Slender-body theory for plasmonic resonance

We develop a slender-body theory for plasmonic resonance of slender metallic nanoparticles, focusing on a general class of axisymmetric geometries with locally paraboloidal tips. We adopt a modal approach where one first solves the plasmonic eigenvalue problem, a geometric spectral problem which governs the surface-plasmon modes of the particle; then, the latter modes are used, in conjunction with spectral-decomposition, to analyse localized-surface-plasmon resonance in the quasi-static limit. We show that the permittivity eigenvalues of the axisymmetric modes are strongly singular in the slenderness parameter, implying widely tunable, high-quality-factor, resonances in the near-infrared regime. For that family of modes, we use matched asymptotics to derive an effective eigenvalue problem, a singular non-local Sturm-Liouville problem, where the lumped one-dimensional eigenfunctions represent axial voltage profiles (or charge line densities). We solve the effective eigenvalue problem in closed form for a prolate spheroid and numerically, by expanding the eigenfunctions in Legendre polynomials, for arbitrarily shaped particles. We apply the theory to plane-wave illumination in order to elucidate the excitation of multiple resonances in the case of non-spheroidal particles.