A band-structure formalism is presented to describe dispersion relations of scalar waves in periodic composite structures. We develop a scalar-wave version of k⋅p theory, a technique that has been successfully used for calculating electronic band structures in semiconductors. Analogs of semiconductor impurities are considered. The scalar-wave analog of semiconductor effective-mass theory is derived to describe the effect of a slowly varying perturbation (shallow impurity). The case of a highly localized perturbation (deep impurity) is discussed via a tight-binding theory. Application of the formalism to describe vector waves and hence photonic band structures in periodic dielectric systems is discussed.

• Received 10 June 1993

DOI:https://doi.org/10.1103/PhysRevB.48.10118