Electromagnetic Resonance in Periodic Structures

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Award abstract, DMS-0505833, S Shipman, Louisiana State UniversityTitle: Electromagnetic Resonance in Periodic StructuresThe subject is motivated by the abundance of analytical problemsgenerated by interesting phenomena of scattering and resonance ofelectromagnetic fields, which comprise a set of basic tools for thedesign of photo-electronic devices. The driving phenomenon is theexistence of photonic frequency stop-bands for periodically structuredmaterials, called photonic crystals, which is the source of therichness of their resonant behavior. One of the salient problems of currentinterest is anomalous transmission of source fields through periodicslabs near resonant frequencies, corresponding to weakly leaky guidedslab modes. The project addresses the mathematics of this and relatedproblems on several fronts: We use the complex-analytic theory ofboundary-integral operators to obtain asymptotic formulas for thestructure of transmission anomalies near non-robust slab modes; wepursue a development of the spectral theory for general open systems,such as leaky slabs, based on the concept of the unique conservativeextension of a dissipative system; and we investigate the existencetheory 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 selectedfrequencies and to trap energy, photonic crystals have become asubject of intense research in recent decades. Applications includehigh-efficiency lasers and filters, waveguides, and directionalantennas. Engineering of these devices relies on precise tuning ofresonant frequencies, transmission-vs.-frequency profiles, and theinteraction of open electromagnetic resonators with their surroundingmedium. This project aims to set the theoretical framework of theseresonant phenomena on a solid mathematical footing. The central themeis the frequency-response theory of open resonators, which providesthe framework for the guiding of light and other radiation bystructures, precise computation of the interaction of guided energywith applied sources, and optimization of resonant properties.Numerical computations and experimental results from the literaturewill guide the theoretical investigations. The project involves anundergraduate student and a graduate student as research assistants.The undergraduate has been assisting the PI in numerical simulationsand will be part of a collaboration with the computationalelectromagnetics group at Cal Tech. The graduate student will pursueher Ph.D. on the subject of resonant scattering by periodic pillars.