Desired Refraction Coefficient Research Articles

Is creating materials with a desired refraction coefficient practically possible?

A theory of many-body wave scattering is developed under the assumption a << d << λ, where a is the characteristic size of the small body, d is the distance between neighboring bodies and λ is the wave-length in the medium in which the bodies are embedded. The multiple scattering is essential under these assumptions. The author’s theory is used for creating materials with a desired refraction coefficient. This theory can be used in practice. A recipe for creating materials with a desired refraction coefficient is formulated. Materials with a desired radiation pattern, for example, wave-focusing materials, can be created.

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.

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.

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.)

Scattering of Electromagnetic Waves by Many Nano-Wires

Electromagnetic wave scattering by many parallel to the z−axis, thin, impedance, parallel, infinite cylinders is studied asymptotically as a → 0. Let Dm be the cross-section of the m−th cylinder, a be its radius and x ^ m = (x m1 , x m2 ) be its center, 1 ≤ m ≤ M , M = M (a). It is assumed that the points, x ^ m , are distributed, so that N(Δ)= 1 2πa ∫ Δ N ( x ^ )d x ^ [1+o(1)] where N (∆) is the number of points, x ^ m , in an arbitrary open subset, ∆, of the plane, xoy. The function, N( x ^ ) ≥0 , is a continuous function, which an experimentalist can choose. An equation for the self-consistent (effective) field is derived as a → 0. A formula is derived for the refraction coefficient in the medium in which many thin impedance cylinders are distributed. These cylinders may model nano-wires embedded in the medium. One can produce a desired refraction coefficient of the new medium by choosing a suitable boundary impedance of the thin cylinders and their distribution law.

Wave scattering by small bodies and creating materials with a desired refraction coefficient

In this paper the author’s invited plenary talk at the 7-th PACOM (Pan African Congress of Mathematicians), is presented. Asymptotic solution to many-body wave scattering problem is given in the case of many small scatterers. The small scatterers can be particles whose physical properties are described by the boundary impedances, or they can be small inhomogeneities, whose physical properties are described by their refraction coefficients. Equations for the effective field in the limiting medium are derived. The limit is considered as the size a of the particles or inhomogeneities tends to zero while their number M(a) tends to infinity. These results are applied to the problem of creating materials with a desired refraction coefficient. For example, the refraction coefficient may have wave-focusing property, or it may have negative refraction, i.e., the group velocity may be directed opposite to the phase velocity. This paper is a review of the author’s results presented in MR2442305 (2009g:78016), MR2354140 (2008g:82123), MR2317263 (2008a:35040), MR2362884 (2008j:78010), and contains new results.

A METHOD FOR CREATING MATERIALS WITH A DESIRED REFRACTION COEFFICIENT

It is proposed to create materials with a desired refraction coefficient in a bounded domain D ⊂ ℝ3 by embedding many small balls with constant refraction coefficients into a given material. The number of small balls per unit volume around every point x ∈ D, i.e., their density distribution, is calculated, as well as the constant refraction coefficients in these balls. Embedding into D small balls with these refraction coefficients according to the calculated density distribution creates in D a material with a desired refraction coefficient.

Scattering by many small particles and creating materials with a desired refraction coefficient

Combining an asymptotic method and computational modelling the authors propose a method for creating materials with the desired electrodynamical characteristics, in particular, with a desired refraction coefficient. The problem of wave scattering by many small particles is solved asymptotically under the assumptions ka > a, where a is the size of the particles and d is the distance between the neighbouring particles. On the wavelength one may have many small particles. Impedance boundary conditions are assumed on the boundaries of small particles. The results of numerical simulation show good agreement with the theory. Constructive conclusions are given for creating materials with a desired refraction coefficient on the basis of the obtained numerical results. Engineering realisation of the theory is of practical interest.

