Derivation of the density of states of leaky photonic bands

This paper presents the formula for the density of states $(\mathrm{DOS})$ of photonic bands ($\mathrm{PB}$'s) in the leaky region of the phase space of a slab-type photonic crystal. It is expressed by the eigenphase shifts of the scattering matrix defined in terms of the complex transmission and reflection amplitudes of plane wave external incident light. The derivation is given for the general case in which a number of diffracted plane wave lights are produced by the incident lights. The $\mathrm{DOS}$ profile calculated as a function of frequency and wave vector enables us to obtain the dispersion relation and lifetime of leaky $\mathrm{PB}$'s. The usefulness of the derived formula is demonstrated by applying it to the $\mathrm{PB}$ structure of dielectric spheres, arrayed periodically to form a photonic crystal of finite thickness.