Electromagnetic Excitation of a Spherical Medium by an Arbitrary Dipole and Related Inverse Problems

A piecewise homogeneous spherical medium is excited by an external or internal electric dipole with arbitrary location and polarization. The dyadic Green's function of the medium is determined analytically. Then, the vector electric fields and far‐field patterns are obtained. Low‐frequency approximations of the far‐field patterns are subsequently derived, which encode the dipole's locations coordinates and polarization components in the different orders of the associated expansions. This fact enables the establishment of far‐field inverse scattering algorithms referring to the electromagnetic interior or exterior excitation of a small sphere by an arbitrary dipole. Inverse medium and inverse source problems are considered concerning, respectively, the determination of the scatterer's material parameters and the dipole's characteristics. The developed inverse algorithms determine exactly the unknown parameters of problems fulfilling the low‐frequency assumption, which is indeed the case in most relevant applications, like, e.g., in biomedical imaging.     

Electromagnetic scattering theorems for interior dipole excitation of a layered obstacle

A layered scatterer is excited by a time‐harmonic spherical electromagnetic wave, generated by a dipole located either in the interior or in the exterior of the scatterer. The scatterer's core may be perfect conducting, impedance or dielectric. This paper initiates the investigation of scattering theorems corresponding to the excitation of a layered scatterer by a dipole in its interior. We establish reciprocity and general scattering theorems relating the total electric fields with the corresponding far‐field patterns. The optical theorem, relating the scattering cross‐section with the electric field in the layer containing the dipole, is recovered as a corollary of the general scattering theorem. Furthermore, for a scatterer excited by a spherical and a plane wave, mixed scattering theorems are derived. Besides, all the derived theorems recover known results, concerning the excitation of a homogeneous scatterer by an exterior dipole. We also analyze numerical implementations of the optical theorem in certain concrete scattering applications. Copyright © 2007 John Wiley & Sons, Ltd.     

Aspects of electromagnetic scattering in chiral media

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon‐type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon‐type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far‐field operator that is defined on a unit sphere and contains the far‐field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Scattering of electromagnetic plane waves by a spheroid uniformly moving in free space

We investigate the scattering process, generated by a plane electromagnetic field that is incident upon a moving perfectly conducting spheroid. An accurate treatment of the electromagnetic waves interaction with scatterers in uniform motion is based on the special relativity principle. In the object's frame the incident wave is assumed to have a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low‐frequency approximation method is applicable to the scattering problem. For the near electromagnetic field we obtain the zeroth‐order low‐frequency coefficients, while in the far field we calculate the leading terms for the scattering amplitude and scattering cross‐section. Finally, using the inverse Lorentz transform, we obtain the same approximations in the observer's frame. Copyright © 2006 John Wiley & Sons, Ltd.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

An enhanced finite difference time domain method for two dimensional Maxwell's equations

An enhanced finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric two-dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour-path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour-path method, the usual staircasing and the volume-average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access.     

A numerical study of electromagnetic waves in periodic waveguides

Here are considered time‐harmonic electromagnetic waves in a quadratic waveguide consisting of a periodic dielectric core enclosed by conducting walls. The permittivity function may be smooth or have jumps. The electromagnetic field is given by a magnetic vector potential in Lorenz gauge, and defined on a Floquet cell. The Helmholtz operator is approximated by a Chebyshev collocation, Fourier–Galerkin method. Laurent's rule and the inverse rule are employed for the representation of Fourier coefficients of products of functions. The computations yield, for known wavenumbers, values of the first few eigenfrequencies of the field. In general, the dispersion curves exhibit band gaps. Field patterns are identified as transverse electric, TE, transverse magnetic, TM, or hybrid modes. Maxwell's equations are fulfilled. A few trivial solutions appear when the permittivity varies in the guiding direction and across it. The results of the present method are consistent with exact results and with those obtained by a low‐order finite element software. The present method is more efficient than the low‐order finite element method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 490–513, 2014     

Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment

Electromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time-harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low-frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three-dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed-form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.     

A stability estimate for a solution to an inverse problem of electrodynamics

We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium. For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ³. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently large we estimate the conditional stability of the solution to this inverse problem.     

Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric

Consider the problem of scattering of electromagnetic waves by a doubly periodic Lipschitz structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure there is a perfect conductor with a partially coated dielectric boundary. We first establish the well‐posedness of the direct problem in a proper function space and then obtain a uniqueness result for the inverse problem by extending Isakov's method. Copyright © 2009 John Wiley & Sons, Ltd.     

The Atkinson-Wilcox expansion theorem for electromagnetic chiral waves

Consider the problem of scattering of a time-harmonic electromagnetic wave by a three-dimensional bounded and smooth obstacle. The infinite space outside the obstacle is filled by a homogeneous isotropic chiral medium. In the region exterior to a sphere that includes the scatterer, any solution of the generalized Helmholtz's equation that satisfies the Silver-Müller radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. The coefficients of the expansion can be determined from the leading coefficient, “the radiation pattern”, by a recurrence relation.     

