# Plasmonic resonances of slender nanometallic rings

Matias Ruiz  School of Mathematics, University of Edinburgh, Edinburgh EH9
3FD, UK  Ory Schnitzer  Department of Mathematics, Imperial College London,
London SW7 2AZ, UK

###### Abstract

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.

##  I Introduction

Localized-surface-plasmon resonance, namely the excitation of collective
electric-field and electron-charge-density oscillations in metallic
nanoparticles and nanostructures, has been instrumental over the past several
decades in enabling new techniques for manipulating visible and near-infrared
electromagnetic waves on nanometric scales, with applications including
sensing, targeted heating and metamaterials Maier (2007); Klimov (2014). An
important motif in fundamental research as well as applications has been the
use of nearly singular, i.e., multiple-scale, geometries to achieve tunable
and high quality resonances, field enhancement in nanoscale ‘hotspots’ and for
probing light-matter interactions and non-classical phenomena Romero  _et al._
(2006); Schuller  _et al._ (2010); Suh  _et al._ (2012); Chen  _et al._
(2013); Toscano  _et al._ (2013).

The class of multiple-scale nanometallic structures most extensively studied
in nanoplasmonics consists of closely spaced particles and similarly particles
very near to a metallic substrate (Tamaru  _et al._ , 2002; Gunnarsson  _et
al._ , 2005; Jain  _et al._ , 2007; Klimov and Guzatov, 2007; Aubry  _et al._
, 2010a, 2011; Lebedev  _et al._ , 2013; Pendry  _et al._ , 2013;
Kadkhodazadeh  _et al._ , 2014; Schnitzer, 2015; Schnitzer  _et al._ , 2016a,
b; Schnitzer, 2020; Yu and Ammari, 2018, 2019). Such structures exhibit a rich
spectrum comprised of several distinct families of modes characterized by
their symmetries, as well as behavior in the limit where the aspect ratio
$\kappa$, say the ratio of particle radius to gap width, is large. Particular
emphasis has been given to bonding-gap modes, which are characterized by
localization of the electric field to the gap, with the field directed across
the gap. For large $\kappa$, the characteristic angular frequencies $\omega$
associated with these modes behave roughly like $\omega/\omega_{p}\simeq
1/\kappa^{1/4}$, wherein $\omega_{p}$ denotes the plasma angular frequency
Schnitzer (2015). For large $\kappa$, the first few bonding-gap modes strongly
couple with incident radiation, giving rise to low-frequency resonances
associated with a giant field enhancement in the gap.

Another class of multiple-scale nanometallic structures that is commonly
employed consists of slender particles such as high-aspect-ratio nanorods and
spheroids (Link and El-Sayed, 1999; Sönnichsen  _et al._ , 2002; Prescott and
Mulvaney, 2006; Khlebtsov and Khlebtsov, 2007; Neubrech  _et al._ , 2008; Lu
_et al._ , 2011; Agha  _et al._ , 2014; Deng  _et al._ , 2021). Parallels can
be drawn between the bonding-gap modes of closely spaced particles and the
longitudinal modes of slender particles, which are characterized by
polarization-charge distributions that for sufficient slenderness vary mainly
along the particle axis. The latter longitudinal modes redshift as the
particle becomes more and more slender, analogously to the redshift of the
bonding-gap modes described above. In the slender-particle case, however, the
plasmon-frequency redshift is far more singular: $\omega/\omega_{p}\simeq
1/\kappa$, where now $\kappa$ represents the ratio of length to thickness
Prescott and Mulvaney (2006). Accordingly, the first few longitudinal modes of
a slender particle typically give rise to remarkably low-frequency plasmonic
resonances (down to the near-infrared regime), which are typically associated
with high quality factors and distinctive directional characteristics.

In comparison with straight slender particles, curved slender particles such
as slender rings and helixes are more subwavelength at their longitudinal
resonances. Indeed, let the length and thickness of some slender particle be
$2l$ and $2b$, respectively; the scaling $\omega/\omega_{p}\simeq 1/\kappa$
implies longitudinal resonances at wavelengths $\lambda$ roughly proportional
to $(l/b)\lambda_{p}$, where $\lambda_{p}=2\pi c/\omega_{p}$ and $c$ is the
speed of light in vacuum. Hence, if the characteristic linear dimension of the
particle is $2a$, we have that $a/\lambda\simeq(a/l)(b/\lambda_{p})$. One
avenue to reducing the latter size factor is to reduce $b$, though this is
ultimately limited by manufacturing capabilities as well as non-classical
effects at subnanometric scales. An alternative avenue is to coil the
centerline so as to decrease $a/l$. In particular, $a/l$ is unity for a
straight particle and $1/\pi$ for a circular ring.

The above characteristics of high-aspect-ratio nanometallic particles can be
indirectly inferred from classical analytical solutions for cylinder and
sphere dimers, ellipsoids and tori, which are exact in the quasi-static
approximation Moussiaux  _et al._ (1977); Aizpurua  _et al._ (2003); Mary  _et
al._ (2005); Dutta  _et al._ (2008); Bohren and Huffman (2008); Klimov (2014);
Voicu and Sandu (2017); Downing and Weick (2020). While these solutions allow
for arbitrary aspect ratio, they are limited to separable geometries. Even
then, exact solutions tend to be cumbersome and sometimes it is not
straightforward to extract singular behaviors in the high-aspect-ratio limit.
For more general geometries, analytical solutions are unavailable, whereas
brute-force finite-element simulations become expensive at high aspect ratios.
An alternative, more insightful and versatile, theoretical approach is to
consider the high-aspect-ratio limit — in which plasmonic resonance is
expected to be most pronounce — from the outset, using tools of asymptotic
analysis and singular-perturbation theory Hinch (1991). The latter tools are
classical, though their application to plasmonics is relatively new and so far
the focus has been on near-touching geometries Schnitzer (2015, 2020).

Recently, we took a first step in applying an asymptotic approach to the
analysis of the plasmonic properties of slender nanometallic structures Ruiz
and Schnitzer (2019). In particular, we developed an approximate theory of the
longitudinal plasmonic resonances of slender bodies of revolution, with the
thickness profile along the symmetry axis being essentially arbitrary. Besides
key scalings, new closed-form approximations for special geometries and
physical insight into the role of slenderness and shape, this work furnished a
versatile semi-analytical scheme which allows one to rapidly calculate the
plasmonic response of a certain family of complex 3D geometries.

The framework in Ruiz and Schnitzer (2019) is underpinned by a spectral theory
which is exact in the quasi-static approximation; it is based on the so-called
plasmonic eigenvalue problem Mayergoyz  _et al._ (2005); Grieser (2014);
Klimov (2014); Davis and Gómez (2017), which defines a set of material- and
frequency-independent modes of a nanometallic structure based on its geometry,
with the relative permittivity of the structure playing the eigenvalue role.
Given this set of modes, calculating the optical response of the structure for
arbitrarily prescribed radiation sources and physical values of the particle
permittivity amounts to the evaluation of normalization and overlap integrals
involving the modes and the incident radiation field. At near-resonance
frequencies, often a single mode dominates the spectral expansion; this modal
approach is accordingly both efficient and insightful in that it provides a
linkage between physical resonances and mathematical eigenfunctions.

In Ruiz and Schnitzer (2019), we used asymptotic tools to calculate modes as
well as overlap integrals. In particular, to analyze the plasmonic eigenvalue
problem, we relied mainly on the method of matched asymptotic expansions,
where the physical domain is decomposed into distinguished regions which are
separately analyzed and then matched together; the use of such methodology in
the context of slender particles is especially popular in fluid dynamics,
where it is known as ‘slender-body theory’ (Tillett, 1970; Batchelor, 1970;
Cox, 1970; Keller and Rubinow, 1976; Van Dyke, 1975; Handelsman and Keller,
1967). This allowed us to reduce the plasmonic eigenvalue problem for the body
of revolution (restricted to the longitudinal — in this case axisymmetric —
modes) to an asymptotically equivalent 1D problem whose domain is a finite
line segment corresponding to the body’s centerline. In this reduced problem,
the eigenvalue is the original permittivity eigenvalue scaled by $\kappa^{2}$.
With that rescaling, the reduced problem involves $\kappa$ only weakly, namely
through its logarithm. The eigenfunctions in the reduced problem represent
effective 1D voltage and polarization-charge profiles. From these profiles,
the corresponding mode distributions can be evaluated in 3D, everywhere in the
interior and exterior of the particle; they can also be used to easily
evaluate the normalization and overlap integrals appearing in the spectral
formulation. The reduced eigenvalue problem consists of a differential
equation which is coupled to an integral equation. The differential equation
represents an effective Gauss law for an infinitesimal longitudinal segment of
the body. (The boundary conditions are dependent on the local geometry of the
tips.) The integral equation represents a spatially nonlocal capacity relation
accounting for electrostatic interactions between different longitudinal
segments of the body.

The purpose of the present work is to extend the plasmonic slender-body in
Ruiz and Schnitzer (2019) to slender nanometallic rings and configurations
thereof. By a slender ring we mean a body whose thickness about a circular
curve is small relative to the radius of that circle. We shall allow azimuthal
variations in thickness and non-circular cross-sectional shapes, thus going
considerably beyond the idealized case of a slender torus; we shall also
consider ring dimers and chains, the separation between the rings being
comparable to their radii. An obvious source of motivation for focusing on
ring structures is that they are commonly employed in nanoplasmonics
Nordlander (2009); they are moderately compact, as discussed above, relatively
easy to fabricate Chow  _et al._ (2020), their hollow center is advantageous
for sensing applications Aizpurua  _et al._ (2003); Genç  _et al._ (2017) and
offer increased tuning through aspect ratio Aizpurua  _et al._ (2003);
Nordlander (2009), azimuthal thickness profile Bochenkov  _et al._ (2017);
Zong  _et al._ (2019) and multiple-ring interactions Zhang  _et al._ (2015);
Pattanayak  _et al._ (2017); Zhang  _et al._ (2019); Dana  _et al._ . We also
have technical reasons to focus on ring geometries. Thus, the absence of tips
eliminates the need to derive boundary conditions to close the reduced
eigenvalue problem, which is generally quite a subtle aspect of the theory.
Furthermore, the fact that rings degenerate to circular curves in the high-
aspect-ratio limit will be seen to enable convenient diagonalizations of the
integral operators appearing in the reduced problem.

The rest of the paper is structured as follows. In §II, we formulate the
general problem of scattering of a plane wave from a nanometallic structure in
the quasi-static approximation and then review the plasmonic eigenvalue
problem and its use for solving that scattering problem. In §III, we employ
asymptotic arguments in the high-aspect-ratio limit to derive a reduced
eigenvalue problem in the case of a single ring; at this stage, we assume
circular cross sections, although the thickness may vary in the azimuthal
direction. To emphasize the physics, the asymptotic arguments are described
intuitively and in dimensional notation; readers requiring a more formal
justification of the approximations can refer to the derivation of the reduced
problem in Ruiz and Schnitzer (2019) and textbooks describing the method of
matched asymptotic expansions and slender-body theory Hinch (1991). In the
remainder of §III, we discuss the singular scaling of the permittivity
eigenvalues, the accuracy of the approximation scheme and develop closed-form
solutions for torus-shaped rings and a semi-analytical scheme for azimuthally
non-uniform rings. In §IV, we generalize the reduced eigenvalue problem to the
case of ring dimers, develop closed-form solutions for coaxial dimers formed
of torus-shaped rings and a semi-analytical scheme for more general dimer
geometries. In §V, we briefly discuss further geometrical extensions to non-
circular cross sections and chains of rings; the latter extension is
illustrated by considering the consequences of a defect in a coaxial chain of
ring dimers. In §VI, we demonstrate the application of the asymptotic modes to
calculate absorption cross sections for ring configurations illuminated by a
plane wave. In §VII, we discuss limits of validity of the quasi-static
approximation specifically in describing the longitudinal resonances of
slender-ring structures, and show comparisons with full-wave simulations. We
give concluding remarks in §VIII.

##  II Quasi-static formulation

###  II.1 Scattering problem

Consider a homogeneous nanometallic structure in vacuum. We assume that the
characteristic linear dimension of the structure is sufficiently small
relative to the free-space wavelength such that a quasi-static approximation
is applicable (Bohren and Huffman (2008), cf. §VII). In what follows, we
formulate a scattering problem governing the electric near-field in the
vicinity of the structure in the scenario where the structure is illuminated
by a plane wave; for simplicity, we do not include in the formulation the
possibility of near-field external sources.

In the quasi-static approximation, the electric field is irrotational in the
vicinity of the structure. Defining this near-field to be the real part of
$\mathbf{E}e^{-i\omega t}$, wherein $\omega$ is angular frequency and $t$
time, we may accordingly introduce an electric potential $\varphi$ such that
$\mathbf{E}=-\boldsymbol{\nabla}\varphi$. Given the absence of near-field
external sources, the potential $\varphi$ satisfies

$\boldsymbol{\nabla}\boldsymbol{\cdot}\left(\epsilon\boldsymbol{\nabla}\varphi\right)=0,$
(1)

where $\epsilon$ denotes the permittivity relative to vacuum. In the exterior
of the structure, $\epsilon=1$. In the interior of the structure, $\epsilon$
is given by the complex-valued and frequency dependent relative permittivity
of the metal, say $\epsilon_{r}(\omega)$, for which empirical data is
available (e.g., (Johnson and Christy, 1972)). Alternatively, we may refer to
the Drude model Maier (2007)

$\epsilon_{r}(\omega)=1-\frac{\omega_{p}^{2}}{\omega^{2}+i\gamma\omega},$ (2)

wherein $\omega_{p}$ is the plasma frequency and $\gamma$ is a parameter
representing ohmic losses. The Drude model is adequate for frequencies
considerably lower than $\omega_{p}$, which as we shall see is the case for
the longitudinal resonances of slender-ring structures. This model will be
used for the sake of illustration in §VI.

