Page 272 - Electromagnetics

These values of α and β are valid for ω> 0. For negative frequencies we must be more
c 1/2
careful in evaluating the square root in k = ω( ˜µ˜ ) . Writing
µ

˜ µ(ω) = ˜µ (ω) + j ˜µ (ω) =| ˜µ(ω)|e jξ (ω) ,

c
c
c
c
˜ (ω) = ˜ (ω) + j ˜ (ω) =|˜ (ω)|e jξ (ω) ,
we have
1
µ

j [ξ (ω)+ξ (ω)]
c
c
˜
k(ω) = β(ω) − jα(ω) = ω µ(ω)˜ (ω) = ω | ˜µ(ω)||˜ (ω)|e 2 .

Now for passive materials we must have, by (4.48), ˜µ < 0 and ˜ c < 0 for ω> 0.

Since we also have ˜µ > 0 and ˜ c > 0 for ω> 0, we find that −π/2 <ξ µ < 0 and

µ

−π/2 <ξ < 0, and thus −π/2 <(ξ + ξ )/2 < 0. Thus we must have β> 0 and α> 0

for ω> 0.For ω< 0 we have by (4.44) and (4.45) that ˜µ > 0, ˜ c > 0, ˜µ > 0, and
µ

˜ > 0.Thus π/2 >(ξ + ξ )/2 > 0, and so β< 0 and α> 0 for ω< 0. In summary,
c
α(ω) is an even function of frequency and β(ω) is an odd function of frequency:
β(ω) =−β(−ω), α(ω) = α(−ω), (4.228)
˜
where β(ω) > 0,α(ω) > 0 when ω> 0. From this we find a condition on E 0 in (4.216).
˜
˜ ∗
Since by (4.47) we must have E(ω) = E (−ω), we see that the uniform plane-wave field
obeys
˜
˜ ∗
E 0 (ω)e [− jβ(ω)−α(ω)] ˆ k·r = E (−ω)e [+ jβ(−ω)−α(−ω)] ˆ k·r
0
or
˜
˜ ∗
E 0 (ω) = E (−ω),
0
since β(−ω) =−β(ω) and α(−ω) = α(ω).
Propagation of a uniform plane wave: the group and phase velocities. We
have derived the plane-wave solution to the wave equation in the frequency domain, but
can discover the wave nature of the solution only by examining its behavior in the time
domain. Unfortunately, the explicit form of the time-domain field is highly dependent on
the frequency behavior of the constitutive parameters. Even the simplest case in which
, µ, and σ are frequency independent is quite complicated, as we discovered in § 2.10.6.
To overcome this difficulty, it is helpful to examine the behavior of a narrowband (but
non-monochromatic) signal in a lossy medium with arbitrary constitutive parameters.
We will find that the time-domain wave field propagates as a disturbance through the
surrounding medium with a velocity determined by the constitutive parameters of the
medium. The temporal wave shape does not change as the wave propagates, but the
amplitude of the wave attenuates at a rate dependent on the constitutive parameters.
For clarity of presentation we shall assume a linearly polarized plane wave (§ ??) with
˜
˜
E(r,ω) = ˆ eE 0 (ω)e − jk(ω)·r . (4.229)
˜
Here E 0 (ω) is the spectrum of the temporal dependence of the wave. For the temporal
dependence we choose the narrowband signal
E 0 (t) = E 0 f (t) cos(ω 0 t)
where f (t) has a narrowband spectrum centered about ω = 0 (and is therefore called a
baseband signal). An appropriate choice for f (t) is the Gaussian function used in (4.52):

π ω 2
2 2
˜
−a t
f (t) = e ↔ F(ω) = e − 4a 2 ,
a 2
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