## Abstract

Lattices of plasmonic nanorings with particular geometries exhibit singular, tunable resonance features in the infrared. This work examined effects of nanoring inner radius, wall thickness, and lattice constant on the spectral response of single nanorings and in Fano resonant square lattices, combining use of the discrete and coupled dipole approximations. Increasing nanoring inner radius red-shifted and broadened the localized surface plasmon resonance (LSPR), while wall thickness modulated the LSPR wavelength and decreased absorption relative to scattering. The square lattice constant was tuned to observe diffractively-coupled lattice resonances, which increased resonant extinction 4.3-fold over the single-ring LSPR through Fano resonance. Refractive index sensitivities of 760 and 1075 nm RIU^{−1} were computed for the plasmon and lattice resonances of an optimized nanoring lattice. Sensitivity of an optimal nanoring lattice to a local change in dielectric, useful for sensing applications, was 4 to 5 times higher than for isolated nanorings or non-coupling arrays. This was attributable to the Fano line-shape in far-field diffractive coupling with near-field LSPR.

© 2014 Optical Society of America

## 1. Introduction

Noble-metal nanostructures can be engineered to support extraordinary electromagnetic (EM) activity such as super-resolution and optical cloaking as well as integration into optoelectronics like biosensing [1] and thermophotovoltaics [2]. The geometry, morphology, and spatial arrangement of nanostructures determines wavelength, intensity, and bandwidth of their spectral features including localized surface plasmon resonance [3–5] (LSPR) and lattice resonances [6–8] (LR) in response to incident EM energy. Isolated ring-shaped nanostructures in particular exhibit near-infrared resonance [9], enhanced sensitivity to surrounding dielectric [9,10], and tunable, intense optical responses [11] relative to spheroids due to geometric features such as higher surface area to volume ratio. However, characteristic features of LRs which arise from Fano resonance [8] and enhance dielectric sensitivity in spheroid lattices have not been evaluated in lattices of nanorings. Accurate prediction of interrelated effects of nanoring geometry and lattice constant on spectral responses of isolated and arrayed nanorings that enhance extinction and sensitivity in the infrared could lead to improved metamaterials for optoelectronic applications.

EM interactions in isolated nanorings have been evaluated by discretizing Maxwell’s equations using finite difference time domain (FDTD) and boundary and finite element methods (BEM; FEM, e.g., COMSOL) [9,11,12]. Interactions in lattices of spheroids have been analyzed using discrete and coupled dipole approximations (DDA; CDA) [13,14] which reduce complexity of target generation and parametric changes relative to higher-order BEM or FEM approaches. Computational expense of element methods generally precludes direct attribution of resonance behavior to changes in structure geometry, lattice constant, and surrounding dielectric across large parametric ranges. LRs from Fano resonant diffractive coupling in lattices of nanospheres have been examined using a rapid semi analytical CDA by treating each nanostructure within the lattice as a polarizable point dipole [15]. This approach was recently expanded to nanoring lattices by using an effective polarizability calculated by the DDA in place of the analytical polarizability for spheres [16]. But only a single lattice constant and a narrow range of radii at a single wall thickness were considered. The CDA permits rapid, large range parametric analysis in lattices of nanostructures to optimize resonance features.

The present work used CDA and DDA to compile the influences of inner radius, wall thickness, and lattice constant on wavelength, intensity, and bandwidth of plasmonic resonance features in isolated nanorings and nanorings in square lattices. Modulation of the lattice constant enabled 4.3-fold enhancement in resonant extinction intensity over isolated nanorings. LSPR wavelength, bandwidth, and extinction intensity of isolated nanorings generally increased as inner radius expanded. Increasing wall thickness expanded LSPR bandwidth and decreased absorption relative to scattering. Arranging nanorings into a square lattice with a lattice constant near the nanoring LSPR produced a diffractively-coupled LR and enhanced intensity of resonance features. DDA solutions for isolated nanorings were used to identify geometric dimensions of gold (Au) nanorings with maximal extinction and absorption. Refractive index (RI) sensitivity of the resulting simulated nanoring lattices was higher than comparable isolated or arrayed nanospheres. RI sensitivity was increased over four-fold relative to spheroid lattices when peak bandwidth was considered. These results support design and implementation of nanoring lattices into optoelectronic and metamaterial systems.

