How Can One Create a Material with a Prescribed Refraction Coefficient?
Published on: 2020-10-07
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.
KeywordsScattering theory; Materials science
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 . 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 .
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:
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 :
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
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