Plasmonic resonances of slender nanometallic rings

We develop an approximate quasi-static theory describing the low-frequency plasmonic resonances of slender nanometallic rings and configurations thereof. In particular, we use asymptotic arguments to reduce the plasmonic eigenvalue problem governing the geometric (material- and frequency-independent) modes of a given ring structure to a 1D-periodic integro-differential problem in which the eigenfunctions are represented by azimuthal voltage and polarization-charge profiles associated with each ring. We obtain closed-form solutions to the reduced eigenvalue problem for azimuthally invariant rings (including torus-shaped rings but also allowing for non-circular cross-sectional shapes), as well as coaxial dimers and chains of such rings. For more general geometries, involving azimuthally non-uniform rings and non-coaxial structures, the reduced eigenvalue problem is solved using a semi-analytical scheme based on Fourier expansions of the reduced eigenfunctions. In conjunction with the quasi-static spectral theory of plasmonic resonance, the geometric modes of a given nanometallic structure explicitly determine its response to external radiation. Here, we employ the asymptotically approximated modes described above to obtain comparable approximations for the frequency response of a wide range of nanometallic slender-ring structures under plane-wave illumination. These examples demonstrate how the theory can be employed to interpret and geometrically tune the plasmonic properties of highly nontrivial 3D nanometallic structures.