Physical optics of photonic crystals

A one-dimensional harmonic model of a photonic crystal is considered using the methods of physical optics. The theoretical formalism is based on the notion of the Fresnel volume reflection and the system of two first-order differential equations, which are equivalent to the wave equation. Using the Rayleigh layer as an example, it is shown that the volume reflection plays a role of the friction, similar to the friction in oscillations of a pendulum, and, in a strongly inhomogeneous medium, can suppress field oscillations and turn the group velocity to zero. In the approximation of small modulation factor, the models of two, four, and six waves are considered. In the two-wave model, the dispersion relation contains a zone of inhomogeneous waves, whose width is determined by the Fresnel reflection coefficient from one period. The refinement in terms of the four-wave or six-wave model yields only a small correction to the position of the zone, retaining its width unchanged. The wavenumber as a function of frequency is described by a circle inside the zone of inhomogeneous waves and by a hyperbola outside this zone. Mathematically, the method used is significantly simpler than those based on the application of the Floquet theorem to the wave equation. It is shown that the notion of forbidden zones is inconsistent with respect to photonic crystals, and the term zones of inhomogeneous waves is proposed instead.