Page 461 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 443

so that
dω ω 0 c
ν g = =
dβ 2n 0 ω
The group velocity at ω 0 is
c
ν g (ω 0 ) =
2n 0
The phase velocity at ω 0 will be
ω c
ν p (ω 0 ) = =
β(ω 0 ) n 0

12.8 PULSE BROADENING
IN DISPERSIVE MEDIA

To see how a dispersive medium affects a modulated wave, let us consider the prop-
agation of an electromagnetic pulse. Pulses are used in digital signals, where the
presence or absence of a pulse in a given time slot corresponds to a digital “one” or
“zero.” The effect of the dispersive medium on a pulse is to broaden it in time. To
see how this happens, we consider the pulse spectrum, which is found through the
Fourier transform of the pulse in time domain. In particular, suppose the pulse shape
in time is Gaussian, and has electric field given at position z = 0by

1
E(0, t) = E 0 e − (t/T ) 2 e jω 0 t (86)
2
where E 0 is a constant, ω 0 is the carrier frequency, and T is the characteristic half-
width of the pulse envelope; this is the time at which the pulse intensity, or magnitude
of the Poynting vector, falls to 1/e of its maximum value (note that intensity is
proportional to the square of the electric field). The frequency spectrum of the pulse
is the Fourier transform of (86), which is

E 0 T 1 2 2
2
E(0,ω) = √ e − T (ω−ω 0 ) (87)
2π
Note from (87) that the frequency displacement from ω 0 at which the spectral intensity
2
(proportional to |E(0,ω)| )falls to 1/e of its maximum is ω = ω − ω 0 = 1/T .
Figure 12.14a shows the Gaussian intensity spectrum of the pulse, centered at
ω 0 , where the frequencies corresponding to the 1/e spectral intensity positions, ω a
and ω b , are indicated. Figure 12.14b shows the same three frequencies marked on
the ω-β curve for the medium. Three lines are drawn that are tangent to the curve at
the three frequency locations. The slopes of the lines indicate the group velocities at
ω a , ω b , and ω 0 , indicated as ν ga , ν gb , and ν g0 .We can think of the pulse spreading
in time as resulting from the differences in propagation times of the spectral energy
packets that make up the pulse spectrum. Since the pulse spectral energy is highest