An Efficient Calculation of Photonic Crystal Band Structures Using Taylor Expansions

Abstract

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.

Higher-order sensitivity analysis of periodic 3-D eigenvalue problems for electromagnetic field calculations

“…Also, there are some ideas to reduce the number of required single solutions to obtain accurate dispersion diagrams, e.g., one method with a forward and backward check has been presented in Klindworth and Schmidt (2014). Another way to compute the band structure is the solution of multiple EVPs at different points with MOR, which has been presented in Scheiber et al (2011).…”

“…In Bandlow (2011) and Bandlow et al (2008), the calculation of the Brillouin diagram for periodic metamaterials is realized using a scattering matrix approach. An approach with a Taylor approximation is shown in Klindworth and Schmidt (2014) for band structures in photonic crystals. Also, for photonic crystals, the band structure is calculated using model order reduction (MOR) in Scheiber et al (2011).…”

Optimization of a Multiphysics Problem in Semiconductor Laser Design

“…Though it is possible that larger eigenvalues and eigenfunctions may also be of interest, the nontrivial multiplicity of even the second eigenvalue vastly complicates any differential sensitivity analysis. For higher eigenvalues some path selections as in [39] are necessary. Even with this choice there are still some challenges.…”

Trapped modes in thin and infinite ladder like domains. Part 1: Existence results

“…where φL is defined in (31). Thus, we reduced the initial problem for a differential operator on the graph to a problem for a finite difference operator acting on sequences {uj} j∈Z .…”

“…Do they move in the complex plane? Let us point out that the use of the sophisticated numerical method (based on an automatic choice of the mesh size) presented in [31] might help to answer these questions.…”