Electromagnetic Resonance in Periodic Structures

奖项摘要，DMS-0505833，S·希普曼，路易斯安那州立大学题目：周期结构中的电磁共振本课题的动机是由有趣的电磁场散射和共振现象产生的大量分析问题，这些问题构成了一套用于光电子器件设计的基本工具。驱动现象是周期性结构材料的光子频率阻带的存在，称为光子晶体，这是其共振行为的温度来源。当前感兴趣的突出问题之一是源场通过共振频率附近的周期性平板的反常传输，对应于弱泄漏导板模式。我们利用边界积分算符的复解析理论，得到了非稳健平板模式附近传输反常结构的渐近公式，基于耗散系统唯一守恒延拓的概念，发展了一般开放系统的谱理论，如漏板；我们研究了模式的存在理论和相关共振现象的优化，以及有效的数值模拟方案。合作伙伴包括S.Venakdes，A.Figotin，D.Volkov，R.Lipton和O.Bruno。由于它们能够在选定的频率上阻挡电磁波并捕获能量，光子晶体在近几十年来已成为密集研究的主题。应用包括高效率激光器和滤光器、波导和定向天线。这些器件的工程设计依赖于谐振频率、传输与频率分布的精确调谐，以及开放式电磁谐振器与其周围介质的相互作用。该项目旨在将这些合理现象的理论框架建立在坚实的数学基础上。本文的中心内容是开放谐振腔的频率响应理论，它为光和其他辐射的结构导引、导能与外加光源相互作用的精确计算和谐振特性的优化提供了理论框架。数值计算和文献中的实验结果将指导理论研究。该项目涉及一名本科生和一名研究生作为研究助理。这名本科生一直在协助PI进行数值模拟，并将成为与加州理工大学计算电磁学小组合作的一部分。这位研究生将继续攻读周期性柱子共振散射的博士学位。

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.