A Fourier transform theory for photon localization and evanescence in photonic bandgap structures

Abstract

We propose an approach to the problem of describing photon behaviour near the bandgap region of photonic crystal structures using a Fourier transform formalism defined in optical reciprocal-space. A modified Debye-Waller approximation in conjunction with the use of a resonant refractive index provides an analytic solution to the coupled-mode equations describing photon propagation in a periodic high dielectric contrast structure. Overall, this represents a new technique for solving the inverse scattering problem. We use this method to consider photon behaviour at the Brillouin zone edges of a one-dimensional photonic crystal structure: our results showing agreement with conventional theory for the standard assumption of an infinitely long grating structure. However, when a photon is located in the bandgap region of a finite (five periods used as an example) high refractive index contrast grating structure, we find that photon velocity becomes superluminal as the photon wave becomes evanescent. This is in agreement with the basic precepts of the uncertainty principle. Additionally, as the bandgap strength increases, the probability of transmission through the dielectric barrier reduces exponentially, with an associated reduction in tunnelling time.