A fast method to compute dispersion diagrams of three-dimensional photonic crystals with rectangular geometry

We propose a method and codes for fast computation of complex dispersion relations in three-dimensional photonic crystals (PCs) with rectangular geometry. The main idea of the method is to convert the eigenproblem to a nonlinear equation equivalent to the zero-determinant condition. This equation is then solved iteratively either by fixed-point iteration or by rational approximation method. Additional mathematical elements include fast-converging continued-fraction expansion to compute the interaction tensor (appearing in the above nonlinear equation) and efficient accounting for the rectangular geometry in matrix-vector multiplications, which are involved in computing the continued fraction coefficients. The method allows one to perform realistic three-dimensional computations on a typical laptop computer, including finding the Bloch wave vector in the band gaps and in evanescent mode bands. This paper is focused on the method and includes its detailed explanation and illustration with examples. The associated computational package contains a detailed user guide and a set of further demonstrations, which can be run with the help of provided scripts.