亚波长等离子体激元晶体的色散关系和有效特性
Sub-Wavelength Plasmonic Crystals: Dispersion Relations and Effective Properties
摘要：
我们得到了一个收敛的幂级数展开的色散关系的第一支等离子体晶体的等离子体激元杆与频率依赖的介电常数嵌入在一个主机介质与单位介电常数。展开参数是$η=kd=2πd/λ$，其中$k$是固定波矢量的范数，$d$是晶体的周期，$λ$是波长，等离子体频率与$d$成反比，使得杆中的介电常数大且为负。级数系数（a.k.a.，动态校正器）和$η$中的收敛半径与高阶单元问题的解和杆的几何形状明确相关。在收敛半径内，我们能够计算色散关系和场，并以数学上严格的方式定义动态有效特性。明确的误差估计表明，一个很好的近似的真实色散关系，得到只有几个条款的扩展。收敛性证明需要使用Catalan数的性质来证明级数系数在$H^1$ Sobolev范数下是指数有界的。
英文摘要：
We obtain a convergent power series expansion for the first branch of the dispersion relation for subwavelength plasmonic crystals consisting of plasmonic rods with frequency-dependent dielectric permittivity embedded in a host medium with unit permittivity. The expansion parameter is $η=kd=2πd/λ$, where $k$ is the norm of a fixed wavevector, $d$ is the period of the crystal and $λ$ is the wavelength, and the plasma frequency scales inversely to $d$, making the dielectric permittivity in the rods large and negative. The expressions for the series coefficients (a.k.a., dynamic correctors) and the radius of convergence in $η$ are explicitly related to the solutions of higher-order cell problems and the geometry of the rods. Within the radius of convergence, we are able to compute the dispersion relation and the fields and define dynamic effective properties in a mathematically rigorous manner. Explicit error estimates show that a good approximation to the true dispersion relation is obtained using only a few terms of the expansion. The convergence proof requires the use of properties of the Catalan numbers to show that the series coefficients are exponentially bounded in the $H^1$ Sobolev norm.
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