Wave Propagation and Scattering in Networks of Thin Waveguides

University Of North Carolina At Charlotte, Charlotte NC

Investigators

Abstract

The objective of this project is the analysis of wave propagation in networks of thin waveguides. Problems of this type cannot be solved explicitly and are poorly amenable to computational methods because of the complicated 3-dimensional geometrical structure of such a network. The goal of the project is to develop methods allowing the reduction of the problem to a much simpler one-dimensional problem on the quantum graph, which is the limit of the network as the diameter of the waveguides tends to zero. The study will include the asymptotic analysis of the scattering solutions, the spectrum, and the resolvent for the Helmholtz equation for all frequencies (at the bottom and in the bulk of the spectrum). The project also includes the analysis of possible mechanisms for slowing down of wave packages in optical networks and the study of stability and intermittency in the system in the presence of random noise. The project will provide mathematical background for the investigation of information flows in complex optical networks. In particular, it will provide an efficient tool to construct necklace type devices for synchronizing the work of very fast optical elements and much slower electronics. The results and the methods to be developed in the project can be applied to the study of photonic crystals, nanotechnology, and other areas of physics and technology.