Rigorous Asymptotics of Surface Plasmon Polariton Wavepackets in Nonlinear Media

• Applicant: Professor Dr. Tomas Dohnal, Ph.D.
• Subject Area: Mathematics
• Term: from 2018 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 407582688

This project studies localized wavepackets of surface plasmon polaritons (SPPs) at the interface of a Kerr nonlinear dielectric and a metal. The material dispersion of the metal is accounted for, and both materials can be either homogeneous or spatially periodic. From the applied point of view such wavepackets are valuable as potential information carriers. Together with the possibility to manipulate SPPs on a much smaller spatial scale than by electromagnetic waves in bulk dielectrics this offers interesting physical phenomena. The governing equations are dispersive nonlinear Maxwell equations. Typically, the full vector valued form of Maxwell equations has to be used to study localized waves. The wavepacket is represented by one or more carrier waves and slowly varying envelopes. Formally, a complex Ginzburg-Landau equation can be derived for the envelope of an asymptotically broad and small wavepacket with a single carrier wave. The aim is to prove that this formal description provides an approximation (in a suitable norm) of a true solution of the Maxwell system. This will be done in several geometrical settings including a two dimensional case, where the field is delocalized in one spatial direction, as well as the case with localization in all three dimensions. Also, both spatial and spatiotemporal wavepackets will be studied. By spatial wavepackets the field is time harmonic and one of the spatial variables plays the role of an evolution variable. The governing equations then can be written as a time independent nonlinear curl-curl problem. By spatiotemporal wavepackets the evolution variable is time. We will consider the interfaces given by a plane as well as by two half-planes forming a corner. Because the material dispersion of the metal leads to losses in the time evolution, the effect of a doped dielectric part of the medium will be also considered. This can lead to the existence of solitary waves in the Ginzburg-Landau equation and thus to an approximately lossless propagation in the original system. The main difficulties for the mathematical analysis are firstly the well-posedness of the governing nonlinear dispersive Maxwell equations, resp. of the curl-curl system posed as Cauchy problem. Secondly, estimates of the residual for the asymptotic wavepacket ansatz have to be provided in a norm compatible with the well-posedness results. The analysis will be accompanied by numerical computations of the wavepacket propagation and by numerical test of the convergence of the asymptotic error.

• DFG Programme: Research Grants