Characterization and Application of Nanomaterials (2023) Volume 6 Issue 1 doi:10.24294/can.v6i1.1953 1 Original Research Article Is creating materials with a desired refraction coefficient practically possible? Alexander G. Ramm Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA. E-mail: ramm@ksu.edu ABSTRACT 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. PACS: 02.30.Rz; 02.30.Mv; 41.20.Jb MSC: 35Q60; 78A40; 78A45; 78A48 Keywords: Wave Scattering by Many Small Bodies; Smart Materials ARTICLE INFO Received: 8 February 2023 Accepted: 2 March 2023 Available online: 16 March 2023 COPYRIGHT Copyright © 2023 by author(s). Characterization and Application of Nano- materials is published by EnPress Publisher LLC. This work is licensed under the Crea- tive Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). https://creativecommons.org/licenses/by- nc/4.0/ 1. Introduction The aim of this paper is to give an affirmative answer to the question in the title of this paper. This brings potentially many possibilities for progress in technology. There is a large literature on wave scattering by small bodies, start- ing from Rayleigh’s work (1871)[1–3]. If the scatterer is small then the scattered field can be calculated analytically for bodies of arbitrary shapes, see reference [4]. The many-body wave scattering problem was discussed in the liter- ature mostly numerically, if the number of scatterers was small, or under the assumption that the influence of the waves, scattered by other parti- cles on a particular particle is negligible[5]. This corresponds to the case when the distance d between neighbouring particles is much larger than the wavelength λ, and the characteristic size a of a small body (particle) is much smaller than λ. Theoretically and practically the assumptions a << λ, d >> λ, (1) are the simplest ones which allow one to neglect multiple scattering. By k = 2𝜋𝜋 𝜆𝜆 , the wave number is denoted. In the author’s theory, the basic assumptions are a << d << λ, (2) and the multiple scattering is of basic importance under these assump- tions[4,6–35]. It is clear that assumption (2) can be practically realized. Its https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ 2 importance comes from the fact that the author gave a rigorous asymptotically exact solution of the many- body scattering problem under assumption (2) when a → 0. This solution can be well approximated nu- merically by the particles of the size a > 30 nm. Prac- tically the size of a can be found by comparison of the solution for some a and for 𝑎𝑎 2 . If these solutions are practically close, then one considers this a as suit- able. The aim of this paper is to show that our theory can be used practically. In reference [36], for the first time the author’s theory was used for solving the scattering problem for 10 billion small particles. This problem was solved numerically and numerical results were pre- sented. Let us formulate the wave scattering problems we deal with. Let D be a bounded domain in ℝ3 with a sufficiently smooth boundary. The scattering problem consists of finding the solution to the prob- lem: (∇2 + 𝑘𝑘2)𝑢𝑢 = 0 in G’ := ℝ3\𝐺𝐺, G := 𝑈𝑈𝑚𝑚=1 𝑀𝑀 𝐷𝐷𝑚𝑚, k = const > 0, (3) where Dm = B(xm, a) is an impedance ball, centered at xm and of small radius a, u = u0 + v, u0 = eikα·x, α ∈ S2, (4) S2 is the unit sphere in ℝ3, u0 is the incident field, v is the scattered field satisfying the radiation condi- tion vr – ikv = o(1 𝑟𝑟 ), r := |𝑥𝑥| → ∞, vr := 𝜕𝜕𝜕𝜕 ∂r , (5) and u satisfies the impedance boundary condition (bc) on the boundary of G: uN – ζmu = 0, on Sm, Imζm ≤ 0, (6) where ζm is a constant, N is the unit normal to S := 𝑈𝑈𝑚𝑚=1 𝑀𝑀 𝑆𝑆𝑚𝑚, pointing out of G := 𝑈𝑈𝑚𝑚=1 𝑀𝑀 𝐷𝐷𝑚𝑚, and Sm is the surface of Dm = B(xm, a). By refraction coefficient n(x) the coefficient in the equation (∇2 + 𝑘𝑘2𝑛𝑛2(𝑥𝑥))𝑢𝑢 = (∇2 + 𝑘𝑘2 − 𝑞𝑞(𝑥𝑥))𝑢𝑢 = 0 (7) is