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

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.

Figures (3)

• Figure 1: Geometric control of surface plasmon polaritons and collective radiance. (a) Comparison between guided optical modes in a curved dielectric waveguide (top) and surface plasmon polaritons (SPPs) at a curved metal-air interface (bottom). In the dielectric case, the transverse mode profile decays symmetrically, producing a geometric potential quadratic in curvature that is insensitive to the sign of curvature. For SPPs, the field decays asymmetrically into the metal and dielectric, yielding a sign-dependent potential that distinguishes convex from concave interfaces: Convex curvature creates a potential well (blue-shift), while concave curvature creates a barrier (red-shift). (b) Covariant Helmholtz equation for the SPP envelope $\psi$ on a weakly curved surface ($R\gg\bar{\lambda}_{\rm spp}$). The geometric potential decomposes into a scalar contribution $C_H H$ and an anisotropic operator $C_\sigma\sigma^{ab}\nabla_a\nabla_b$. The right column shows the resulting local momentum-space dispersion: On a flat surface ($H=0$, $\sigma^{ab}=0$), the allowed wavevectors form a circle of radius $k_{\rm spp}$ (shown as the dashed line reference for all cases); extrinsic curvature ($H\neq 0$, $\sigma^{ab}=0$) uniformly shifts this circle to a larger (convex, $k > k_{\rm ssp}$, as shown) or smaller (concave, $k < k_{\rm ssp}$) radius; anisotropic curvature ($\sigma^{ab}\neq 0$) deforms the flat-SPP circle into an ellipse, producing direction-dependent SPP momentum. (c) Curvature-mediated control of collective radiance. Varying the eccentricity of a metallic spheroid modifies the SPP-mediated interactions between quantum emitters (red dipoles oriented normally to the surface), redistributing the collective spectrum between superradiant (bright) and subradiant (dark) modes.
• Figure 2: Curvature-dependent modulation of collective eigenvalues. Collective energy shifts [left lower panels in (a) and (b)] and decay rates [right lower panels in (a) and (b)] for a ring of $N=9$ emitters positioned near the pole of a silver--air spheroidal interface at $\lambda_0=600\,\mathrm{nm}$. All quantities are normalized to the corresponding single-emitter decay rate $\gamma_0^{\rm curved}$ on the same surface. The nearest-neighbor emitter spacing (geodesic distance) is set to $3\,\bar{\lambda}_{\rm spp}$ and is kept constant while the surface geometry is varied. (a) Collective eigenvalues for emitters on a spherical surface as a function of the dimensionless curvature $H\bar{\lambda}_{\rm spp}$. Negative (positive) values of $H\bar{\lambda}_{\rm spp}$ correspond to convex (concave) spherical interfaces, while $H\bar{\lambda}_{\rm spp}\!=\!0$ equals the planar limit, indicated by the vertical solid lines. (b) Collective eigenvalues for emitters on spheroidal surfaces as a function of the aspect ratio $c/a$, ranging from oblate ($c/a<1$) to prolate ($c/a>1$), with $c/a=1$ corresponding to the sphere, indicated by the vertical solid lines. The spherical case in (b) corresponds to $H\bar{\lambda}_{\rm spp}=-0.016$ in (a). Black dashed horizontal lines in (a) and (b) denote the analytical reference values for the planar interface. The full collective spectrum is shown in dark gray, while a selected eigenmode (solid arrow) is highlighted with a color gradient (blue to red), indicating its evolution from convex to concave curvature in (a) and from oblate to prolate shape in (b).
• Figure 3: The intrinsic curvature does only weakly change the collective radiance. Collective energy shifts [left lower panels] and decay rates [right lower panel] for a ring of $N=9$ emitters positioned near the pole of a silver--air spherical interface at $\lambda_0=600\,\mathrm{nm}$. The nearest-neighbor emitter spacing (geodesic distance) is set to $3\,\bar{\lambda}_{\rm spp}$ and is kept constant while the sphere radius is varied. The scalar potential term is artificially set to $V_H=0$ such that all effects of curvature originate from the intrinsic curvature contributions in $\Delta_\gamma$ only. All quantities are normalized to the corresponding single-emitter decay rate $\gamma_0^{\rm curved}$ on the same surface. The nearest-neighbor emitter spacing (arc length) is kept constant while the surface geometry is varied. The eigenvalues are plotted as a function of the dimensionless curvature $H\bar{\lambda}_{\rm spp}$. Negative (positive) values correspond to convex (concave) spherical interfaces, while $H\bar{\lambda}_{\rm spp}\!=\!0$ equals the planar limit, indicated by the vertical solid lines.