Propagation of Waves in Complex Media, and Analysis of Small Noises

University Of North Carolina At Charlotte, Charlotte NC

Investigators

Abstract

The investigators develop a mathematical approach for studying optical processes in networks of thin fibers (waveguides) and mathematical models describing optical waveguides on a silicon wafer (so-called leaking wires). The problem is described by the Helmholtz equation in web-like three-dimensional geometric structures with the thickness parameter going to zero. The investigators also extend these results to more profound mathematical models involving Maxwell equations. The problem does not have an explicit solution. A numerical solution, difficult to obtain due to a complicated structure of the domain and coefficients of the equations, would not allow determination of parameters that are needed for applications. Therefore, the asymptotic analysis is applied. The original three-dimensional problem is reduced to a much simpler one-dimensional problem on the limiting graph. The latter problem admits a detailed analysis. Periodic networks are very often used in applications. These periodic structures have some technological errors due to the nano-scale of the device. The investigators study the wave problem in periodic media with small random noise. One of the outputs of this study is an estimate of the stochastic stability of optical systems (i.e., an estimate of admissible values of the random errors). The motivation and practical applications of the project concern the mathematical background for creating a compact (nano-optical scale) and efficient optical delay device which can be used to synchronize very fast optical communication lines with much slower electronics. The analysis of the limiting one-dimensional problem obtained at the first stage of the investigation suggests a network which can be used to create such a device.