## 2. Methods

#### 2.1 Discrete Dipole Approximation

The spectral response of an isolated nanoring was modeled using the DDA package DDSCAT (v. 7.3.0) in order to identify optimal geometric features. DDA provides solutions to Maxwell’s equations for an arbitrary target geometry, e.g. rings, by approximating the geometry as a cubic lattice of point dipoles resulting in the absorption and scattering of EM waves [17,18]. Nanoring geometries were discretized into targets for DDSCAT using Matlab (v. 7.14, MathWorks, Natick, MA, USA), as illustrated in Fig. 1. Ring targets were generated by discretization of two concentric circles using an implementation of a Bresenham algorithm [19]. An inter-dipole spacing of 3.33 nm was chosen for all targets, and resulted in discretization without deviation from specified dimensions [20]. The considered ring geometries featured dipole counts on the order of 10^{3}-10^{4}, depending on target volume.

DDSCAT was used to evaluate absorption and scattering efficiencies of isolated nanorings over 400 nm to 1400 nm wavelengths, with 1 nm resolution, in a vacuum (*n* = 1). RI data for Au reported by Palik was used [21]. The polarizability of each dipole point within the target was chosen according to a “Lattice Dispersion Relation” for cubic arrays [22]. Effective polarizability of the target geometry was calculated [Eq. (1)] according to its interdipole spacing (${d}_{di}$), the complex polarization of the *j*th dipole per unit volume (${P}_{j}$), and the incident complex electric field (${E}_{o}$).

Effective complex polarizability as a function of wavelength for each geometry was extracted for input to a coupled dipole approximation (CDA), which was used for incorporation of diffraction modes supported by square lattices of these geometries [16]. Polarizability of rings has been described analytically [23] and extracted from FDTD simulations [24]. However, reported analytic methods are limit rings sizes much smaller than incident wavelength, and the FDTD method requires numerical integration of the charge distribution throughout the ring. DDCSAT can periodically distribute target geometries to couple diffraction with solutions of absorption and scattering efficiency [13]. However this approach was not pursued in evaluating full lattices due to computational requirements. Instead, CDA was used to model effects of diffraction, resulting in a 40,000-fold reduction in computation time [16].

DDSCAT was run under the same conditions for the optimized geometry immersed in a water RI environment (*n* = 1.33) in order to test dielectric sensitivity by using polarizability values in water. This approach is detailed below.

#### 2.2 Coupled Dipole Approximation

A rapid semi-analytic CDA algorithm (rsa-CDA) [14,15] was used to evaluate the influence of lattice constant on LSPR and lattice resonance features in square lattices of the optimized nanoring geometry (identified from DDSCAT). The rsa-CDA treats each element within a lattice as a single point with a scalar, dipolar polarizability. Calculation of Fano resonant interactions between narrow-band diffraction and dipolar scattering within the lattice was performed under the assumption of an infinite lattice. Reduction in function calls by taking advantage of 4-fold rotational symmetry in square lattices allowed a net reduction in required computing time [14,15]. This simplification mitigates computation of parameter changes, such as lattice constant, for large scale parametric predictions not feasible with methods such as FEM due to memory requirements and computation time. Use of Eq. (1) to obtain polarizability of isolated nanorings for CDA is valid for rings which exhibit primarily dipolar/quadrupolar behavior in the far-field [16], as considered in this work.

Effective polarizability values of a nanoring with optimized geometry (from DDA per Eq. (1) were used to model the spectral response between 400 nm to 1400 nm with lattice constant variations between 700 nm and 1200 nm. RI data for Au reported by Palik was maintained [21]. Results reported from CDA were multiplied by a factor of 0.7 to match single nanoring extinction magnitudes obtained with DDSCAT. CDA calculates extinction efficiency according to the cross sectional area of a spherical particle. The effective cross sectional area for the CDA was calculated as the cross sectional area of the ring set equal to that of a circle [16]. A simulation was performed to investigate the RI sensitivity of the optimized nanoring lattice, requiring simulations from both DDA and the rsa-CDA in a water environment ($n$ = 1.33). RI sensitivity ($S$) was calculated as the change in a peak location ($\Delta \lambda $) as a result of a change in local RI ($\Delta RI$), i.e. $S=\Delta \lambda /\Delta RI,$ and is given in units of nm RIU^{−1} (refractive index unit).

