Motivated by the physics literature on “photonic doping” of scatterers made from “epsilon-near-zero” (ENZ) materials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region $\mathrm{\Omega }\times \mathbb{R}$ is affected by the presence of a “dopant” $D\subset \mathrm{\Omega }$ in which the dielectric permittivity is not near zero. Mathematically, this reduces to analysis of a 2D Helmholtz equation $\mathrm{div}(a(x)\nabla u)+k^{2}u=f$ with a piecewise-constant, complex valued coefficient a that is nearly infinite (say $a=\frac{1}{\delta }$ with $\delta \approx 0$) in $\mathrm{\Omega }\setminus \overline{D}$. We show (under suitable hypotheses) that the solution u depends analytically on δ near 0, and we give a simple PDE characterization of the terms in its Taylor expansion. For the application to photonic doping, it is the leading-order corrections in δ that are most interesting: they explain why photonic doping is only mildly affected by the presence of losses, and why it is seen even at frequencies where the dielectric permittivity is merely small. Equally important: our results include a PDE characterization of the leading-order electric field in the ENZ region as $\delta \to 0$, whereas the existing literature on photonic doping provides only the leading-order magnetic field.

中文翻译：

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ENZ 材料中介电常数的复杂分析依赖性：光子掺杂示例

受有关“ε 近零”(ENZ) 材料制成的散射体“光子掺杂”的物理文献的启发，我们考虑圆柱 ENZ 区域如何散射时谐 TM 电磁波$\mathrm{\Omega }\times \mathbb{右}$受到“掺杂剂”存在的影响$D\subset \mathrm{\Omega }$其中介电常数不接近于零。从数学上讲，这可以简化为二维亥姆霍兹方程的分析 $\mathrm{分区}（A（X）\nabla 你）+k^2你=F$具有接近无穷大的分段常数、复值系数a （例如$A=\frac{1}{\delta }$和$\delta \approx 0$） 在$\mathrm{\Omega }\setminus \stackrel{￣}{D}$。我们表明（在适当的假设下）解u在分析上依赖于接近 0 的 δ，并且我们给出了泰勒展开式中各项的简单 PDE 表征。对于光子掺杂的应用，最有趣的是 δ 的前阶校正：它们解释了为什么光子掺杂仅受到损耗存在的轻微影响，以及为什么即使在介电常数仅为小的。同样重要的是：我们的结果包括 ENZ 区域中前阶电场的偏微分方程表征：$\delta \to 0$，而现有的光子掺杂文献仅提供了前阶磁场。