The quasi-static formulation is closed by the far-field condition

$\mathbf{E}\to\mathbf{E}_{\infty}\quad\text{as}\quad|\mathbf{x}|\to\infty,$
(3)

where the constant vector $\mathbf{E}_{\infty}$ represents the incident plane-
wave. Specifically, keeping in mind that the near-field domain is small
relative to the free-space wavelength, $\mathbf{E}_{\infty}$ is the electric-
field phasor of the incoming wave evaluated at the location of the structure.

###  II.2 Plasmonic eigenvalue problem

Upon setting $\mathbf{E}_{\infty}$ to zero, (1) and (3) give the so-called
‘plasmonic eigenvalue problem’ governing the localized-surface-plasmon
resonances of the nanometallic structure. Since frequency enters this problem
solely via the metal permittivity $\epsilon_{r}$, it is natural to take
$\epsilon_{r}$ as the eigenvalue, henceforth denoted $\mathcal{E}$, rather
than the frequency. The permittivity eigenvalues are scale invariant and
independent of material and frequency; they are determined solely by the
geometry of the structure. For a smooth geometry, there are infinitely many
negative-real permittivity eigenvalues which accumulate at $-1$ Mayergoyz  _et
al._ (2005); Klimov (2014); Grieser (2014); Ando and Kang (2016). The
associated field distributions, namely eigenfunctions, are also scale-
invariant up to a linear stretching and can always be chosen real-valued.

Physically, the eigenfunctions can be interpreted as perpetual localized-
surface-plasmon oscillations in the absence of material losses and external
forcing. (Radiation losses are effectively neglected in the quasi-static
approximation.) Mathematically, the set of eigenfunctions possesses
completeness and orthogonality properties that can be used to explicitly solve
arbitrary scattering problems. In particular, the solution to the scattering
problem formulated above can be written as the spectral expansion (see
(Bergman, 1979; Klimov, 2014; Grieser, 2014; Ammari  _et al._ , 2018; Davis
and Gómez, 2017))

$\varphi(\mathbf{x})=-\mathbf{E}_{\infty}\boldsymbol{\cdot}\mathbf{x}+\sum_{I\in\mathcal{I}}\frac{\epsilon_{r}(\omega)-1}{\epsilon_{r}(\omega)-\mathcal{E}^{(I)}}\frac{\int
dV\,\mathbf{E}_{\infty}\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi^{({I})}}{\int
dV\,\boldsymbol{\nabla}\varphi^{({I})}\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi^{({I})}}\,\varphi^{({I})}(\mathbf{x}),$
(4)

where $\mathcal{I}$ is an index set for the eigenvalues and eigenfunctions,
$dV$ is a volume element and the integrals are over the interior of the
structure. The integrals in the numerator and denominator are called overlap
and normalization integrals, respectively. We note that scattering problems
involving near-field sources can be treated similarly, with
$\mathbf{E}_{\infty}$ replaced by the total field in the absence of the
structure. From (4), quasi-static approximations for the optical cross
sections can be readily calculated, as we shall see in §VI.

For frequencies such that the complex-valued permittivity
$\epsilon_{r}(\omega)$ is close to a permittivity eigenvalue, the
corresponding term in the expansion (4) — or terms, in the case of perfect or
near-perfect degeneracy — may become dominant. In that case, we say that the
associated eigenfunctions (or eigenmodes) are resonantly excited by the
incident radiation. Thus, for such near-resonance frequencies, the spectral
solution can be asymptotically simplified, revealing a connection between the
physical and mathematical perspectives described above. The existence and
characteristics of such resonances in practice, however, depend on several
additional factors including the relative smallness of the imaginary component
of $\epsilon_{r}(\omega)$, the overlap between the incident field and the
eigenfunctions participating in the resonance, as well as interference
effects. All of these dependencies can be studied using (4) once the
eigenvalues and eigenfunctions of the geometry have been calculated.

##  III Longitudinal modes of a single slender ring

###  III.1 Geometry

In this section we consider the plasmonic eigenvalue problem (cf. §II.2) in
the case of a single nano-sized ring of radius $a$ and characteristic
thickness $b$. For now, we assume that the ring is formed of circular cross
sections of radius $bf(\phi)$, which are centered about a circular centerline
of radius $a$; the thickness profile $f(\phi)$ is a dimensionless function of
the azimuthal angle $\phi$. (Non-circular cross-sections are treated in §V.1.)
In Fig. 1, we define the Cartesian $(x,y,z)$, cylindrical $(\rho,z,\phi)$ and
‘cross-sectional’ $(r,\theta,\phi)$ coordinate systems. Accordingly, the
position measured from the center of the ring can be written
$\mathbf{x}=\mathbf{y}+r\mathbf{\hat{e}}_{r}$, where
$\mathbf{\hat{e}}_{r}=\mathbf{\hat{e}}_{\rho}\cos\theta+\mathbf{\hat{e}}_{z}\sin\theta$
and $\mathbf{y}(\phi)=a\mathbf{\hat{e}}_{\rho}$ is the circular centerline,
with
$\mathbf{\hat{e}}_{\rho}=\mathbf{\hat{e}}_{x}\cos\phi+\mathbf{\hat{e}}_{y}\sin\phi$.
Furthermore, the ring boundary can be written $r=bf(\phi)$.

Figure 1: Schematic of a slender ring with circular cross sections as
considered in §III.

###  III.2 Reduced eigenvalue problem

Henceforth, we consider slender rings for which the aspect ratio $\kappa=a/b$
is large and $f(\phi)$ is of order unity. In particular, our interest is in
modes, i.e., solutions of the plasmonic eigenvalue problem, for which
$\mathcal{E}\to-\infty$ as $\kappa\to\infty$. (According to the Drude model
(2), large-negative permittivity eigenvalues imply low frequencies
$\omega\ll\omega_{p}$.) For any mode with this asymptotic property, scaling
arguments suggest that, as $\kappa\to\infty$, cross-sectional variations of
the interior potential become negligible relative to azimuthal variations. We
accordingly label such modes ‘longitudinal’. We note that in the case of a
body of revolution Ruiz and Schnitzer (2019), those modes can be identified
based on symmetry as they simply correspond to the axisymmetric modes.

In light of the above, we approximate the interior potential as

$\varphi(\mathbf{x})=v(\phi),$ (5)

where the azimuthal function $v(\phi)$ represents the voltage relative to
infinity. (Without loss of generality, we take the potential to vanish at
infinity.) Another key azimuthal function is the polarization-charge line
density

$q(\phi)=\oint\limits_{r=bf(\phi)}dl\,\sigma,$ (6)

where
$\sigma=\epsilon_{0}[\mathbf{E}\boldsymbol{\cdot}\mathbf{\hat{n}}]_{i}^{e}$ is
the polarization-charge surface density, $dl$ is a differential length element
and the integral is over the circular cross-sectional interface $r=bf(\phi)$
at constant $\phi$; in the expression for $\sigma$, $\epsilon_{0}$ is the
permittivity of vacuum, $\mathbf{\hat{n}}$ an outward normal unit vector and
the square brackets stand for the jump across the interface in that direction,
the subscript $i$ and superscript $e$ indicating the interior and exterior
sides, respectively. For $|\mathcal{E}|\gg 1$, continuity of electrical
displacement indicates that the interior normal field is negligible compared
to the exterior normal field. We therefore make the approximation

$\sigma=\epsilon_{0}(\mathbf{E}\boldsymbol{\cdot}\mathbf{\hat{n}})_{e},$ (7)

the $e$ subscript indicating evaluation on the exterior side of the interface.

Consider now the potential distribution in the exterior of the ring,
specifically at distances from the centerline comparable with the
characteristic thickness $b$. In that neighborhood of the ring boundary, the
exterior potential can be approximated as

$\varphi(\mathbf{x})=-\frac{q(\phi)}{2\pi\epsilon_{0}}\ln\frac{r}{bf(\phi)}+v(\phi).$
(8)

Namely, in any given cross-sectional plane, the exterior potential is locally
provided by the potential distribution around an infinite, perfectly
conducting, cylinder, which coincides with the ring in that plane and is held
at a potential $v(\phi)$. In light of (7), the electrostatic surface-charge
density at the cylinder boundary is nothing but the polarization surface-
charge density $\sigma$, thence the cylinder’s net apparent surface charge,
per unit length, is $q(\phi)$. Lastly, the absence of $\theta$-dependent
solutions in (8) can be justified based on the circular shape of the boundary,
the uniform potential prescribed on that boundary, as well as the anticipation
that the potential gradient is negligible in the azimuthal direction and
decays away from the ring’s centerline.

We shall now derive two relations between $v(\phi)$ and $q(\phi)$. The first
is based on Gauss law, which in the absence of free charge reduces to (cf.
(1))

$\oint
dA\,\epsilon_{0}\epsilon\mathbf{E}\,\boldsymbol{\cdot}\mathbf{\hat{n}}=0,$ (9)

where $dA$ is a differential area element, $\mathbf{\hat{n}}$ is an outward
normal unit vector and the integral is over an arbitrary closed surface.
Consider that closed surface to be the boundary of a curved tube, which
closely encloses a segment of the ring between the cross-sectional planes
$\phi=\phi^{\prime}$ and $\phi=\phi^{\prime}+\Delta\phi$. From (6) and (7),
the contribution to the integral from the part of the tube boundary that lies
outside the ring is $aq(\phi^{\prime})\Delta\phi$. Since the interior
azimuthal field is approximately uniform over the cross section, we find that
the contribution from the part of the tube boundary that lies inside the ring,
namely the end faces, is

$\epsilon_{0}\mathcal{E}\left\\{(A\mathbf{E}\boldsymbol{\cdot}\mathbf{\hat{e}}_{\phi})_{\phi^{\prime}+\Delta\phi}-(A\mathbf{E}\boldsymbol{\cdot}\mathbf{\hat{e}}_{\phi})_{\phi^{\prime}}\right\\},$
(10)

where $A$ denotes the cross-sectional area. By taking the limit $\Delta\phi\to
0$, using (5), we find the effective Gauss law

$\frac{q}{\epsilon_{0}}=\frac{\mathcal{E}}{\kappa^{2}}\frac{d}{d\phi}\left(\bar{A}\frac{dv}{d\phi}\right),$
(11)

where we define the scaled cross-sectional area $\bar{A}=A/b^{2}$. In the
present case, where the cross-sectional shape is circular, $\bar{A}=\pi
f^{2}$.

The second relation arises from the need to resolve the logarithmic growth of
the exterior potential away from the centerline of the ring. This is done by
matching (8) with an approximation for the exterior potential that holds at
distances from the centerline that are comparable to $a$. Thus, on the latter
scale, the finite thickness of the ring is indiscernible and the potential is
approximated as that of a circular wire of charge line density $q(\phi)$,
viz.,

$\varphi(\mathbf{x})=\frac{a}{4\pi\epsilon_{0}}\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q(\phi^{\prime})}{|\mathbf{x}-\mathbf{y}(\phi^{\prime})|}.$
(12)

We then compare (8) with the behavior of (12) close to the centerline, which
is derived in Appendix A, to derive the matching condition

$v(\phi)=\frac{q(\phi)}{2\pi\epsilon_{0}}\ln\frac{8\kappa}{f(\phi)}+\frac{1}{4\pi\epsilon_{0}}\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q(\phi^{\prime})-q(\phi)}{2\sin\frac{|\phi-\phi^{\prime}|}{2}},$
(13)

which is an integral capacitance relation between the voltage at a given
azimuthal angle $\phi$ and the azimuthal distribution of polarization charge
over the entire ring.

The differential Gauss law (11) and the integral capacitance relation (13)
together constitute a ‘reduced eigenvalue problem’ for the ‘reduced
eigenvalue’ $\mathcal{E}/\kappa^{2}$ and ‘reduced eigenfunctions’ $v(\phi)$
and $q(\phi)/\epsilon_{0}$, which are $2\pi$-periodic functions of $\phi$.
Like the exact plasmonic eigenvalue problem, this reduced problem is purely
geometric; in particular, the geometry enters through the thickness profile
$f(\phi)$ and the logarithm of the aspect ratio $\kappa$.

###  III.3 ‘Logarithmically approximated’ solutions to the reduced eigenvalue
problem

Before attempting to solve the reduced eigenvalue problem in its present form,
it is insightful to discuss a coarse ‘logarithmic’ approximation of the
reduced problem which is based on the formal largeness of $\ln\kappa$ as
$\kappa\to\infty$. In this approximation, the integral capacitance relation
(13) reduces to the algebraic relation

$q/\epsilon_{0}\approx\frac{2\pi}{\ln\kappa}v.$ (14)

Substituting (14) into the Gauss law (11) then gives

$-\frac{\mathcal{E}}{\kappa^{2}}\frac{d}{d\phi}\left(\bar{A}\frac{dv}{d\phi}\right)+\frac{2\pi}{\ln\kappa}v\approx
0,$ (15)

to be solved together with the periodic boundary conditions on $v$.

