OP: Heterogeneous Optical Media: Boundary Effects, Spectral Properties, and Inversion

Drexel University, Philadelphia PA

Investigators

Abstract

Despite advances in computing speed, the simulation of the propagation of optical waves through complex materials continues to pose significant challenges. Fundamentally, the difficulty arises because variations in the material structure are on the same scale as the optical wavelength. Brute-force approaches thus require very fine discretization of the materials and a very large number of parameters, which is frequently computationally infeasible. The investigator and collaborators aim to develop new mathematical techniques that allow extraction of important properties and features of a system without the need for full simulation. In medical imaging, an initial image could thus be reconstructed in real time. Such an initial image can be useful in itself, or used as input to speed up the run time of high-resolution methods. In design of materials, the energy losses of a material can be estimated to provide qualitative insight into the potential suitability of the material and thus accelerate the construction of the next generation of optical devices. Several graduate and undergraduate students will participate in related research projects. The objective of this project is to solve several open problems related to optics and photonics of heterogeneous materials. The project will involve four main components. The first is to find a complete characterization of boundary effects in periodic optical materials. While boundary corrections in homogenization theory are notoriously difficult to understand, they have a large effect on the resulting fields. A large class of periodic optical materials can be studied by new asymptotic techniques that will allow us to fully characterize the role of the boundary. The second is the derivation of explicit formulae for various spectral properties of periodic scatterers. This includes the transmission eigenvalues, which can be read in the far field and provide information about the medium. The third is to develop a new reduced order model approach to inversion of elliptic operators. The method employs ideas from model reduction theory, and originates from a breakthrough in a previous spectrally matched grid approach that allows application to higher dimensions and general geometries. The last goal is to derive explicit calculations of the resonance values for multiple linear and nonlinear scatterer interactions. The investigator and collaborators use spectral asymptotic perturbation methods to understand the behavior of multiple-scatterer resonant interactions, both for linear materials and for Kerr effects.