Small Impedance Particles Research Articles

Asymptotic method of solution to the problem of electromagnetic field scattering on a set of small impedance particles

Asymptotic method of solution to the problem of electromagnetic field scattering on a set of small impedance particles

Wave scattering by many small impedance particles and applications

Formulas are derived for solutions of many-body wave scattering problem by small impedance particles embedded in a homogeneous medium. The limiting case is considered, when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. The basic physical assumption is a << d << λ , where d is the minimal distance between neighboring particles, λ is the wavelength, and the particles can be impedance balls B ( x m , a ) with centers x m located on a grid. Equations for the limiting effective (self-consistent) field in the medium are derived. It is proved that one can create material with a desired refraction coefficient by embedding in a free space many small balls of radius a with prescribed boundary impedances. The small balls can be centered at the points located on a grid. A recipe for creating materials with a desired refraction coefficient is formulated. It is proved that materials with a desired radiation pattern, for example, wave-focusing materials, can be created.

Asymptotic regularisation of the solution to the problem of electromagnetic field scattering from a set of small impedance particles

The solution to the problem of electromagnetic (EM) wave scattering from a set of small-size impedance particles of arbitrary shape is derived by the asymptotic approach. Particles are located in a homogeneous domain with constant ɛ0 and μ0. The solution is derived under the condition b → 0, where b is the characteristic size of the particle; further, the number M(b) of particles tends to infinity at a specific rate. The regularising procedure consists of the derivation of the explicit form of a solution that excludes the necessity to solve the respective integral equation for determination of the fields at the surface of particles and thus avoids integrating the Green function derivatives, which are in the kernel of this boundary integral equation. Practical application of this approach yields an ability to create media with the desired inhomogeneous distribution of effective refractive index n and magnetic permeability μ(x). Explicit analytical formulas are derived for these physical parameters and supported by computations.

Моделювання матеріалів із бажаним коефіцієнтом рефракції на основі асимптотичного розв’язку задачі розсіювання

The explicit solution to the diffraction problem on a set of small particles, supplemented into homogeneous material, is used for modeling the materials with the desired refractive index. The closed form solution is reduced for the scattering problem. This allows to obtain an explicit formula for the refractive index of the resulting inhomogeneous material. The numerical calculations show the possibility to get the specific values of refractive index. References Veselago, V. G. (1967). The electrodynamics of substances with simultaneously negative values of e and μ. Sov. Phys. Usp., 10(4), 509–514.DOI https://doi.org/10.1070/pu1968v010n04abeh003699 Ogier, R., Fang, Y. M., Svedendahl, M. (2015). Near-complete photon spins electivity in a metasurface of anisotropic plasmonic antennas. Phys. Rev., 10(5). Pendry, J. B., Schurig, D., Smith, D. R. (2006). Controlling electromagnetic fields. (em)Science, 312, 1780–2. Yang, Y., Da Costa, R. C., Fuchter, M. J., Campbell, A. J. (2013). Circularly polarized light detection by a chiral organic semiconductor transistor. Nat. Photon., 7, 634–8.DOI https://doi.org/10.1038/nphoton.2013.176 Chalabi, H., Schoen, D, Brongersma, M. L. (2014). Hot-electron photodetection with a plasmonic nanostripe antenna. Nano Lett., 14, 1374–80.DOI https://doi.org/10.1021/nl4044373 Ramm, A. G. (2008). Wave scattering by many small particles embedded in a medium. Physics Letters A, 372, 3064-3070.DOI https://doi.org/10.1016/j.physleta.2008.01.006 Ramm, A. G. (2013). Electromagnetic wave scattering by small impedance particles of an arbitrary shape. J. of Appl. Math. and Comput. (JAMC), 43(1), 427–444.DOI https://doi.org/10.1007/s12190-013-0671-3 Ramm, A. G. (2007). Many body wave scattering by small bodies and applications. Journal of Mathematical Physics, 48(10), 1035-1–1035-6. Ramm, A. G. (2009). A Collocation method for solving integral equations. Intern. Journ. of Comput. Sci. and Mathem., 3(2), 122–128. Andriychuk, M. I., Ramm, A. G. (2010). Scattering by many small particles and creating materials with a desired refraction coefficient. Intern. Journ. of Computing Science and Mathematics, 3, 102–121.DOI https://doi.org/10.1007/s12190-013-0671-3

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(Δ)∼1/a2−κ∫ΔN(x)dx as a → 0. Here, N(x) ≥ 0 is an arbitrary continuous function that can be chosen by the experimenter and N(Δ) is the number of particles in an arbitrary sub-domain Δ. 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.