The simplified model (15) clearly implies the asymptotic scaling

$\mathcal{E}\simeq\frac{\kappa^{2}}{\ln\kappa}\quad\text{as}\quad\kappa\to\infty,$
(16)

which slightly differs from the usually quoted scaling for slender geometries
mentioned in the introduction by the logarithmic factor; we note that the same
scaling was found in Ruiz and Schnitzer (2019) for the longitudinal modes of
slender bodies of revolution. In particular, consider the case of a torus-
shaped ring, with $b$ identified as the cross-sectional radius. Then
$\bar{A}(\phi)\equiv\pi$ and solving (15) yields the approximate eigenvalues

$\mathcal{E}^{({m})}\approx-\frac{2\kappa^{2}}{m^{2}\ln\kappa},\quad
m=1,2,\ldots,$ (17)

with corresponding pairs of independent eigenfunctions

$v^{({m,c})}\approx\cos m\phi,\quad v^{({m,s})}\approx\sin m\phi,$ $None$

the degeneracy being induced by symmetry. The corresponding polarization-
charge eigenfunctions $q^{({m,c})}$ and $q^{({m,s})}$ can readily be deduced
from (14).

The above scheme corresponds to a leading-order approximation in a
perturbative expansion in inverse powers of $\ln\kappa$. In particular, the
relative error in (17) is on the order of $1/\ln\kappa$. In slender-body
theory, such logarithmic approximations are often referred to as ‘local
slender-body theory’. This should be contrasted with the non-approximated
reduced eigenvalue problem, which constitutes a ‘nonlocal slender-body
theory,’ the term ‘nonlocal’ here referring to the integral capacitance
relation (13). It can be shown that the reduced eigenvalue problem effectively
sums the logarithmic expansion mentioned above, resulting in solutions which
are ‘algebraically’ accurate, i.e., involving relative errors which are
asymptotically smaller than some negative power of $\kappa$. The distinction
between local and nonlocal slender-body theory should not be confused with the
notion of spatial nonlocality of the metal’s dielectric function, which we do
not address in this paper.

###  III.4 ‘Exact’ solutions to the reduced eigenvalue problem

####  III.4.1 Diagonalization of the self-interaction integral operator

We now proceed to derive ‘exact’ solutions to the reduced eigenvalue problem
(cf. (11) and (13)), i.e., without exploiting the formal largeness of
$\ln\kappa$. An essential step is to note that the integral operator appearing
in the capacitance relation (13) is diagonalized by the set of Fourier
eigenfunctions. Specifically, for any integer $m$,

$\int_{0}^{2\pi}d\phi^{\prime}\,\frac{e^{im\phi^{\prime}}-e^{im\phi}}{2\sin\frac{|\phi^{\prime}-\phi|}{2}}=\lambda_{m}e^{im\phi},$
(19)

with $\lambda_{0}=0$ and

$\lambda_{m}=-4\sum_{k=1}^{|m|}\frac{1}{2k-1}\quad\text{for}\quad m=\pm 1,\pm
2,\ldots$ (20)

A proof of this result is given in Appendix B.

####  III.4.2 Torus-shaped rings

We first consider the case of torus-shaped rings, for which $f$ and $\bar{A}$
are independent of $\phi$. Identifying the cross-sectional radius as $b$, we
have $f(\phi)\equiv 1$ and $\bar{A}(\phi)\equiv\pi$. (Henceforth, we will
always adopt this convention when referring to torus-shaped rings.) For
constant $\bar{A}$, the same set of Fourier eigenfunctions that diagonalizes
the integral operator in the capacitance relation (13) as in (19) also
diagonalizes the differential operator in the Gauss law (11), since
$d^{2}e^{im\phi}/d\phi^{2}=-m^{2}e^{im\phi}$. Furthermore, for constant $f$,
the logarithmic factor multiplying $q(\phi)$ in (13) reduces to a constant.
Accordingly, it is immediate to deduce the eigenvalues

$\mathcal{E}^{({m})}=-\frac{2\kappa^{2}}{m^{2}}\left(\ln
8\kappa-2\sum_{k=1}^{m}\frac{1}{2k-1}\right)^{-1},\quad\text{for}\quad
m=1,2,\ldots,$ (21)

with associated voltage eigenfunctions

$v^{({m,c})}=\cos m\phi,\quad v^{({m,\,s})}=\sin m\phi.$ $None$

The closed-form expression (21) constitutes an algebraically accurate slender-
body approximation for the reduced eigenvalues of a torus-shaped ring. It is
asymptotically consistent with, and improves on, the corresponding
logarithmically accurate approximation (17). Surprisingly, the algebraically
accurate voltage eigenfunctions ($None$) are of the same form as the
logarithmically accurate eigenfunctions ($None$). The algebraically accurate
relation between the voltage and polarization-charge eigenfunctions, however,
reads

$q^{({m,c})}/\epsilon_{0}=2\pi\left(\ln
8\kappa-2\sum_{k=1}^{m}\frac{1}{2k-1}\right)^{-1}v^{({m,c})},$ (23)

which should be compared with the corresponding logarithmically accurate
relation (14).

In the present case of a torus-shaped ring, exact solutions of the plasmonic
eigenvalue problem can be obtained by separation of variables in toroidal
coordinates Mary  _et al._ (2005). In Fig. 2, we compare the first three
eigenvalues, not counting multiplicities, computed in this manner with the
corresponding values predicted by the slender-body approximation (21). As
expected, the slender-body approximations approach the computed eigenvalues as
$\kappa\to\infty$.

At the same time, the comparison in Fig. 2 demonstrates the fact that, for
fixed $\kappa$, the slender-body approximation deteriorates in accuracy with
increasing mode number; conversely, as the mode number increases, accuracy can
be retained only by increasing $\kappa$. This is not surprising given that our
slender-body approximation corresponds to a limit process where
$\kappa\to\infty$ for a fixed mode; there is therefore no reason to expect the
approximation to hold in the double limit where both $\kappa$ and $m$ are
large Ruiz and Schnitzer (2019). In fact, the scale on which the voltage and
charge eigenfunctions vary along the ring — assumed large compared to the ring
thickness in the slender-body approximation — becomes comparable to the ring
thickness for $m\simeq\kappa$; moreover, for such large $m$, (21) gives
$|\mathcal{E}^{({m})}|\simeq 1$, whereas the theory relies on the permittivity
contrast being large. We note that this limitation of the theory does not
interfere with our goal of describing the low-order (low-frequency)
longitudinal resonances of slender nanometallic rings.

The remainder of this section and the next two sections are concerned with
extensions of the slender-body theory derived so far to the calculation of the
longitudinal modes of more general ring geometries, including azimuthally non-
uniform rings, ring dimers and chains and rings with non-circular cross-
sections. Some readers may prefer at this point to jump to §VI, where we
illustrate the application of the slender-body theory to the solution of the
quasi-static scattering problem formulated in §II.

Figure 2: Three lowest permittivity eigenvalues, not counting multiplicities,
of a torus-shaped ring as a function of the reciprocal of its aspect ratio
$\kappa$: exact solutions of the quasi-static plasmonic eigenvalue problem
Mary  _et al._ (2005) vs. the slender-body approximation (21).

####  III.4.3 Rings of non-uniform thickness

Consider now rings for which the thickness profile $f$ depends on the
azimuthal angle $\phi$. In this more general case, we solve the reduced
eigenvalue problem using a semi-analytical method, which is based on
approximating the reduced eigenfunctions $v(\phi)$ and $q(\phi)$ by the
truncated Fourier series

$v=\sum_{k=0}^{K}\left\\{\alpha_{k}\cos k\phi+\beta_{k}\sin
k\phi\right\\},\quad\frac{q}{\epsilon_{0}}=\sum_{k=1}^{K}\left\\{\tilde{\alpha}_{k}\cos
k\phi+\tilde{\beta}_{k}\sin k\phi\right\\},$ $None$

where $K\in\mathbb{N}$ is a truncation parameter discussed below. Note that
the zeroth harmonic is omitted from the charge representation ($None$b) in
accordance with the zero-net-charge constraint $\int_{0}^{2\pi}d\phi\,q=0$,
which readily follows from integration of Gauss law (11). With ($None$),
Fourier projection of the capacitance relation (13) and Gauss law (11) yields
a $2K\times 2K$ generalized eigenvalue problem of the form

$\mathsf{q}=-\bar{\mathcal{E}}\,\mathsf{M}\boldsymbol{\cdot}\mathsf{v},\quad\mathsf{v}=\mathsf{U}\boldsymbol{\cdot}\mathsf{q},$
$None$

where
$\mathsf{v}=(\alpha_{1},\ldots,\alpha_{K},\beta_{1},\ldots,\beta_{K})^{T}$ and
$\mathsf{q}=(\tilde{\alpha}_{1},\ldots,\tilde{\alpha}_{K},\tilde{\beta}_{1},\ldots,\tilde{\beta}_{K})^{T}$,
with $T$ denoting the vector transpose; we have introduced the notation
$\bar{\mathcal{E}}=\mathcal{E}/\kappa^{2}$ for the reduced eigenvalue; and the
matrices $\mathsf{M}$ and $\mathsf{U}$ are provided in Appendix C, along with
an expression for the coefficient $\alpha_{0}$, which is computed a
posteriori. As already mentioned, the slender-body approximation cannot be
expected to correctly capture thickness-scale variations in the longitudinal
direction. In fact, the reduced formulation can be shown to be ill posed for
$K\simeq\kappa$ Tornberg and Shelley (2004). Accordingly, the truncation
parameter $K$ should be chosen much smaller than $\kappa$. Of course, $K$
should also be chosen larger than the number of modes to be resolved.

In contrast to the case of a torus-shaped ring, it is clear from the above
scheme that for a non-uniform ring each mode generally contains multiple
Fourier harmonics. In particular, whereas for a torus-shaped ring only modes
$(1,c)$ and $(1,s)$, which share the degenerate eigenvalue
$\mathcal{E}^{({1})}$, are ‘dipolar’, i.e., include a first Fourier harmonic,
for non-uniform rings generally all modes include a dipolar component. As we
shall see in §VI, this observation is important for interpreting the
differences between the resonant response of uniform and non-uniform rings
under plane-wave illumination.

As an example, we consider an azimuthally non-uniform ring whose thickness
profile is $f=1+0.5\cos\phi$. The voltage eigenfunctions can be classified as
being either even or odd with respect to the symmetry plane of the geometry
which is normal to the plane of the ring; referring to ($None$a), these modes
are respectively comprised of either only cosine or only sine Fourier
components. Consistently with our notation for the longitudinal modes of a
torus-shaped ring (cf. ($None$)), we denote the $m$th even mode by the
superscript $(m,c)$; similarly, the $m$th odd mode by the superscript $(m,s)$.
In Fig. 3, we show for $\kappa=10$ the ‘dipolar’ $(1,c)$ and ‘quadrupolar’
$(2,c)$ modes of a torus-shaped ring, along with the corresponding modes of
the azimuthally non-uniform ring. In the latter case, the modes are computed
by solving ($None$) with $K=6$.

Figure 3: Slender-body approximations for the ‘dipolar’ $(1,c)$ and
‘quadrupolar’ $(2,c)$ voltage eigenfunctions and associated eigenvalues of a
torus-shaped ring and an azimuthally non-uniform ring with thickness profile
$f(\phi)=1+0.5\cos\phi$; for both geometries $\kappa=10$.

##  IV Longitudinal models of ring dimers

In this section we consider the longitudinal modes of ring dimers. In §IV.1,
we generalize the reduced eigenvalue problem derived in §III.2 for a single
ring to the dimer case. In §IV.2, we obtain exact closed-form solutions to the
generalized reduced problem in the case of a dimer formed of coaxial,
generally dissimilar, torus-shaped rings. In §IV.3, we consider more general
ring dimers using a semi-analytical scheme (we also present an ad hoc
approximation which is in some cases suitable). As we will demonstrate in §V,
configurations of more than two rings can be handled similarly.

###  IV.1 Reduced eigenvalue problem for ring dimers

We adopt the same notation as in §III, only with subscripts added to indicate
to which ring a given quantity is associated with. By revisiting the
derivation in §III, we find that in the thickness-scale vicinity of the rings
we have the local approximation

$\varphi(\mathbf{x})=-\frac{q_{n}(\phi)}{2\pi\epsilon_{0}}\ln\frac{r}{b_{n}f_{n}(\phi)}+v_{n}(\phi),\quad
n=1,2,$ (26)

for the exterior potential, which generalizes (8). Similarly, the effective
Gauss law (11) now applies to each ring separately:

$\frac{q_{n}}{\epsilon_{0}}=\frac{\mathcal{E}}{\kappa_{n}^{2}}\frac{d}{d\phi}\left(\bar{A}_{n}\frac{dv_{n}}{d\phi}\right),\quad
n=1,2.$ (27)

It remains to generalize the integral capacitance relation (13). To this end,
consider the ring-scale exterior potential (12), which generalizes as

$\varphi(\mathbf{x})=\sum_{n=1,2}\frac{a_{n}}{4\pi\epsilon_{0}}\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q_{n}(\phi^{\prime})}{|\mathbf{x}-\mathbf{y}_{n}(\phi^{\prime})|}.$
(28)

Using the results of Appendix A to match (28) and (26), we find the coupled
pair of integral capacitance relations

$v_{n}(\phi)=\frac{q_{n}(\phi)}{2\pi\epsilon_{0}}\ln{\frac{8\kappa_{n}}{f_{n}(\phi)}}+\frac{1}{4\pi\epsilon_{0}}\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q_{n}(\phi^{\prime})-q_{n}(\phi)}{2\sin\frac{\left|\phi^{\prime}-\phi\right|}{2}}\\\
+\frac{a_{k}}{4\pi\epsilon_{0}}\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q_{k}(\phi^{\prime})}{|\mathbf{y}_{n}(\phi)-\mathbf{y}_{k}(\phi^{\prime})|},$
(29)

for $(n,k)=(1,2)$ and $(2,1)$.