Creating materials with a desired refraction coefficient: numerical experiments

A recipe for creating materials with a desired refraction coefficient is implemented numerically. An error estimate is given for the approximate solution of the many-body scattering problem in the case of small scatterers. This result is used for the estimate of the minimal number of small particles to be embedded in a given domain D in order to get a material whose refraction coefficient approximates the desired one with the relative error not exceeding a desired small quantity.

A collocation method for solving integral equations

A collocation method is formulated and justified for numerical solution of the Fredholm second-kind integral equations. As an application the Lippmann-Schwinger equation is considered. The results obtained provide an error estimate and a justification of the limiting procedure used in the earlier papers by the author, dealing with many-body scattering problems in the case of small scatterers, and with creating materials with a desired refraction coefficient: Ramm, A.G. 'Many-body wave scattering by small bodies and applications', J. Math. Phys., Vol. 48, No. 10, 103-511; Ramm, A.G. (2008) 'Wave scattering by many small particles embedded in a medium', Phys. Lett. A, Vol. 372, No. 17, (2008), pp.3064-3070

Preparing materials with a desired refraction coefficient

A recipe is given for creating material with a desired refraction coefficient by embedding many small particles in a given material. To implement this recipe practically, some technological problems have to be solved. These problems are formulated.

Electromagnetic wave scattering by small bodies

A reduction of the Maxwell's system to a Fredholm second-kind integral equation with weakly singular kernel is given for electromagnetic (EM) wave scattering by one and many small bodies. This equation is solved asymptotically as the characteristic size of the bodies tends to zero. The technique developed is used for solving the many-body EM wave scattering problem by rigorously reducing it to solving linear algebraic systems, completely bypassing the usage of integral equations. An equation is derived for the effective field in the medium, in which many small particles are embedded. A method for creating a desired refraction coefficient is outlined.

Wave scattering by many small particles embedded in a medium

Theory of wave scattering by many small bodies is developed under various assumptions concerning the ratio a d , where a is the characteristic dimension of a small body and d is the distance between neighboring bodies d = O ( a κ 1 ) , 0 < κ 1 < 1 . On the boundary S m of every small body an impedance-type condition is assumed ∂ u M ∂ N = ζ m u M on S m , 1 ⩽ m ⩽ M , ζ m = h m a − κ , 0 < κ , h m are constants independent of a. The behavior of the field in the region in which M = M ( a ) ≫ 1 small particles are embedded is studied as a → 0 and M ( a ) → ∞ . Formulas for the refraction coefficient of the limiting medium are derived under the assumptions: (a) κ 1 = ( 2 − κ ) / 3 , 0 < κ ⩽ 1 , and (b) κ 1 = 1 / 3 , κ > 1 . A method for creating materials with a desired refraction coefficient is proposed and justified theoretically on the basis of the above results.

Many-body wave scattering by small bodies and applications

A rigorous reduction of the many-body wave scattering problem to solving a linear algebraic system is given bypassing solving the usual system of integral equation. The limiting case of infinitely many small particles embedded into a medium is considered and the limiting equation for the field in the medium is derived. The impedance boundary conditions are imposed on the boundaries of small bodies. The case of Neumann boundary conditions (acoustically hard particles) is also considered. Applications to creating materials with a desired refraction coefficient are given. It is proved that by embedding a suitable number of small particles per unit volume of the original material with suitable boundary impedances, one can create a new material with any desired refraction coefficient. The governing equation is a scalar Helmholtz equation, which one obtains by Fourier transforming the wave equation.

Materials with a desired refraction coefficient can be made by embedding small particles

Abstract A method is proposed to create materials with a desired refraction coefficient, possibly negative one. The method consists of embedding into a given material small particles. Given n 0 ( x ) , the refraction coefficient of the original material in a bounded domain D ⊂ R 3 , and a desired refraction coefficient n ( x ) , one calculates the number N ( x ) of small particles, to be embedded in D around a point x ∈ D per unit volume of D, in order that the resulting new material has refraction coefficient n ( x ) .