An Efficient Calculation of Photonic Crystal Band Structures Using Taylor Expansions (Communications in Computational Physics, 2014)
In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Ourmethod relies on the fact that the dispersion curves of the band structure are smoothfunctions of the quasi-momentum in the one-dimensional Brillouin zone. We show thederivation and computation of the group velocity, the group velocity dispersion, andany higher derivative of the dispersion curves. These derivatives are then employedin a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for theselection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. Theproposed algorithm is advantageous not only as it decreases the computational effortto compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is notpossible with the standard approach of solving the underlying eigenvalue problem ata discrete set of values of the quasi-momentum without taking the mode parity intoaccount.