The differential equations (27) and integral equations (29) together
constitute a generalized reduced eigienvalue problem for ring dimers. We see
that the rings interact solely through the last ‘coupling’ integral on the
right-hand side of (29), whereby the polarization-charge distribution of one
ring induces a voltage disturbance in the other ring, and vice versa.

###  IV.2 Coaxial rings

####  IV.2.1 Diagonalization of the coupling integral operator

For coaxial rings, the coupling-integral operator in (29) is diagonalized by
the same Fourier basis that diagonalizes the self-interaction integral
operator as in (19). Indeed, in this case, the distance between the point on
the centerline of ring $1$ at azimuthal angle $\phi_{1}$ and the point on the
centerline of ring $2$ at azimuthal angle $\phi_{2}$ can be written
$|\mathbf{y}_{1}(\phi_{1})-\mathbf{y}_{2}(\phi_{2})|=D(\phi_{1}-\phi_{2})$,
where $D(u)=\sqrt{h^{2}+a_{1}^{2}+a_{2}^{2}-2a_{1}a_{2}\cos u}$, $h$ being the
vertical distance between the rings. Since $D(u)$ is even, periodic and
positive, its reciprocal can be expanded as a cosine Fourier series. It then
readily follows that

$\int_{0}^{2\pi}d\phi^{\prime}\,\frac{e^{im\phi^{\prime}}}{|\mathbf{y}_{1}(\phi)-\mathbf{y}_{2}(\phi^{\prime})|}=\tau_{m}e^{im\phi},\quad\text{for}\quad
m=0,\pm 1,\pm 2,\ldots$ (30)

where

$\tau_{m}=\int_{0}^{2\pi}du\,\frac{\cos mu}{D(u)}.$ (31)

Figure 4: Schematic of a bilayer dimer of torus-shaped rings. The rings have
centerline radii $a_{1}$ and $a_{2}$, and aspect ratios
$\kappa_{1}=a_{1}/b_{1}$ and $\kappa_{2}=a_{2}/b_{2}$. The horizontal and
vertical separation between the rings is $d$ and $h$, respectively. The
coaxial case considered in §IV.2 corresponds to the case $d=0$. The non-
coaxial case $d\neq 0$ is considered in §IV.3.

With (30), it is straightforward to generalize the semi-analytical scheme of
§III.4.3 to the case of arbitrary coaxial ring dimers. Instead, we shall focus
in the present subsection on the case of coaxial dimers formed of torus-
shaped, not necessarily identical, rings (see Fig. 4 for $d=0$), where (30)
actually facilitates the derivation of closed-form solutions. Later, in §IV.3,
we will present a more general semi-analytical scheme, not relying on (30),
that applies to arbitrary ring dimers, including non-coaxial dimers formed of
non-uniform rings.

####  IV.2.2 Coaxial homodimers

Consider first the case of a coaxial homodimer, namely a coaxial pair of
identical torus-shaped rings ($a_{1}=a_{2}=a$,
$\kappa_{1}=\kappa_{2}=\kappa$). Given the symmetries of the geometry, we
anticipate eigenvalues $\mathcal{E}^{({m,\pm})}$, for $m=1,2,\ldots$, with
corresponding eigenfunctions

$\begin{pmatrix}v_{1}^{({m,c,\pm})}\\\
v_{2}^{({m,c,\pm})}\end{pmatrix}=\begin{pmatrix}1\\\ \pm 1\end{pmatrix}\cos
m\phi,\quad\begin{pmatrix}v_{1}^{({m,s,\pm})}\\\
v_{2}^{({m,s,\pm})}\end{pmatrix}=\begin{pmatrix}1\\\ \pm 1\end{pmatrix}\sin
m\phi.$ $None$

The $\pm$ modes are, respectively, even and and odd about the plane
equidistance between the rings; we shall also refer to these as in- and out-
of-phase modes, respectively, as this will allow a generalized interpretation
in the heterodimers case considered next. Writing the eigenfunctions as in
($None$), use of the diagonalization identities (19) and (30) readily yields

$\mathcal{E}^{({m,\pm})}=-\frac{2\kappa^{2}}{m^{2}}\left(\ln
8\kappa-2\sum_{k=1}^{m}\frac{1}{2k-1}\pm\frac{\Delta_{m}}{2}\right)^{-1},$
(33)

where $\Delta_{m}=a\tau_{m}$ is a dimensionless function of the ratio
$h^{\prime}=h/a$ defined by the quadrature (cf. (31))

$\Delta_{m}(h^{\prime})=\int_{0}^{2\pi}\,du\frac{\cos mu}{\sqrt{h^{\prime
2}+2-2\cos u}},$ (34)

which determines the eigenvalue splitting induced by the interaction between
the rings (cf. (21)).

The functions $\Delta_{m}(h^{\prime})$ are positive. Hence, for any $m$, the
in-phase modes are higher energy (less negative permittivity), as one would
expect. Furthermore, the functions $\Delta_{m}(h^{\prime})$ are monotonically
decreasing, asymptotically like $\Delta_{m}=O(1/h^{2m+1})$ as
$h^{\prime}\to\infty$; this represents the approach to the eigenvalues (21) of
a single torus-shaped ring. We also note that $\Delta_{m}(h^{\prime})$ is
logarithmically singular as $h^{\prime}\to 0$, which is acceptable given that
the theory only holds for $h\gg b$, i.e., $h^{\prime}\gg 1/\kappa$. In
particular, for $m=1$ we find from (34) the behaviours

$\Delta_{1}\sim-2\ln h^{\prime}+6\ln 2-4\quad\text{as}\quad h^{\prime}\to
0,\quad\Delta_{1}\sim\frac{\pi}{h^{\prime 3}}\quad\text{as}\quad
h^{\prime}\to\infty.$ $None$

In Fig. 5, the eigenvalues $\mathcal{E}^{({1,\pm})}$ of the in- and out-of-
phase dipolar modes, calculated using (33), are depicted by the solid curves
as a function of $h/a$, for $\kappa=10$. Also shown are the asymptotic
behaviors of the eigenvalues for small and large $h/a$, which follow from the
behaviors ($None$).

####  IV.2.3 Coaxial heterodimers

The above results are easily generalized to allow for dissimilar torus-shaped
rings. Since the geometry remains azimuthally invariant, we look for voltage
eigenfunctions of the form $\\{v_{1},v_{2}\\}=\\{c_{1},c_{2}\\}\times\cos
m\phi$, in which $c_{1}$ and $c_{2}$ are constant prefactors and
$m=1,2,\ldots$ There are also $\pi/2$-rotations of these modes having the same
eigenvalues. Using the diagonalization identities (19) and (30), we find that,
for $m=1,2,\ldots$, the reduced eigenvalue problem of §IV.1 is transformed
into the $2\times 2$ matrix eigenvalue problem

$\begin{pmatrix}c_{1}\\\
c_{2}\end{pmatrix}=-\mathcal{E}\begin{pmatrix}\gamma_{1,1}&\gamma_{1,2}\\\
\gamma_{2,1}&\gamma_{2,2}\end{pmatrix}\begin{pmatrix}c_{1}\\\
c_{2}\end{pmatrix},$ (36)

in which the matrix on the right-hand side has the diagonal and non-diagonal
elements

$\gamma_{n,n}=\frac{m^{2}}{2\kappa_{n}^{2}}\left(\ln
8\kappa_{n}-2\sum_{k=1}^{m}\frac{1}{2k-1}\right),\quad\gamma_{n,k}=\frac{m^{2}}{4\kappa_{k}^{2}}(a_{k}\tau_{m}),$
$None$

respectively, where the products $a_{k}\tau_{m}$ are dimensionless geometric
factors which determine the coupling between the rings (cf. (31)). Solving the
above system gives the eigenvalues

$\displaystyle\mathcal{E}^{({m,\pm})}=-\frac{2}{\gamma_{1,1}+\gamma_{2,2}\pm\sqrt{(\gamma_{1,1}-\gamma_{2,2})^{2}+4\gamma_{1,2}\gamma_{2,1}}}$
(38)

and corresponding eigenvectors

$\begin{pmatrix}c_{1}^{({m,\pm})}\\\
c_{2}^{({m,\pm})}\end{pmatrix}=\begin{pmatrix}\gamma_{1,2}\\\
\frac{\gamma_{2,2}-\gamma_{1,1}\pm\sqrt{(\gamma_{1,1}-\gamma_{2,2})^{2}+4\gamma_{1,2}\gamma_{2,1}}}{2}\end{pmatrix}.$
(39)

In summary, for $m=1,2,\ldots$, there are two degenerate plasmonic eigenvalues
$\mathcal{E}^{({m,\pm})}$, given by (38), with corresponding eigenfunctions

$\begin{pmatrix}v_{1}^{({m,c,\pm})}\\\
v_{2}^{({m,c,\pm})}\end{pmatrix}=\begin{pmatrix}c_{1}^{({m,\pm})}\\\
c_{2}^{({m,\pm})}\end{pmatrix}\cos
m\phi,\quad\begin{pmatrix}v_{1}^{({m,s,\pm})}\\\
v_{2}^{({m,s,\pm})}\end{pmatrix}=\begin{pmatrix}c_{1}^{({m,\pm})}\\\
c_{2}^{({m,\pm})}\end{pmatrix}\sin m\phi,$ $None$

where $c_{1}^{({m,\pm})}$ and $c_{2}^{({m,\pm})}$ are given by (39).

It can be verified from the above results that
$\mathcal{E}^{({m,+})}>\mathcal{E}^{({m,-})}$, namely that for any given $m$
the $+$ mode is higher energy than the $-$ mode. Furthermore,
$c_{1}^{({m,\pm})}$ is positive, whereas $c_{2}^{({m,+})}$ and
$c_{2}^{({m,-})}$ are positive and negative, respectively. The latter
observation suggests referring to the $\pm$ modes as in- and out-of-phase
modes, respectively, consistently with the coaxial homodimer case. In contrast
to the latter case, where the in- and out-of-phase labeling is associated with
a mirror symmetry, here the voltage profile in one ring is not simply the same
as or negative of that in the other. We shall see in §VI that this distinction
is important for interpreting the resonant response of coaxial dimers.

As an example, consider a coaxial heterodimer with $a_{1}/a_{2}=2$ and
$b_{1}=b_{2}$ (i.e., $\kappa_{1}=10$ and $\kappa_{2}=5$). In Fig. 5a, the
dashed curves depict the eigenvalues $\mathcal{E}^{({1,\pm})}$, calculated
using (38), of the in- and out-of-phase dipolar modes of this coaxial
heterodimer as a function of $h/a_{1}$. The voltage profiles for the
corresponding cosine-dipolar modes, for $h/a_{1}=0.3$, are shown in Fig. 5b.
Unlike in the coaxial-homodimer case also presented in that figure, the rings
of the heterodimer do not become arbitrarily close as $h\to 0$, hence the
eigenvalues of the heterodimer approach finite values in that limit. Of
course, we could have also considered a coaxial heterodimer formed of rings of
similar radius yet different thickness, in which case the eigenvalues would be
singular as $h\to 0$, as in the coaxial-homodimer case.

Figure 5:  (a) Eigenvalues of in- and out-of-phase dipolar modes of a a
coaxial homodimer ($\kappa=10$) and heterodimer ($\kappa_{1}=10$,
$\kappa_{2}=5$, $a_{1}/a_{2}=2$) as a function of the scaled vertical
separation $h/a_{1}$. The eigenvalues are calculated based on formulas (21)
and (38) for the homodimer and heterodimer, respectively. The dash-dotted
lines depict the asymptotic behaviors implied by ($None$). (b) Corresponding
cosine-dipolar voltage eigenfunctions in the heterodimer case, for
$h/a_{1}=0.3$.

###  IV.3 Non-coaxial dimers

####  IV.3.1 Semi-analytical scheme

We now present a semi-analytical scheme to solve the reduced eigenvalue
problem of §IV.1 in the case of an arbitrary ring dimer, meaning that the
rings need not be coaxial, identical nor azimuthally uniform. Following the
derivation of the single-ring semi-analytical scheme in §III.4.3, we represent
the voltage and polarization-charge distributions in each ring by truncated
Fourier series as in ($None$), with the Fourier coefficients now denoted
$\alpha_{n,k},\beta_{n,k}$, etc., with the first and second subscripts
corresponding to the ring number and Fourier harmonic, respectively. With that
representation, projection of the reduced eigenvalue problem on the truncated
Fourier basis yields the $4K\times 4K$ generalized matrix-eigenvalue problem

$\begin{pmatrix}\mathsf{q}_{1}\\\
\mathsf{q}_{2}\end{pmatrix}=-{\bar{\mathcal{E}}}\begin{pmatrix}\dfrac{\kappa_{2}}{\kappa_{1}}\mathsf{M}_{1}&0\\\
0&\dfrac{\kappa_{1}}{\kappa_{2}}\mathsf{M}_{2}\end{pmatrix}\boldsymbol{\cdot}\begin{pmatrix}\mathsf{v}_{1}\\\
\mathsf{v}_{2}\end{pmatrix},\quad\begin{pmatrix}\mathsf{v}_{1}\\\
\mathsf{v}_{2}\end{pmatrix}=\begin{pmatrix}\mathsf{U}_{1}&\sqrt{\frac{a_{2}}{a_{1}}}\mathsf{V}\\\
\sqrt{\frac{a_{1}}{a_{2}}}\mathsf{V}^{T}&\mathsf{U}_{2}\end{pmatrix}\boldsymbol{\cdot}\begin{pmatrix}\mathsf{q}_{1}\\\
\mathsf{q}_{2}\end{pmatrix},$ $None$

where
$\mathsf{v}_{n}=(\alpha_{n,1},\ldots,\alpha_{n,K},\beta_{n,1},\ldots,\beta_{n,K})^{T}$
and
$\mathsf{q}_{n}=(\tilde{\alpha}_{n,1},\ldots,\tilde{\alpha}_{n,K},\tilde{\beta}_{n,1},\ldots,\tilde{\beta}_{n,K})^{T}$;
$\bar{\mathcal{E}}=\mathcal{E}/(\kappa_{1}\kappa_{2})$ is a rescaled
eigenvalue; $\mathsf{M}_{n}$ and $\mathsf{U}_{n}$ are identical to the
matrices $\mathsf{M}$ and $\mathsf{U}$, respectively, used in the single-ring
scheme ($None$), with the geometric parameters being those of ring $n$; and
$\mathsf{V}$ is a coupling matrix whose form is provided in Appendix §C.2.

