WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Phononic and photonic bandgaps 3
Frequency (kHz)
80
60
40
20
bandgap
0
M Γ
Reduced wavevector
Figure 1.2 A band diagram for an infinite periodic material. The solid lines show propagating
modes with the wave frequency along the vertical axis and the reduced wave vector
(property of wave length and propagation direction) along the horizontal axis. A bandgap
appears as a frequency range for which no waves can propagate through the material regardless
of its direction. The inserted figure shows the irreducible Brillouin zone for the
periodic material which is spanned by the triangle Γ−X−M−Γ. The wave propagation
can be completely described by analyzing the wavevectors corresponding to the triangular
path.
a significant impact and intensive research on the photonic bandgap phenomenon
continues today in connection with applications in photonic crystal fibres and possible
future applications in integrated optical circuits. Many books and papers have
been published on the subject, e.g. the monograph by Joannopoulos et al. (1995)
who provide a comprehensive set of band diagrams for different configurations of
photonic bandgap materials – also known as photonic crystals.
The discovery of photonic bandgaps led to a ”rediscovery” of the bandgap phenomenon
for elastic waves. Band diagrams were produced and bandgaps found for
manydifferentperiodicmaterialconfigurationsin1D,2Dand3D,seee.g.Sigalasand
Economou (1992), Kushwaha (1996) and Suzuki and Yu (1998). Gaps in the band
diagrams were now often referred to as phononic bandgaps or sometimes acoustic
or sonic bandgaps. Parallel to this work, research on structures with a periodic-like
nature was carried out including the works by Elachi (1976) and Mead (1996). Here,
themainfocuswasnotonthebandgappropertiesofaninfinite periodicmaterialbut
instead on the effect of the periodicity on the dynamic performance of engineering
X
Γ
M
X
M