Inverse diffraction by a doubly periodic structure

Let us consider the scattering of electromagnetic waves by a doubly periodic structure. Above the structure, the medium is assumed to be homogeneous with a fixed real dielectric coefficient. The medium is a perfect conductor below the structure. For a given incident plane wave, the tangential electric field is measured away from the structure. An inverse problem arises: To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of an eigenvalue problem.     

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

In this paper, we consider the inverse problem of recovering a doubly periodic Lipschitz structure through the measurement of the scattered field above the structure produced by point sources lying above the structure. The medium above the structure is assumed to be homogeneous and lossless with a positive dielectric coefficient. Below the structure is a perfect conductor partially coated with a dielectric. A periodic version of the linear sampling method is developed to reconstruct the doubly periodic structure using the near field data. In this case, the far field equation defined on the unit ball of ?³ is replaced by the near field equation which is a linear integral equation of the first kind defined on a plane above the periodic surface. Copyright © 2010 John Wiley & Sons, Ltd.     

On the scattering by a biperiodic structure

Consider scattering of electromagnetic waves by a nonmagnetic biperiodic structure. The structure separates the whole space into three regions: above and below the structure the medium is assumed to be homogeneous. Inside the structure, the medium is assumed to be defined by a bounded measurable dielectric coefficient. Given the structure and a time-harmonic electromagnetic plane wave incident on the structure, the scattering (diffraction) problem is to predict the field distributions away from the structure. In this note, the problem is reduced to a bounded domain and solved by a variational method. The main result establishes existence and uniqueness of the weak solutions in .

     

On the Existence of an Infinite Number of Leaky Complex Waves in a Dielectric Layer

Doklady Mathematics - The problem of propagation of TE-polarized electromagnetic waves in a dielectric layer is considered. The waves existing in the structure under study are classified. It is...     

Insulated antenna in a layered medium

The article presents a generalization of the integral equation of an insulated linear antenna immersed in a cylindrically layered lossy dielectric medium. The insulation is provided by a lossless dielectric layer. The kernel of the integral equation is represented as a superposition of the fundamental solutions of the wave equation with equivalent propagation constants for the given media. A generalization to a plane-layered medium is proposed. The problem of a vertical radiator above a layered half-space is considered. Translated from Chislennye Metody v Matematicheskoi Fiziki, Published by Moscow University, Moscow, 1996, pp. 80–88.     

A hybrid method for numerical solution of Poisson’s equation in a domain with a dielectric corner

An electrostatic problem of determining a potential in a domain containing an incoming dielectric corner, which reduces to solving Poisson’s equation in this domain, is considered. A specific feature of the solution of this problem is that it is bounded in a neighborhood of the dielectric corner but its gradient increases without limit. An efficient hybrid algorithm for the numerical solution of the problem, based on the finite element method and taking into account the known asymptotic representation of the solution in the neighborhood of the dielectric corner, is proposed.     

Diffraction of an electromagnetic signal on an ideally conducting cone with a dielectric coating

The paper examines the scattering of a Gaussian video signal by an ideally conducting circular cone coated with a dielectric layer. The problem is solved by the method of integral equations for monochromatic waves at low and resonance frequencies with subsequent Fourier transformation to the time domain. The response is investigated with signals propagating along the symmetry axis from the direction of the apex and from the direction of the base for dielectric layers of various permittivities. Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 46–52, 1998.     

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.     

Temperature fields in a semiconductor film heated by a laser beam

We consider the heating of a thin semiconductor film on a dielectric substrate by a laser beam. The problem is solved using an implicit symmetric difference scheme. The results of a numerical experiment using real data are presented.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 53–59, 1986.     

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.     

Homogenization of the 3D Maxwell system near resonances and artificial magnetism

It is now well known that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a possibly negative frequency dependent effective permeability. However this result based on a two-dimensional micro resonator problem on the section of the fibers holds merely in the case of polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this Note we propose a full 3D extension of previous asymptotic analysis based on a new averaging method for the magnetic field. We evidence a vectorial spectral problem on the periodic cell which accounts for micro-resonance effects and leads to a 3D negative effective permeability tensor. This suggests that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry. To cite this article: G. Bouchitté et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

Resolving Operators for a Three-Dimensional Periodic Waveguide

We construct the solving operators of a three-dimensional periodic dielectric waveguide with absorption. The results obtained can be applied to the problem describing a dropping wave onto the interface of two periodic dielectric waveguides with absorption. Bibliography: 6 titles.     

On the Diffraction by Biperiodic Anisotropic Structures

This article studies the scattering of electromagnetic waves by a nonmagnetic biperiodic structure. The structure consists of anisotropic optical materials and separates two regions with constant dielectric coefficients. The time harmonic Maxwell equations are transformed to an equivalent strongly elliptic variational problem for the magnetic field in a bounded biperiodic cell with nonlocal boundary conditions. This guarantees the existence of quasiperiodic magnetic fields in H 1 and electric fields in H (curl) solving Maxwell's equations. The uniqueness is proved for all frequencies excluding possibly a discrete set. The analytic dependence of these solutions on frequency and incident angles is studied.     

