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

## TL;DR

This paper develops a theory describing how the density of states in photonic bandgap crystals approaches the ideal infinite-crystal limit as the crystal size increases, revealing a universal 1/L scaling in all dimensions.

## Contribution

It introduces a finite-size scaling theory for the density of states in photonic crystals, accounting for boundary-induced Bloch-mode broadening.

## Key findings

- DOS inside the band gap scales as 1/L with crystal size
- The 1/L scaling applies to 1D, 2D, and 3D photonic crystals
- Finite crystal effects significantly influence the DOS near the band gap

## Abstract

The famous vanishing of the density of states (DOS) in a band gap, be it photonic or electronic, pertains to the infinite-crystal limit. In contrast, all experiments and device applications refer to finite crystals, which raises the question: Upon increasing the linear size $L$ of a crystal, how fast does the DOS approach the infinite-crystal limit? We present a theory for finite crystals that includes Bloch-mode broadening due to the presence of crystal boundaries. Our results demonstrate that the DOS for frequencies inside a band gap has a $1/L$ scale dependence for crystals in one, two and three dimensions.