Wave Propagation in Linear Media | re-examined

2.1 Superluminal wave propagation
2.1 Superluminal wave propagation
Causality states that the propagation of signals and energy is upper bounded by c, the velocity
of light invacuum. This does, of course, not a ect the phase velocity ofawave, and numerous
thought experiments have been invented to illustrate how vp can exceed c (perhaps the most
lucid example is that of a rotating light source being observed in a su ciently large distance).
Some of them have been described in an introductory article by Rothman [52]. He resolved
the puzzles in the usual way by pointing out the di erence between phase and group velocity.
Furthermore, he mentioned that in electric wave guides, the group velocity is always smaller
than c due to the relation vp vg = c 2 . This is, however, not universally true and applies only
to waves governed by very simple boundary conditions in lossless media. We have already
mentioned that the group velocity may grow beyond the velocity of light in an absorptive
medium. Such examples eventually led to Sommerfeld's proof that a wave front always travels
at c, and to the formulation of the signal velocity.
Remark (Phase and group velocity) We can most easily derive a criterion necessary
for the relation vp vg = c 2 to hold. If we replace vp and vg with their de nitions from the
previous chapter and separate the variables, we obtain
Integration of both sides gives
!d!=c 2 kdk : (2.1)
! 2 = c 2 k 2 + C; (2.2)
with some integration constant C that must be determined from the boundary conditions.
The dispersion relation therefore is of the form ! = c p k 2 + C, which is true for hollow
wave guides and a lossless plasma. The expression vp vg = c 2 is not valid, on the other
hand, in delay lines, as we shall see in the following chapter. In an article by Borgnis [42],
this relation appears as vp ve = c 2 , which would be equivalent whenever the velocity of
energy transport equals the group velocity. In delay lines, the group and energy velocities
are indeed identical, and hence the above relation is false in these cases.
For some decades, all remained quiet. With the invention of lasers and thus emissive media,
experiments were carried out, the results of which allowed interpretations of `superluminal'
pulse velocities. Anderson et al. [49] refer to such publications. As they were considering the
motion of the maximum of a pulse, they also noted that this velocity can exceed c for certain
parameter values in an atomic medium. At the same time, however, they conceded that in
this case (when the peak moves faster than the pulse front) the underlying assumption of
their analysis | i. e. that the amplitude of the pulse envelope is only slowly varying | will
eventually be violated.
Only recently, Enders and Nimtz [53, 54] carried out microwave experiments to study an
analogy of the quantum mechanical tunnel e ect | the propagation of evanescent electromagnetic
waves in a wave guide. It is well known that the basic mode in a rectangular wave
guide is governed by the dispersion relation
k = 1
c
p ! 2 , !c 2 ; (2.3)
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