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Asymptotic approximations for the plasmon resonances of nearly touching spheres

Part of: Representations of solutions Spectral theory and eigenvalue problems Optics, electromagnetic theory - Basic methods

Published online by Cambridge University Press:  09 January 2019

O. SCHNITZER
O. SCHNITZER*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: [email protected]
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Abstract

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.

Keywords

Surface plasmonsasymptotic analysiseigenvalue problems

MSC classification

Primary: 78M35: Asymptotic analysis
Secondary: 35C20: Asymptotic expansions 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 2 , April 2020 , pp. 246 - 276
Copyright
© Cambridge University Press 2019 

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Footnotes

The author acknowledges funding from EPSRC New Investigator Award EP/R041458/1.

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