(Microsoft Word - SEPBG-prepri\311)

I consider an exact model of atomic spontaneous dipole emission and classical dipole radiation in a finite photonic band-gap structure. The full 3D or 2D problem is reduced to a finite 1D model, and then this is solved for analytically using algebraic matrix transfer techniques. The results give insight to the electromagnetic emission process in periodic dielectrics, quantitative predictions for emission in 1D dielectric stacks, and qualitative formulas for the 2D and 3D problem. 1. INTRODUCTION From the beginning, one of the principal potential applications of photonic band-gap (PBG) materials was to the control of atomic spontaneous dipole emission [1, 2]. In particular, it was predicted that emission could be suppressed for dipoles located inside the PBG material when their resonant emission frequency ω 0 was in the photonic band gap. In this frequency range the electromagnetic density of modes ρ(ω) is very small. Resonant enhancement of emission was expected at the photonic band edges where the density of modes (DOM) was anomalously large. These phenomena are a result of the so-called Purcell Effect. Purcell concluded that nontrivial boundary conditions on the electromagnetic field surrounding a dipole emitter can alter the emission rate. They do so by altering the DOM and the electromagnetic modal field e(ω 0 ,r 0) at the position r 0 of the dipole [3]. (This original paper of Purcell was actually a short abstract for a talk to be given at an annual meeting of the American Physical Society. He apparently never wrote up anything more about it nor even gave the talk. Nevertheless, he gives in this short abstract the first quantitative formula for the alteration of spontaneous emission in cavities.) In 1992, Bowden and I gave a general formalism for computing atomic or classical dipole radiation rates in an inhomogeneous dielectric medium, of which a PBG material is a specific example [4]. Although our theory was completely general, it takes a supercomputer to figure out the DOM and electromagnetic modal structure of a full 2D or 3D PBG structure. Hence, after presenting the general formalism, we gave an exactly solvable model in terms of an infinite 1D Kronig-Penney model of the full 2D or 3D structure. This approximation is due to John and Wang. It amounts to replacing the Brilloun zone (BZ) of the 2D structure by a perfect circle, or of the 3D one by a perfect sphere—the same circle or sphere for …