具有Drude-Lorentz折射率的Helmholtz方程远场宽带近似隐身
Far field broadband approximate cloaking for the Helmholtz equation with a Drude-Lorentz refractive index
AI总结：
本文利用变换光学和Drude-Lorentz折射率模型构造Helmholtz方程的近似隐身方案，证明其无法实现完美隐身，但散射场在ε→0时于任意有界频带上一致趋于零，从而实现宽带近似隐身。
摘要：
本文研究在${\mathbb R}^d$（$d=2$或$d=3$）中Helmholtz方程的一种无源宽带近似隐身方案的分析。利用变换光学思想，我们通过将一个半径为$ε>0$的小球“膨胀”为半径为$1$的球来构造近似隐身。在由“膨胀”变量变换产生的各向异性隐身层中，我们引入Drude-Lorentz型折射率模型，并假设被隐身物体是一个软（完全导电）障碍物。我们首先证明（对任意固定的$ε$）与表示隐身的非均匀性相关的实传输特征值不存在，这意味着我们创建的隐身装置在任何频率下都不会产生完美隐身，即使对于单个入射时谐波也是如此。其次，我们建立了由任意时谐入射波引起的散射场的估计。这些估计表明，当$ε$趋近于$0$时，隐身层外部散射场的$L^2$范数及其远场模式在任何有界频带上一致趋近于$0$。换句话说：我们的方案对任意入射时谐波实现了宽带近似隐身。
英文摘要：
This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in ${\mathbb R}^d$ for $d=2$ or $d=3$. Using ideas from transformation optics, we construct an approximate cloak by ``blowing up" a small ball of radius $ε>0$ to one of radius $1$. In the anisotropic cloaking layer resulting from the ``blow-up" change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed $ε$) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as $ε$ approaches $0$, the $L^2$-norm of the scattered field outside the cloak, and its far field pattern, approach $0$ uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves.
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