Confined modes in finite-size photonic crystals

2005 
The real band structure of finite-size photonic crystals (PCs) is different from the band structure of infinite PCs derived using Bloch's theorem. There are isolated instead of a continuum of modes around a band edge. We use the envelope function approximation (EFA) method to derive simple formulas to describe the frequency and field pattern of the modes in finite-size PCs. The results are compared with those from finite-difference time-domain (FDTD) method. We observe that the agreement between the results from EFA with a vanishing boundary condition and FDTD is sensitive to the boundary orientation. In addition, although a simple group-velocity-based analysis explains the long lifetime of the confined modes, this explanation is inappropriate for certain structures. In particular, we find boundary orientations, for which the lifetime of the confined modes can be enhanced by orders of magnitude compared with that anticipated from group velocity analysis. We explain this by total internal reflection. Our results demonstrate the utility of the EFA approach in studying the modes in finite-size PCs and reveal the importance of boundary conditions to develop an EFA-based tool for cavity lasers and for studying nonlinear interactions.
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