Introduction to Photonic Crystals: Bloch’s Theorem, Band Diagrams, and Gaps (But No Defects)

Photonic crystals are periodically structured electromagnetic media, generally possessing photonic band gaps: ranges of frequency in which light cannot propagate through the structure. This periodicity, whose lengthscale is proportional to the wavelength of light in the band gap, is the electromagnetic analogue of a crystalline atomic lattice, where the latter acts on the electron wavefunction to produce the familiar band gaps, semiconductors, and so on, of solid-state physics. The study of photonic crystals is likewise governed by the BlochFloquet theorem, and intentionally introduced defects in the crystal (analogous to electronic dopants) give rise to localized electromagnetic states: linear waveguides and point-like cavities. The crystal can thus form a kind of perfect optical “insulator,” which can confine light losslessly around sharp bends, in lower-index media, and within wavelength-scale cavities, among other novel possibilities for control of electromagnetic phenomena. Below, we introduce the basic theoretical background of photonic crystals in one, two, and three dimensions (schematically depicted in Fig. 1), as well as hybrid structures that combine photonic-crystal effects in some directions with more-conventional index guiding in other directions. (Line and point defects in photonic crystals are discussed elsewhere.) Electromagnetic wave propagation in periodic media was first studied by Lord Rayleigh in 1887, in connection with the peculiar reflective properties of a crystalline mineral with periodic “twinning” planes (across which the dielectric tensor undergoes a mirror flip). These correspond to one-dimensional photonic crystals, and he identified the fact that they have a narrow band gap prohibiting light propagation through the planes. This band gap is angle-dependent, due to the differing periodicities experienced by light propagating at non-normal incidences, producing a reflected color that varies sharply with angle. (A similar effect is responsible for many other iridescent colors in nature, such as butterfly wings and abalone shells.) Although multilayer films received intensive study