In this chapter we shall meet examples of electromagnetic wave propagation in systems containing fine dielectric structure on a scale of the order of the wavelength, where the scalar-wave approximation is inadequate. Clearly, in these cases we have to solve Maxwell's equations directly. On writing the equations, we shall discover that they bear a close similarity to those of quantum mechanics, where the dielectric constant in Maxwell's equations is analogous to the potential in Schrödinger's equation. This opens up a vast arsenal of methods, both analytical and numerical, which have been developed for their solution.
We first discuss the optical waveguide, already familiar in everyday life as the optical fibre, which has caused a revolution in the communications industry (Agrawal (2002)). The second topic is the dielectric multilayer system which, in its simplest form (the quarter-wave anti-reflection coating) has been with us for more than a century, but can today be used to make optical filters of any degree of complexity (MacLeod (2001)).
Following these examples, we shall briefly discuss their application to photonic crystals, structures with periodic refractive index leading to optical band gaps, whose behaviour can immediately be understood in terms of the quantum analogy (Joannopoulos et al. (2008)). Photonic crystals have always existed.