Over the past years, plasmonic systems, i.e., nano-structured metal particles or films, have experienced a significantly increased interest world-wide. In such systems, the freely movable conduction electrons lead to strong field enhancement and strong field gradients. In turn, this allows for the strong modification of a number of basic physical processes such as the emission of light from and the absorption of light by quantum dots or molecules, optical forces, and fluctuation-induced phenomena such as Casimir forces or thermal radiation. A quantitative description of such nano-technological processes has to be based on appropriate material models. While the linear behavior of simple metals may adequately be described by the (spatially local) Drude model and/or Drude-Lorentz models, their nonlocal and/or nonlinear behavior requires more advanced models – in the simplest version a hydrodynamic extension of the Drude model. Similarly, transition metals such as Ni, Fe, and Co feature linear behavior that cannot be accounted for within Drude-Lorentz models. In this talk, an attempt is made to provide a (necessarily biased) overview of some of the above material models and the associated physical effects they allow (and fail) to describe.
Spectral analysis of periodic operators has been thoroughly studied but the propagation of light in heterogenous periodic structures still offers interesting problems. We focus on an ideal waveguide made of a weakly nonlinear material and we discuss the propagation of a wave train along the guide paying attention to the mathematical issues.
The endeavour of using photonic crystals to create all-optical computational devices motivated a lot of research in the area of wave equations with periodically varying coefficients. Of particular interest is the dynamics of localized structures which model light pulses. This talk illustrates two techniques which allow a surprisingly thorough description of the dynamics of such solutions: the derivation of modulation equations and the use of a blend of center manifold and inverse spectral theory. The presented work was initiated by a project of the RTG 1294 within the research group of Guido Schneider.