Wave Propagation in Linear Media | re-examined

1.2 A few notes on dispersion
v/c
6
4
2
-2
2 4 6 8 10 12 ω
Figure 1.3: Index of refraction (solid line), phase velocity vp=c (dashed line), and group velocity vg=c
(dotted line) of the resonant dielectric in g. 1.2 .
leads to an attenuation of the wave in the direction of energy propagation, it seems sensible
to insert only the real part of the wave number into the velocity de nitions. We thus obtain
and
vp = !
Re k
vg = 1
Re dk
d!
= c
Re n
(1.26)
: (1.27)
The results for the dispersive dielectric, scaled to the vacuum velocity of light, are depicted
in g. 1.3 together with the refractive index. We note that the group velocity exhibits two
areas where it grows excessively, and becomes negative between the poles.
This behaviour extends far beyond the region where anomalous dispersion in the above sense
occurs. Consequently, many authors gave alternative de nitions. According to Sommerfeld
[12] and Stratton [6], this is the region where vp < vg, which holds only for a narrow
range about the poles. Jackson [11] and Schulz-DuBois [13] require dn(!)=d! < 0, which
is tantamount to the most common de nition of anomalous dispersion in modern literature,
dvp=d! > 0 (Baldock and Bridgeman [14], Ramo et al. [15], Piefke [16]).
Seen purely mathematically, the imaginary part of the wave number gives rise to an attenuation
of the wave along the direction of propagation. Physically, this e ect can be traced back
to two distinct causes:
Absorption is associated with various losses in the medium like the dielectric resonance
in the above example. It is characterised by a complex wave number. Hence wave
propagation is possible, but a part of the wave's energy is dissipated.
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