Direct and Inverse Scattering Problems in Near-Field Optics Modeling

本提案的研究目标是研究数学问题并开发计算方法来解决近场光学建模中出现的重要类别的直接和逆散射问题。该方法是先处理正散射问题，再处理逆散射问题，并尽可能增加建模系统的复杂性。主要研究内容如下：(1)基于扫描隧道显微镜的全局模型，提出了一种基于后验误差估计误差控制的有限元与边界积分自适应耦合方法来解决直接问题；(2)将有限元与边界积分法的自适应耦合推广到电磁散射矢量理论；(3)开发自适应树码算法，加速边界积分计算，从而提供更有效的直接求解；(4)提出了一种新的求解定波数逆问题的延拓方法。所提出的研究将产生一套良好的建模和计算技术，适用于近场光学系统中各种实验配置的定性和定量研究。特别是，这些技术将有助于更好地理解近场光学中复杂的物理和数学问题，并为工业设计和制造新的光学器件提供有价值的信息。近场光学作为突破衍射极限、获得亚波长分辨率图像的有效手段，近年来得到了迅速发展，在生物、化学、材料科学、信息存储等现代科学技术领域有着广泛的应用。在越来越精确和逼真的数值模拟的指导下，近场光学显微镜的重大进展导致了光学器件的集成化和小型化，以及在复杂的光刻设计纳米结构附近进行了许多原始和可重复的测量。反过来，实际应用和科学发展推动了对严格的数学模型和分析的需求，以描述复杂结构的散射，准确计算电磁矢量场，从而预测给定结构在近场和纳米光学中的性能，以及进行新结构的优化设计。这项研究是数学、物理、工程和材料科学的交叉领域。它在推进应用数学和计算数学的前沿，以及发展新的数学和科学方面具有重要的潜力。

The research objective of this proposal is to examine mathematical issues and develop computational methods for solving important classes of direct and inverse scattering problems that arise in near-field optics modeling. The approach is to treat first the direct scattering problem and then the inverse scattering problem, and increase the complexity of the modeled system as far as possible. The proposed research concerns the following topics: (1) based on a global model for scanning tunneling microscopy, develop an adaptive coupling of the finite element and boundary integral method with error control by an a-posteriori error estimate to solve the direct problem; (2) extend the adaptive coupling of the finite element and boundary integral method to the vector theory of electromagnetic scattering; (3) develop an adaptive treecode algorithm to accelerate the boundary integral evaluations and thus to provide more efficient direct solvers; (4) develop a novel continuation method to solve the inverse problem at a fixed wavenumber. The proposed research will result in a suite of nice modeling and computational techniques, suitable for qualitative and quantitative study of various experimental configurations innear-field optical systems. Particularly, these techniques will contribute towards better understandings of the complex physical and mathematical problems in near-field optics, and provide valuable information for industry to design and fabricate new optical devices.Near-field optics has developed dramatically in recent years as an effective approach to breaking the diffraction limit and obtaining images with subwavelength resolution, which leads to vast applications in modern science and technology, including biology, chemistry, materials science, and information storage. Guided by the increasingly accurate and realistic numerical simulations, the significant advances of the near-field optical microscopies have led to integration and miniaturization of optical devices, and many original and reproducible measurements in the vicinity of complex lithographically designed nanostructures. Reciprocally, the practical applications and scientific developments have driven the need for rigorous mathematical models and analysis to describe the scattering of complicated structures, and to accurately compute electromagnetic vector fields and thus to predict the performance of a given structure in near-field and nano optics, as well as to carry out optimal design of new structures. The research lies at the interface of mathematics, physics, engineering, and materials science. It has significant potential for advancing the frontiers of applied and computational mathematics, and for evolving new mathematics and science.