Two-dimensional treatment of the level shift and decay rate in photonic crystals

We present a comprehensive treatment of the level shift and decay rate of a model line source in a two-dimensional photonic crystal (2D PC) composed of circular cylinders. The quantities in this strictly two-dimensional system are determined by the two-dimensional local density of states (2D LDOS), which we compute using Rayleigh-multipole methods. We extend the critical point analysis that is traditionally applied to the 2D DOS (or decay rate) to the level shift. With this, we unify the crucial quantity for experiment--the 2D LDOS in a finite PC--with the band structure and the 2D DOS, 2D LDOS, and level shift in infinite PC's. Consistent with critical point analysis, large variations in the level shift are associated with large variations in the 2D DOS (and 2D LDOS), corroborating a giant anomalous Lamb shift. The boundary of a finite 2D PC can produce resonances that cause the 2D LDOS in a finite 2D PC to differ markedly from the 2D LDOS in an infinite 2D PC.