During the first funding period of the SPP 1391 Ultrafast Nanooptics, we have developed advanced material models for quantitative computations of nanosplasmonic systems and have applied them to a number of challenging problems. In the course of this, several methodical advances of the Discontinuous Galerkin Time-Domain (DGTD) approach have been realized. Below, we list a few highlights of this research. 

Advanced excitations: Electron energy losss spectra via DGTD:

During the past two to three years, there has been a considerable increase in activities to characterize plasmonic nanostructures via electron energy loss spectroscopy (EELS) as this, e.g., provides unprecedented spatial resolution. Corresponding computational approaches that can handle such problems are scarce and mostly based on frequency-domain boundary element methods. Much the same can be said regarding experiments and computations of cathodoluminescence signals.

On the other hand, we feel that spatial resolution provided by EELS might be highly advantageous for addressing the questions regarding nonlocal behavior of metallic nanoparticles (as discussed below). Therefore, we have augmented our initially proposed research plan by the extension of the DGTD method to EELS computations (PNFA 9, 367 (2011). As a matter of fact, this work has been the first to demonstrate how to determine electron energy loss spectra via a time-domain approach and thus opens the avenue of analyzing nonlinear effects both within EELS and cathode-luminescence.

Together with the group of Stefan Linden, we have thus been to characterize both experimentally and theoretically, the resonances of metallic C-shaped nano-particles and complementary C-shaped nano-apertures, thus providing an interesting elucidation of Babinet's principle (Opt. Mater. Expr. 1, 1009 (2011); see also Fig. 1).

Hydrodynamic model for the ultrafast nonlinear optical response of metallic nanostructures:

Within this model, the conduction electrons within a metallic nanoparticle are treated as a plasma in confined geometry. Consequently, the Maxwell equations have to be solved simultaneously together with the corresponding plasma equations. This results in a highly nonlinear set of PDEs which requires considerable care.

For instance, the plasma equations alone already admit shock-wave type solutions within certain parameter ranges. Clearly, such shock waves are unphysical in the present situation of conduction electrons in a rather lossy metal. Nevertheless, numerical errors may build-up and lead to the formation of shock waves, especially - as we had to find out very painfully - in three dimensional systems with rather sharp corners or edges. Consequently, we have implemented an automated shock-capturing scheme for our DGTD package.

In addition, the nonlinear nature of hydrodynamical model also leads to considerable increases of computational time relative to linear calculations on the same mesh. Consequently, we have developed optimized time-stepping schemes (J. Comput. Phys. 231, 364 (2012)). Nevertheless, the speed-up provided by this approach can be further increased by moving from a CPU-based code to a GPU-based version (GPU = graphic processing unit). Since extensive parameter studies such as those required in the ongoing and future collaborations with experimental partners have to be conducted in a finite amount of time, we have chosen to port our DGTD package to GPU-based platforms. The first results (see Fig. 2) have exceeded our wildest expectations - which have been around a speed-up of at least 10 for realistic computations. Note, that the speed-up factors reported in Fig. 3 are for a dual six-core CPU computation, i.e., we obtain a speed-up factor of about 350 relative to a single-core computation. These improvements in computational efficiency will be a real asset in our future work both regarding more advanced computational schemes as described in the research program as well as with regard to the collaboration with the experimental groups within the SPP.

Notwithstanding these technical improvements, we have implemented, tested and applied the hydrodynamic model. First, we have considered nonlinear effects, in particular, the case of wave mixing phenomena

In Fig. 3, we display the result of a computation where a gold sphere of radius 200 nm has been illuminated with a high-intensity pulse that contains two closely-spaced carrier frequencies. Obviously and as expected, the hydrodynamic model exhibits a rather rich nonlinear response which we are presently exploiting for the analysis of resonant SHG processes.

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