Site and lattice resonances in metallic hole arrays

F. J. García de Abajo; J. J. Sáenz; I. Campillo; J. S. Dolado

doi:10.1364/OPEX.14.000007

Optics Express
Vol. 14,
Issue 1,
pp. 7-18
(2006)
•https://doi.org/10.1364/OPEX.14.000007

Site and lattice resonances in metallic hole arrays

F. J. García de Abajo, J. J. Sáenz, I. Campillo, and J. S. Dolado

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F. J. García de Abajo, J. J. Sáenz, I. Campillo, and J. S. Dolado, "Site and lattice resonances in metallic hole arrays," Opt. Express 14, 7-18 (2006)

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Abstract

A powerful analytical approach is followed to study light transmission through subwavelength holes drilled in thick perfect-conductor films, showing that full transmission (100%) is attainable in arrays of arbitrarily narrow holes as compared to the film thickness. The interplay between resonances localized in individual holes and lattice resonances originating in the array periodicity reveals new mechanisms of transmission enhancement and suppression. In particular, localized resonances obtained by filling the holes with high-index-of-refraction material are examined and experimentally observed through large enhancement in the transmission of individual holes.

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Fig. 1. (a) The field scattered by a subwavelength hole drilled in a perfect-conductor film in response to external electric (E ⁰) and magnetic (H ⁰) fields is equivalent (at a large distance compared to the radius a) to that of effective electric (p) and magnetic (m) dipoles, which allow defining polarizabilities (α_E and α_M, respectively) both on the same side as the external fields (α_v) and on the opposite side (α´_v). Only the perpendicular component of the electric field and the parallel component of the magnetic field induce dipoles. (b)- (c) Real part of the hole response functions g^±_v for ε = μ = 1.

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Fig. 2. (a) Light transmittance through a circular hole drilled in a perfect-conductor film and filled with dielectric material for different values of the permittivity ε (see labels). The transmitted power is normalized to the incoming flux times the hole area. The ratio of the film thickness to the hole radius is 0.1. The left inset shows the transmission for ε = 50 in log scale. (b) Ratio of transmission for ε = 10.2 dielectric filling and ε = 1 (air) under the same conditions as in (a): theory (solid curve) vs experiment (symbols). The transmission through an infinite ε = 10.2 dielectric of the same thickness is shown as a dashed curve for comparison (dashed and solid curves approach each other in the high-frequency limit).

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Fig. 3. Lattice sum G_z [Eq. (2)] for a square lattice of period d as a function of parallel momentum k _∣ and wavelength.

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Fig. 4. Zeroth order light transmittance through square arrays of circular holes drilled in perfect-conductor films as a function of parallel momentum k _∥ and wavelength λ. The ratio of the hole radius to the lattice spacing is taken as a/d = 0.2. Different values of the film thickness t and the dielectric constant inside the holes ε are considered, as shown by labels. Both p-polarized (H parallel to the film) and s-polarized (E parallel to the film) incident light are examined.

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Fig. 5. Normal-incidence transmittance (top), lattice sums and hole response functions (middle), and hole polarizability [bottom; see Fig. 1(a)] under the same conditions as in Fig. 4(j) (a/d = 0.2, t/a = 0.1, ε = 100).

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Equations (19)

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Im {g_{v}^{\pm}} = Im {\frac{1}{α_{v} \pm {α'}_{v}}} = \frac{- {2 k}^{3}}{3},

P = α_{E} (E_{z}^{ext} + G_{z} P - Hm) + {α'}_{E} (G_{z} P' - Hm')

P' = {α'}_{E} (E_{z}^{ext} + G_{z} P - Hm) + α_{E} (G_{z} P' - Hm')

m = α_{M} (H_{y}^{ext} + G_{y} m - Hp) + {α'}_{M} (G_{y} m' - Hp')

m' = {α'}_{M} (H_{y}^{ext} + G_{y} m - Hp) + α_{M} (G_{y} m' + Hp'),

G_{j} = \sum_{R \neq 0} e^{- i k_{∣ ∣} x} (k^{2} + \partial_{j} \partial_{j}) \frac{e^{i kR}}{R}

H = - i k \sum_{R \neq 0} e^{- i k_{∣ ∣} x} \partial_{x} \frac{e^{i kR}}{R} .

p \pm p' = - \frac{2 [\frac{(g_{M}^{\pm} - G_{y}) k_{∥}}{k + H}]}{Δ_{\pm}}

m \pm m' = \frac{2 [\frac{(g_{E}^{\pm} - G_{z}) + H k_{∥}}{k}]}{Δ_{\pm}}

Δ_{\pm} = (g_{E}^{\pm} - G_{z}) (g_{M}^{\pm} - G_{y}) - H^{2} .

E^{trans} (r_{1}) = [(p' (k^{2} + \nabla_{1} \partial_{z 1}) - ikm' \hat{y} \times \nabla_{1}] \frac{e^{i k ∣ r_{1} - r ∣}}{∣ r_{1} - r ∣}

= [(p' (k^{2} + \nabla_{1} \partial_{z 1}) - ikm' \hat{y} \times \nabla_{1}] \int d^{2} {k'}_{∥} \frac{i}{2 π {k'}_{z}} e^{ik' ∙ (r_{1} - r)}

= \int d^{2} {k'}_{∥} \frac{i}{2 π {k'}_{z}} e^{ik' ∙ (r_{1} - r)} [({k'}_{z} \hat{x} - {k'}_{x} \hat{z}) km' - ({k'}_{x} {k'}_{z} \hat{x} + {k'}_{y} {k'}_{z} \hat{y} - {k'}_{∥}^{2} \hat{z}) p'],

E^{trans} (r_{1}) = \frac{2 πi}{A k_{z}} e^{ik ∙ r_{1}} (k_{z} \hat{x} - k_{∥} \hat{z}) (km' - k_{∥} p') .

T = {(\frac{2 π k_{z}}{A})}^{2} {∣ \frac{1}{g_{M}^{+} - G_{x}} - \frac{1}{g_{M}^{-} - G_{x}} ∣}^{2}

= {∣ \frac{1}{1 + \frac{iA}{2 π k_{z}} Re {g_{M}^{+} - G_{x}}} - \frac{1}{1 + \frac{iA}{2 π k_{z}} Re {g_{M}^{-} - G_{x}}} ∣}^{2} .

G_{j} \propto \frac{1}{\sqrt{{(\frac{k_{∥} + 2 πn}{d})}^{2} + {(\frac{2 πl}{d})}^{2} - k^{2}}} .

1 + {(\frac{A}{2 π k_{z}})}^{2} Re {g_{M}^{+} - G_{x}} Re {g_{M}^{-} - G_{x}} = 0

\frac{A}{2 π k_{z}} ∣ g_{M}^{+} - g_{M}^{-} ∣ > 1,

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Equations (19)

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