Photonic crystals, also known as photonic band gap (PBG) structures, are structures with a periodically modulated dielectric constant. Main reason for the study of photonic crystals is that, in analogy to the case of an electron moving in a periodic potential, certain photon frequencies can become forbidden, independent of photon polarization and the direction of propagation - a complete photonic bandgap (CPBG). There is a common belief that, in the near future, photonic crystals systems will allow us to perform many functions with light that ordinary crystals do with electrons. At the same time, photonic structures are of great promise to become a laboratory for testing fundamental processes involving interactions of radiation with matter in novel conditions. A bandgap can also occur in acoustic crystal structures, which are structures charaterized by a periodically modulated elastic constant. Its obvious use is that of an acoustic isolator.
A study of wave propagation in periodic structures has a long history,
which stretches back to, at least, Lord Rayleigh classical article
on the influence of obstacles arranged
in rectangular order upon the properties of a medium . This work was soon followed
by work of Kasterin , who studied reflection and refraction of low frequency
sound by an orthorombic grid of hard spheres (an acoustic crystal). Another
early paper on photonic crystals is that of Ewald .
Ewald studied if the optical birefringence can be caused by lattice structure
alone, i.e. if it requires anisotropy of the atomic scatterers.
Later on, wave propagation in
periodic structures was a subject of the book  of the same title by
Brillouin and Parodi. The use of a two-dimensional distributed feedback
periodic structures in thin film optical devices and lasers is disclosed in the
US patent publication US-3,884,549 . (See especially Figs. 1-2
therein.) In the second half of seventies and early eighties a popular subject
was an investigation of absorption spectrum of clusters of spheres.
Within this subject, results were also obtained for regular finite
and infinite linear chains of metal spheres . Without knowing that,
the authors basically rediscovered and applied Beeby's multiple-scattering theory
 to electromagnetic waves.
Some of early history of acoustic and photonic crystals can also be found
in a review  by Korringa. See also
Photonic Crystal Headlines at least 25 Years Old.