The subject is motivated by the abundance of analytical problems generated by interesting phenomena of scattering and resonance of electromagnetic fields, which comprise a set of basic tools for the design of photo-electronic devices. The driving phenomenon is the existence of photonic frequency stop-bands for periodically structured materials, called photonic crystals, which is the source of the richness of their resonant behavior. One of the salient problems of current interest is anomalous transmission of source fields through periodic slabs near resonant frequencies, corresponding to weakly leaky guided slab modes. The project addresses the mathematics of this and related problems on several fronts: We use the complex-analytic theory of boundary-integral operators to obtain asymptotic formulas for the structure of transmission anomalies near non-robust slab modes; we pursue a development of the spectral theory for general open systems, such as leaky slabs, based on the concept of the unique conservative extension of a dissipative system; and we investigate the existence theory of modes and optimization of the associated resonant phenomena, as well as efficient numerical schemes for their simulation. Collaborators include S. Venakides, A. Figotin, D. Volkov, R. Lipton, and O. Bruno.
Because of their capacity to block electromagnetic waves at selected frequencies and to trap energy, photonic crystals have become a subject of intense research in recent decades. Applications include high-efficiency lasers and filters, waveguides, and directional antennas. Engineering of these devices relies on precise tuning of resonant frequencies, transmission-vs.-frequency profiles, and the interaction of open electromagnetic resonators with their surrounding medium. This project aims to set the theoretical framework of these resonant phenomena on a solid mathematical footing. The central theme is the frequency-response theory of open resonators, which provides the framework for the guiding of light and other radiation by structures, precise computation of the interaction of guided energy with applied sources, and optimization of resonant properties. Numerical computations and experimental results from the literature will guide the theoretical investigations. The project involves an undergraduate student and a graduate student as research assistants. The undergraduate has been assisting the PI in numerical simulations and will be part of a collaboration with the computational electromagnetics group at Cal Tech. The graduate student will pursue her Ph.D. on the subject of resonant scattering by periodic pillars.
