Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers
Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers
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针对局部扰动无限周期层的散射问题的直接和逆求解器
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发表时间：
2017
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通讯作者：
Thi
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作者：
Thi
本文通过分析局部扰动周期无限大层在固定频率下的散射和逆散射问题而引起我们的兴趣。这个问题与光子结构、光纤、光栅等周期性介质的无损检测有关。我们首先分析了前向散射问题，并建立了一些不存在导模的条件。这种类型的条件很重要，因为它表明可以在结构上方的一层上进行测量，而不会丢失波传播部分的大量信息。在此基础上，提出了一种基于Floquet-Bloch变换的背景介质周期方向直接散射问题的数值求解方法。我们将问题均匀地离散在Floket-Bloch变量中，并在空间变量中使用谱方法。在空间上的离散化利用了网格中问题的体积重构(Lippmann-Schwinger积分方程)和在与周期垂直的方向上的核的周期化。后者允许使用FFT技术来加速矩阵向量乘积的迭代求解线性系统。最后得到一个耦合积分方程组，它可以用雅可比分解来求解。对有吸收的情况进行了收敛分析，并在二维情况下进行了数值验证。对于反问题，我们推广了三种采样方法来解决从与入射近场平面波相关的多基地数据中恢复缺陷的问题。我们分析了这些方法在Floquite-Bloch变量下的半离散化问题。然后，我们提出了一种新的方法，能够在既不知道背景材料属性也不知道缺陷属性的情况下直接恢复缺陷。我们提出的这种所谓的微分成像泛函是基于对单个Floquite-Bloch模的采样方法的分析，以及与解的关系，即所谓的内传输问题。理论研究与合成数据的数值实验结果相吻合。我们首先对标量波动方程进行分析，其中对比度是Helmholtz算符的低阶项。然后，我们将对主运算符中也存在对比度的情况进行扩展。我们用两个关于负折射率周期材料散射问题的分析结果来补充我们的论文。在对比度等于-1的情况下，我们在2D中证明了问题的适定性。在对比度不同于-1的情况下，我们还用T-矫顽力方法证明了问题的体势公式的Fredholm性。
We are interested in this thesis by the analysis of scattering and inverse scattering problems for locally perturbed periodic infinite layers at a fixed frequency. This problem has connexions with non destructive testings of periodic media like photonics structures, optical fibers, gratings, etc. We first analyze the forward scattering problem and establish some conditions under which there exist no guided modes. This type of conditions is important as it shows that measurements can be done on a layer above the structure without loosing substantial informations in the propagative part of the wave. We then propose a numerical method that solves the direct scattering problem based on Floquet-Bloch transform in the periodicity directions of the background media. We discretize the problem uniformly in the Floquet-Bloch variable and use a spectral method in the space variable. The discretization in space exploits a volumetric reformulation of the problem in a cell (Lippmann-Schwinger integral equation) and a periodization of the kernel in the direction orthogonal to the periodicity. The latter allows the use of FFT techniques to speed up Matrix-Vector product in an iterative to solve the linear system. One ends up with a system of coupled integral equations that can be solved using a Jacobi decomposition. The convergence analysis is done for the case with absorption and numerical validating results are conducted in 2D. For the inverse problem we extend the use of three sampling methods to solve the problem of retrieving the defect from the knowledge of mutistatic data associated with incident near field plane waves. We analyze these methods for the semi-discretized problem in the Floquet-Bloch variable. We then propose a new method capable of retrieving directly the defect without knowing either the background material properties nor the defect properties. This so-called differential-imaging functional that we propose is based on the analysis of sampling methods for a single Floquet-Bloch mode and the relation with solutions toso-called interior transmission problems. The theoretical investigations are corroborated with numerical experiments on synthetic data. Our analysis is done first for the scalar wave equation where the contrast is the lower order term of the Helmholtz operator. We then sketch the extension to the cases where the contrast is also present in the main operator. We complement our thesis with two results on the analysis of the scattering problem for periodic materials with negative indices. Weestablish the well posedness of the problem in 2D in the case of a contrast equals -1. We also show the Fredholm properties of the volume potential formulation of the problem using the T-coercivity approach in the case of a contrast different from -1.