#### 2.3 Nanoring Geometry

Geometric parameters which describe a rectilinear lattice of ideal nanorings are illustrated in Fig. 1 as inner radius (${r}_{in}$), thickness ($t$), height ($h$), and lattice constant (${d}_{x}$ and ${d}_{y}$). Inner radii from 20 nm to 100 nm simulated in 20 nm increments. Ring thicknesses of 10 nm and 20 nm at a single ring height of 50 nm were considered. Selected thicknesses were within a range of 10 to 30 nm consistent with previously reported fabricated nanoring thicknesses [9,11,25,26]. Such thicknesses were modeled by DDA targets using a number of dipoles whose resultant spectra proved reproducible at different dipole counts (order of 10^{3}-10^{4}) with minimum computational expense. A dielectric function developed by Aizpurua *et al*. showed that LSPR wavelength is predominately a function of the ring aspect ratio, concluding that height variations have a less dramatic influence in the quasi-static case [11]. Increasing height has been shown to red-shift the LSPR due to a mode splitting polarization scheme between the upper and lower ring surfaces [11]. Nanorings were simulated as isolated rings and in square lattices (${d}_{x}={d}_{y}$) with lattice constants ranging from 700 nm to 1200 nm.

Nanorings lithographed by electron beam lithography and deposited by Au electroless plating shown in Fig. 1 have been fabricated to dimensions comparable to those analyzed by computation [27]. No spectra were obtained from this nanoring array before it was annealed to form spheres [27]. Fabrication challenges limited prior comparisons of experimental and computed spectra to evaluating isolated nanorings (not LR in lattices) for which either radius or thickness were adjusted [9,11,25,26,28]. The present work supports rapid, concomitant characterization of radius, wall thickness, and inter-particle spacing on the optical responses from lattices of nanorings *in silico,* to support development of optimal metamaterials.

## 3. Results and discussion

#### 3.1 Isolated nanorings

Increasing nanoring radius primarily increased LSPR wavelength and bandwidth, while increasing wall thickness decreased absorption relative to scattering and tuned the LSPR wavelength. Figure 2 shows modeled extinction spectra for isolated nanorings as the inner radius increased from 20 nm to 100 nm at ring thickness of 10 nm and 20 nm. LSPR wavelengths increased from about 650 nm to 1150 nm as the inner radius increased. These results are consistent with other reported LSPRs from 800 nm to 1400 nm reported for nanorings with inner radii of 45-50 nm with wall thicknesses of 10-20 nm [9,11,12,25]. This behavior is also consistent with spheroids where different electron oscillation modes (dipole, quadrupole, etc.) cause phase retardation as particle size is increased [29]. Spectral features observed at 564 nm, 1265 nm, and 1378 nm in the simulated spectra are artifacts of the Au RI data reported by Palik that was used in the simulations.

Peak location, breadth, and symmetry in Fig. 2 are comparable to simulated and experimental data over a limited parametric range reported by Huang *et al*. [9]. Costly top-down lithography approaches like electron beam lithography (EBL) [30] performed in extreme environments limit the number and range of parametric changes evaluated for isolated nanorings [10,28]. So nanosphere lithography (NSL) has been preferred to template isolated nanoring structures [9,11,25,31,32]. However, finite availability of nanosphere radii and imprecise metal film etching during subsequent metallization has precluded large-range parametric evaluation of radius and wall thickness by this approach. Alternatives such as nano-imprint lithography [33] and/or electrodeposition [34] support fabrication of ordered nanorings, but lack the ease and repeatability of NSL. Recently, bottom-up electroless plating has been introduced to deposit Au nanorings onto EBL-templated square lattices of electron resist, as shown in Fig. 1 [27,35]. Other recent methods show promise, but remain to be broadly examined [36]. The present *in silico* approach provides understanding of parametric changes over large ranges that complements the paucity of experimental results.

