Photonic crystals are inhomogeneous materials that consist of a periodic arrangement of (usually) dielectric scatterers which create a periodic modulation of the dielectric function in space. Photonic crystals are one of the very few inhomogeneous materials that can be treated exactly within the framework of the electromagnetic theory due to their inherent periodicity. The latter is at the heart of their working principle and is responsible for some unusual properties, such as the appearance of a photonic bandgap (John 1987, Yablonovitch 1987), i.e., a frequency band within which propagating waves are excluded from the crystal.∗

Most photonic crystals, studied theoretically and realized experimentally, are two-dimensional, where the inclusions are infinitely long (very long compared to the transverse dimensions in experiments) cylinders of circular crosssection. One-dimensional photonic crystals or periodically layered materials have been extensively studied as well, especially since the work of Lord Rayleigh (see Chapter 1), but have been less popular because of the fewer degrees of freedom they offer. Nonetheless, due to such well-established grounds, one-dimensional photonic crystals have benefited from the resurgence of interest in the search for negative indices of refraction. Three-dimensional photonic crystals are less common because they are more complex to analyze theoretically and more challenging to realize experimentally. Here also, however, the regained interest in negative refraction phenomena and in particular in flat lens imaging has given the study of three-dimensional photonic crystals a fillip and various studies have been reported in the literature.