NSP · Nonlinear spectral problems and wave propagation in crystals

Nonlinear spectral problems and wave propagation in crystals

Non-linear spectral problems frequently occur when damping effects or frequency dependencies are included in a mathematical model. Prominent examples are plasmon resonances and band gaps in dielectric and metallic photonic crystals. Such crystal structures that are designed for controlling propagation of electromagnetic waves have numerous applications including telecommunication, solar cells and integrated circuits. The fellow studied photonic crystals theoretically and developed mathematical theory and numerical software. In the lossless case (no dispersion / absorption) the spectral properties are known and the convergence of a Galerkin approximation of the underlying operator is well established. However, the impact of material losses and the frequency dependence of the material parameters are in many cases crucial for the value of metallic photonic crystals and metamaterials at optical frequencies. A model with frequency dependent material parameters leads to a nonlinear dependence on the spectral parameter. That is, the resonances in the system are given by a nonlinear operator function. Basic questions such as the discreetness of the spectrum and convergence of a Galerkin approximation of the operator function was not known before the start of the project. In his work, Christian Engström addresses these and other problems based on operator theory and approximation theory for non-compact operators. Moreover he applies a novel set of numerical techniques, which make it possible to simulate a broad range of problems in nano-optics. The software provides physicists and engineers with a new simulation tool for dielectric and metallic photonic crystals. The mathematical analysis and the numerical methods in the project are highly relevant since they assure that the numerical algorithms are reliable.

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