Some rings in Fig. 2 exhibit peak asymmetry attributed to charge density interactions between the inner and outer ring walls. Charge distribution on the walls of the ring can support two dipolar plasmon modes [10,11,28,37], commonly denoted as high-energy anti-bonding and low-energy bonding [32]. Anti-bonding occurs when a dipole appears on opposite walls in the direction of polarization; bonding is analogous to a sphere where all positive charge and all negative charge are collectively on opposite walls. Control of incident polarization can elicit different bonding modes in nanoring ellipsoids [28], dimers/trimers [26,37], and overlapped rings [38]. Electron charge density interactions responsible for these modes can ultimately split the LSPR, as evident by the double-peak spectra for the ${r}_{in}$ = 40 nm, $t$ = 10 nm ring in Fig. 2(a). Other considered geometries did not support an independent anti-bonding mode. Lesser interactions may broaden one side of the peak, as seen in the ${r}_{in}$ = 60 nm with $t$ = 10 nm nanoring. This work focuses on the bonding mode for its tenability and increased optical extinctions. Electric near-field plots confirm bonding mode resonance patterns underlying the spectra shown (data not shown). A higher order oscillation mode began to appear at a wavelength of ~700 nm for ${r}_{in}$ = 80 nm and 100 nm rings with $t$ = 10 nm. Higher order modes have been observed in FDTD simulations for rings of comparable size [25]. This may explain the observed decrease in extinction efficiency magnitude of the dipolar bonding peak from ${r}_{in}$ = 80 nm to 100 nm. Higher order oscillation modes did not yet appear for the $t$ = 20 nm rings at the considered radii, thus preserving a consistent trend in increasing extinction efficiency with larger radii.

Full width at half maximum (FWHM) of the LSPR tended to increase as ${r}_{in}$ increased from 20 nm to 100 nm, as shown in Fig. 2. As an example, the FWHMs of the LSPR for ${r}_{in}$ = 40 nm, 60 nm, and 80 nm with $t$ = 20 nm were 66, 146, and 178, respectively. LSPR broadening has been attributed to phase retardation and/or reduction in phase coherence from interactions between different electron oscillation modes in nanospheres [29]. Rings may behave analogously to spheres in this regard because coupling between the inner and outer ring walls is similar to the inter-shell coupling in a hollow nanosphere [11].

The influence of widening the ring wall on the LSPR wavelength is dependent on the overall dimensions of the ring, as observed in Fig. 2. Blue-shifting of the bonding mode with increasing ring aspect ratio was observed for small inner radii, as observed for the 664 nm LSPR for ${r}_{in}$ = 20 nm nanoring with $t$ = 10 nm, which blue-shifts to 626 nm after increasing $t$ to 20 nm. Drude-like models under an electrostatic assumption for rings much smaller than the incident wavelength predict this blue-shift in LSPR wavelength as the aspect ratio is increased [11,31]. As wall thickness increases, the bonding mode produced by this inner/outer wall coupling converges to that of a planar surface plasmon mode. Huang *et al*. experimentally showed an ~80 nm LSPR blue-shift for a constant inner radius nanoring when increasing its wall thickness [9]. Aizpurua *et al*. similarly showed a blue-shift in LSPR for constant outer radius rings resulting from increasing wall thickness by 5 nm [11].

