Introduction

Our method for creating materials with a desired refraction coefficient consists of distributing many small impedance particles with prescribed boundary impedance. Let us define first the notion of the refraction coefficient. Let $u(x,k)$ be a wave field satisfying the equation

$$\begin{eqnarray}&&\left[{{\rm{\nabla }}}^{2}+{k}^{2}{n}_{0}^{2}(x)\right] u=0 {\rm{in}}\quad {{\mathbb{R}}}^{3},\end{eqnarray} \tag{ 1.1 }$$

where ${{\mathbb{R}}}^{3}$ is the three-dimensional space, $x\in {{\mathbb{R}}}^{3}$, $k\gt 0$ is a wave number.

The coefficient ${n}_{0}(x)$ is called the refraction coefficient of the medium.

We assume that

$$\begin{eqnarray}&&{\rm{Im}}\;{n}_{0}^{2}(x)\geqslant 0.\end{eqnarray} \tag{ 1.2 }$$

Physically, this means that the medium does not produce energy. We assume that outside of an arbitrary large but fixed area, for example, a ball ${B}_{R}=\{x:| x| \leqslant R\}$, ${n}_{0}(x)=1$.

The basic question is:

Suppose ${n}_{0}(x)$ is the refraction coefficient of some material in ${{\mathbb{R}}}^{3}$. How many particles of the characteristic dimension a and the boundary impedance $\zeta (x)$ should be distributed in B_R in order to get a new material in B_R with the desired refraction coefficient $n(x)$?

The wavelength ${\lambda }_{0}$ in the medium with the refraction coefficient ${n}_{0}(x)=1$ can be calculated by the formula ${\lambda }_{0}=\frac{2\pi }{k}$. The smallness of a particle of the characteristic dimension a is described by the relation

$$\begin{eqnarray}&&a\ll \frac{{\lambda }_{0}}{{n}_{0}},\qquad {n}_{0}:= \underset{x\in {{\mathbb{R}}}^{3}}{\mathrm{max}}\quad |{n}_{0}(x)|.\end{eqnarray} \tag{ 1.3 }$$

Wave scattering by the impedance particles in a medium is described by the equations

$$\begin{eqnarray}&&\left({{\rm{\nabla }}}^{2}+{k}^{2}{n}_{0}^{2}(x)\right)u=0\quad {\rm{in}}\quad {{\mathbb{R}}}^{3}\backslash D,\quad D:= {\cup }_{m=1}^{M}{D}_{m},\end{eqnarray} \tag{ 1.4 }$$

$$\begin{eqnarray}&&{u}_{N}={\zeta }_{m}u\quad {\rm{on}}\quad {S}_{m},\quad 1\leqslant m\leqslant M,\end{eqnarray} \tag{ 1.5 }$$

$$\begin{eqnarray}&&u={u}_{0}+v,\end{eqnarray} \tag{ 1.6 }$$

$$\begin{eqnarray}&&\frac{\partial v}{\partial r}-{ikv}=o\left(\frac{1}{r}\right),\quad r=| x| \to \infty .\end{eqnarray} \tag{ 1.7 }$$

Here ${u}_{0}(x)$ is the incident field. It satisfies the equation:

$$\begin{eqnarray}&&\left({{\rm{\nabla }}}^{2}+{k}^{2}{n}_{0}^{2}(x)\right)\quad {u}_{0}=0\quad {\rm{in}}\quad {{\mathbb{R}}}^{3},\end{eqnarray} \tag{ 1.8 }$$

the dependence on k is dropped since $k\gt 0$ is constant, v is the scattered field, D_m is mth small particle, S_m is its sufficiently smooth boundary (surface), N is a unit normal to S_m pointing out of D_m , ${\zeta }_{m}$ is the boundary impedance of D_m , Im ${\zeta }_{m}\leqslant 0$, M is the number of small particles. For simplicity only we assume in what follows that ${n}_{0}(x)=1$. In a separate section at the end of this book we explain why this assumption is not important for our theory.

