When a quantum or classical wave propagates in a periodic structure, in any number of dimensions, the dispersion curves that relate the frequencies of the wave to the wave vector characterizing its propagation, possess a number of branches. These branches form bands that are separated by frequency gaps at points of symmetry in the corresponding Brillouin zones. In some cases, an absolute gap occurs (i.e., a frequency range in which no wave can propagate) that exists for all values of the wave vector in the Brillouin zone. This also gives rise to a gap in the density of states of the waves propagating through the structure.
These general features are common to all types of waves that propagate in appropriate lattices of periodic scaterers. Thus, acoustic, electronic and neutronic as well as electromagnetic waves exhibit a similar behaviour.
The idea that singly, doubly and triply periodic dielectric lattices can be designed to posses photonic band gaps has attracted wide attention, both theoretically and experimentally. The absence of electromagnetic modes inside a photonic band gap can lead to unusual physical phenomena. Thus, atoms or molecules embedded in such a structure, called a dielectric crystal, can be locked in an excited state if the energy of this state, relative to the ground state, falls within the photonic band gap. In this case, the atoms (or molecules) are also expected to exhibit an anomalous Lamb shift. At the same time, in a dielectric crystal new types of electron - photon interactions appear leading to a specific behaviour of light.
We are investigating the problem of diffraction of a plane electromagnetic wave by a periodic structure, using the Extended Rayleigh Method (ERM). This work should reveal the photonic band structure of periodic composite materials. It is complementary to most theoretical work in the field, which is limited to dielectric contrasts between inclusions and the background material which are not too large.