Lower and upper bounds to photonic band gap edges

Abstract

A photonic band gap is determined by its edges, which are frequently calculated by the Rayleigh-Ritz method, producing a sequence of upper bounds. Since there are no error estimates available on these approximations, the level of accuracy of the computed band-gap edges, and thus of the extent of the band gap, remains undetermined. We adopt the method of intermediate problems to develop a procedure to calculate the lower bounds to the photonic band-gap edges. Lower bounds, supplemented with the Rayleigh-Ritz approximations, determine a band gap with arbitrary and known degree of accuracy, as well as provide error bounds on approximate calculations by other methods. Calculation of the lower bounds by the present method requires slightly more computational effort than the Rayleigh-Ritz method with a plane-wave basis. A parallel method to compute the alternative upper bounds is also developed in the process.