understood, where q(x) := k2(n2(x) – 1). Let g(x, y) = 𝑒𝑒𝑖𝑖𝑖𝑖|𝑥𝑥−𝑦𝑦| 4π|𝑥𝑥−𝑦𝑦| . Then (∇2 + k2)g(x, y) = – δ(x – y), where δ(x) is the delta function. Let us distribute small impedance particles Dm = B(xm, a) in D so that ℕ(∆) = aκ–2|∆|[1 + o(1)], a → 0, (8) where ∆ ⸦ D is an arbitrary connected open subset of D, |∆| is its volume, κ ∈ (0, 1) is a number the experimenter may choose arbitrarily and ℕ(∆) is the number of particles in ∆. Throughout this paper the important assumptions a << d << λ and (8) are satisfied. As a → 0, the number of small particles ℕ(∆) in (8) tends to infinity since κ – 2 < 0. We assume in this paper (for simplicity only) that the small particles are distributed in the domain D and the refraction coefficient in D equals to 1. In the monograph [31], it is assumed that D is filled with the material whose refraction coefficient n0(x) is known and we wanted to create in D the material with the desired refraction coefficient n(x). The boundary impedances ζm are chosen by the formula ζm = a–κh(xm), (9) where h(x) is a continuous function in D, Imh ≤ 0. It will be clear from Section 3 that the function h(x) can be determined by choosing a suitable bound- ary impedance ζ(x). When a → 0, the ζm and h(xm) can be considered as continuous functions ζ(x) and h(x). 2. Solution of many-body scattering problem We look for the solution of the form (10) where σm(s) are unknown, Qm := ∫ 𝜎𝜎𝑚𝑚(𝑠𝑠)𝑑𝑑𝑠𝑠𝑆𝑆𝑚𝑚 . One may think about σm as of charge densities on Sm and 3 of Qm as of total charge on the surface Sm. We prove that 𝐽𝐽 ≔ � � [𝑔𝑔(𝑥𝑥, 𝑠𝑠) − 𝑔𝑔(𝑥𝑥, 𝑥𝑥𝑚𝑚)]𝜎𝜎𝑚𝑚(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑆𝑆𝑚𝑚 𝑀𝑀 𝑚𝑚=1 (11) is negligible compared to (12) so J << I as a → 0. (13) We prove that the field u satisfies the following inte- gral equation as a → 0: u(x) = u0(x) – 4π∫ 𝑔𝑔(𝑥𝑥,𝑦𝑦)ℎ(𝑦𝑦)𝑢𝑢(𝑦𝑦)𝑑𝑑𝑦𝑦𝐷𝐷 , (14) where h(xm) = 𝜁𝜁𝑚𝑚 𝑎𝑎𝜅𝜅 , and, since there are sufficiently many points xm ∈ D, the function h(x) is uniquely de- termined in D if the boundary impedances are known. Apply the operator to ∇2 + k2 to both sides of equation (14) and get �∇2 + 𝑘𝑘2 − 4𝜋𝜋ℎ(𝑥𝑥)�𝑢𝑢(𝑥𝑥) ∶= �∇2 + 𝑘𝑘2𝑛𝑛2(𝑥𝑥)�𝑢𝑢(𝑥𝑥) = 0 (15) Therefore, n2(x) = 1 – 4πk–2h(x). (16) We omit details since they can be found in the au- thor’s publications listed in the References, in partic- ular, in monograph [31]. If originally in D were material with the known refraction coefficient n0(x), then formula (16) were n2(x) = 𝑛𝑛02(𝑥𝑥) – 4πh(x)N(x)k–2, where N(x) is the distribution density for the small particles, see refer- ence [31]. In this paper, we assume (for simplicity only) that N(x) = 1, see formula (8). 3. Recipe for creating materials with a desired refraction coefficient Let us formulate a recipe for creating materials with a desired refraction coefficient. Formula (16) shows that if h(x) is chosen properly, then any n(x) can be obtained in D. Recipe for creating materials with a desired re- fraction coefficient: a) Calculate by formula (16) the function h(x); b) Distribute small impedance balls in the do- main D by the distribution law (8). The boundary im- pedances of these balls are defined by the function h(x). Theorem 1. The refraction coefficient of the re- sulting medium tends to the desired coefficient n(x) as a → 0. Let us show that practically negative refraction coefficient n(x) can be obtained by the above recipe. Denote b := 4πk–2 > 0 and write equation (16) as n(x) = (1 – bh(x))1/2 = |1 − 𝑏𝑏ℎ(𝑥𝑥)|1/2𝑒𝑒𝜙𝜙/2, (17) where ϕ is the argument of 1 – bh(x). Since the oper- ator in (14) is of Fredholm type, it remains Fredholm type under small perturbations. Therefore one can take h – i𝜖𝜖, where 𝜖𝜖 > 0 is sufficiently small, and equation (14) will still have a unique solution. By choosing h so that Re(1 – bh) > 0 and Im(1 – bh) < 0 and small, one gets the argument ϕ = 2π – δ, where δ > 0 is arbitrarily small if 𝜖𝜖 is sufficiently small. Then n(x) will be nearly negative: its argument will be π – δ/2. 