EM Wave Scattering by Many Small Impedance Particles and Applications to Materials Science

EM Wave Scattering by Many Small Impedance Particles and Applications to Materials Science

Electromagnetic Wave Scattering by Small Impedance Particles of an Arbitrary Shape and Applications

The proposal deals with electromagnetic (EM) wave scattering by one and many small impedance particles of an arbitrary shape. Analytic formula is derived for EM wave scattering by one small impedance particle of an arbitrary shape and an integral equation for the effective field in the medium where many such particles are embedded. These results are applied for creating a medium with a desired refraction coefficient. The proposed theory has no analogs in the literature. (Mathematical Subject Classiffication: 35J05, 35J25, 65N12, 78A25, 78A48.)

Calculation of Electromagnetic Wave Scattering by a Small Impedance Particle of an Arbitrary Shape

Scattering of electromagnetic (EM) waves by small (ka ≪ 1) impedance particle D of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic, closed form, formula for the scattered field is derived. The scattered field is of the order O(a2 − κ), where κ ∈ [ 0,1) is a number. This field is much larger than in the case of Rayleigh-type scattering. The numerical results demonstrate a wide range of applicability of the analytic formula for the scattered field. Comparison with Mie-type solution is carried out for various boundary impedances and radii of the particle.

Application of the asymptotic solution to EM field scattering problem for creation of media with prescribed permeability

Scattering of electromagnetic (EM) waves by many small impedance particles (bodies), embedded in a homogeneous medium, is studied. Physical properties of the particles are described by their boundary impedances. The limiting equation is obtained for the effective EM field in the limiting medium, in the limit a→0, where a is the characteristic size of a particle and the number M(a) of the particles tends to infinity at a suitable rate. The proposed theory allows one to create a medium with a desirable spatially inhomogeneous permeability. The main new physical result is the explicit analytical formula for the permeability μ(x) of the limiting medium. The computational results confirm a possibility to create the media with various distributions of μ(x).

Electromagnetic wave scattering by small impedance particles of an arbitrary shape

Scattering of electromagnetic (EM) waves by one and many small (ka≪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−κ), where κ∈[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.

Electromagnetic wave scattering by a small impedance particle: Theory and modeling

Electromagnetic (EM) waves, scattered by a small impedance particle of arbitrary shape, embedded in a homogeneous medium, are calculated by a new analytic formula. The range of applicability and the accuracy of this formula are illustrated by numerical results. The formula was derived in (*) A.G.Ramm, Optics Communications, 284,(2011), 3872–3877. The accuracy of the new formula is estimated by a comparison with the Mie-type solution for an impedance sphere.The novelty of our paper is in the demonstration of the range of applicability of the new formula and its practical value, by the numerical results and their comparison with the exact solution for EM wave scattering by impedance spheres. The exact solution is obtained in the form of Mie-type series, and is new. Estimate of the error of this series, in which five terms are kept, shows that the relative error of this solution is less than 10−3 for the parameters' range considered. The numerical results obtained are of interest to a wide audience, and the novelty of the formula from (*) is in its applicability to wave scattering by small particles of arbitrary shapes, when Mie-type solution is not applicable.

Electromagnetic wave scattering by a small impedance particle of arbitrary shape

Scattering of electromagnetic (EM) waves by one small ( ka ≪ 1) impedance particle (body) D of arbitrary shape, embedded in a homogeneous medium, is studied. Physical properties of the particle are described by its boundary impedance. The problem is of interest because scattering of light by colloidal particles, or by dust in the air is an example of the scattering theory discussed in this paper. An analytic formula is obtained for the EM field in the far zone without usage of boundary integral equation. If a monochromatic incident field of frequency ω is E 0( x, ω), then the scattered field v in the zone r : = | x| ≫ a, where a = 0.5 diamD is the characteristic size of D, is calculated by the formula v = [ ∇ e ikr 4 π r , Q ] , where [ A, B] is the cross product of two vectors, ( Q, e j ) is the dot product, e j , 1 ≤ j ≤ 3, are orthonormal basis vectors in R 3 , Q j : = ( Q , e j ) = − i ζ | S | ω μ 0 τ jp ( ∇ × E 0 ( O ) ) p , over the repeated index p summation is understood from Eqs. (1) to (3), ζ is the boundary impedance and | S| is the surface area of the particle, O ∈ D is the origin, the tensor τ jp : = δ jp − | S| − 1 ∫ S N j ( s) N p ( s) ds, where N j ( s) is the j-th component of the unit normal N(s) to the surface S at the point s ∈ S, k = ω( ε 0 μ 0) 1/2 is the wave number.

Wave scattering by small impedance particles in a medium

The theory of acoustic wave scattering by many small bodies is developed for bodies with impedance boundary condition. It is shown that if one embeds many small particles in a bounded domain, filled with a known material, then one can create a new material with the properties very different from the properties of the original material. Moreover, these very different properties occur although the total volume of the embedded small particles is negligible compared with the volume of the original material.