Dinh-Liem Nguyen

Factorization Method for Electromagnetic Inverse Scattering from Biperiodic Structures

We investigate the Factorization method as an analytical as well as a numerical tool for solving inverse electromagnetic wave scattering problems from penetrable biperiodic structures in three dimensions. This method constructs a simple criterion whether a given point in space lies inside or outside the penetrable biperiodic structure, yielding a fast imaging algorithm. The required data consists of tangential components of Rayleigh sequences corresponding to (measured) scattered electromagnetic fields. In our setting, the incident electromagnetic fields causing these scattered waves are plane incident electromagnetic waves. We propose, on the one hand, a rigorous analysis for the Factorization method in this electromagnetic plane wave setting, building upon existing results for the method in the context of inverse electromagnetic scattering from bounded objects and of scalar periodic inverse scattering problems. On the other hand, we provide, to the best of our knowledge, the first three-dimensional nume...

Numerical solution of a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm

Abstract We analyze in this paper the performance of a newly developed globally convergent numerical method for a coefficient inverse problem for the case of multi-frequency experimental backscatter data associated to a single incident wave. These data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. The challenges for the inverse problem under the consideration are not only from its high nonlinearity and severe ill-posedness but also from the facts that the amount of the measured data is minimal and that these raw data are contaminated by a significant amount of noise, due to a non-ideal experimental setup. This setup is motivated by our target application in detecting and identifying explosives. We show in this paper how the raw data can be preprocessed and successfully inverted using our inversion method. More precisely, we are able to reconstruct the dielectric constants and the locations of the scattering objects with a good accuracy, without using any advanced a priori knowledge of their physical and geometrical properties.

Scattering of Herglotz waves from periodic structures and mapping properties of the Bloch transform

When an incident Herglotz wave function scatteres from a periodic Lipschitz continuous surface with Dirichlet boundary condition, then the classical (quasi-)periodic solution theory for scattering from periodic structures does not apply since the incident field lacks periodicity. Relying on the Bloch transform, we provide a solution theory in H for this scattering problem: We first prove conditions guaranteeing that incident Herglotz wave functions propagating towards the periodic structure have traces in H on the periodic surface. Second, we show that the solution to the scattering problem can be decomposed by the Bloch transform into its periodic components that solve a periodic scattering problem. Third, these periodic solutions yield an equivalent characterization of the solution to the original non-periodic scattering problem, which allows, for instance, to prove new characterizations of the Rayleigh coefficients of each of the periodic components. A corollary of our results is that under the conditions mentioned above the operator mapping densities to the restriction of their Herglotz wave function on the periodic surface is always injective; this result generally fails for bounded surfaces.

Volume integral equations for scattering from anisotropic diffraction gratings

We analyze electromagnetic scattering of transverse magnetic polarized waves from a diffraction grating consisting of a periodic, anisotropic, and possibly negative index dielectric material. Such scattering problems are important for the modelization of, for example, light propagation in nano-optical components and metamaterials. The periodic scattering problem can be reformulated as a strongly singular volume integral equation, a technique that attracts continuous interest in the engineering community but has rarely received rigorous theoretic treatment. In this paper, we prove new (generalized) Garding inequalities in weighted and unweighted Sobolev spaces for the strongly singular integral equation. These inequalities also hold for materials for which the real part of the material parameter takes negative values inside the diffraction grating, independently of the value of the imaginary part. Copyright

Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method

Abstract This paper is concerned with the numerical solution to a three-dimensional coefficient inverse problem for buried objects with multi-frequency experimental data. The measured data, which are associated with a single direction of an incident plane wave, are backscatter data for targets buried in a sandbox. These raw scattering data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. We develop a data preprocessing procedure and exploit a newly developed globally convergent inversion method for solving the inverse problem with these preprocessed data. It is shown that dielectric constants of the buried targets as well as their locations are reconstructed with a very good accuracy. We also prove a new analytical result which rigorously justifies an important step of the so-called “data propagation” procedure.

Shape identification of anisotropic diffraction gratings for TM-polarized electromagnetic waves

This paper concerns the shape identification problem of anisotropic periodic structures which are known as diffraction gratings. We study the so-called Factorization method as a tool for reconstructing the anisotropic periodic media from measured spectral data involving scattered electromagnetic waves in TM modes. This sampling method provides a simple criterion to compute a picture of shape of diffraction gratings in a rapid way. We propose a rigorous analysis for the method as well as numerical experiments to examine its performance.