This thesis presents work that I have done with Egor Muljarov and Wolfgang Langbein in order to extend an existing perturbation theory for open systems describable by a scalar equation to 3D systems which cannot be reduced to effectively lower dimensions. This perturbation theory is called the resonant-state expansion (RSE). The RSE is derived from properties of the dyadic tensor Green’s function (GF) of the unperturbed system written in terms of resonant states (RSs). Hence to extend the RSE it was necessary for us to derive this spectral form of the GF in terms of normalised RSs for arbitrary 3D systems. To process the numerical output of the RSE, we develop and evaluated algorithms for error estimation and their reduction by extrapolation. In the case of planar systems the RSE can be compared with other methods such as the scattering matrix or transfer matrix methods. It is also possible to solve the boundary conditions analytically to provide transcendental equations that can be solved by the Newton-Raphson method. We study these systems for that reason since we can validate the numerical calculations of the RSE by showing the convergence of perturbed solutions to the exact result found from these other methods. We study the planar systems both zero and non-zero in-plane wavevector. As an intermediate step to a fully 3D perturbation theory for open systems we make an implementation of the RSE in 2D. We use as a basis the analytically known RSs of the infinitely extended homogeneous dielectric cylinder. We find that the unperturbed GF contains a cut in the complex frequency plane, which must be included in the RSE basis for the accuracy of the perturbation theory. Zero frequency longitudinal modes are found to be formal solutions of Maxwell’s wave equation which also must be included in the basis for the accuracy of the method. Zero frequency modes occur for systems of all dimensionality when considering the TM modes, modes with electric field component normal to the interfaces. In the penultimate chapter of this thesis we apply the RSE to fully 3D open systems. We use as a basis the analytically known RSs of the homogeneous dielectric sphere. This advance was non-trivial due to a general mixing of transversal and longitudinal electro-magnetic modes. We compare the performance of the RSE with available commercial electromagnetic solvers. In the case of 3D perturbations, we find that the RSE provides a higher accuracy than the finite element method (FEM) and finite difference in time domain (FDTD) for a given computational effort, demonstrating its potential to supersede presently used methods. At the end of the penultimate chapter we introduce a local perturbation method for RSE, which is a unique capability of the RSE compared to FEM or FDTD, and allows to calculate small perturbations of a system with a small computational effort.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:611061 |
| Date | January 2014 |
| Creators | Doost, Mark |
| Publisher | Cardiff University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://orca.cf.ac.uk/61445/ |
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