Spreading Waves in a Three-Dimensional Dielectric Periodic Waveguide Described by the Generalized Reduced Wave Equation

The boundedness of the resolvent of an operator similar to the wave operator is proved in the case of the three-dimensional periodic dielectric waveguide with absorption. Bibliography: 4 titles.     

Projection Operators of a Three-Dimensional Periodic Waveguide

We construct the projection operators onto invariant subspaces of a 3-dimensional periodic dielectric waveguide with absorption corresponding to the parts of the spectrum located to the left and to the right of the imaginary axis. Bibliography: 7 titles.     

Optimization of Periodic Composite Structures for Sub-Wavelength Focusing

Recently, there has been plenty of work in designing and fabricating materials with an effective negative refractive index. Veselago realized that a slab of material with a refractive index of −1 would act as a lens. Pendry suggested that the Veselago lens would act as a superlens, providing a perfect image of an object in contrast to conventional lenses which are only able to focus a point source to an image having a diameter of the order of the wavelength of the incident field. Recent work has shown that similar focusing effects can be obtained with certain slabs of “conventional” periodic composite materials: photonic crystals. The present work seeks to answer the question of what periodic dielectric composite medium (described by dielectric coefficient with positive real part) gives an optimal image of a point source. An optimization problem is formulated and it is shown that a solution exists provided the medium has small absorption. Solutions are characterized by an adjoint-state gradient condition, and several numerical examples illustrate both the plausibility of this design approach, and the possibility of obtaining smaller image spot sizes than with typical photonic crystals. This work was partially supported by NSF grant DMS-0537015.     

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.     

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.     

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).     

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.     

Imaging of periodic dielectrics

We consider imaging of periodic penetrable structures from measurements of scattered electromagnetic waves. The importance of this problem stems from the decreasing size of periodic structures in photonic devices, together with an increasing demand in fast non-destructive testing. This demand makes qualitative inverse scattering techniques particularly attractive since they do not use time consuming optimization techniques for reconstruction but rather directly transform measured data into a picture of the scattering object. We present the Factorization method as an algorithm for imaging of a special class of periodic dielectric structures known as diffraction gratings. Our sampling method computes a picture of the shape of the periodic structure from measured near-field data in a rapid way. We provide numerical examples illustrating this imaging technique.     

Homogenization of magneto-electro-elastic multilaminated materials

In this work, based on the periodic unfolding homogenizationtechnique, the limiting equations modelling the behaviour ofthree-dimensional magneto-electro-elastic periodic structuresare rigorously established. The local problems and the correspondinghomogenized coefficients of the elastic, dielectric, magneticpermittivity, piezoelectric, piezomagnetic and magneto-electric(ME) tensors are explicitly described. The homogenization modelis exemplified for laminated composites and a unified generalformula for all effective properties of periodic multilaminatedmagneto-electro-elastic composites is obtained. This formulais applied to investigate the global behaviour for the importantcase of transversely isotropic constituents and any finite numberof layers in each periodic cell. Examples that provide theoreticalevidence of the presence of both a product property and theME effect are given.     

Projecting and resolving operators of a periodic waveguide

A dielectric periodic waveguide with almost arbitrary smooth boundary is studied. The existence of projecting and resolving operators is established. Bibliography: 9 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 47–57     

On the Bethe-Sommerfeld conjecture for certain periodic Maxwell operators

The Bethe-Sommerfeld conjecture states that the spectrum of the stationary Schrödinger operator with a periodic potential in dimensions higher than 1 has only finitely many gaps. After work done by many authors, it has been proven by now in full generality. Another case of a significant interest, due to its importance for the photonic crystal theory, is of a periodic Maxwell operator, where apparently no results of such kind are known. We establish here that in the case of a 2D photonic crystal, i.e. of the medium periodic in two variables and homogeneous in the third one, if the dielectric function is separable, the number of spectral gaps of the corresponding Maxwell operator is indeed finite. It is also shown that, as one would expect, when the medium is near to being homogeneous, there are no spectral gaps at all.     

Dielectric properties of polyacrylamide and its utilization as a hydrogel

Some of the dielectric properties of polyacrylamide are given such as the variation of the dielectric constant, ε^′, and the dielectric loss, ε^′′, and the power factor tan δ with the applied frequency from 10⁵ to 10⁷ Hz within the temperature range 20–50°C. The dielectric dispersion and absorption bands appear within these ranges of frequencies and temperatures. The relaxation process is explained and discussed, also the relaxation times are calculated and the activation energy and entropy change of dielectric relaxation are calculated from the usual rate equation and discussed. The process of polymerization of polyacrylamide is explained through comparing the dielectric properties of the monomer (acrylamide) and the polymer. Out of two approaches to obtain a hydrogel from polyacrylamide, the physical approach is thoroughly investigated and discussed.     