####  IV.3.2 Non-coaxial homodimers

Figure 6: (a) Eigenvalues $\mathcal{E}^{({1,c,+})}$ and
$\mathcal{E}^{({1,s,+})}$ of the non-coaxial bilayer dimer shown in Fig. 4, in
the case where the rings are identical ($\kappa=10$). The eigenvalues,
obtained by solving the semi-analytical scheme ($None$), are plotted as a
function of the scaled horizontal displacement $d/a$ and for the indicated
values of the relative vertical displacement $h/a$. Dash-dotted curves depict
the ad hoc analytical approximation (42). (b) Corresponding cosine-dipolar and
sine-dipolar voltage eigenfunctions in the non-coaxial bilayer dimer for
$d/a=2.5$ and $h/a=0.3$.

As an example of a non-coaxial dimer configuration, we consider the bilayer
configuration shown in Fig. 4. It consists of a pair of torus-shaped rings
whose centerlines define parallel planes separated by the vertical distance
$h$, and whose symmetry axes are separated by the horizontal distance $d$. The
case where $d$ vanishes corresponds to the coaxial configuration considered in
§IV.2, while the limit $h^{2}+d^{2}\to\infty$ corresponds to that of non-
interacting rings. We have seen that both of these extreme cases are
analytically solvable, with each longitudinal mode involving only a single
Fourier harmonic. In contrast, in the general case each mode is expected to
consist of a combination of Fourier harmonics. Symmetry still allows, however,
the modes of the bilayer geometry to be classified based on whether the
voltage eigenfunctions are even or odd about the $x$–$z$ mirror plane. With
the azimuthal angle $\phi$ measured from the $x$ direction, these even and odd
modes involve only cosine or sine Fourier harmonics, respectively. Thus, the
semi-analytical scheme ($None$) is reduced in this case to two uncoupled
$2K\times 2K$ matrix problems.

We first consider the homodimer case where the rings are identical
($a_{1}=a_{2}=a$, $\kappa_{1}=\kappa_{2}=\kappa$). In particular, we focus
attention on the two modes continuated, as $d$ is increased from zero, from
the degenerate in-phase dipolar modes $(1,c,+)$ and $(1,s,+)$ found in the
coaxial case (§IV.2.2). Fig. 4 depicts the variation with $d/a$ of the
respective eigenvalues, say $\mathcal{E}^{({1,c,+})}$ and
$\mathcal{E}^{({1,s,+})}$, calculated using the semi-analytical scheme
($None$) for two values of $h/a$. Note that the ‘$c$’ and ‘$s$’ modes are even
and odd about the $x$–$z$ plane, respectively, as in the coaxial case, but are
no longer degenerate rotations of each other. The insets show the voltage
eigenfunctions for the indicated values of $d/a$ and $h/a$.

A numerical study based on the semi-analytical scheme ($None$) suggests that
the modes of this bilayer homodimer configuration are, by visual inspection,
dominated by a single Fourier harmonic. This harmonic corresponds to that of
the mode of the corresponding coaxial (or isolated-ring) configuration from
which the bilayer mode is continuated from. This observation suggests an
intuitive and ad hoc approximation in which the voltage profiles in both rings
are constructed from just the apparently dominant harmonic. With that
assumption, we obtain

$\mathcal{E}^{({m,c,\pm})}\approx-\frac{2\kappa^{2}}{m^{2}}\left(\ln
8\kappa-\sum_{k=1}^{m}\frac{2}{2k-1}\pm\frac{a}{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}d\phi_{1}d\phi_{2}\,\frac{\cos
m\phi_{1}\cos
m\phi_{2}}{|\mathbf{y}_{1}(\phi_{1})-\mathbf{y}_{2}(\phi_{2})|}\right)^{-1}$
(42)

along with a similar expression for $\mathcal{E}^{({m,s,\pm})}$ where $\cos
m\phi_{1}\cos m\phi_{2}$ is replaced by $\sin m\phi_{1}\sin m\phi_{2}$. In
Fig. 6, the dash-dotted curves depict approximation (42) for
$\mathcal{E}^{({1,c,+})}$ and $\mathcal{E}^{({1,s,+})}$. Despite the
approximation being heuristic, the agreement with the semi-analytical slender-
body scheme ($None$) is reasonably good for all $d/a$ and especially at
moderately large $d/a$. It is clear that this approximation is exact for $d=0$
and asymptotically correct as $d/a\to\infty$.

####  IV.3.3 Non-coaxial heterodimers

Consider now a bilayer heterodimer where the torus-shaped rings are not
identical. Unlike in the bilayer homodimer scenario considered above, now the
spectrum of Fourier harmonics comprising each mode undergoes significant
evolution as the horizontal displacement $d$ is increased from the coaxial
case $d=0$, or, alternatively, decreased from the isolated-ring case
$d=\infty$. Moreover, we find that this evolution can continuously link two
different Fourier harmonics in the latter limiting cases, in which each mode
is composed of a single Fourier harmonic. We demonstrate this in Fig. 7 by
considering the evolution of the two modes continuated from the coaxial in-
phase dipolar mode $({1,c,+})$ and out-of-phase quadrupolar mode $({2,c,-})$,
respectively, with the geometric parameters of the two rings chosen such that
the eigenvalues are close in the coaxial configuration. In the coaxial
configuration, the mode $(2,c,-)$ is dominated by a quadrupolar distribution
in the larger ring, whereas the mode $(1,c,+)$ is dominated by a dipolar
distribution in the smaller ring. As $d$ is continuously increased to
$\infty$, the mode $(2,c,-)$ is ultimately dominated by the dipolar mode
$(1,c)$ of the smaller ring; similarly, the mode $(1,c,+)$ is ultimately
dominated by the quadrupolar mode $(2,c)$ of the larger ring. This evolution
is seen to involve multiple stages during which the Fourier harmonics of each
mode undergo mixing and the eigenvalues split apart and then re-approach each
other twice. We will see in §VI some of the implications of this evolution in
the context of the scattering problem.

Figure 7:  Evolution as a function of $d/a_{1}$ of two eigenvalues of the
bilayer heterodimer configuration schematically shown in Fig. 4, in the case
$\kappa_{1}=10$, $\kappa_{2}=5$, $a_{1}/a_{2}=2$ and $h/a_{1}=0.3$. The modes
are those continuated from the in-phase dipolar mode $(1,c,+)$ and out-of-
phase quadrupolar mode $(2,c,-)$, respectively, of the coaxial configuration
obtained for $d=0$. Also shown are the voltage eigenfunctions at the indicated
values of $d/a_{1}$.

##  V Further geometric extensions

###  V.1 Arbitrary cross-sectional shapes

It is straightforward to extend our theoretical framework to allow for non-
circular cross-sectional shapes. In particular, let us briefly revisit the
derivation in §III.2 of the single-ring reduced eigenvalue problem. Given our
focus on longitudinal modes of slender rings, we still expect that the
interior potential is approximately uniform over the ring’s cross section and
varies mainly in the azimuthal direction. Thus, the representation (5) of the
interior potential by an azimuthal voltage profile still holds. The exterior
potential in the vicinity of the ring, however, can no longer be approximated
as in (8), since that radially symmetric distribution assumes that the cross
sections are circular. Nonetheless, it can be shown that (8) still holds at
intermediate radial distances from the centerline, i.e., $b\ll r\ll a$, if
only the cross-sectional radius $bf(\phi)$ is replaced by the so-called
‘conformal radius’, say $bf^{*}(\phi)$, of the cross-sectional geometry at the
azimuthal angle $\phi$. Working in the corresponding cross-sectional plane,
the conformal radius $bf^{*}(\phi)$ can be extracted from a conformal mapping
from the exterior of a circle of that radius to the domain exterior to the
true cross section (see, e.g., (Hinch, 1991, Chapter 5) and Schnitzer and
Craster (2017)). In particular, for elliptical cross sections with semi-
diameters $b\sigma_{1}(\phi)$ and $b\sigma_{2}(\phi)$, one finds
$bf^{*}(\phi)=(b\sigma_{1}(\phi)+b\sigma_{2}(\phi))/2$. Expressions for
several other geometries can be found in (Paquin-Lefebvre  _et al._ , 2020,
Table 1). As a consequence, $f(\phi)$ should be replaced by $f^{*}(\phi)$ in
the capacitance relation (13), while (11) remains unchanged, with
$\bar{A}(\phi)$ still denoting the scaled cross-sectional area
$A(\phi)/b^{2}$. The significance of the present extension is that, in
contrast to the case of circular cross sections, $\bar{A}(\phi)$ can now be
tuned independently from ${f^{*}}(\phi)$. The generalization of this extension
to the case of ring dimers is evident.

Consider for example an azimuthally uniform ring whose cross-sectional shape
is arbitrary. The eigenvalues follow from the result (21) for a torus-shaped
ring as

$\mathcal{E}^{({m})}=-\frac{2\pi\kappa^{2}}{m^{2}\bar{A}}\left(\ln\frac{8\kappa}{f^{*}}-2\sum_{k=1}^{m}\frac{1}{2k-1}\right)^{-1},\quad\text{for}\quad
m=1,2,\ldots$ (43)

Note the logarithmically weak influence of the ring’s cross-sectional shape,
which we emphasize is a specific feature of the longitudinal modes considered
herein. This prediction is consistent with the experimental and numerical
results in Hao  _et al._ (2008), which show only slight differences between
the plasmon-resonance frequencies of rings of different cross-sectional shape
yet similar cross-sectional area.

###  V.2 Chain of rings

For clarity of exposition, we have so far considered either single rings or
ring dimers. It is straightforward, however, to extend our approximation
scheme to a system consisting of an arbitrary number of interacting rings. In
particular, here we consider a coaxial chain of $N$ not necessarily identical
azimuthally invariant rings. (In light of the preceding generalization,
azimuthally invariant does not necessarily imply torus-shaped.) In this case,
the appropriate reduced eigenvalue problem can be treated analytically. The
analysis closely follows the derivation in §§IV.2 for a coaxial dimer of
torus-shaped rings. Similar to the latter case, symmetry implies collective
modes, with azimuthal number $m=1,2,\ldots$, in which the voltage
eigenfunction in the $n$th ring is $v_{n}=c_{n}\cos m\phi$. (There are also
$\pi/2$ rotations of these collective modes having the same eigenvalues.)
Following the steps in §§IV.2, for given $m$ we find the $N\times N$
generalized matrix-eigenvalue problem (cf. (36))

$\mathsf{c}=-\mathcal{E}\,\mathsf{G}\boldsymbol{\cdot}\mathsf{c}$ (44)

where $\mathsf{c}=(c_{1},c_{2},\ldots,c_{N})^{T}$ and $\mathsf{G}$ is a
$m$-dependent $N\times N$ matrix whose components are defined analogously to
($None$).

Figure 8: (a) Eigenvalues of the ‘dipolar’ ($m=1$) modes of a coaxial chain of
$N=43$ identical torus-shaped rings ($\kappa=10$). The chain consists of two
identical sub-chains, each formed of $10$ dimers (spacing between rings and
dimers being $0.3$ and $0.7$ times the ring radius, respectively), which are
linked by a trimer (spacings similar to dimer sub-chains). Each eigenvalue is
doubly degenerate, corresponding to a $\pi/2$ rotation of the eigenfunctions
about the symmetry axis. The spectrum is calculated in the slender-body
approximation by solving (44). It consists of two nearly continuous bands that
cross the eigenvalues $\mathcal{E}^{({1,\pm})}$ of the dimers in isolation.
Additionally, there are two isolated eigenvalues which are associated with
localized ‘defect modes’. The defect mode marked in (a) is visualized in terms
of the dipole intensities $c_{n}$ (cf. (44)) in (b) and voltage profiles in
(c). The horizontal lines in (a) mark the eigenvalues $\mathcal{E}^{({1,-})}$
and $\mathcal{E}^{({1,+})}$ of the dimers in isolation.

