  We develop an approximate quasi-static theory describing the low-frequency
plasmonic resonances of slender nanometallic rings and configurations thereof.
First, 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. Second, 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,
we solve the reduced eigenvalue problem using a semi-analytical scheme based on
Fourier expansions of the reduced eigenfunctions. Third, we used the
asymptotically approximated modes, in conjunction with the quasi-static
spectral theory of plasmonic resonance, to study and interpret the frequency
response of a wide range of nanometallic slender-ring structures under
plane-wave illumination.