On domain switching in deformable ferroelectrics, seen as continua with microstructure

We obtain a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by a spontaneous polarization. This is accomplished by assuming the body as comprised of a continuum with vectorial microstructure: in each point of the body therefore a gross and a fine structure are superposed, the gross structure representing a non linear polarizable elastic body and the vectorial fine structure describing the spontaneous polarization.¶Among the distinctive features of ferroelectric materials, the most interesting is represented by the organization of spontaneous polarization into a domain structure, which minimizes electrostatic energy and which can be modified by the application of electric and deformation fields. This process, called "polarization reversal" or "domain switching", is associated with various hysteresis loops, the most typical being those between spontaneous polarization and electric field (dielectric hysteresis), and between strain and electric field ("butterfly" loop).¶From the balance laws of continua with vectorial microstructure and dissipation inequality we arrive at the evolution equation for the spontaneous polarization which allows for both inertial and dissipative terms and describes domain switching. We postulate a simple interaction mechanism between the spontaneous polarization and the pair electric field, deformation and arrive at, in the quasi-static case, to a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics.¶For linearized kinematics we study, in the one-dimensional case, stable relative minimizers and give a simple justification for dielectric hysteresis and butterfly loops: under the hypothesis that the domain walls are sharp interfaces, the solutions we find explain the banded twin domains morphology which is typical of many ferroelectrics.     

High contrast microwave tomography using topology optimization techniques

Microwave tomography for medical applications leads to a difficult reconstruction problem for the dielectric properties of biological tissue due to strongly diffracting waves in combination with large dielectric contrasts. We apply the material distribution technique used for topology optimization of elastic structures in order to solve the nonlinear least-squares problem underlying the reconstruction problem. Using simulated numerical data with an approximate signal-to-noise ratio of 40 dB and geometrical a priori information on the unknown objects, we obtain good estimates of the dielectric properties corresponding to biological objects.     

Closed-form solutions for two collinear dielectric cracks in a magnetoelectroelastic solid

The problem of two collinear electromagnetically dielectric cracks in a magnetoelectroelastic material is investigated under in-plane electro-magneto-mechanical loadings. The semi-permeable crack-face boundary conditions are adopted to simulate the case of two collinear cracks full of a dielectric interior. Utilizing the Fourier transform technique, the boundary-value problem is reduced to solving singular integral equations with Cauchy kernel, which then are solved explicitly. The intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD) and the energy release rates near the inner and outer crack tips are determined in closed forms for two cases of possible far-field electro-magneto-mechanical loadings respectively. Numerical results for a BaTiO₃–CoFe₂O₄ composite are carried out to show the effects of applied mechanical loadings on the crack-face electric displacement and magnetic induction, the stress intensity factor and the COD intensity factor, respectively. The obtained results reveal that when the applied mechanical loading is stress, applied electromagnetic loadings have no influences on the stress intensity factor. When the applied mechanical loadings is train, the applied positive electromagnetic loadings decrease the intensity factors of stress and COD, and the applied negative ones increase the intensity factors of stress and COD. The variations of energy release rates are also given to show the effects of the geometry of two collinear dielectric cracks.     

Axisymmetric excitation of a dielectric torus: Numerical solution

The article considers the mathematical model of a lens antenna in the shape of a homogeneous dielectric torus excited by an axial magnetic dipole. The boundary-value problem for an elliptical differential equation is posed and reduced by the fundamental solution method to an integral equation of 2nd kind over the torus section. A numerical algorithm is proposed, and the numerical results are compared with experimental findings. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 109–115.     

A nonlinear boundary eigenvalue problem for TM-polarized electromagnetic waves in a nonlinear layer

We consider the propagation of TM-polarized electromagnetic waves in a nonlinear dielectric layer located between two linear media. The nonlinearity in the layer is described by the Kerr law. We reduce the problem to a nonlinear boundary eigenvalue problem for a system of ordinary differential equations. We obtain a dispersion relation and a first approximation for eigenvalues of the problem. We compare the results with those obtained for the case of a linear medium in the layer.     

Microscopic model of a non-Debye dielectric relaxation: The Cole-Cole law and its generalization

Based on a self-similar spatial-temporal structure of the relaxation process, we construct a microscopic model for a non-Debye (nonexponential) dielectric relaxation in complex systems. In this model, we derive the Cole-Cole expression for the complex dielectric permittivity and show that the exponent ?? involved in that expression is equal to the fractal dimension of the spatial-temporal self-similar ensemble characterizing the structure of the medium and the relaxation process occurring in it. We find a relation between the macroscopic relaxation time and the micro- and mesoparameters of the system. We obtain a generalized Cole-Cole expression for the complex dielectric permittivity involving log-periodic corrections that occur because of a discrete scaling invariance of the fractal structure generating the relaxation process on the mesoscopic scale. The found expression for the dielectric permittivity can be used to interpret dielectric spectra in disordered dielectrics.