Abstract
We investigate the transmission properties of a metallic layer with narrow slits. Recent measurements and numerical calculations concerning the light transmission through metallic sub-wavelength structures suggest that an unexpectedly high transmission coefficient is possible. We analyze the time harmonic Maxwell's equations in the $H$-parallel case for a fixed incident wavelength. Denoting by $\eta>0$ the typical size of the complex structure, effective equations describing the limit $\eta\to 0$ are derived. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the channels. When these waves are in resonance with the height of the layer, the result can be perfect transmission through the layer.
Highlights
The interest to construct small scale optical devices for technical applications has initiated much research in the fields of micro- and nano-optics
One example is the behavior of metamaterials with a negative index, see [17] for a first investigation and [7] for a mathematical analysis
Another interesting and technically relevant example of the astonishing behavior of small scale structures is the high transmission of light through metallic layers with thin holes
Summary
The interest to construct small scale optical devices for technical applications has initiated much research in the fields of micro- and nano-optics. As a result we obtain an effective scattering problem in which the metallic layer is replaced by an effective material with frequency dependent permittivity εeff and permeability μeff The formulas for these effective parameters allow to evaluate the transmission coefficient T = T (k) of the total structure in terms of the incident wave number k. The mathematical tools of this contribution are related to those of [2, 3, 5], and [7], where the Maxwell equations in other singular geometries have been investigated Another application where the negative real part of the permittivity becomes relevant is cloaking by anomalous localized resonance, see [16] and the rigorous results in [6]
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