Explicit solutions for the band edges of multi-layer photonic crystals with integer ratios of optical path lengths

Abstract

The tangent formulation for photonic bandgap materials is used to derive explicit expressions for the energy eigenvalues at the centre and the edge of the Brillouin zone of multi-layer photonic bandgap materials with integer ratios of optical path lengths (e.g. any combination of quarter-wave, half-wave, etc., stacks). Unlike the transfer matrix formalism?in which such solutions are implicit functions of photon energy and require a root search?here, all the eigenfrequencies and wave amplitudes are found through a single diagonalization of a Hermitian eigenvalue problem. By reformulating the general nonlinear problem as an algebraic eigenvalue problem, the present solution replaces the search for resonance frequencies with a single eigenproblem, which considerably reduces the computational cost and provides a more accurate solution and a better insight for structure resonances.