Scattering of electromagnetic waves by small impedance particles of an arbitrary shape

An explicit formula is derived for the electromagnetic (EM) field scattered by one small impedance particle $D$ of an arbitrary shape. If $a$ is the characteristic size of the particle, $\lambda$ is the wavelength, $a<<\lambda$ and $\zeta$ is the boundary impedance of $D$, $[N,[E,N]]=\zeta [N,H]$ on $S$, where $S$ is the surface of the particle, $N$ is the unit outer normal to $S$, and $E$, $H$ is the EM field, then the scattered field is $E_{sc}=[\nabla g(x,x_1), Q]$. Here $g(x,y)=\frac{e^{ik|x-y|}}{4\pi |x-y|}$, $k$ is the wave number, $x_1\in D$ is an arbitrary point, and $Q=-\frac{\zeta |S|}{i\omega \mu}\tau \nabla \times E_0$, where $E_0$ is the incident field, $|S|$ is the area of $S$, $\omega$ is the frequency, $\mu$ is the magnetic permeability of the space exterior to $D$, and $\tau$ is a tensor which is calculated explicitly. The scattered field is $O(|\zeta| a^2)>> O(a^3)$ as $a\to 0$ when $\lambda$ is fixed and $\zeta$ does not depend on $a$. Thus, $|E_{sc}|$ is much larger than the classical value $O(a^3)$ for the field scattered by a small particle. It is proved that the effective field in the medium, in which many small particles are embedded, has a limit as $a\to 0$ and the number $M=M(a)$ of the particles tends to $\infty$ at a suitable rate. Thislimit solves a linear integral equation. The refraction coefficient of the limiting medium is calculated analytically. This yields a recipe for creating materials with a desired refraction coefficient

Electromagnetic Wave Scattering by Small Impedance Particles of an Arbitrary Shape

Scattering of electromagnetic (EM) waves by one and many small ($ka\ll 1$) impedance particles $D_m$ of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic formula for the field, scattered by one particle, is derived. The scattered field is of the order $O(a^{2-\kappa})$, where $\kappa \in [0,1)$ is a number. This field is much larger than in the Rayleigh-type scattering. An equation is derived for the effective EM field scattered by many small impedance particles distributed in a bounded domain. Novel physical effects in this domain are described and discussed

Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies

A theory of electromagnetic (EM) wave scattering by many small particles of an arbitrary shape is developed. The particles are perfectly conducting or impedance. For a small impedance particle of an arbitrary shape, an explicit analytical formula is derived for the scattering amplitude. The formula holds as a → 0, where a is a characteristic size of the small particle and the wavelength is arbitrary but fixed. The scattering amplitude for a small impedance particle is shown to be proportional to a2−κ, where κ ∈ [0,1) is a parameter which can be chosen by an experimenter as he/she wants. The boundary impedance of a small particle is assumed to be of the form ζ = ha−κ, where h = const, Reh ≥ 0. The scattering amplitude for a small perfectly conducting particle is proportional to a3, and it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions a ≪ d ≪ λ, where d is the minimal distance between neighboring particles and λ is the wavelength. The distribution law for the small impedance particles is N(∆) ∼ 1/a2−κ∆ N(x)dx as a → 0. Here, N(x) ≥ 0 is an arbitrary continuous function that can be chosen by the experimenter and N(∆) is the number of particles in an arbitrary sub-domain ∆. It is proved that the EM field in the medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as a → 0 and a differential equation is derived for the limiting field. On this basis, a recipe is given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4929965

A Surface Admittance Equivalence Principle for Non-Radiating and Cloaking Problems

In this paper, we address non-radiating and cloaking problems exploiting the surface equivalence principle, by imposing at any arbitrary boundary the control of the admittance discontinuity between the overall object (with or without cloak) and the background. After a rigorous demonstration, we apply this model to a non-radiating problem, appealing for anapole modes and metamolecules modeling, and to a cloaking problem, appealing for non-Foster metasurface design. A straightforward analytical condition is obtained for controlling the scattering of a dielectric object over a surface boundary of interest. Previous quasi-static results are confirmed and a general closed-form solution beyond the subwavelength regime is provided. In addition, this formulation can be extended to other wave phenomena once the proper admittance function is defined (thermal, acoustics, elastomechanics, etc.).Comment: 7 page

Metamaterials: supra\textit{supra}-classical dynamic homogenization

Metamaterials are artificial composite structures designed for controlling waves or fields, and exhibit interaction phenomena that are unexpected on the basis of their chemical constituents. These phenomena are encoded in effective material parameters that can be electronic, magnetic, acoustic, or elastic, and must adequately represent the wave interaction behaviour in the composite within desired frequency ranges. In some cases -- for example, the low frequency regime -- there exist various efficient ways by which effective material parameters for wave propagation in metamaterials may be found. However, the general problem of predicting frequency-dependent dynamic effective constants has remained unsolved. Here, we obtain novel mathematical expressions for the effective parameters of two-dimensional metamaterial systems valid at higher frequencies and wavelengths than previously possible. By way of an example, random configurations of cylindrical scatterers are considered, in various physical contexts: sound waves in a compressible fluid, anti-plane elastic waves, and electromagnetic waves. Our results point towards a paradigm shift in our understanding of these effective properties, and metamaterial designs with functionalities beyond the low-frequency regime are now open for innovation.Comment: 14 pages (including 4 figures and 1 table) in New Journal of Physics, 201

Scattering of electromagnetic waves by many thin cylinders: theory and computational modeling

Electromagnetic (EM) wave scattering by many parallel infinite cylinders is studied asymptotically as a tends to 0, where a is the radius of the cylinders. It is assumed that the centres of the cylinders are distributed so that their numbers is determined by some positive function N(x). The function N(x) >= 0 is a given continuous function. An equation for the self-consistent (limiting) field is derived as a tends to 0. The cylinders are assumed perfectly conducting. Formula for the effective refraction coefficient of the new medium, obtained by embedding many thin cylinders into a given region, is derived. The numerical results presented demonstrate the validity of the proposed approach and its efficiency for solving the many-body scattering problems, as well as the possibility to create media with negative refraction coefficients.Comment: 21 pages, 13 figure

On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body

The interface problem describing the scattering of time-harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface $\Gamma$. In this paper, following an idea developed by Kleinman and Martin \cite{KlMa} for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over $\Gamma$ for a single unknown density. One knows that such boundary integral formulations of the Maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior Maxwell boundary value problem. We obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way, they become uniquely solvable for all real frequencies. We analyze the well-posedness of the integral equations in the space of finite energy on smooth and non-smooth boundaries