the
spatial coordinate. In contrast to the following gures, this one shows no modulation, and
so the vanishing voltage outside the actual pulse also denotes the least possible value of the
eld. Therefore the area surrounding the contour of the pulse is shaded in black.
When we use a carrier with = 0:5, the behaviour of the wave is much more interesting.
Fig. 3.26 shows rst of all a dominating evanescent signal that resembles the unmodulated case
treated before. It is particularly noteworthy that the carrier frequency increases gradually
with X. This is manifested in the graph by the narrowing of the black and white stripes
and can be explained as a pulse reshaping e ect. Since the dispersion in the plasma causes
higher frequency components to be less attenuated than lower ones, the centre frequency of
the spectrum is slightly raised. Note that although it looks as if the pulse became shorter with
increasing X, this is not the case. It is only the attenuation in combination with the limited
plotting range that creates this deceptive impression. The overall shape of the pulse remains
more or less the same, albeit with reduced amplitude. Direct comparison of the unmodulated
and the present results reveal another detail: the plot ranges of the two graphs are identical,
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