Page 280 - Electromagnetics

Since v p >v g , this model of a plasma demonstrates normal dispersion at all frequencies
above cutoff.
For the case of a plasma with collisions we retain ν in (4.76) and find that

ω
ω 2 p ω 2 p
k = 1 − − jν .
2
2
2
c ω + ν 2 ω(ω + ν )
When ν = 0 a true cutoff effect is not present and the wave maypropagate at all
frequencies. However, when ν ω p the attenuation for propagating waves of frequency
ω< ω p is quite severe, and for all practical purposes the wave is cut off. For waves of
frequency ω> ω p there is attenuation. Assuming that ν ω p and that ν ω,wemay
approximate the square root with the first two terms of a binomial expansion, and find
that to first order

2
ω ω 2 p 1 ν ω /ω 2
p
β = 1 − , α = .
c ω 2 2 c ω p 2
1 −
ω 2
Hence the phase and group velocities above cutoff are essentiallythose of a lossless
plasma, while the attenuation constant is directlyproportional to ν.
4.11.4 Monochromatic plane waves in a lossy medium
Manyproperties of monochromatic plane waves are particularlysimple. In fact, cer-
tain properties, such as wavelength, onlyhave meaning for monochromatic fields. And
since monochromatic or nearlymonochromatic waves are employed extensivelyin radar,
communications, and energytransport, it is useful to simplifythe results of the preceding
section for the special case in which the spectrum of the plane-wave signal consists of a
single frequencycomponent. In addition, plane waves of more general time dependence
can be viewed as superpositions of individual single-frequencycomponents (through the
inverse Fourier transform), and thus we mayregard monochromatic waves as building
blocks for more complicated plane waves.
We can view the monochromatic field as a specialization of (4.230) for a → 0. This
˜
results in F(ω) → δ(ω), so the linearly-polarized plane wave expression (4.232) reduces
to
ˆ
ˆ eE(r, t) = ˆ eE 0 e −α(ω 0 )[ ˆ k·r] cos(ω 0 t − jβ(ω 0 )[k · r]). (4.243)
It is convenient to represent monochromatic fields with frequency ω = ˇω in phasor form.
The phasor form of (4.243) is
ˇ
e
E(r) = ˆ eE 0 e − jβ( ˆ k·r) −α( ˆ k·r) (4.244)
where β = β( ˇω) and α = α( ˇω). We can identifya surface of constant phase as a locus of
points obeying

ˆ
ˇ ωt − β(k · r) = C P (4.245)
for some constant C P . This surface isa plane, as shown in Figure 4.10, with its normal
ˆ
in the direction of k. It is easyto verifythat anypoint r on this plane satisfies (4.245).
ˆ
ˆ
Let r 0 = r 0 k describe the point on the plane with position vector in the k direction, and
let d be a displacement vector from this point to anyother point on the plane. Then
ˆ
ˆ
ˆ ˆ
ˆ
k · r = k · (r 0 + d) = r 0 (k · k) + k · d.
© 2001 by CRC Press LLC