Wave Propagation in Linear Media | re-examined

3.7 Turn-on e ects in a lossless plasma
a carrier oscillation on and o , which is in fact a crude amplitude modulation also known as
on-o -keying. Such apulse can then transport one bit of information. Owing to its sharply
limited duration, however, its spectrum now spans a broad range of frequencies. Irrespective
of any measures taken to limit the spectrum, it still widens as the pulse is shortened. Hence
the narrow-band approximation quickly becomes senseless.
Remark (Relation between spectral and pulse width) Like in quantum mechanics,
an inequality can be given that establishes a lower bound for the product of a pulse
p x 2 , x 2
duration t and the width of the dominant peak of its spectrum !. If x =
denotes the variance of the respective intensities (i. e. the squared signal value and the
power spectrum), then this uncertainty principle reads
holds for Gaussian distributions [11].
t ! 1=2. The equal sign
We now turn to a fairly classical problem: the propagation of a pulsed wave in a dispersive
medium. This has been scrutinised over and over again, beginning with Sommerfeld [12] and
Brillouin [19]. Yet we shall add a new facet to the plethora of results.
The model we examine is again that of a lossless plasma like in the previous section. But
this time we leave it unbounded for x>0 so that we need not bother about re ections. We
also adhere to the transmission line model of g. 3.1 . In principle we could as well calculate
the electric and magnetic elds in the plasma directly, but as we are concerned only with the
basic TEM wave, we prefer to use the convenient and proven equivalent circuit. The voltage
then corresponds for instance to the electric eld vector Ey and the current to the magnetic
component Hz, ifxis the direction of propagation. All other components do not exist.
The plasma shall be excited by a monochromatic current source being switched on at t =0
and turned o an integer multiple of periods later, at t = =2n ,
I(0;t)=
8
><
>:
0 if t
: (3.67)
The latter requirement entails no loss of generality but only facilitates the analytical treatment.
It is evident that due to the linearity of the model, the pulse can also be described as
superposition of two delayed step functions of opposite sign,
I(0;t)=I0 (t) cos !0t , I0 (t , ) cos !0t ; (3.68)
such that we may restrict our analysis to the step response of the system. Similar investigations
were carried out for example by Knop [96] or Haskell and Case [97]. In contrast to
the approach pursued here, they used modulated sine waves, whereas we consider a cosine
wave, which has the convenient side bene t that the wave front of the signal can be identi ed
without any doubt.
Remark (A glance at the literature) The original articles of Sommerfeld and Brillouin
were concerned with the propagation of light through a resonant Lorentz-type medium
49