A Helmholtz Equation for Surface Plasmon Polaritons on Curved Interfaces: Controlling Cooperativity with Geometric Potentials

TL;DR

This paper derives a covariant Helmholtz equation for surface plasmon polaritons on curved interfaces, revealing geometry-induced effects and predicting curvature-dependent phenomena in plasmonic systems.

Contribution

The work introduces a first-order curvature-dependent Helmholtz equation for surface plasmons, including geometric potentials that distinguish convex from concave interfaces.

Findings

Reproduces known results for spherical and cylindrical interfaces.

Predicts vanishing anisotropic potential when permittivity ratio equals the square of the golden ratio.

Demonstrates curvature-driven redistribution of quantum emitter decay rates.

Abstract

Surface plasmon polaritons propagating along curved metal-dielectric interfaces experience geometry-induced modifications absent on flat surfaces. In this work, we derive a covariant, effective two-dimensional wave equation for the transverse magnetic surface plasmon mode on weakly curved smooth interfaces. By perturbatively expanding Maxwell's equations with curvature-adapted boundary conditions, we find a Helmholtz equation with two geometric potential terms that enter at first order in the extrinsic curvature: an isotropic contribution proportional to the extrinsic curvature, and an anisotropic operator arising from the traceless part of the second fundamental form. These linear-in-curvature potentials distinguish convex from concave interfaces, in contrast to the quadratic potentials known from symmetrically confined systems such as dielectric waveguides. We show that our equation reproduces established results for spherical and cylindrical interfaces. We furthermore predict that the anisotropic contribution vanishes when the ratio of the material permittivities equals the square of the golden ratio. As an application, we demonstrate sign-dependent cooperative frequency shifts as well as a curvature-driven redistribution of superradiant and subradiant decay rates for a ring of quantum emitters on a curved metallic spheroid interacting through the surface plasmons.