A linear sampling method for inverse problems of diffraction gratings of mixed type

Abstract

This paper is concerned with the direct and inverse problem of scattering of a time‐harmonic wave by a Lipschitz diffraction grating of mixed type. The scattering problem is modeled by the mixed boundary value problem for the Helmholtz equation in the unbounded half‐plane domain above a periodic Lipschitz surface on which a mixed Dirichlet and impedance boundary condition is imposed. We first establish the well‐posedness of the direct problem, employing the variational method, and then extend Isakov's method to prove uniqueness in determining the Lipschitz diffraction grating profile by using point sources lying above the structure. Finally, we develop a periodic version of the linear sampling method to reconstruct the diffraction grating. In this case, the far field equation defined on the unit circle is replaced by a near field equation defined on a line above the surface, which is a linear integral equation of the first kind. Numerical results are also presented to illustrate the efficiency of the method in the case when the height of the unknown grating profile is not very large and the noise level of the near field measurements is not very high. Copyright © 2012 John Wiley & Sons, Ltd.

Factorization Method for Electromagnetic Inverse Scattering from Biperiodic Structures

“…One can find recent developments of the linear sampling method in [10,11]. The linear sampling method has been extended to inverse scattering involving periodic media, in [15,16,33]. However, in spite of the advantages of the method, a full mathematical justification still remains open, see [10].…”

The linear sampling method for the inverse electromagnetic scattering by a partially coated bi-periodic structure

“…The numerical reconstruction results presented in [9] have shown the efficiency of the algorithm. The implementation of the above algorithm for the three-dimensional case of the full Maxwell equations is still in progress.…”

“…For F ×h ∈ B * := Y( D ) ×L 2 t ( I ), we are going to prove that F = 0,h= 0 under the assumption that <Hg, F ×h> B,B * = 0 for any g ∈ L 2 t ( b ). Recalling that the duality between Y( D ) and Y( D ) is defined by (9) and the duality between L 2 t ( I ) and L 2 t ( I ) is the L 2 scalar product, we have…”

“…The application of the linear sampling method to the inverse electromagnetic scattering problems can be found in [9,10,12]. Recently in [16], a periodic version of the linear sampling method was proposed and implemented for the two-dimensional TE polarization case of the inverse problem considered in this paper, where the Maxwell equations are replaced by the scalar Helmholtz equation and the boundary conditions on Γ D and Γ I are replaced with the Dirichlet and impedance conditions, respectively. In [17], Kirsch proposed a mathematically-justified version of the linear sampling method, the so-called factorization method.…”

A novel method in determining a layered periodic structure

“…There also exist some numerical methods in reconstructing periodic structures. For example, a linear sampling method was developed in [20,22] for determining the shape of partially coated bi-periodic structures, and in [35] a novel linear sampling method was introduced for simultaneously reconstructing dielectric grating structures in an inhomogeneous periodic medium. See also [10] for a finite element method or [3,4,17] for the factorization method in determining the periodic structures, or [30] for the uniquely reconstruction of a locally perturbed infinite plane.…”