WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

2 Chapter 1 Introduction
incident wave
t = 0.62s
t = 1.00s
reflected wave
t = 1.25s
reflected wave
transmitted wave
incident wave
t = 0.62s
t = 1.00s
reflected wave
t = 1.25s
reflected wave
transmitted wave
Figure 1.1 Illustration of the bandgap phenomenon for wave propagation in a onedimensional
layered structure. The plots show the incident/transmitted wave pulses as
well as the reflected wave pulse at three different time instances. Left plots: a single inclusion
with different material properties. Right plots: Four equidistantly placed inclusions,
where the distance between the inclusions combined with the material properties leads
to in-phase reflections of the incident wave. With infinitely many inclusions present, a
complete reflection of the wave will occur – known as the bandgap phenomenon.
given along the vertical axis and the reduced wave vector given along the horizontal
axis. Afrequency rangeappearsforwhich nomodesexist. Inthisgap(thebandgap)
waves cannot propagate regardless of the direction. The small inserted figure shows
theirreducible Brillouin zone (Brillouin,1953)spannedbythetriangleΓ−X−M−Γ.
The zone relates to the smallest repetitive unit in the periodic material – the unit
cell (it is here a quadratic cell) – as well as its symmetry properties. The irreducible
Brillouin zone indicates the wave vectors which are necessary to investigate in order
to completely describe wave propagation through the material. It turns out to be
sufficient to examine the wave vectors corresponding to the triangular path 1 .
About half a century later, in 1987, the discovery of the bandgap phenomenon
for optical waves – photonic bandgaps – by John (1987) and Yablonovitch (1987)
marks a major breakthrough in the field of electromagnetics. This finding had
1 In principle the entire triangular area must be examined, but it is commonly accepted that it
is sufficient to examine the wavevectorson the exterior triangular path. To the author’s knowledge
no proof of this has been presented yet.