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

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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.