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PhD Theses
| Zitatschlüssel | KlindworthDiss |
|---|---|
| Autor | D. Klindworth |
| Jahr | 2015 |
| Schule | Technische Universität Berlin |
| Zusammenfassung | In this thesis, we develop numerical schemes for the accurate and efficient computation of band structures of two-dimensional photonic crystal waveguides, which are periodic nanostructures with a line defect. The perfectly periodic medium on both sides of the line defect has to be modelled mathematically. For this, we employ Dirichlet-to-Neumann and Robin-to-Robin transparent boundary conditions. These boundary conditions are transparent in the sense that they do not introduce a modelling error, which is in contrast to the well-known supercell method. The numerical realization of these transparent boundary conditions in terms of high-order finite element discretizations addresses the first objective of this work, i.e. to improve the accuracy of photonic crystal waveguide band structure calculations. The realization of Robin-to-Robin transparent boundary conditions is more involved than the realization of Dirichlet-to-Neumann boundary conditions. However, in contrast to Dirichlet-to-Neumann boundary conditions, they do not exhibit any forbidden frequencies for which the boundary conditions are not well-defined or their computation is ill-posed. Since the eigenvalue problems with Dirichlet-to-Neumann or Robin-to-Robin transparent boundary conditions are nonlinear, efficient numerical schemes for their solution are crucial. We propose an indirect scheme based on Newton's method that is ideally suited for the eigenvalue problems under consideration. Moreover, we develop a path following algorithm, which we apply for the efficient approximation of the eigenpaths of the nonlinear eigenvalue problems, the so-called dispersion curves of the photonic crystal waveguide band structures. This path following algorithm is based on the fact that the dispersion curves are analytic, and hence, a Taylor expansion can be applied. For this, we introduce formulas for the derivatives of the dispersion curves and an adaptive selection of nodes at which a Taylor expansion is computed. With this adaptive selection we can resolve the dispersion curves in full detail while saving computation time. Our proposed numerical scheme, that includes these two ingredients, i.e. the high-order finite element discretization of the transparent boundary conditions for periodic media and the adaptive path following algorithm, allows for efficiently resolving physical phenomena with high accuracy. For example, we show how to identify mini-stopbands, i.e. avoided crossings of dispersion curves, and we discuss the behaviour of dispersion curves at band edges, which is not possible with standard methods such as the supercell method and an equidistant sampling of dispersion curves. |
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