Page 460 - Engineering Electromagnetics, 8th Edition

442 ENGINEERING ELECTROMAGNETICS

Of interest to us are the phase velocities of the carrier wave and the modulation
envelope. From (82), we can immediately write these down as:
ω 0
ν pc = (carrier velocity) (83)
β 0
ω
ν pe = (envelope velocity) (84)
β
Referring to the ω-β diagram, Figure 12.12, we recognize the carrier phase velocity
as the slope of the straight line that joins the origin to the point on the curve whose
coordinates are ω 0 and β 0 .We recognize the envelope velocity as a quantity that
approximates the slope of the ω-β curve at the location of an operation point specified
by (ω 0 ,β 0 ). The envelope velocity in this case is thus somewhat less than the carrier
velocity. As ω becomes vanishingly small, the envelope velocity is exactly the slope
of the curve at ω 0 .We can therefore state the following for our example:

ω dω
lim = = ν g (ω 0 ) (85)
ω→0 β dβ
ω 0
The quantity dω/dβ is called the group velocity function for the material, ν g (ω). When
evaluated at a specified frequency ω 0 ,it represents the velocity of a group of frequen-
cies within a spectral packet of vanishingly small width, centered at frequency ω 0 .In
stating this, we have extended our two-frequency example to include waves that have a
continuous frequency spectrum. Each frequency component (or packet) is associated
with a group velocity at which the energy in that packet propagates. Since the slope
of the ω-β curve changes with frequency, group velocity will obviously be a function
of frequency. The group velocity dispersion of the medium is, to the first order, the
rate at which the slope of the ω-β curve changes with frequency. It is this behavior
that is of critical practical importance to the propagation of modulated waves within
dispersive media and to understanding the extent to which the modulation envelope
may degrade with propagation distance.

EXAMPLE 12.10
Consider a medium in which the refractive index varies linearly with frequency over
a certain range:
ω
n(ω) = n 0
ω 0
Determine the group velocity and the phase velocity of a wave at frequency ω 0 .
Solution. First, the phase constant will be
ω n 0 ω 2
β(ω) = n(ω) =
c ω 0 c
Now
dβ 2n 0 ω
=
dω ω 0 c