An important physical quantity d is defined as the minimal distance between neighboring particles

$$\begin{eqnarray}&&d=\underset{j,m}{\mathrm{min}}\quad {{\rm{dist}}}_{m\ne j}({D}_{m},{D}_{j}).\end{eqnarray} \tag{ 1.9 }$$

Let us assume that

$$\begin{eqnarray}&&{\zeta }_{m}=\frac{h\left({x}_{m}\right)}{{a}^{\kappa }},\quad {\rm{Im}}\;h(x)\leqslant 0,\end{eqnarray} \tag{ 1.10 }$$

where ${x}_{m}\in {D}_{m}$, $h(x)$ is a continuous function in B_R , $\kappa \in [0,1)$ is a number, $a=\frac{1}{2}{\mathrm{max}}_{m}{\rm{diam}}{D}_{m}$.

The distribution of the small particles is described by the function

$$\begin{eqnarray}&&{ \mathcal N }({\rm{\Delta }})=\frac{1}{{a}^{2-\kappa }}{\int }_{{\rm{\Delta }}}N(x){dx}(1+o(1)),\quad a\to 0,\end{eqnarray} \tag{ 1.11 }$$

where ${\rm{\Delta }}\subset {B}_{R}$ is an arbitrary open set, $N(x)\geqslant 0$ is a continuous function, ${ \mathcal N }({\rm{\Delta }})$ is the number of small particles in Δ. In particular,

$$\begin{eqnarray}&&M=\frac{1}{{a}^{2-\kappa }}{\int }_{{B}_{R}}N(x){dx}(1+o(1))=O\left(\frac{1}{{a}^{2-\kappa }}\right),\quad a\to 0.\end{eqnarray} \tag{ 1.12 }$$

We prove that if the small particles are distributed by the law equation (1.11), then one can choose $h(x)$ that is boundary impedances, so that the solution to the many-body scattering problem has a limit u, as $a\to 0$, and this u solves the equation

$$\begin{eqnarray}&&[{{\rm{\nabla }}}^{2}+{k}^{2}{n}^{2}(x)]u=0,\end{eqnarray} \tag{ 1.13 }$$

where the refraction coefficient $n(x)$ can be an arbitrary function satisfying the condition Im ${n}^{2}(x)\leqslant 0$. Moreover, the limiting function ${n}^{2}(x)$ will be calculated explicitly:

$$\begin{eqnarray}&&{n}^{2}(x)={n}_{0}^{2}(x)-c\frac{h(x)N(x)}{{k}^{2}},\end{eqnarray} \tag{ 1.14 }$$

the constant c is defined by the formula

$$\begin{eqnarray}&&| {S}_{m}| ={{ca}}^{2},\end{eqnarray} \tag{ 1.15 }$$

where $| {S}_{m}|$ is the surface area of S_m , $h(x)$ is the function from equation (1.10) and $N(x)$ is the function from equation (1.11).

Our basic physical assumption is

$$\begin{eqnarray}&&a\ll d\ll \lambda ,\end{eqnarray} \tag{ 1.16 }$$

where d is defined in equation (1.9) and $\lambda =\frac{2\pi }{k}{\mathrm{min}}_{x\in {B}_{R}}| {n}_{0}(x)|$. Assumption equation (1.16) means that multiple scattering is essential: one cannot assume that the field acting on a particle D_m is the incident field u₀.

From equation (1.14) it is already clear that if $h(x)$ can be chosen arbitrarily with Im $h\leqslant 0$ then $n(x)$ can be obtained fairly arbitrarily.

In chapter 2 our theory of many-body wave scattering is presented for small impedance particles and formulas equations (1.13) and (1.14) are derived.

In chapter 3 a recipe for creating materials with a desired refraction coefficient is formulated and justified theoretically.

In chapter 4 we will discuss creating wave-focusing materials. These are materials such that a plane wave, scattered by such material, has a desired radiation pattern.

In chapter 5 some inverse scattering problems with non-over-determined scattering data are studied.