Time-harmonic Electromagnetic Waves Research Articles

A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering

We consider an inverse scattering problem for time-harmonic acoustic or electromagnetic waves with sparse multifrequency far field data-sets. The goal is to localize several small penetrable objects embedded inside an otherwise homogeneous background medium from observations of far fields of scattered waves corresponding to incident plane waves with one fixed incident direction but several different frequencies. We assume that the far field is measured at a few observation directions only. Taking advantage of the smallness of the scatterers with respect to wavelength we utilize an asymptotic representation formula for the far field to design and analyze a MUSIC-type reconstruction method for this setup. We establish lower bounds on the number of frequencies and receiver directions that are required to recover the number and the positions of an ensemble of scatterers from the given measurements. Furthermore we briefly sketch a possible application of the reconstruction method to the practically relevant case of multifrequency backscattering data. Numerical examples are presented to document the potentials and limitations of this approach.

Uncertainty Principles for Three-Dimensional Inverse Source Problems

Novel uncertainty principles for far fields of time-harmonic acoustic or electromagnetic waves radiated by collections of well-separated localized sources in two-dimensional free space have recently been established in [R. Griesmaier and J. Sylvester, SIAM J. Appl. Math., 77 (2017), pp. 154--180]. These uncertainty principles have been utilized to develop stability estimates and reconstruction algorithms for the recovery of the far field radiated by each of the individual sources, and the simultaneous restoration of missing data segments. In this paper, we present generalizations of these uncertainty principles for a relevant three-dimensional setting. We consider extensions of the reconstruction schemes for far field splitting and data completion, including their stability analysis, and we discuss the sharpness of our results in the three-dimensional case. We briefly comment on the numerical implementation of the reconstruction algorithms and include a numerical example.

A Factorization Method for Multifrequency Inverse Source Problems with Sparse Far Field Measurements

We consider a multifrequency inverse source problem for time-harmonic acoustic or electromagnetic waves with a limited set of far field data. Assuming that far field measurements of the wave radiat...

Uncertainty Principles for Inverse Source Problems, Far Field Splitting, and Data Completion

Starting with far field data of time-harmonic acoustic or electromagnetic waves radiated by a collection of compactly supported sources in two-dimensional free space, we develop criteria and algorithms for the recovery of the far field components radiated by each of the individual sources, and the simultaneous restoration of missing data segments. Although both parts of this inverse problem are severely ill-conditioned in general, we give precise conditions relating the wavelength, the diameters of the supports of the individual source components and the distances between them, and the size of the missing data segments, which guarantee that stable recovery in the presence of noise is possible. The only additional requirement is that a priori information on the approximate location of the individual sources is available. We give analytic and numerical examples to confirm the sharpness of our results and to illustrate the performance of corresponding reconstruction algorithms, and we discuss consequences for stability and resolution in inverse source and inverse scattering problems.

Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to $0$. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.

Topological Sensitivity for Solving Inverse Multiple Scattering Problems in Three-dimensional Electromagnetism. Part I: One Step Method

In this paper we compute closed-form expressions for the topological derivative for three-dimensional time-harmonic electromagnetic waves for perfect conductors (Dirichlet condition), electromagnet...

A Galerkin approximation for integro-differential equations in electromagnetic scattering from a chiral medium

We consider the scattering of time-harmonic electromagnetic waves from a chiral medium. It is known for the Drude–Born–Fedorov model that the forward scattering problem can be described by an integro-differential equation. In this paper we study a Galerkin finite element approximation for this integro-differential equation. Our Galerkin scheme, which relies on a suitable periodization of the integral equation, enables the use of the fast Fourier transform and a simple numerical implementation. We establish a quasi-optimal convergence analysis for the Galerkin method. Explicit formulas for the discrete scheme are also provided.

Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems

This paper deals with the material derivative analysis of the boundary integral operators arising from the scattering theory of time-harmonic electromagnetic waves and its application to inverse problems. We present new results using the Piola transform of the boundary parametrisation to transport the integral operators on a fixed reference boundary. The transported integral operators are infinitely differentiable with respect to the parametrisations and simplified expressions of the material derivatives are obtained. Using these results, we extend a nonlinear integral equations approach developed for solving acoustic inverse obstacle scattering problems to electromagnetism. The inverse problem is formulated as a pair of nonlinear and ill-posed integral equations for the unknown boundary representing the boundary condition and the measurements, for which the iteratively regularized Gauss-Newton method can be applied. The algorithm has the interesting feature that it avoids the numerous numerical solution of boundary value problems at each iteration step. Numerical experiments are presented in the special case of star-shaped obstacles.

