Wave Propagation in Linear Media | re-examined

2.1 Superluminal wave propagation
this argument intwo recent publications [71, 72]. Real signals are always band-limited, they
argued, so there exists no actual wave front to run ahead of the pulse at the speed of light.
Consequently, superluminal transport of both signals and energy is possible.
Apart from the misconception regarding the actual pulse width, such e ects are well known as
`pulse reshaping'. This peculiar behaviour of the evanescent region can be explained in a very
illustrative manner [61]. Consider a Gaussian wave packet that has already travelled some
distance in the medium outside the tunnel. Owing to dispersion, the leading tail of the pulse
will be dominated by higher-frequency components that propagate faster. In addition, these
components are less attenuated when moving through the evanescent region. Hence the pulse
appearing on the far side of the barrier will be caused mainly by the front of the wave packet.
The trailing tail, on the other hand, consists of lower-frequency components that are mostly
absorbed or re ected. Consequently, the peak of the transmitted pulse seems to be shifted
forward in time, and in addition, its centre frequency has increased. The overall impression is
that the pulse has moved through the barrier at an abnormally great velocity. In the extreme
case of a medium with gain, the velocity of the pulse may even become negative (see Chiao
et al. [73, 74, 75]).
Remark (Negative pulse velocities) Formally, a medium that provides gain can be
described analogous to the Lorentz medium with a negative oscillator strength. This
medium in uences the wave inversely to the e ect just described, such that the peak of
the pulse at the end of the evanescent region leaves the medium even before the peak of
the incident pulse enters it. Owing to the fact that the wave can temporarily borrow
energy from the inverted atomic states in the medium, the energy velocity in the classical
sense is negative. This does, however, not interfere with causality, and a wave front due
to a turn-on of the signal still propagates with the velocity of light [74].
Another approach was pursued by Yun-ping and Dian-lin [76], who explained the reshaping
e ect with interference between di erent possible paths along which aphoton can travel. If
an incident Gaussian wave 0(t) can take several paths, each characterised by a delay time
i and a probability j ij2 , then the output wave can be written as
(t) = X
i 0(t, i): (2.8)
i
If the delay times do not di er too much, the original shape of the pulse is approximately
preserved despite the interference, and the peak of the pulse is advanced with respect to that
of a reference pulse propagating through free space. The only problem with this formulation
is that it obviously neglects the dispersion e ects of the medium. Yet, the superposition of
undistorted waves yields an overall dispersive transfer function.
Remark (Multipath propagation) The concept of electromagnetic waves travelling
along di erent paths from a sender to a receiver plays an important role in the modelling
of mobile communication systems. Particularly in urban areas, the signal at the receiver
antenna is a combination of waves scattered from and di racted around the surrounding
buildings. Depending on the phase delays, the interference can be constructive or
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