Finite size scaling of density of states in photonic bandgap crystals.

The famous vanishing of the density of states (DOS) in a band gap, be it photonic or electronic, pertains to crystals in the infinite-size limit. In contrast, all experiments and applications pertain to finite crystals, which raises the question: Upon increasing the size L of a crystal, how fast does the DOS approach the infinite-crystal limit? Answering this question, however, requires an understanding of how linewidth of the modes in crystal with finite support scales as a function of crystal length L. We develop such a theory in a finite support crystal using Bloch-mode broadening due to the crystal boundaries. Our results suggest that total DOS inside a bandgap has the same scale dependence irrespective of the number of dimensions which the crystal is defined in. This can pave the way to establishing design rules for the usage of vanishing density of states, notably to cavity QED, quantum information processing, and Anderson localization.