A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles

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SUMMARYWe consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above and below the structure. We discuss convergence and implementation of an optimization method for solving the inverse TE transmission problem, following an approach ÿrst developed by Kirsch and Kress for acoustic obstacle scattering. The convergence analysis includes the case of Lipschitz grating proÿles and relies on variational methods and solvability properties of periodic boundary integral equations. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm.

“…We refer to Reference [12] in the case of the inverse Dirichlet problem. To reduce computational e orts, we here propose a two-step procedure as in Reference [13].…”

Section: Implementation As a Two-step Methods And Numerical Resultsmentioning

confidence: 99%

“…Since the singular value decomposition of the ÿrst kind integral operators S ± b deÿned in (17) is known explicitly, the solutions ' ± can be represented as (cf. Reference [13])…”

Section: Implementation As a Two-step Methods And Numerical Resultsmentioning

confidence: 99%

“…Note that the Jacobi matrix J = (@r j =@p ) can be easily computed for several important classes of grating proÿles; we refer to Reference [13] for the perfectly re ecting case.…”

Section: Implementation As a Two-step Methods And Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The numerical solution of an inverse periodic transmission problem

Bruckner

Elschner

2004

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Uniqueness results in the inverse scattering problem for periodic structures

Yang

Zhang

2012

This paper is concerned with the inverse electromagnetic scattering by a 2D (impenetrable or penetrable) smooth periodic curve. Precisely, we establish global uniqueness results on the inverse problem of determining the grating profile from the scattered fields corresponding to a countably infinite number of quasiperiodic incident waves. For the case of an impenetrable and partially coated perfectly reflecting grating, we prove that the grating profile and its physical property can be uniquely determined from the scattered field measured above the periodic structure. For the case of a penetrable grating, we show that the periodic interface can be uniquely recovered by the scattered field measured only above the interface. A key ingredient in our proofs is a novel mixed reciprocity relation that is derived in this paper for the periodic structures and seems to be new.

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

Zhang

2012

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.