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Near-field imaging of locally perturbed periodic surfaces
This paper concerns the inverse scattering problem to reconstruct a locally
perturbed periodic surface. Different from scattering problems with
quasi-periodic incident fields and periodic surfaces, the scattered fields are
no longer quasi-periodic. Thus the classical method for quasi-periodic
scattering problems no longer works. In this paper, we apply a Floquet-Bloch
transform based numerical method to reconstruct both the unknown periodic part
and the unknown local perturbation from the near-field data.
By transforming the original scattering problem into one defined in an
infinite rectangle, the information of the surface is included in the
coefficients. The numerical scheme contains two steps. The first step is to
obtain an initial guess, i.e., the locations of both the periodic surfaces and
the local perturbations, from a sampling method. The second step is to
reconstruct the surface. As is proved in this paper, for some incident fields,
the corresponding scattered fields carry little information of the
perturbation. In this case, we use this scattered field to reconstruct the
periodic surface. Then we could apply the data that carries more information of
the perturbation to reconstruct the local perturbation. The Newton-CG method is
applied to solve the associated optimization problems. Numerical examples are
given at the end of this paper to show the efficiency of the numerical method
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