Direct and Inverse Electromagnetic Scattering Problems for Complex Periodic Media

纳米尺度周期结构与光的相互作用和散射是纳米光子学和超材料技术领域的一个重要课题。随着这一使能技术的迅速发展，对周期性复杂介质或周期性超材料光散射正问题和逆问题的数学理论和计算算法提出了更高的要求。本项目旨在开发此类理论和算法。该项目的成果将推进纳米光子学和超材料技术的知识，理解和成像技术。例如，为正问题发展的数学理论和计算算法对于光学器件的模拟、制造和维护是非常有用的。为逆问题开发的成像技术可以潜在地用于非破坏性测试，这有助于检测和表征光学元件和设备中的离散缺陷。该项目包括两个主要研究领域：直接散射问题的数学和数值分析，以及逆散射问题的高效反演算法的开发。更准确地说，对于直接问题，PI和他的同事将致力于以下主题：1）开发周期性复杂介质直接散射问题的体积积分方程公式; 2）开发直接散射问题的快速积分方程数值求解器; 3）研究周期性复杂介质的内部传输本征值; 4）研究手性超材料散射的适定性。本文提出的逆问题研究内容包括：1）发展周期性复杂介质成像的采样方法; 2）发展光子晶体局部缺陷检测的采样方法;第三章该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

The interaction and scattering of light by periodic structures at the nanoscale is a topic of great significance in the area of nanophotonics and metamaterials technology. With the current rapid development of this enabling technology, there is a high demand on new mathematical theories and computational algorithms for both direct and inverse problems of the light scattering by periodic complex media or periodic metamaterials. This project aims to develop such theories and algorithms. Results from this project will advance knowledge, understanding and imaging techniques in the technology of nanophotonics and metamaterials. For instance, the mathematical theories and the computational algorithms developed for the direct problems are extremely useful for the simulation, fabrication and maintaining of optical devices. The imaging techniques developed for the inverse problems can be potentially of practical use for non-destructive tests, which help to detect and characterize discrete flaws in optical components and devices.The project contains two main areas of study: the mathematical and numerical analysis for direct scattering problems, and the development of efficient inversion algorithms for inverse scattering problems. More precisely, for the direct problem, the PI and his colleagues will work on the following topics: 1) develop volume integral equation formulations for direct scattering problems for periodic complex media; 2) develop fast integral equation-based numerical solvers for the direct scattering problems; 3) study interior transmission eigenvalues for periodic complex media; 4) study well-posedness of the scattering by chiral metamaterials. The proposed research for the inverse problem includes the topics: 1) develop sampling methods for imaging of periodic complex media; 2) develop sampling methods for the detection of local defects in photonic crystals; 3) develop globally convergent methods for the identification of material parameters for photonic crystals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.