US-China-Germany Planning Visits: Direct and Inverse Scattering Methods for Periodic Structures with Arbitrary Profiles and Defects

摘要具有任意轮廓和缺陷的周期结构的直接散射法和逆散射法在太阳能、半导体、光子晶体、医学成像和衍射光栅等领域有着广泛的应用。在太阳能中，太阳能集热器是周期性的阵列，衍射光栅的表面由周期性的沟槽阵列组成。光波导有许多相同结构的单元。拟议的国际合作的主要主题是为具有任意轮廓和缺陷的周期结构开发快速而严格的数值模拟和逆散射技术。周期结构的研究对周期结构的分析、设计和优化具有重要意义，一直是许多数学家、科学家和工程师的研究热点。所提出的研究活动将导致具有任意轮廓和埋入缺陷的周期结构的正反向散射问题的计算方法和数学理论的重要进展。研究成果将为物理学家和工程技术人员提供有用的工具来处理实际问题，如入射角的影响和衍射光栅的遮挡现象。研究的第一个目标是发展有效和精确的数值方法来计算周期结构和带有局部缺陷的双周期结构中的波传播。由于缺陷引起的散射场扰动部分，经典的方法，如微分法，一般不起作用。基于极限吸收原理，即无吸收的周期结构的解是吸收参数趋于零时解的唯一极限，PI和他的合作者提出了一种有效的递归倍增技术。由于该方法可以在超级计算机上进一步加速，因此对于具有大量相同单元的周期结构的实际工业应用是现实的。周期结构中紧致不均匀的检测一直是一个具有挑战性的问题，这是许多工程师感兴趣的。例如，存储芯片和LCD电极的缺陷多年来一直是一个严重的问题。高效、有效的逆散射检测/重构缺陷算法具有重要的实用价值。作为第二个目标，PI计划开发有效的定性方法来重建周期性结构中的缺陷，定性方法避免了错误的模型假设，并且可以提供极快的重建速度。据PI所知，这项研究是逆散射领域首次尝试有效地重建具有任意几何形状和物理性质的周期结构中的缺陷。

AbstractDirect and Inverse Scattering Methods for Periodic Structures with Arbitrary Profiles and DefectsPeriodic structures have many applications, such as solar energy, semiconductors, photonic crystals, medical imaging, and diffraction gratings. In solar energy, the solar thermal collectors are periodic arrays.The surface of a diffraction grating is made up with a periodic array of grooves. Optical waveguides have many cells of the same structure. The main subject of the proposed international collaboration is to develop fast and rigorous numerical simulations and inverse scattering techniques for periodic structures with arbitrary profiles and defects. The research is important to the analysis, design, and optimization of periodic structures and has been an active area for many mathematicians, scientists and engineers. The proposed research activities will lead to important progress for computational methods and mathematical theory for direct and inverse scattering problems for periodic structures with arbitrary profiles and imbedded defects. The research outcome will provide physicists and engineers useful tools to treat practical problemssuch as effect of incidence angle and the shadowing phenomenon of diffraction gratings. The first goal of the proposed research is to develop efficient and accurate numerical methods for wave propagation in periodic structures and bi-periodic structures with local defects. Due to the induced perturbed part of the scattered field by the defects, classic methods, such as the differential method, do not work in general. Based on the Limiting Absorption Principle, i.e., the solution for the periodic structure without absorption is the unique limit of the solutions as the absorption parameter tends to zero, the PI and his collaborators propose to employ an effective recursive doubling technique. Since the method can be further accelerated using super computers, it is realistic for practical industry applications with legions of identical cells for periodic structures.Detection of a compact inhomogeneity in a periodic structure has been a challenging problem, which is of interest by many engineers. For example, defects in memory chips and LCD electrodes have been a serious problem for years. Efficient and effective inverse scattering algorithms to detect/reconstruct defects are of great practical value. As the second goal, the PI plans to develop effective qualitative methods for the reconstruction of defects embedded in periodic structures.The qualitative methods avoid incorrect model assumptions and can provide extremely fast reconstruction. To the PI's knowledge, the proposed research is one the first attempts in the inverse scattering communityto effectively reconstruct defects in periodic structures with arbitrary geometric profiles and physical properties.