A Stability Estimate for a Solution to a Three-Dimensional Inverse Problem for the Maxwell Equations

We consider the problem of determining dielectric permittivity and conductivity in the Maxwell equations. As additional information we prescribe the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindric domain. We establish a stability estimate for a solution to the inverse problem and a uniqueness theorem.     

The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem

The interior transmission problem appears naturally in the study of the inverse scattering problem of determining the shape of a penetrable medium from a knowledge of the time harmonic incident waves and the far field patterns of the scattered waves. We propose a variational study of this problem in the case of Maxwell's equations in an inhomogeneous anisotropic medium. Then we apply the obtained results to build an ‘extented far field’ operator and give a characterization of the medium from the knowledge of the range of this operator. We then show how the linear sampling method can be viewed as an approximation of this characterization. Copyright © 2004 John Wiley & Sons, Ltd.     

Uniqueness of inverse scattering problem for a penetrable obstacle with rigid core

In this paper, we discuss the inverse scattering problem for a penetrable obstacle with an impenetrable rigid core. Using a generalization of Schiffer's method to nonsmooth domains due to Ramm, we prove that the rigid core is uniquely determined by the far field patterns for a range of interior wavenumbers.     

Some results on electromagnetic transmission eigenvalues

The electromagnetic interior transmission problem is a boundary value problem, which is neither elliptic nor self-adjoint. The associated transmission eigenvalue problem has important applications in the inverse electromagnetic scattering theory for inhomogeneous media. In this paper, we show that, in general, there do not exist purely imaginary electromagnetic transmission eigenvalues. For constant index of refraction, we prove that it is uniquely determined by the smallest (real) transmission eigenvalue. Finally, we show that complex transmission eigenvalues must lie in a certain region in the complex plane. The result is verified by examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Remarks on the Born approximation and the Factorization Method

In this paper, we compare the far-field operators for the full nonlinear inverse scattering problem with the Born approximation as its linearization. The Factorization Method shows that both operators share the same behavior with respect to illposedness of the inverse problem. The results are derived for acoustic and electromagnetic scattering problems and the corresponding problem in electrical impedance tomography.     

An overfilled cavity problem for Maxwell's equations

This paper is concerned with the mathematical analysis of the scattering of a time‐harmonic electromagnetic plane wave by an open and overfilled cavity that is embedded in a perfect electrically conducting infinite ground plane, where the electromagnetic wave propagation is governed by the Maxwell equations. Above the flat ground surface and the open aperture of the cavity, the space is assumed to be filled with a homogeneous medium with a constant permittivity and permeability, whereas the interior of the cavity is filled with some inhomogeneous medium with a variable permittivity and permeability. The scattering problem is modeled as a boundary value problem over a bounded domain, with transparent boundary condition proposed on the hemisphere enclosing the inhomogeneity represented by the cavity. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. The perfectly matched layer (PML) method is investigated to truncate the unbounded electromagnetic cavity scattering problem. It is shown that the truncated PML problem attains a unique solution. An explicit error estimate is given between the solution of the original scattering problem and that of the truncated PML problem. The error estimate implies that the PML solution converges exponentially to the original cavity scattering problem by increasing either the PML medium parameter or the PML layer thickness. The convergence result is expected to be useful for determining the PML medium parameter in the computational electromagnetic scattering problem. Copyright © 2012 John Wiley & Sons, Ltd.     

A volume integral equation method for periodic scattering problems for anisotropic Maxwell's equations

This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.     

A Stability Estimate for a Solution of the Problem of Determining the Dielectric Permittivity

We consider the problem of determining the dielectric permittivity for a nonconducting and nonmagnetic medium. As information we take the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindrical domain. These traces correspond to a solution to some direct problem for the Maxwell system. The impulse source of the current flux lies outside the domain in which the coefficient is sought. The main result of the article is a stability estimate for a solution to the inverse problem in question.     

A mathematical formulation of the random phase approximation for crystals

This works extends the recent study on the dielectric permittivity of crystals within the Hartree model [E. Cancès, M. Lewin, Arch. Ration. Mech. Anal. 197 (1) (2010) 139–177] to the time-dependent setting. In particular, we prove the existence and uniqueness of the nonlinear Hartree dynamics (also called the random phase approximation in the physics literature), in a suitable functional space allowing to describe a local defect embedded in a perfect crystal. We also give a rigorous mathematical definition of the microscopic frequency-dependent polarization matrix, and derive the macroscopic Maxwell–Gauss equation for insulating and semiconducting crystals, from a first order approximation of the nonlinear Hartree model, by means of homogenization arguments.