许多具有负虚表面阻抗的小孔的簇可以产生负折射率
A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index
摘要：
我们处理的散射声介质的折射率$n$在有界区域$Ω$的$\mathbb{R}^3$和等于单位外$Ω$变化。该区域被穿孔有极大量的最大半径$a$、$a<<1$的小孔$D_m$，其由表面阻抗函数建模。准确地说，我们是在政权所描述的数量孔的顺序$M:=O(a^{β-2})$，孔之间的最小距离是$d\sim a^t$和表面阻抗函数的形式$λ_m \sim λ_{m,0} a^{-β}$与$β>0$和$λ_{m,0}$是常数，最终复数。在参数$β, t$和$λ_{m,0}$的某些自然条件下，我们描述了产生近似与原始穿孔声学介质相同的散射波的等效介质。我们给出了一个显式的误差估计之间的散射波的穿孔介质和等效的一个分别，作为$a \rightarrow 0$。作为这些结果的应用，我们讨论了以下发现：1.如果我们选择负值虚表面阻抗函数，附加到每个孔的表面，那么等效介质仅当它是具有折射率$\tilde{n}(x)=-n(x),\; x \in Ω$和$\tilde{n}(x)=1,\; x \in \mathbb{R}^3\setminus{\overlineΩ}$的声学超材料时才表现为无源声学介质。这意味着，通过这个过程，我们可以将折射率的符号从正值切换到负值。2.我们可以选择附着到孔的每个表面的表面阻抗函数，使得等效折射率$\tilde{n}$为$\tilde{n}(x)=1,\; x \in \mathbb{R}^3$。这意味着由原始折射率$n$建模的区域$Ω$近似地被遮盖。
英文摘要：
We deal with the scattering of an acoustic medium modeled by an index of refraction $n$ varying in a bounded region $Ω$ of $\mathbb{R}^3$ and equal to unity outside $Ω$. This region is perforated with an extremely large number of small holes $D_m$'s of maximum radius $a$, $a<<1$, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order $M:=O(a^{β-2})$, the minimum distance between the holes is $d\sim a^t$ and the surface impedance functions of the form $λ_m \sim λ_{m,0} a^{-β}$ with $β>0$ and $λ_{m,0}$ being constants and eventually complex numbers. Under some natural conditions on the parameters $β, t$ and $λ_{m,0}$, we characterize the equivalent medium generating, approximately, the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one respectively, as $a \rightarrow 0$. As applications of these results, we discuss the following findings: 1. If we choose negative valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde{n}(x)=-n(x),\; x \in Ω$ and $\tilde{n}(x)=1,\; x \in \mathbb{R}^3\setminus{\overlineΩ}$. This means that, with this process, we can switch the sign of the index of the refraction from positive to negative values. 2. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde{n}$ is $\tilde{n}(x)=1,\; x \in \mathbb{R}^3$. This means that the region $Ω$ modeled by the original index of refraction $n$ is approximately cloaked.
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