On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients

We investigate in a 2D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non-dissipative bounded and penetrable obstacle with a negative permittivity. Using the T-coercivity approach, we first prove that the problem is well-posed in the classical framework Hloc1 if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods.

Vector problem of electromagnetic wave diffraction by a system of inhomogeneous volume bodies, thin screens, and wire antennas

The vector problem of time-harmonic electromagnetic wave diffraction by a system of non-intersecting solid inhomogeneous bodies, infinitely thin perfectly conducting screens and wire antennas in the semi-classical formulation is considered. The original boundary value problem for Maxwell’s equations is reduced to a new system of singular integral equations over the volume domains, the screen surfaces and antennas, that is, over compact domains of dimension 3, 2, and 1. To obtain the equations, the fields scattererd by the bodies and the screens were represented via volume and surface potentials, respectively. To solve the integral equations approximately, the Galerkin method is applied; basis functions on the body, the screens and antennas are introduced as well as formulas for matrix elements in Galerkin method. Combining potential theory and pseudodifferential calculus allowed to obtain results on solvability of new system of integro-differential equations, as well as to theoretically justify the Galerkin method. To solve the problem of diffraction by obstacles of complex shape, novel subhierarchical approach is applied.

GetDDM: An open framework for testing optimized Schwarz methods for time-harmonic wave problems

We present an open finite element framework, called GetDDM, for testing optimized Schwarz domain decomposition techniques for time-harmonic wave problems. After a review of Schwarz domain decomposition methods and associated transmission conditions, we discuss the implementation, based on the open source software GetDP and Gmsh. The solver, along with ready-to-use examples for Helmholtz and Maxwell’s equations, is freely available online for further testing. Program summaryProgram title: GetDDMCatalogue identifier: AEZZ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZZ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU GPL v2No. of lines in distributed program, including test data, etc.: 1426800No. of bytes in distributed program, including test data, etc.: 12362781Distribution format: tar.gzProgramming language: Gmsh (http://gmsh.info) and GetDP (http://getdp.info).Computer: PC, Mac, Tablets, Computer clusters.Operating system: Linux, Windows, MacOSX.Has the code been vectorized or parallelized?: YesRAM: From 512 Megabytes upwardsClassification: 4.3, 4.12, 6.5, 10.Nature of problem: Computing the solution of large scale time-harmonic acoustic and electromagnetic wave problems.Solution method: Finite element method with optimized Schwarz domain decomposition method.Running time: From a few seconds for simple problems to several days for large-scale simulations.

Domain Decomposition Methods for Time-Harmonic Electromagnetic Waves With High-Order Whitney Forms

Classically, the domain decomposition methods (DDMs) for time-harmonic electromagnetic wave propagation problems make use of the standard, low-order, Nedelec basis functions. This paper analyzes the convergence rate of DDM when higher order finite elements are used for both volume and interface discretizations, in particular when different orders are used in the volume and on the interfaces.

Near-field imaging of periodic interfaces in multilayered media

This paper is concerned with the inverse problem of scattering of time-harmonic electromagnetic waves by a penetrable multilayered periodic structure. The structure separates the whole space into three regions: the medium above and below the structure is assumed to be homogeneous but with different wave numbers, and the medium inside the structure is assumed to be inhomogeneous characterized by the refractive index. We prove, under certain conditions, that the factorization method can be used to reconstruct the upper interface from the scattered near-field data measured only above the structure and generated by a countably infinite number of downward propagating incident waves merely from the top region, leading to a fast imaging algorithm. A similar result can be obtained for reconstructing the lower interface from the scattered near-field data measured only below the structure, corresponding to a countably infinite number of upward propagating incident waves also merely from the bottom region. Thus, our approach is applicable, even if one merely has access to the object under investigation from one side. Finally, numerical examples are presented to illustrate the effectiveness of the inversion algorithm.