Projecting and Resolving Operators for a Three-Dimensional Waveguide

We prove the existence of projecting and resolving operators of a three-dimensional periodic waveguide for discontinuous media with absorption. Bibliography: 7 titles.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 31—45.     

Asymptotic analysis of the diffusion-absorption equation with fast and strongly oscillating absorbtion coefficient in the two-dimensional case

The Helmholtz equation with fast oscillating absorbtion coefficient and constant reflection coefficient is considered. The equation models light absorption in a medium containing a periodic set of fine blood vessels. It is assumed that the absorption takes place only inside the vessels. It is also assumed that the reflection coefficient is constant whereas the absorbtion coefficient is small everywhere except for a set of periodic thin strips modeling blood vessels, where the absorption coefficient equals a large parameter ω. There are two other parameters in the problem: ? is the ratio of the distance between the vessel axes to the characteristic macroscopic size, and δ is the ratio of the width of the fine vessels to the period. Both parameters ? and δ are assumed to be small. The main result is the construction of an asymptotic solution.     

Homogenization of the layered medium equation

A new partial differential equation to be called the layered medium equation is introduced, and it is proved that certain relevant initial or periodic boundary conditions give well-posed problems. Then, the homogenized limit of the layered medium equation is studied. It is shown to be preserved in limit in the limit in the physical problem in which the coefficients that arise from the dielectric layer are both proportional to thickness. Otherwise, a non-local problem is obtained as the limiting form     

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) Gårding 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 © 2012 John Wiley & Sons, Ltd.     

Electrohydrodynamic instability of two superposed viscous miscible streaming fluids

The linear stability of two dielectric viscous fluids separated by a horizontal interface is investigated. The interface admits heat and mass transfer. The system is stressed by a normal periodic electric field producing surface charges at the interface. The effect of surface tension, small viscosity, velocity streaming and gravity on the critical surface charges density and on corresponding electric field are analyzed. The contribution of viscosity with the existence of surface charges and streaming are discussed. The investigation includes the stability analysis of the presence of the periodic electric field as well as the constant one. It is found that the presence of the surface charges made by the normal electric field play a dual role in the stability criterion, which shows some analogy with the nonlinear theory of stability. Some previous studies are compared using appropriate data. The marginal state of stability is also considered. It is found that the surface charges vanish under certain conditions. This study shows that the mass and heat transfer parameter has a destabilizing effect whether the electric field is static or periodic. Parametric excitation of the electrohydrodynamic (EHD) surface waves is analyzed in the case of Rayleigh–Taylor (R–T) instability. The transition curves are obtained by means of Whittaker's technique. The analytical results are numerically confirmed.     

On surface waves in diffraction gratings

We discuss the existence criterion of surface waves based on the augmented scattering matrices. Such matrices arise if one takes into account not only oscillating waves but also those which grow (attenuate) in amplitude far from the grating. A family of planar dielectric gratings with periodic modulation of the refraction index is considered. Asymptotic and numerical analysis of the model are given. We represent various examples of gratings which support (or do not support) surface waves. Copyright © 2000 John Wiley & Sons, Ltd.     

Infra-red spectral studies of CHF2-CHCl2 and CBrCl2-CBrCl2

The infra-red spectra of two polyhalogenated ethanes, 1, 1-difluoro-2, 2-dichloro ethane and 1, 2-dibromo-1, 1, 2, 2-tetrachloro ethane were obtained in the region of 2·5–38 microns. The spectra of CHF₂-CHCl₂ were obtained in gas, liquid and solid states and also in solvents of widely varying dielectric constants. These spectral studies showed that the molecule exists in the form of two rotational isomers. The fundamental modes of the two isomers were identified on the basis of intensity changes from gas to liquid to solid and in solvents of different dielectric constants. The observed infra-red absorption bands have been assigned in terms of the fundamental absorption bands. Spectra of CBrCl₂-CBrCl₂ were obtained in the solid and vapour phase and in various solvents. These spectra showed that CBrCl₂-CBrCl₂ exists only in the trans form. The infra-red spectra, combined with Raman data available from literature was used for complete assignment of the observed absorption bands.     

On a hierarchy of models for electrical conduction in biological tissues

In this paper we derive a hierarchy of models for electrical conduction in a biological tissue, which is represented by a periodic array of period ε of conducting phases surrounded by dielectric shells of thickness εη included in a conductive matrix. Such a hierarchy will be obtained from the Maxwell equations by means of a concentration process η → 0, followed by a homogenization limit with respect to ε. These models are then compared with regard to their physical meaning and mathematical issues. Copyright © 2005 John Wiley & Sons, Ltd.