Sub-wavelength plasmonic crystals: dispersion relations and effective properties
Sub-wavelength plasmonic crystals: dispersion relations and effective properties
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亚波长等离子体晶体：色散关系和有效特性
DOI：
10.1098/rspa.2009.0542
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
S. Shipman
中科院分区：
文献类型：
--
作者：
Santiago P. Fortes;R. Lipton;S. Shipman
我们得到了由介电常数随频率变化的等离子体棒组成的亚波长等离子体晶体色散关系第一分支的收敛级数展开式。展开参数为η=kd=2πd/λ，其中k是固定波矢的范数，d是晶体的周期，λ是波长，等离子体频率与d成反比，使棒中的介电常数变大且为负值。η中的级数系数(也称为动态校正器)和收敛半径的表达式与高阶胞元问题的解和杆的几何形状明确相关。在收敛半径内，我们能够以严格的数学方式计算色散关系和场，并定义动态有效性质。显式误差估计表明，仅用几个展开式就能很好地逼近真正的色散关系。收敛证明要求利用加泰罗尼亚数的性质来证明级数系数在H1Soblev范数下指数有界。
We obtain a convergent power series expansion for the first branch of the dispersion relation for sub-wavelength 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 (also called 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 H1 Sobolev norm.