As an example, we employ (44) to calculate the eigenfunctions of a finite
coaxial chain of torus-shaped rings. Specifically, we consider a chain
consisting of a series of identical, equally spaced, homodimers, where the
center dimer is replaced by a homotrimer. In Fig. 8, we focus our attention on
the dipolar modes of the chain ($m=1)$, for which the permittivity-eigenvalue
spectrum is seen to consist of two disjoint, nearly continuous, bands formed
of a large number of densely distributed eigenvalues; the lower- and higher-
energy bands respectively cross the eigenvalues $\mathcal{E}^{({1,-})}$ and
$\mathcal{E}^{({1,+})}$ of the dimers in isolation (§IV.2). Additionally,
there are two isolated eigenvalues, one in the spectral gap between the two
bands and one at an energy lower than the lower-energy band. As demonstrated
in the figure, the isolated eigenvalues correspond to modes that are localized
around the defect. We remark that (44) predicts localized modes for all $m$,
not only for the dipolar modes. In retrospect, the existence of localized
modes rationalizes the application of our quasi-static theory to the case of
an extended chain of rings, whose total length in any realistic scenario would
be at least comparable to the free-space wavelength. Accordingly, we expect
non-localized modes of the chain to be significantly affected by retardation
Pocock  _et al._ (2018).

##  VI Plane-wave illumination

Armed with our slender-body approximations for the longitudinal modes of
slender-ring structures, we return to the quasi-static scattering problem
formulated in §II.1. There, the near-field potential $\varphi(\mathbf{x})$
generically possesses the asymptotic far-field behavior Jackson (2007)

$\varphi(\mathbf{x})\sim-\mathbf{E}_{\infty}\boldsymbol{\cdot}\mathbf{x}+\mathbf{E}_{\infty}\boldsymbol{\cdot}\boldsymbol{\alpha}\boldsymbol{\cdot}\frac{\mathbf{x}}{4\pi|\mathbf{x}|^{3}}\quad\text{as}\quad|\mathbf{x}|\to\infty,$
(45)

in which $\boldsymbol{\alpha}$ is the polarizability tensor of the structure.
In terms of that tensor, quasi-static approximations for the extinction and
absorption cross sections in the direction of the applied field, say
$\hat{\textbf{{\i}}}$, are given by Bohren and Huffman (2008)

$C_{\text{ext}},C_{\text{abs}}=\frac{2\pi}{\lambda}\,\hat{\textbf{{\i}}}\hat{\textbf{{\i}}}\boldsymbol{:}\text{Im}\,\boldsymbol{\alpha},$
(46)

where $\lambda$ denotes the wavelength of the incident plane-wave. We shall
use the approximation scheme developed in this paper to calculate
$\boldsymbol{\alpha}$ and hence $C_{\text{abs}}$ for a range of slender-ring
geometries.

A slender-body approximation for $\boldsymbol{\alpha}$ can be extracted by
considering the large $|\mathbf{x}|$ expansion of the spectral solution (4)
and substituting the slender-body approximations for the voltage
eigenfunctions and eigenvalues. For a structure formed of $N$ arbitrarily
shaped rings, we find

$\boldsymbol{\alpha}=\frac{1}{\epsilon_{0}}\sum_{I\in\mathcal{I}}\frac{\epsilon_{r}(\omega)-1}{\epsilon_{r}(\omega)-\mathcal{E}^{(I)}}\frac{\left(\sum_{n=1}^{N}a_{n}\int_{0}^{2\pi}d\phi\,\mathbf{y}_{n}q_{n}^{({I})}\right)\left(\sum_{n=1}^{N}a_{n}\int_{0}^{2\pi}d\phi\,\mathbf{y}_{n}q_{n}^{({I})}\right)}{\sum_{n=1}^{N}a_{n}\int_{0}^{2\pi}d\phi\,q_{n}^{({I})}v_{n}^{({I})}}.$
(47)

A derivation of this result is given in Appendix D. We note that it only
includes the longitudinal modes studied in this paper, which for low
frequencies $\omega\ll\omega_{p}$ dominate the plasmonic response as further
discussed in Ruiz and Schnitzer (2019) and the concluding section §VIII.

To evaluate (47) for general ring geometries, we employ the semi-analytical
scheme ($None$), whose generalization from the case of two to $N$ rings is
straightforward; furthermore, non-circular cross sections can be included in
that scheme as discussed in §V.1.

In the special cases of single azimuthally invariant rings, as well as
azimuthally invariant coaxial dimers, it is possible to evaluate the integrals
appearing in (47) in closed form. In those cases, symmetry dictates that the
polarizability tensor possesses the form
$\boldsymbol{\alpha}=(\mathbf{\hat{e}}_{x}\mathbf{\hat{e}}_{x}+\mathbf{\hat{e}}_{y}\mathbf{\hat{e}}_{y})\alpha$,
where $\mathbf{\hat{e}}_{x}$ and $\mathbf{\hat{e}}_{y}$ are orthogonal unit
vectors parallel to the plane defined by the rings and $\alpha$ is a scalar
polarizability. Furthermore, each mode involves a single Fourier harmonic, and
since the centerlines $\mathbf{y}_{n}$ are circular curves, the overlap
integrals in the numerator of (47) vanish for all except the dipolar modes.
Assuming for simplicity that the azimuthally invariant rings are torus-shaped,
we use the results of §III to find

$\alpha=\frac{\epsilon_{r}(\omega)-1}{\epsilon_{r}(\omega)-\mathcal{E}^{(1)}}\frac{2\pi^{2}a^{3}}{\ln
8\kappa-2},$ (48)

with the slender-body approximation for $\mathcal{E}^{({1})}$ provided by
(21). Similarly, using the results of §IV.2, we find for a coaxial dimer of
torus-shaped rings

${\alpha}=\sum_{\pm}\frac{1-\epsilon_{r}(\omega)}{\epsilon_{r}(\omega)-\mathcal{E}^{({1,\pm})}}\frac{\pi\mathcal{E}^{({1,\pm})}\left(A_{1}c_{1}^{({1,\pm})}+A_{2}c_{2}^{({1,\pm})}\right)^{2}}{\frac{A_{1}}{a_{1}}\left(c_{1}^{({1,\pm})}\right)^{2}+\frac{A_{2}}{a_{2}}\left(c_{2}^{({1,\pm})}\right)^{2}},$
(49)

with $c_{1}^{({1,\pm})}$ and $c_{1}^{({2,\pm})}$ provided by (39) and a
slender-body approximation for $\mathcal{E}^{({1,\pm})}$ provided by (38). In
the more specific coaxial homodimer case, the numerator in (49) vanishes
identically for the out-of-phase dipolar mode, namely because the induced
dipole excited in the two rings cancel. Thus, (49) degenerates to

$\alpha=\frac{\epsilon_{r}(\omega)-1}{\epsilon_{r}(\omega)-\mathcal{E}^{(1,+)}}\frac{4\pi^{2}a^{3}}{\ln
8\kappa-2+\frac{1}{2}\Delta_{1}},$ (50)

where $\mathcal{E}^{({1,+})}$ now possesses the slender-body approximation
(33) and $\Delta_{1}$ is given by (34).

In formulas (47)–(50), resonance of mode ${I}$ is associated with cancellation
in the denominator of the frequency-dependent factor
$F^{({I})}=(\epsilon_{r}(\omega)-1)/(\epsilon_{r}(\omega)-\mathcal{E}^{({I})})$.
Let $\mathcal{E}^{({I})}_{0}$ denote the slender-body approximation for
$\mathcal{E}^{({I})}$. We stress that the approximation
$F\approx(\epsilon_{r}(\omega)-1)/(\epsilon_{r}(\omega)-\mathcal{E}^{({I})}_{0})$
only holds if
$|\mathcal{E}^{({I})}-\mathcal{E}^{({I})}_{0}|\ll\mathrm{Im}(\epsilon_{r}(\omega))$,
namely if the loss is large relative to the error in the slender-body
approximation for the eigenvalue. On the one hand, the algebraic order of the
latter error is smaller than $\kappa^{2}$, though we do not know whether it is
unity, $\kappa$ or some other scaling $\ll\kappa^{2}$. On the other hand, the
Drude model (2) and eigenvalue scaling (16) together imply
$\mathrm{Im}(\epsilon_{r}(\omega))\simeq\kappa^{3}(\gamma/\omega_{p})$,
suggesting that the condition is easily met for the longitudinal resonances.
This is confirmed by the comparison below with exact quasi-static solutions.
Even if the condition
$|\mathcal{E}^{({I})}-\mathcal{E}^{({I})}_{0}|\ll\mathrm{Im}(\epsilon_{r}(\omega))$
is not met, formulas (47)–(50) still give a valid approximation for the
resonance curve only shifted in frequency; moreover, the approximation
$F^{({I})}\approx\mathcal{E}^{({I})}_{0}/\mathrm{Im}(\epsilon_{r})$ still
holds at resonance, i.e., at $\omega$ such that
$\mathrm{Im}(\epsilon_{r})=\mathcal{E}^{({I})}$.

We now apply the above results for the scattering problem to several ring
structures whose plasmonic eigenvalues and eigenfunctions we have analyzed
earlier in the paper. In particular, we consider single torus-shaped and
azimuthally non-uniform rings, as well as coaxial and non-coaxial dimers of
torus-shaped rings.

Figure 9: Absorption cross section (46), normalized by $S=\pi a^{2}$, for the
torus-shaped and azimuthally non-uniform rings defined in the caption of Fig.
3. The incident-field direction $\boldsymbol{\hat{\imath}}$ is in the plane of
the ring, pointing in the maximum-thickness direction in the case of the
azimuthally non-uniform ring. For the torus-shaped ring, the slender-body
approximation (48) is compared with an exact quasi-static solution Mary  _et
al._ (2007). The slender-body approximation for the azimuthally nonuniform
ring is evaluated numerically from (47). We assume the Drude model (2) with
$\omega_{p}=1.196\times 10^{16}\,\text{rad}/\text{s}$ and $\gamma=8.05\times
10^{13}\,\text{rad}/\text{s}$. Figure 10: Absorption cross section (46),
normalized by $S=\pi a_{1}^{2}$, for the coaxial homodimer of Fig. 5; the
coaxial heterodimer of Fig. 5 and Fig. 7 for $d=0$; and the bilayer
heterodimer of Fig. 7 for $d/a_{1}=0.5$. In all cases $a_{1}\geq a_{2}$ and
the incident-field direction $\boldsymbol{\hat{\imath}}$ is in the plane of
the rings, pointing in the direction of the horizontal displacement of the
smaller ring in the non-coaxial case. The curves represent slender-body
approximations evaluated using (50) for the coaxial homodimer, (49) for the
coaxial heterodimer, and (47) for the non-coaxial heterodimer. The vertical
lines mark the frequencies corresponding to the eigenvalues
$\mathcal{E}^{({1})}$ of the larger and smaller torus-shaped rings in
isolation. We assume the Drude model (2) with $\omega_{p}=1.196\times
10^{16}\,\text{rad}/\text{s}$ and $\gamma=8.05\times
10^{13}\,\text{rad}/\text{s}$.

In Fig. 9, we show $C_{\text{abs}}$, normalized by the planar area $\pi
a^{2}$, for a torus-shaped ring alongside that for the azimuthally non-uniform
ring from Fig. 3. The incident-field direction $\boldsymbol{\hat{\imath}}$ is
parallel to the plane of the ring; in the case of the azimuthally non-uniform
ring, $\boldsymbol{\hat{\imath}}$ is further specified as pointing towards the
direction of maximum thickness of the ring. For the torus-shaped ring, the
slender-body approximation (48) is compared with the exact quasi-static
computations in Mary  _et al._ (2007) and good agreement is found despite the
aspect ratio $\kappa=10$ being only moderately large. The torus-shaped ring
exhibits a single resonance peak, which is associated with the excitation of
the cosine-dipolar mode $(1,c)$ shown at the top-right corner of Fig. 3. (With
$\phi$ measured from $\boldsymbol{\hat{\imath}}$, the corresponding sine mode
is not excited.) In contrast, for the azimuthally non-uniform ring, we observe
multiple resonance peaks. As explained in §III.4.3, in that case all modes
generally include a dipolar component (first Fourier harmonic). Accordingly,
the overlap integrals in (47) do not vanish identically as they do in the
torus case for $m>1$. In particular, the lowest- and second-lowest-frequency
peaks seen in Fig. 9 are due to excitation of the ‘dipolar’ mode $(1,c)$ and
‘quadrupolar’ mode $(2,c)$ shown on the left of Fig. 3.

In Fig. 10, we show $C_{\text{abs}}$, normalized by the planar area $\pi
a_{1}^{2}$, for several bilayer dimer configurations formed of torus-shaped
rings: a coaxial homodimer (same as in Fig. 5), a coaxial heterodimer (same as
in Fig. 5 and Fig. 7 for $d=0$) and a bilayer non-coaxial heterodimer (same as
in Fig. 7 for $d/a_{1}=0.5$). The incident-field direction
$\boldsymbol{\hat{\imath}}$ is taken parallel to the planes of the rings; in
the non-coaxial-heterodimer case, $\boldsymbol{\hat{\imath}}$ is further
specified as pointing along the direction of the horizontal displacement of
the smaller ring.

For the coaxial homodimer, we observe a single resonant peak, a prediction
which is clearly inferred from the closed-form slender-body approximation
(50). This is because only the in-phase cosine-dipolar mode $(1,c,+)$ is
excited by the plane wave. (With $\phi$ measured from
$\boldsymbol{\hat{\imath}}$, the corresponding sine mode is not excited.)
While this is not demonstrated here, we know from the analytical expression
(33) for $\mathcal{E}^{({1,+})}$, or from Fig. 5a, to expect this peak to
blueshift with decreasing distance between the rings. This is opposite to the
trend observed for the analogous low-frequency bonding-gap modes of sphere
dimers Klimov and Guzatov (2007); Pendry  _et al._ (2013); Schnitzer (2015,
2020).

