Page 18 - Fall 2008

systems with periodic elements. By the 1970’s virtually all
aspects of waves in periodic systems were understood and
applied in many clever devices. An article reviewing the sub-
ject, written in 1976 by C. Elachi,9 contains nearly 300 refer-
ences. With the advent of semiconductor devices in the
1960’s, accurate methods for calculating the band structure of
complicated three-dimensional periodic systems were devel-
4
oped. Currently, there is little that could be added to the
understanding of wave propagation in periodic systems.
New possibilities arise, however, if one studies systems that deviate from the confines of static, linear and exactly
periodic systems. For example:
(1) No real system can be perfectly periodic, so it is
important to understand wave propagation in arrays of scat-
terers that are disordered. The disorder may range from
slight, where random deviations from perfect periodicity are
small, to significant, where disorder is characterized by a
broad distribution function. The appendix shows how the
modulus of the modes of Fig. 7 may be derived. The modu-
lus is illustrated in Fig. 9 and is the same in every unit cell
throughout an infinite system. If a system deviates randomly
from perfect periodicity, then one might expect that the new
wavefunctions would show random variations in the moduli
in different unit cells. However, this is not what happens.
Instead, the wavefunctions become exponentially localized;
that is, the modulus of the wavefunction has a maximum
value at one unit cell, and then the modulus decays exponen-
tially with distance from that site. This quite unexpected
behavior is called Anderson localization, and it was cited in
the 1977 Nobel Prize of Philip Anderson and Sir Neville
10
(2) There are configurations of scatterers, referred to as quasicrystalline,3 that are intermediate between periodic and
Thus wave propagation in disordered arrays of scat- terers is a significant matter indeed.
Mott.
Fig. 9. Plots of the modulus of the modes from Fig. 7. Taking the modulus results in just the periodic part of the mode, |Uk,n(x)|, with the period being the lattice con- stant (one-fourth of the length of string shown in the plot). Double letters refer to degenerate modes which were combined to give the modulus.
Fig. 8. Band structure diagram for the example with l/a = 4. The letters refer to the modes in Fig. 7. The frequencies and their symbols are taken from the modes in Fig. 6.
disordered. For one-dimensional disordered or quasicrys- talline configurations, there are rigorous theorems, analo- gous to Floquet’s theorem, that govern the behavior of wave propagation in such systems. However, in two and three dimensions, no one has been able to prove any theorems, so the behavior of waves in higher dimensional disordered or
2,3
quasicrystalline systems has been a viable field of research. (3) In the article so far we have assumed that the posi- tions of the scatterers (periodic or disordered) were static, i.e., they did not change in time. However, there are impor-
Wave Propagation in Scatterers 17