4. Creating materials with a desired radiation pattern Let us define what we mean by the radiation pattern. Consider the scattering problem for the equation: ∇2𝑢𝑢 + 𝑘𝑘2𝑢𝑢 − 𝑞𝑞(𝑥𝑥)𝑢𝑢 = 0, 𝑢𝑢 = 𝑒𝑒𝑖𝑖𝑖𝑖𝑎𝑎·𝑥𝑥 + 𝑣𝑣, (18) where v satisfies the radiation condition. Assume that k > 0 and α ∈ S2 are fixed. Then the scattering am- plitude A(β, α, k) = A(β), where the dependence on k, α is dropped since k and α are fixed. The formula for the scattering amplitude is known, see, e.g., refer- ence [35]: A(β): = Aq(β) = − 1 4𝜋𝜋 ∫ 𝑒𝑒 −𝑖𝑖𝑖𝑖𝑖𝑖·𝑦𝑦𝑞𝑞(𝑦𝑦)𝑢𝑢(𝑦𝑦)𝑑𝑑𝑦𝑦. (19) We call A(β) the radiation pattern. Consider an inverse problem (IP): Given an arbitrary f(β) ∈ L2(S2) and an arbi- trary small 𝜖𝜖 > 0, can one find a q ∈ L2(D) such that �𝑓𝑓(𝛽𝛽) − 𝐴𝐴𝑞𝑞(𝛽𝛽)� 𝐿𝐿2(𝑆𝑆2) < 𝜖𝜖. (20) 4 This inverse problem was not formulated and was not studied in the works of other authors, to our knowledge. Our result is stated in Theorem 2. Theorem 2. For any f(β) ∈ L2(S2) and an ar- bitrary small 𝜖𝜖 > 0 there is a q ∈ L2(D) such that (20) holds. Since small perturbations of q result in small perturbations of A(β), there are infinitely many po- tentials q for which inequality (20) holds. The conclusion of Theorem 2 follows from lem- mas 3 and 4. Lemma 3. The set �∫ 𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖·𝑥𝑥ℎ(𝑥𝑥)𝑑𝑑𝑥𝑥𝐷𝐷 � ∀ℎ∈𝐿𝐿2(𝐷𝐷) is dense in L2(S2). Corollary 1. Given f ∈ L2(S2) and 𝜖𝜖 > 0, one can find h ∈ L2(D) such that �𝑓𝑓(𝛽𝛽) + 1 4𝜋𝜋 ∫ 𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖·𝑥𝑥ℎ(𝑥𝑥)𝑑𝑑𝑥𝑥𝐷𝐷 � < 𝜖𝜖. Lemma 4. The set {𝑞𝑞(𝑥𝑥)𝑢𝑢(𝑥𝑥,𝛼𝛼)}∀𝑞𝑞∈𝐿𝐿2(𝐷𝐷) is dense in L2(D). Corollary 2. Given h ∈ L2(D) and 𝜖𝜖 > 0, one can find q ∈ L2(D) such that ‖ℎ(𝑥𝑥) − 𝑞𝑞(𝑥𝑥)𝑢𝑢(𝑥𝑥,𝛼𝛼)‖𝐿𝐿2(𝐷𝐷) < 𝜖𝜖. Since the scattering amplitude A(β) = − 1 4𝜋𝜋 ∫ 𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖·𝑥𝑥ℎ(𝑥𝑥)𝑑𝑑𝑥𝑥𝐷𝐷 depends continuously on h, the inverse problem IP is solved by Lemmas 3 and 4. Proofs are omitted. They can be found in refer- ence [31]. 5. Discussion How is the theory, outlined in the previous sec- tions, can be used practically? To create a material with a desired refraction coefficient, or a material with a refraction coefficient close to the desired, is practically very important. To my knowledge, there were no general methods for creating material with a desired refraction coefficient. To use the theory, outlined in this paper and in the monographs[31–33], one has to solve a technological problem: how to prepare a small particle, say, a ball of radius a, with the prescribed boundary impedance ζ. This problem should be solvable, see reference [33] for arguments supporting this conclusions. If this technological problem is solved, then the recipe outlined in this paper (and in the author’s mono- graphs[31–33] can be immediately used in practice. The problem of creating materials with a de- sired radiation pattern, the wave focusing materials, for example, was not investigated earlier. This prob- lem is of great practical interest. The usual bodies scatter waves mostly backwards, somewhat sidewise and a little forwards. If one creates a body which scatters waves, for example, in a given solid angle, this would be of great practical interest. Such a body can be created as follows from the theory outlined in the previous Section. The author wrote this paper in an attempt to draw attention of the specialists in material sciences to the theory he has developed for creating materials with the desired refraction coefficient. The author is not aware of the experimental re- sults based on his theory. Such results are very desir- able. There are numerical results, based on his theory, see references [37] and [38]. Conflict of interest Author declares no conflict of interest. References 1. Rayleigh J. Scientific papers. 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