For the coaxial heterodimer, we observe an additional resonance peak at higher
frequency. This is because the net dipole induced in the rings for the out-of-
phase dipolar modes does not cancel out between the two rings as it does in
the coaxial-homodimer case. This feature is easily inferred from the slender-
body approximation (49), which explicitly shows two resonances associated with
the in-phase and out-of-phase dipolar modes. (With $\phi$ measured from
$\boldsymbol{\hat{\imath}}$, it is again only the ‘cosine’ dipolar modes that
are excited.) As seen in Fig. 5b, for this configuration the out-of-phase
dipolar mode is dominated by a dipolar distribution in the larger ring,
whereas the in-phase dipolar mode is dominated by a dipolar distribution in
the smaller ring. From Fig. 5a, we expect the in-phase and out-of-phase
resonance to respectively blueshift and redshift as the vertical distance
between the rings is reduced.

Consider next the non-coaxial-heterodimer configuration. Similarly to the
azimuthally non-uniform ring, we observe multiple resonance peaks owing to the
absence of axial symmetry. In particular, we note that the higher-frequency
resonance in the coaxial-heterodimer case, which is associated with the in-
phase-dipolar mode $(1,c,+)$ of that configuration, is replaced by two
distinct peaks. This can be understood from the study carried out in Fig. 7.
Thus, this pair of resonances are associated with the excitation of the two
modes continuated from the in-phase-dipolar $(1,c,+)$ and out-of-phase-
quadrupolar $(2,c,-)$ modes of the corresponding coaxial-heterodimer
configuration. While, owing to symmetry, in the coaxial case only dipolar
modes are excited, in the non-coaxial case both modes include dipolar
components and can therefore be excited. Specifically, the non-coaxial
configuration is as shown in insets (b) and (g) of Fig. 7, in which case the
eigenvalues are farthest apart as a function of the horizontal displacement,
and both modes have a visible dipolar component in the smaller ring.

##  VII Validity of quasi-static formulation

The theory in this paper is predicated on the quasi-static approximation,
which is widely used to describe the localised-surface-plasmon resonances of
nanometallic structures Klimov (2014). In this approximation, radiation
damping is neglected on the basis of the smallness of the particle relative to
the free-space wavelength, and the near-field of the particle is assumed to be
irrotational and forced by the field locally associated with the incident
radiation. For sufficiently weak material loss, however, radiative corrections
must in fact be appreciable even for particles which are much smaller than the
free-space wavelength Aubry  _et al._ (2010b); Ruan and Fan (2010). In the
concluding section §VIII, we discuss opportunities to extend the present
theory to incorporate radiative corrections for low-loss subwavelength
particles, as well as to a full-wave theory for slender rings whose diameter
is comparable to the free-space wavelength. Here, we derive a scaling
condition for the quasi-static formulation to adequately describe the
longitudinal resonances of a slender ring. Furthermore, we illustrate these
constraints by comparing with full-wave simulations.

Figure 11: Extinction cross section for torus-shaped rings for the indicated
values of the ring radius $a$ and aspect ratio $\kappa$. The incident field is
parallel to the ring plane. Solid lines: solutions exact in the quasi-static
approximation Mary  _et al._ (2007). Dash-dotted lines: slender-body
approximation (48). Circles: full-wave numerical computations using the MNPBEM
toolbox Hohenester and Trügler (2012). We assume the Drude model (2) with
$\omega_{p}=1.196\times 10^{16}\,\text{rad}/\text{s}$ and $\gamma=8.05\times
10^{13}\,\text{rad}/\text{s}$.

Consider first the condition for a slender ring to be subwavelength at its
longitudinal resonances. At the level of scalings, and ignoring logarithmic
factors, (2) and (16) give the free-space wavelength
$\lambda\simeq\kappa\lambda_{p}$, where $\lambda_{p}=2\pi c/\omega_{p}$. Thus,
the condition $a\ll\lambda$ implies $a\ll\lambda_{p}\kappa$, or, for the
typical value $\omega_{p}=1.196\times 10^{16}\,\text{rad}/\text{s}$,
$a\ll\kappa\times 150\text{nm}$. Under this condition, which is met even for
quite large slender rings, the field is approximately irrotational in the
vicinity of the ring, whereby the quasi-static modes remain relevant to the
expansion of that near field. Nonetheless, the quasi-static scattering problem
formulated in §II, and in particular its spectral solution (4), is valid under
a more stringent condition. Thus, the scattering cross section, which scales
like $k^{4}\alpha^{2}$ in the quasi-static approximation, must be negligible
compared to the absorption cross-section. Using (46), this implies
$\alpha\ll\lambda^{3}$. Estimating $\lambda$ at the longitudinal resonances as
before, and using (48) (with (16)), we find
$a^{3}/\mathrm{Im}(\epsilon_{r})\ll\kappa\lambda_{p}^{3}$. More explicitly,
using the Drude model (2) to estimate $\mathrm{Im}(\epsilon_{r})$ (with (16))
we can write this condition as

$\frac{a}{\lambda_{p}}\ll\left(\frac{\gamma}{\omega_{p}}\right)^{1/3}\times\kappa^{4/3}.$
(51)

For typical parameters (same $\omega_{p}$ as above and $\gamma=8.05\times
10^{13}\,\text{rad}/\text{s}$), we have $a\ll\kappa^{4/3}\times 30\text{nm}$.

In Fig. 11, we show a comparison between quasi-static and full-wave
predictions for the extinction cross-section of a torus-shaped ring
($\kappa=10$, typical values of $\omega_{p}$ and $\gamma$ as above), with the
incident field in the ring plane. The full-wave results are obtained by
boundary-element simulation using the the MNPBEM toolbox Hohenester and
Trügler (2012). As for the quasi-static results, both exact Mary  _et al._
(2007) and slender-body approximations are shown (cf. (48)). This comparison
demonstrates the improved agreement between full-wave and quasi-static
predictions with decreasing $a$ and increasing $\kappa$.

##  VIII Concluding remarks

We have developed an approximate theory to describe the longitudinal
localized-surface-plasmon resonances of slender metallic nanorings of
virtually arbitrary shape, as well as more involved nanometallic structures
formed of two or more such rings, the separation between the rings being
comparable with their centerline radii. At the heart of the theory is an
asymptotic reduction of the 3D plasmonic eigenvalue problem governing the
material- and frequency-independent longitudinal modes of the structure to a
reduced 1D formulation, in which the plasmonic eigenmodes are represented by
azimuthal voltage and polarization-charge profiles attached to each ring. Once
this reduced eigenvalue problem has been solved for a given geometry,
approximations for the near-field eigenfunctions in 3D can be extracted from
their associated 1D eigenfunctions. When joined with the standard spectral
theory of plasmonic resonance, this provides the means to obtain any quantity
of interest in a physical scattering problem (that involves the same
geometry), such as scattering cross sections. Overall, the present theoretical
framework allows to rapidly gauge the plasmonic properties of unprecedentedly
complex 3D structures, which may foster the study of new plasmonic structures
and phenomena.

Our approximation approach heavily draws on a transfer of knowledge from the
field of fluid dynamics, where approximations of the sort derived here are
known as slender-body theory. We stress that the slender-body theory developed
herein is of the more accurate ‘nonlocal’ type, which in the present context
means that we account for electrostatic interactions between any two azimuthal
segments of any of the rings forming the structure. The inclusion of these
interactions can be shown to ensure an ‘algebraic’ rather than ‘logarithmic’
asymptotic accuracy of the scheme in the large-aspect-ratio limit pertinent to
slender rings. This claimed accuracy is demonstrated by the good agreement in
Figs. 2 and 9 with known solutions for single torus-shaped rings.

To demonstrate the versatility of the theory, we have applied it to the
calculation of the longitudinal modes and absorption cross sections of a
number of configurations. Still, it is clear that we only considered a small
sample of the wide range of geometries that can be studied. For several
families of geometries, namely azimuthally invariant rings (not necessarily
torus-shaped) and coaxial dimers and chains thereof, the reduced formulation
was solved in closed form, thus generating several apparently new analytical
approximations. We also considered a range of other geometries, including
azimuthally nonuniform rings and non-coaxial multiple-ring configurations. In
the latter cases, we solved the reduced formulation using straightforward
semi-analytical schemes, in which the 1D eigenfunctions are represented by
their Fourier coefficients.

We have focused on the description of the longitudinal (low-frequency) modes
of slender-ring structures, as defined in the introduction and in §III.2, as
well as the excitation of these modes by incident plane waves. Slender rings
also possess non-longitudinal, or transverse modes, characterised by
polarization-charge distributions which, in contrast to the longitudinal case,
exhibit significant variation in the cross-sectional planes. As we discussed
in Ruiz and Schnitzer (2019), and others have observed in theory and
experiments Link  _et al._ (1999); Murphy  _et al._ (2005), the contribution
of non-longitudinal modes to the resonant response of slender bodies is
generally negligible, except in specialised excitation scenarios involving
near-field sources located very close to the structure. To treat those cases,
our slender-body theory could be extended to describe the transverse modes of
slender-ring structures. A leading-order analysis, as done in Ruiz and
Schnitzer (2019) for axisymmetric bodies, shows that those modes essentially
correspond to the in-plane modes of an infinite cylinder, hence with the
permittivity eigenvalues satisfying the asymptotic behaviour
$\mathcal{E}\rightarrow-1$ as $\kappa\to\infty$. (Alluding to Drude’s model
(2), we see that transverse modes are high frequency relative to longitudinal
modes.) To consider corrections to this leading behaviour, one would need to
match the dipolar (or quadruople, etc.) cross-sectional potential
distributions with an approximation of the potential on the ring scale formed
by superposition of the corresponding potential singularities along the
centreline of the body.

Another desirable extension to this work would be to consider the effects of
retardation and radiation damping. In particular, there exists a
generalization of the spectral approach employed herein based on permittivity
eigenvalues to the full-wave regime Bergman and Stroud (1980); Agranovich  _et
al._ (1999); Farhi and Bergman (2015); Chen  _et al._ (2019). In that
generalized formulation, the permittivity eigenvalues become complex valued
and frequency dependent. To describe the regime of low-loss and moderately
subwavelength slender-ring structures, where deviations from the quasi-static
can be substantial (as demonstrated in §VII), it would be necessary to carry
out a low-frequency analysis of the generalized eigenvalue problem up to the
order of the permittivity eigenvalue that is comparable to the imaginary part
of the physical permittivity. (This regime could also be described by applying
suitable corrections to the quasi-static predictions following Wokaun  _et
al._ (1982); Draine and Flatau (1994); Carminati  _et al._ (2006); Aubry  _et
al._ (2010b).) Another avenue would be to develop a full-wave slender-body
theory in the case where the ring diameter is comparable to the free-space
wavelength; the cross-sectional scale would still be subwavelength, while the
ring-scale approximation could be formed by superposition of fundamental
solutions of Maxwell’s equations along the centerline of the ring. Lastly, we
note that it is possible but not straightforward to generalize the
permittivity-eigenvalue formulation to multi-constituent photonic structures
Bergman  _et al._ (2020), e.g., hybrid metal-dielectric structures. For such
ring structures, a reformulation of the theory based on quasi-normal modes and
complex frequency eigenvalues may be more convenient Sauvan  _et al._ (2013).

Acknowledgement. The authors acknowledge funding from EPSRC UK (New
Investigator Award EP/R041458/1).

##  Appendix A Matching the ring and cross-sectional scales

In this appendix, we shall derive the small $r$ behavior of the ring-scale
potential (12). A preliminary step is to write that solution in the form

$\varphi=\frac{q(\phi)}{4\pi\epsilon_{0}}\int_{0}^{2\pi}\,ad\phi^{\prime}\,\frac{1}{|\mathbf{x}(r,\theta,\phi)-\mathbf{y}(\phi^{\prime})|}+\frac{1}{4\pi\epsilon_{0}}\int_{0}^{2\pi}\,ad\phi^{\prime}\,\frac{q(\phi^{\prime})-q(\phi)}{|\mathbf{x}(r,\theta,\phi)-\mathbf{y}(\phi^{\prime})|},$
(52)

where the geometry gives

$\displaystyle|\mathbf{x}(r,\theta,\phi)-\mathbf{y}(\phi^{\prime})|=\sqrt{r^{2}+4(a^{2}+ar\cos\theta)\sin^{2}\frac{\phi-\phi^{\prime}}{2}}.$
(53)

Using (53), it is readily seen that the second integral in (52) is regular as
$r\rightarrow 0$:

$\int_{0}^{2\pi}\,ad\phi^{\prime}\,\frac{q(\phi^{\prime})-q(\phi)}{|\mathbf{x}(r,\theta,\phi)-\mathbf{y}(\phi^{\prime})|}=\int_{0}^{2\pi}\,d\phi^{\prime}\,\frac{q(\phi^{\prime})-q(\phi)}{2\sin\frac{\left|\phi-\phi^{\prime}\right|}{2}}+o(1)\quad\text{as}\quad
r\rightarrow 0.$ (54)

To treat the first integral in (52), which is singular as $r\to 0$, we use the
method of splitting the range of integration Hinch (1991). Thus, we write

$\int_{0}^{2\pi}\,ad\phi^{\prime}\,\frac{1}{|\mathbf{x}(r,\theta,\phi)-\mathbf{y}(\phi^{\prime})|}=\left\\{\int_{0}^{\delta}\,dt+\int_{\delta}^{\pi}\,dt\right\\}\frac{2}{\sqrt{\bar{r}^{2}+4(1+\bar{r}\cos\theta)\sin^{2}\frac{t}{2}}},$
(55)

where we introduce the normalized radial coordinate $\bar{r}=r/a$ and an
auxiliary parameter in the range $\bar{r}\ll\delta\ll 1$. Denote the first
integral on the right-hand side of (55) by $I_{1}$. Making the change of
variables $\tau=t/\bar{r}$, we have

