Theory of chirped photonic crystals

Abstract

In chirped photonic crystals, the structural parameters describing a unit cell are progressively varied from a unit cell to the nearest ones. Geometric and dielectric response functions can be affected by this modulation, but here we only investigate the effect of a long-range, slowly varying, modulation of the refractive index. The Bloch modes are modified by essentially being modulated by an envelope function which adapts to the long-range dielectric function perturbation. It is shown that this envelope function obeys a simple linear Schrödinger equation of classical (non-quantum) origin. Close to a band extremum, at a gap edge, the envelope functions can be interpreted as wave functions of particles, called “energy carriers”. These carriers have a mass and come as two species, referred to as “effective photons” (for positive band curvatures) or “photonic holes” (for negative band curvatures). The energy transfer through the chirped structure can be viewed as resulting from the migration of these particles under forces implied by the long-range dielectric function modulation.