The direct scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder

In this paper we consider the direct scattering problem of obliquely incident time-harmonic electromagnetic plane waves by an infinitely long dielectric cylinder. We assume that the cylinder and the outer medium are homogeneous and isotropic. From the symmetry of the problem, Maxwell's equations are reduced to a system of two 2D Helmholtz equations in the cylinder and two 2D Helmholtz equations in the exterior domain where the fields are coupled on the boundary. We prove uniqueness and existence of this differential system by formulating an equivalent system of integral equations using the direct method. We transform this system into a Fredholm type system of boundary integral equations in a Sobolev space setting. To handle the hypersingular operators we take advantage of Maue's formula. Applying a collocation method we derive an efficient numerical scheme and provide accurate numerical results using as test cases transmission problems corresponding to analytic fields derived from fundamental solutions.

Optimized Schwarz algorithms for solving time-harmonic Maxwell’s equations discretized by a hybridizable discontinuous Galerkin method

This work is concerned with the development of numerical methods for the simulation of time-harmonic electromagnetic wave propagation problems. A hybridizable discontinuous Galerkin (HDG) method is adopted for the discretization of the two-dimensional time-harmonic Maxwell’s equations on a triangular mesh. A distinguishing feature of the present work is that this discretization method is employed at the subdomain level in the framework of a Schwarz-type domain decomposition algorithm (DDM). We show that HDG method naturally couples with a Schwarz method relying on optimized transmission conditions. The presented numerical results show the effectiveness of the optimized DDM-HDG method.

Volume integral equations for electromagnetic scattering in two dimensions

We study the strongly singular volume integral equation that describes the scattering of time-harmonic electromagnetic waves by a penetrable obstacle. We consider the case of a cylindrical obstacle and fields invariant along the axis of the cylinder, which allows the reduction to two-dimensional problems. With this simplification, we can refine the analysis of the essential spectrum of the volume integral operator started in a previous paper (Costabel et al., 2012) and obtain results for non-smooth domains that were previously available only for smooth domains. It turns out that in the transverse electric (TE) case, the magnetic contrast has no influence on the Fredholm properties of the problem. As a byproduct of the choice that exists between a vectorial and a scalar volume integral equation, we discover new results about the symmetry of the spectrum of the double layer boundary integral operator on Lipschitz domains.

Electromagnetic scattering by periodic structures with sign-changing coefficients

We analyze the well-posedness of a scattering problem of time-harmonic electromagnetic waves by periodic structures with sign-changing coefficients. Transmission problems for Maxwell's equations with sign-changing coefficients in bounded domains have been recently studied by Bonnet-Ben Dhia and co-workers in the so-called T-coercivity framework. In this article, we generalize such a framework for periodic scattering problems relying on an integral equation approach. The periodic scattering problem is formulated by a hypersingular integral equation of Lipmann–Schwinger type. We prove that the integral equation satisfies a Gårding-type estimate, which allows us to establish the well-posedness of the problem in the sense of Fredholm.

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane. A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases. Given the incident field, the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity. In this paper, both the direct and inverse scattering problems are discussed based on a symmetric coupling method. Variational formulations for the direct scattering problem are presented, existence and uniqueness of weak solutions are studied, and the domain derivatives of the field with respect to the cavity shape are derived. Uniqueness and local stability results are established in terms of the inverse problem.

Electromagnetic scattering from a topological insulator cylinder placed in chiral medium

We study the scattering of time harmonic electromagnetic plane wave from a topological insulator circular cylinder in chiral medium. Variation of scattering behavior for different values of radii and permittivity of the cylinder is noted when time reversal symmetry is protected and we also studied the scattering behavior when time reversal symmetry is broken. Although scattering behavior of co-polarized component of field is not significantly different from chiral cylinder in chiral medium yet it is found that the same, when time reversal symmetry broken, is similar to scattering from a perfect conductors. Scattering behaviour of cross-polarized field component of topological cylinder in chiral medium is remarkably different from chiral cylinder i.e. it has stronger scattering possibly due to the presence of conducting states near the surface. Moreover, it is noted that the scattering in forward direction for cross-polarized and co-polarized components are not correlated with permittivity but for carefully selected values of permittivity, we can control the scattering behavior of topological insulator circular cylinder.

Variational approach to scattering by unbounded rough surfaces with Neumann and generalized impedance boundary conditions

This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an innite penetrable medium with a certain non-trapping and geometric conditions.