Asymptotic approximations for the plasmon resonances of nearly touching spheres

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Title: Asymptotic approximations for the plasmon resonances of nearly touching spheres
Authors: Schnitzer, O
Item Type: Journal Article
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.
Issue Date: 1-Apr-2020
Date of Acceptance: 13-Dec-2018
DOI: 10.1017/S0956792518000712
ISSN: 0956-7925
Publisher: Cambridge University Press (CUP)
Start Page: 246
End Page: 276
Journal / Book Title: European Journal of Applied Mathematics
Volume: 31
Issue: 2
Copyright Statement: © 2019 Cambridge University Press. This paper has been accepted for publication and will appear in a revised form, subsequent to peer-review and/or editorial input by Cambridge University Press.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/R041458/1
Keywords: Applied Mathematics
0102 Applied Mathematics
Publication Status: Published
Online Publication Date: 2019-01-09
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences

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