In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in 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 the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. These derivatives are then employed in 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 the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

A computational study on the enhancement of the second harmonic generation (SHG) in one-dimensional (1D) photonic crystals is presented. The mathematical model is derived from a nonlinear system of Maxwell’s equations, which partly overcomes the shortcoming of some existing models based on the undepleted pump approximation. We designed an iterative scheme coupled with the finite element method which can be applied to simulate the SHG in one dimensional nonlinear photonic band gap structures in our previous work. For the case that the nonlinearity is strong which is desirable to enhance the conversion efficiency, a continuation method is introduced to ensure the convergence of the iterative procedure. The convergence of our method is fast. Numerical experiments also indicate the conversion efficiency of SHG can be significantly enhanced when the frequencies of the fundamental and the second harmonic wave are tuned at the photonic band edges. The maximum total conversion efficiency available reaches more than 50% in all the cases studied.