Local density of optical states in the band gap of a finite one-dimensional photonic crystal

We study the local density of states (LDOS) in a finite photonic crystal, in particular in the frequency range of the band gap. We propose an original point of view on the band gap, which we consider to be the result of vacuum fluctuations in free space that tunnel in the forbidden range in the crystal. As a result, we arrive at a model for the LDOS that is in two major items modified compared to the well-known expression for infinite crystals. First, we modify the Dirac delta functions to become Lorentzian with a width set by the crystal size. Second, building on characterization of the fields versus frequency and position we calculated the fields in the band gap. We start from the fields at the band edges, interpolated in space and position, and incorporating the exponential damping in the band gap. We compare our proposed model to exact calculations in one dimension using the transfer matrix method and find very good agreement. Notably, we find that in finite crystals, the LDOS inside the band gap depends on frequency, on position, and on crystal size, in contrast to the well-known results for infinite crystals where the LDOS is zero, independent of frequency and position.