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How Can One Create a Material with a Prescribed Refraction Coefficient?

Ramm AG

Published on: 2020-10-07

Abstract

A recipe is given for creating materials with prescribed refraction coefficient. This recipe would be practically applicable if there is method for preparing small particles with prescribed boundary impedance. The problem of practical preparing small impedance particles with prescribed boundary impedance is formulated and its importance in physics and technology is emphasized. It is shown that if this problem is solved then one can easily prepare materials with a desired refraction coefficient, in particular, meta-materials. One can also prepare materials with a desired radiation pattern.

Keywords

Scattering theory; Materials science

Introduction

This is a brief review of the author's work cited in the references. The most important references are [1-4]. The author's theory consists of an elective solution to the many-body scattering problem for small particles of an arbitrary shape, see [5,6]. The practical importance of a method for producing small particles with prescribed boundary impedance is emphasized in [7,8]. Some portions of the text of this paper is borrowed. The authors theory of inverse scattering is presented [9,10]. Monographs present the theory of wave scattering by obstacles and potentials. In section 1 of this paper the basic problem of practical preparing (producing, manufacturing) of small impedance particles is discussed. Wave scattering by many small impedance particles is developed in, where the basic physical assumption is a<<d<<. Here a is the characteristic size of the small particles, d is the minimal distance between neighboring particles, and _ is the wave length in the medium. We refer for this theory [4]. It is proved there that if one prepares many small particles with prescribed boundary impedance and embed these particles (with a specified distribution density, see formula (7) below) into a given material, then one obtains a material whose refraction coefficient approximates any desired refraction coefficient with an arbitrary small error. In particular, one can create meta-materials. How to embed small particles into a given material physicists know. Therefore the basic practical problem of preparing small impedance particles with a prescribed boundary impedance is of great interest both technologically and physically. In section 2 of this paper the basic definitions are given. It is explained what a small particle is, what an impedance particle is and what boundary impedance is. In section 3 of this paper a recipe for creating materials with a desired refraction coefficient is formulated [1].

Basic Definitions

Let  be a bounded domain with a connected smooth boundary: = be the unbounded exterior domain and  be the unit sphere in.

Consider the scattering problem:

Where k>0 is the wave number, a constant, is a unit vector in the direction of the propagation of the incident plane wave  is the unit normal to pointing out of,  is the normal derivative of  is the boundary impedance,  is the refraction coefficient of the small impedance particle  is a constant, the scattered field  satisfies the radiation condition

The scattering amplitude

Where is the direction of the scattered wave, is the direction of the incident wave. A particle of a characteristic size  is called small if it is much smaller than the wave length, that is,

The function that the solution to the scattering problem does exist and is unique [11-13]. If there are many  small impedance particles  embedded in a bounded domain  filled with material whose refraction coefficient is, then the wave scattering problem can be formulated as follows:

                                                          In      (4)

Where the scattered field  satisfies the radiation condition,  is the total number of the embedded particles. The incident field  is assumed known. It satisfies equation (4) in  and, as was noted earlier,  in. For simplicity we assume that all small particles are of the same characteristic size

Let  be the Green's function of the scattering problem in the absence of the embedded particles. Outside  the refraction coefficient  is assumed to be equal to 1.Assume that the distribution of small particles is given by the formula

Where  is the number of particles in an arbitrary open subset  of  is a given continuous function,  is a number, and the boundary impedance is defined as follows [14]:

Where),  is an arbitrary point and  is a given continuous function in  this function, number  and the function  can be chosen by the experimenter. The field  in satisfies, as  the following integral equation:

Where  is the surface area of a small particle,  and  are defined in (7)-(8). For simplicity we assume here that the surface area  is the same for all small particles. It follows from (9) that the new refraction coefficient in  which one gets after embedding many small impedance particles [15-17].

 

Since  and  are at our disposal, one can get by formula (10) any desirable refraction coefficient such that

Why should the equation (5) make sense physically regardless of the size of the particle?

Because a problem whose solution exists and is unique must have sense physically.

Why should the small impedance particles with a prescribed boundary impedance exist?

Because the particles with  acoustically soft particles, do exist, and the particles with  acoustically hard particles, do exist, we conclude that small particles with any "intermediate" value of the boundary impedance should also exist.

The problem we raise is:

How can one produce practically (fabricate) such particles?

Recipe for Creating Material with a Desired Refraction Coefficient

Let us formulate the result in the following theorem:

Theorem 1: Given  and a bounded domain  one can create in D a material with a desired refraction Coefficient  by embedding  small impedance particles according to the distribution law (7). The refraction coefficient  corresponding to a finite a, approximates the desired refraction coefficient  in the sense

The functions  and  defined in (7)-(8) are found by the Steps 1, 2 of the Recipe formulated below. Finally, let us discuss briefly the possibility to create material with negative refraction coefficient

one gets  if the argument of  is equal to  Assume that we know that

Let us take, where  is very small, that is,  and is very small. Equation (9) is uniquely solvable for  sufficiently small. For such  one concludes that the argument (x)- is very close to and the square root in (10) is negative, provided that (x)- This argument shows that it is possible to create materials with negative refraction coefficient  by embedding in a given material many small particles with properly chosen boundary impedances.

A recipe for creating materials with a desired refraction coefficient

Problem 1: Given a material with known  in a bounded domain  and  in  one wants to create in  a material with a desired

Step 1: Given and (x) calculate (x)]. This is a trivial step.

Step 2: Given  calculate  and  from the equation

The constant  one can take to be  if the small particles are balls of radius. This we can assume without loss of generality if we are interested in creating materials with a desired refraction coefficient.

There are infinitely many solutions and  to the above equation. For example, one can fix arbitrarily  in  in  and find  by the formulas  . Here

Note that  implies  so our assumption is satisfied. Step 2 is also trivial.

Step 3: Given  and  distribute  small impedance balls of radius in the bounded region  according to the distribution law (7), where  is the number (parameter) that can be chosen by the experimenter. Note that condition  is satisfied automatically for the distribution law (7). Indeed, if  is the minimal distance between neighboring particles, then there are at most  particles in a cube with the unit side, and since  is a bounded domain there are at most  of small particles in . On the other hand, by the distribution law (7) one has  Thus. Therefore  Consequently, condition  is satisfied,  moreover

References

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  8. Ramm AG. Scattering by obstacles. 1986.
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  13. Ramm AG. Uniqueness theorem for inverse scattering problem with non-overdetermined data. J Phys A. 2010; 43: 112001.
  14. Ramm AG. Uniqueness of the solution to inverse scattering problem with backscattering data. Eurasian Math J. 2010; 1: 97-111.
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