Geometric quantization of localized surface plasmons

摘要: We consider the quasi-static problem governing localized surface plasmon modes and permittivity eigenvalues $\epsilon$ of smooth, arbitrarily shaped, axisymmetric inclusions. develop an asymptotic theory for dense part spectrum, i.e., close to accumulation value $\epsilon=-1$ at which a flat interface supports plasmons; in this regime, field oscillates rapidly along decays exponentially away from it on comparable scale. With $\tau=-(\epsilon+1)$ as small parameter, we surface-ray description eigenfunctions narrow boundary layer about interface; fast phase variation, well slowly varying amplitude geometric phase, rays are determined functions local geometry. focus most moderately azimuthal direction, case meridian arcs that two poles. Asymptotically matching diverging ray solutions with expansions valid inner regions vicinities poles yields quantization rule $$\frac{1}{\tau} \sim \frac{\pi n }{\Theta}+\frac{1}{2}\left(\frac{\pi}{\Theta}-1\right)+o(1),$$ where $n\gg1$ is integer $\Theta$ parameter given by product inclusion length reciprocal average its cross-sectional radius symmetry axis. For sphere, $\Theta=\pi$, whereby formula returns exact $\epsilon_n=-1-1/n$. also demonstrate good agreement prolate spheroids.