Wave Propagation in Linear Media | re-examined

1.2 A few notes on dispersion
He could then demonstrate that the motion of the wave energy can be looked at as the
superposition of the spectral energy components, each moving at its respective group velocity.
So far, we considered the media to be energy conservative or non-dissipative, which implies
that the dispersion relation is a real function. When this condition holds, the concept of group
velocity is clearly de ned. As soon as dissipation or evanescence enter the stage, however,
the dispersion relation becomes complex, the imaginary part describing the attenuation of
the respective monochromatic component. In such a case, the validity of the de nition of
group velocity is at least questionable. For only weak absorption, Brillouin [2] argued that
the group velocity may safely be determined based on the real part of the wave number,
vg = @ Re !=@k. As absorption becomes more marked, this approach is no longer justi ed,
and alternatives must be sought.
1.2 A few notes on dispersion
In general, there are two e ects that cause dispersion:
Parameters of the medium may be frequency-dependent. This accounts in particular
for resonance e ects in dielectrics. In fact the characteristics of all real media depend
on the frequency. The only exception to this rule is vacuum.
Boundary conditions may impose constraints on the relation between wave number
and frequency even if no medium is present atall. Atypical example are hollow wave
guides where waves can propagate only in certain frequency ranges.
Let us rst explore the e ects of frequency-dependent parameters and consider a dielectric
with losses and a distinct resonance frequency !0 caused by elastically bound polarisable
electrons. The relative dielectric constant of such a medium can be written as [2, 11]
!p 2
"r(!) =1,
! 2 ,!0 2 +2j!
; (1.22)
where the e ects of the losses are contracted into a phenomenological attenuation constant
. Fig. 1.2 shows the respective curves for the lossy and lossless ( = 0) case. It is important
to notice that without losses, "r is monotonically increasing over the entire range with a pole
at resonance, whereas in the presence of losses, the pole degenerates to a zero at !0 and "r
consequently decreases in the neighbourhood of this point. For the range where d"r=d! < 0,
the dispersion is said to be anomalous (Jackson [11]). This is, however, not the only de nition
for this term, as we shall see shortly.
Remark (A word of caution) In the derivation of (1.2), the time dependence of the
physical quantities was taken to be e ,j!t , which is a common assumption in physics.
In electrical engineering, however, the sign of the phase function is usually reversed, so
that the time factor reads e j!t . With this setting, the denominator of (1.2) would have
been ! 2 , !0 2 , 2j! . Consequently, if"ras given in (1.2) is used in standard electrical
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