Geometric quantization of localized surface plasmons

Abstract

We consider the quasi-static problem governing the localized surface plasmon modes and permittivity eigenvalues ǫ of smooth, arbitrarily shaped, axisymmetric inclusions. We develop an asymptotic theory for the dense part of the spectrum, i.e., close to the accumulation value ǫ = −1 at which a flat interface supports surface plasmons; in this regime, the field oscillates rapidly along the surface and decays exponentially away from it on a comparable scale. With τ = −(ǫ + 1) as the small parameter, we develop a surface-ray description of the eigenfunctions in a narrow boundary layer about the interface; the fast phase variation, as well as the slowly varying amplitude and geometric phase, along the rays are determined as functions of the local geometry. We focus on modes varying at most moderately in the azimuthal direction, in which case the surface rays are meridian arcs that focus at the two poles. Asymptotically matching the diverging ray solutions with expansions valid in inner regions in the vicinities of the poles yields the quantization rulewhere n ≫ 1 is an integer and Θ a geometric parameter given by the product of the inclusion length and the reciprocal average of its cross-sectional radius along its symmetry axis. For a sphere, Θ = π, whereby the formula returns the exact eigenvalues ǫ n = −1 − 1/n. We also demonstrate good agreement with exact solutions in the case of prolate spheroids.

Slender-body theory for plasmonic resonance

“…Conversely, fixing h, the longitudinal modes vary faster and faster along the particle axis, on a scale O(a/n), as we increase the mode number n; accordingly, the scale disparity between the longitudinal variations and the thickness is reduced and we expect the slender-body approximation to quickly deteriorate (as indeed observed in § §4(a)). In the latter high-mode-number limit, one of us recently derived a two-term eigenvalue quantisation rule for a general class of smooth axisymmetric particles, using WKBJ methods and matched asymptotic expansions [68]. In order to connect the present theory, for n fixed, with the latter one, for h fixed, it would be necessary to consider the intermediate limit…”

Modal approximation for plasmonic resonators in the time domain: the scalar case

“…Remark 3 (On high-order modes) The oscillation rate of high-order modes has been quantified in the case of smooth axisymmetric inclusions [39]. Moreover, it has been also shown in that case that the contribution of high-order modes to the electric field outside the particle decays exponentially as the distance from the inclusion increases.…”

Scattering resonances in unbounded transmission problems with sign-changing coefficient

“…It turns out, scattering resonances are associated to localized waves corresponding to surface plasmons waves. This study has been carried out without reducing to the quasi-static case, and the considered spectral parameter is the wavenumber in contrast to [23,41,1,2]. Our asymptotic analysis revealed that, given some incident source associated to k > 0, surface plasmon waves can only be excited when a c < −1 (in the case −1 < a c < 0 the scattering resonances are purely imaginary).…”