International audienceThis paper is dedicated to the design and analysis of sampling methods to reconstruct the shape of a local perturbation in a periodic layer from measurements of scattered waves at a fixed frequency. We first introduce the model problem that corresponds with the semi-discretized version of the continous model with respect to the Floquet-Bloch variable. We then present the inverse problem setting where (propagative and evanescent) plane waves are used to illuminate the structure and measurements of the scattered wave at a parallel plane to the periodicity directions are performed. We introduce the near field operator and analyze two possible factorizations of this operator. We then establish sampling methods to identify the defect and the periodic background geometry from this operator measurement. We also show how one can recover the geometry of the background independently from the defect. We then introduce and analyze the single Floquet-Bloch mode measurement operators and show how one can exploit them to built an indicator function of the defect independently from the background geometry. Numerical validating results are provided for simple and complex backgrounds
International audienceWe investigate the scattering problem for the case of locally perturbedperiodic layers in $\R^d$, $d=2,3$. Using the Floquet-Bloch transform in theperiodicity direction we reformulate this scattering problem as an equivalentsystem of coupled volume integral equations. We then apply a spectral method todiscretize the obtained system after periodization in the direction orthogonalto the periodicity directions of the medium. The convergence of this method is established and validating numerical results are provided
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