$I_{1}=2\int_{0}^{\delta/\bar{r}}\frac{d\tau}{\sqrt{1+4(1+\bar{r}\cos\theta)\bar{r}^{-2}\sin^{2}(\bar{r}\tau/2)}}.$
(56)

Using $\bar{r}\tau<\delta\ll 1$, the integrand can be expanded to show that

$I_{1}=2\int_{0}^{\delta/\bar{r}}\frac{d\tau}{(1+\tau^{2})^{1/2}}+o(1)=2\ln\frac{2\delta}{\bar{r}}+o(1)\quad\text{as}\quad\bar{r}\to
0,$ (57)

where for the last step we used $\delta/\bar{r}\gg 1$. Similarly, denote the
second integral on the right-hand side of (55) by $I_{2}$. Using
$t>\delta\gg\bar{r}$, the integrand of that integral can be expanded to show
that

$I_{2}=\int_{\delta}^{\pi}\frac{dt}{\sin\frac{t}{2}}+o(1)=2\ln\frac{4}{\delta}+o(1)\quad\text{as}\quad\bar{r}\to
0.$ (58)

Combining (54), (57) and (58), we find the requisite behavior

$\varphi(\mathbf{x})=\frac{q(\phi)}{2\pi\epsilon_{0}}\ln\frac{8a}{r}+\frac{1}{4\pi\epsilon_{0}}\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q(\phi^{\prime})-q(\phi)}{2\sin\frac{|\phi-\phi^{\prime}|}{2}}+o(1)\quad\text{as}\quad
r\to 0.$ (59)

##  Appendix B Proof of identity (19)

In this appendix we prove identity (19), which constitutes a diagonalization
of the integral operator appearing in the capacitance relation (13). To this
end, it is convenient to denote that integral operator as

$\mathcal{N}[q(\phi)]=\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q(\phi^{\prime})-q(\phi)}{2\sin\frac{\left|\phi-\phi^{\prime}\right|}{2}}.$
(60)

We shall show that, for any non-negative integer $n$, there exists a constant
$\lambda_{n}$ such that

$\mathcal{N}[\cos n\phi]=\lambda_{n}\cos n\phi,\quad\mathcal{N}[\sin
n\phi]=\lambda_{n}\sin n\phi.$ $None$

Our method of proof will also furnish the constants, which constitute the
eigenvalues of the operator $\mathcal{N}$. In what follows, we consider only
the cosine eigenfunctions, the corresponding result for the sine
eigenfunctions readily following from a straightforward change of variables.

Direct calculation shows that $\mathcal{N}[1]=0$ and
$\mathcal{N}[\cos\phi]=-4\cos\phi$, thence that ($None$a) holds for $n=0$ and
$n=1$ with $\lambda_{0}=0$ and $\lambda_{1}=-4$. Now, let $m$ be any integer
$>1$ and assume that ($None$a) also holds for $n=1,\ldots,m-1$, with

$\lambda_{n}=-\frac{4}{2n-1}+\lambda_{n-1}.$ (62)

This assumed difference relation is consistent with the result for $n=1$ and
implies that

$\lambda_{n}=-4\sum_{k=1}^{n}\frac{1}{2k-1}\quad\text{for}\quad
n=1,\ldots,m-1.$ (63)

Next, we use the definition (60) and ($None$a) for $n=m-2$ to write

$\displaystyle\mathcal{N}[\cos m\phi]$ $\displaystyle=$ $\displaystyle
2\mathcal{N}[\cos\left((m-1)\phi\right)\cos\phi]-\lambda_{m-2}\cos\left((m-2)\phi\right).$
(64)

Using (60) ($None$a) for $n=m-1$, we have

$\mathcal{N}[\cos\left((m-1)\phi\right)\cos\phi]=\int_{0}^{2\pi}d\phi^{\prime}\,\frac{\cos\left((m-1)\phi^{\prime}\right)\cos\phi^{\prime}-\cos\left((m-1)\phi\right)\cos\phi}{2\sin\frac{|\phi^{\prime}-\phi|}{2}}\\\
=\int_{0}^{2\pi}d\phi^{\prime}\,\frac{\cos\left((m-1)\phi^{\prime}\right)\left(\cos\phi^{\prime}-\cos\phi\right)}{2\sin\frac{|\phi^{\prime}-\phi|}{2}}+\cos\phi\,\mathcal{N}[\cos\left((m-1)\phi\right)]\\\
=-\int_{0}^{2\pi}d\phi^{\prime}\,\frac{\cos\left((m-1)\phi^{\prime}\right)\sin{\frac{\phi^{\prime}-\phi}{2}}\sin{\frac{\phi^{\prime}+\phi}{2}}}{\sin\frac{|\phi^{\prime}-\phi|}{2}}+\lambda_{m-1}\cos\phi\cos\left((m-1)\phi\right)\\\
=2\frac{\cos((m-2)\phi)}{2m-3}-2\frac{\cos
m\phi}{2m-1}+\frac{1}{2}\lambda_{m-1}\left(\cos m\phi+\cos((m-2)\phi)\right).$
(65)

So that combining (64) and (65) gives

$\displaystyle\mathcal{N}[\cos
m\phi]=\left(\lambda_{m-1}-\frac{4}{2m-1}\right)\cos
m\phi+\left(\lambda_{m-1}-\lambda_{m-2}+\frac{4}{2m-3}\right)\cos((m-2)\phi),$
(66)

where the second term on the right-hand side vanishes owing to (62), while the
first term on the right-hand side shows that (62), and thence (63), holds also
for $n=m$. It follows by induction that ($None$) holds for any non-negative
integer $n$, with $\lambda_{0}=0$ and $\lambda_{n}$ given by (63) for $n>0$.

##  Appendix C Semi-analytical scheme

###  C.1 Single ring

For a single ring of non-uniform thickness, the reduced eigenvalue problem of
§III.2 is approximated by the generalized matrix-eigenvalue problem ($None$).
In that problem, $\mathsf{M}=\\{M_{n,k}\\}$ and $\mathsf{U}=\\{U_{n,k}\\}$ are
$2K\times 2K$ matrices with components

$M_{n,k}=\frac{nk}{\pi}\times\begin{dcases}\int_{0}^{2\pi}d\phi\,\bar{A}(\phi)\sin
n\phi\sin k\phi,\quad n\leq K,\,k\leq K\\\
-\int_{0}^{2\pi}d\phi\,\bar{A}(\phi)\sin n\phi\cos k\phi,\quad n\leq
K,\,k>K\\\ -\int_{0}^{2\pi}d\phi\,\bar{A}(\phi)\cos n\phi\sin k\phi,\quad
n>K,\,k\leq K\\\ \int_{0}^{2\pi}d\phi\,\bar{A}(\phi)\cos n\phi\cos k\phi,\quad
n>K,\,k>K\end{dcases}$ (67)

and

$U_{n,k}=-\frac{1}{\pi}\delta_{n,k}\sum_{k=1}^{n}\frac{1}{2k-1}+\frac{1}{2\pi^{2}}\times\begin{dcases}\int_{0}^{2\pi}d\phi\,\cos
n\phi\cos k\phi\ln{\frac{8\kappa}{f(\phi)}},\quad n\leq K,\,k\leq K\\\
\int_{0}^{2\pi}d\phi\,\cos n\phi\sin k\phi\ln{\frac{8\kappa}{f(\phi)}},\quad
n\leq K,\,k>K\\\ \int_{0}^{2\pi}d\phi\,\sin n\phi\cos
k\phi\ln{\frac{8\kappa}{f(\phi)}},\quad n>K,\,k\leq K\\\
\int_{0}^{2\pi}d\phi\,\sin n\phi\sin k\phi\ln{\frac{8\kappa}{f(\phi)}},\quad
n>K,\,k>K.\end{dcases}$ (68)

Furthermore, the coefficient $\alpha_{0}$ can be calculated a posteriori as

$\alpha_{0}=\frac{1}{4\pi^{2}}\sum_{k=1}^{K}\int_{0}^{2\pi}d\phi\,\left(\tilde{\alpha}_{k}\cos
k\phi+\tilde{\beta}_{k}\sin k\phi\right)\ln{\frac{8\kappa}{f(\phi)}}.$ (69)

###  C.2 Ring dimer

For an arbitrary ring dimer, the generalized eigenvalue problem of §IV.1 is
approximated by the generalized matrix-eigenvalue problem ($None$). In that
problem, the $2K\times 2K$ matrices $\mathsf{M}_{n}$ and $\mathsf{U}_{n}$ are
like above with $n=1,2$ indicated the ring number, whereas
$\mathsf{V}=\\{V_{n,k}\\}$ is a $2K\times 2K$ coupling matrix with components

$V_{n,k}=\frac{\sqrt{a_{1}a_{2}}}{4\pi^{2}}\begin{dcases}\int_{0}^{2\pi}\int_{0}^{2\pi}d\phi_{1}d\phi_{2}\,\frac{\cos
n\phi_{1}\cos
k\phi_{2}}{|\mathbf{y}_{1}(\phi_{1})-\mathbf{y}_{2}(\phi_{2})|},\quad n\leq
K,\,k\leq K\\\ \int_{0}^{2\pi}\int_{0}^{2\pi}d\phi_{1}d\phi_{2}\,\frac{\cos
n\phi_{1}\sin
k\phi_{2}}{|\mathbf{y}_{1}(\phi_{1})-\mathbf{y}_{2}(\phi_{2})|},\quad n\leq
K,\,k>K\\\ \int_{0}^{2\pi}\int_{0}^{2\pi}d\phi_{1}d\phi_{2}\,\frac{\sin
n\phi_{1}\cos
k\phi_{2}}{|\mathbf{y}_{1}(\phi_{1})-\mathbf{y}_{2}(\phi_{2})|},\quad
n>K,\,k\leq K\\\ \int_{0}^{2\pi}\int_{0}^{2\pi}d\phi_{1}d\phi_{2}\,\frac{\sin
n\phi_{1}\sin
k\phi_{2}}{|\mathbf{y}_{1}(\phi_{1})-\mathbf{y}_{2}(\phi_{2})|},\quad
n>K,\,k>K.\\\ \end{dcases}$ (70)

##  Appendix D Far-field expansion

In this appendix we derive the far field behavior of the scattered field
starting from the spectral solution (4), with the eigenvalues and eigenmodes
approximated by their slender-body approximations. As in §VI, we only include
longitudinal modes in this calculation. For generality, we consider a system
of $N$ rings.

Consider first the far-field behavior of the eigenpotentials
$\varphi^{({I})}(\mathbf{x})$, whose slender-body approximations are (cf.
(28))

$\varphi^{({I})}(\mathbf{x})=\sum_{n=1}^{N}\frac{a_{n}}{4\pi\epsilon_{0}}\int_{0}^{2\pi}d\phi^{\prime}\,\frac{q_{n}^{({I})}(\phi^{\prime})}{|\mathbf{x}-\mathbf{y}_{n}(\phi^{\prime})|}.$
(71)

Using the asymptotic behavior

$\frac{1}{|\mathbf{x}-\mathbf{y}_{n}|}\sim\frac{1}{|\mathbf{x}|}+\frac{\mathbf{x}\boldsymbol{\cdot}\mathbf{y}_{n}}{|\mathbf{x}|^{3}}\quad\text{as}\quad|\mathbf{x}|\to\infty,$
(72)

for $n=1,2,\ldots,N$, and the zero-charge constraints
$\int_{0}^{2\pi}d\phi\,q_{n}^{({\mathcal{I}})}=0$, (71) gives

$\varphi^{({\mathcal{I}})}(\mathbf{x})\sim\frac{\mathbf{x}}{4\pi\epsilon_{0}|\mathbf{x}|^{3}}\boldsymbol{\cdot}\sum_{n=1}^{N}a_{n}\int_{0}^{2\pi}d\phi\,\mathbf{y}_{n}\,q_{n}^{({\mathcal{I}})}\quad\text{as}\quad|\mathbf{x}|\to\infty.$
(73)

We next note that, for the sake of evaluating the overlap and normalization
integrals in (4), we may approximate the eigenfield inside the $n$th ring as

$\boldsymbol{\nabla}\varphi_{n}^{({\mathcal{I}})}=\frac{1}{a_{n}^{2}}\frac{dv_{n}^{({I})}}{d\phi}\frac{\partial\mathbf{y}_{n}}{\partial\phi}.$
(74)

Thus, using integration by parts together with the effective Gauss law (27),
we find

$\frac{\int
dV\,\mathbf{E}_{\infty}\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi^{({I})}}{\int
dV\,\boldsymbol{\nabla}\varphi^{({I})}\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi^{({I})}}=\frac{\sum_{n=1}^{N}a_{n}\int_{0}^{2\pi}d\phi\,\mathbf{E}_{\infty}\boldsymbol{\cdot}\mathbf{y}_{n}\,q_{n}^{({I})}}{\sum_{n=1}^{N}a_{n}\int_{0}^{2\pi}d\phi\,v_{n}^{({I})}q_{n}^{({I})}}.$
(75)

We emphasize that this result is valid even for azimuthaly non-uniform rings
and arbitrary cross-sectional shapes.

With the above slender-body approximations, the spectral solution (4) gives
the far-field behavior (45), with the polarization tensor
$\boldsymbol{\alpha}$ given by (47).

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