However, it appears variance in wall thickness becomes less important to LSPR wavelength at higher radii (which dominate the aspect ratio), as blue-shifts are not observed for increases in aspect ratio as Drude-like models predict. For example, increasing $t$ in ${r}_{in}$ = 40 nm and 60 nm rings in Fig. 2 did not affect LSPR wavelength, which remained within < 1%. At higher inner radii, effects from increased wall thickness changed to give an LSPR red-shift (${r}_{in}\ge $ 80 nm), broadening (${r}_{in}\ge $ 60 nm), and extinction magnitude increase (${r}_{in}$ = 100 nm). For example, a ring with ${r}_{in}$ = 100 nm and $t$ = 10 nm has an LSPR at 1038 nm and FWHM of 142, with an extinction efficiency of 31. Increasing $t$ to 20 nm red-shifted the LSPR to 1138 nm, broadened the FWHM to 179, and increased extinction efficiency to 39. Available analytic methods which predict LSPR blue-shifts for increasing aspect ratio are limited to the quasi-static approximation. This approximation diminishes in validity for rings with an outer radius beyond 80 nm at the thicknesses considered. As particle radius begins to dominate the aspect ratio, electric field retardation across the particle and dynamic effects increase in importance. Previous studies that report LSPR blue-shifts at higher radii consider thicker rings, in which the radius does not dominate the aspect ratio (0.2 to 0.8) [10]; conversely, nanorings examined herein have aspect ratios less than 0.2. The effect of increasing $t$ on a constant outer radius nanoring was observed by comparing spectra of a ring with ${r}_{in}$ = 90 nm and $t$ = 20 nm to that of a ring with ${r}_{in}$ = 100 nm with $t$ = 10 nm (data not shown). Both rings featured a constant 110 nm outer radius, and a red-shift of approximately 4% was observed after increasing the wall thickness.

Plasmonic absorption intensity was highest in thinner ring geometries. Rings with $t$ = 10 nm exhibited 18-48% higher absorption efficiencies than those with $t$ = 20 nm, though extinction magnitudes were generally higher for $t$ = 20 nm at each ${r}_{in}$. Absorption peak shapes remained the same as extinction peak shapes (data not shown). Highest extinction magnitudes are achieved with two nanoring geometries in Fig. 2: ${r}_{in}$ = 80 nm and $t$ = 10 nm [purple line, Fig. 2(a)], and ${r}_{in}$ = 100 nm and $t$ = 20 nm [orange line, Fig. 2(b)]. However, a nanoring with ${r}_{in}$ = 80 nm and $t$ = 10 nm has a 19% higher plasmonic absorption. Absorption of EM energy into plasmons allows localization of light for intense electric near-fields, which are the basis of surface-enhanced Raman spectroscopy [39] and heating [40] applications. Balancing extinction and absorption efficiencies meets needs of various opto-plasmonic applications. High absorption increases local field intensity and photothermal response, while high scattering provides opportunity for in-phase constructive interference from scattering structures to obtain a diffractive coupling lattice resonance useful for RI sensing. The product of extinction and absorption efficiency (area under the curve) could be useful to identify a ring geometry with the greatest cumulative extinction and absorption potential.

Subsequent examination of nanoring lattices consider the geometry with ${r}_{in}$ = 80 nm, $t$ = 10 nm, and $h$ = 50 nm, because these specifications resulted in the highest product of extinction and absorption among examined geometries. This geometry exhibits a spectral extinction maximum at 919 nm comprised of 56% plasmonic absorption and 44% scattering efficiency (scattering data not shown). Larger rings exhibited higher scattering components: 60% in the case of ${r}_{in}$ = 100 nm and $t$ = 20 nm. Scattering similarly increases with size in nanospheres.

#### 3.2 Fano resonance in nanoring lattices

Periodic ordering of the optimized nanoring (identified in section 3.1) enhanced extinction intensity more than 4-fold as a result of Fano resonant coupling between LSPR and far-field lattice diffraction. Fano resonance describes a coupling between two independent, overlapped resonances: a discrete state (e.g., diffraction at the Rayleigh anomaly) and a broadband resonance (e.g., LSPR). Diffractive coupling can support Fano resonance, or plasmon-photon coupling, when the inter-particle spacing exceeds the single-particle LSPR wavelength. Scattered light from nanoring antennae oscillates in phase with plasmon oscillation causing increased intensity of the resonance. A detailed discussion of Fano resonance in nanostructure lattices can be found in Refs [7,8,16]. Figure 3(a) shows spectra of arrayed nanorings (solid lines) with lattice constant $d$ ranging from 800 nm to 1100 nm. For comparison, isolated nanoring spectra with ${r}_{in}$ = 80 nm, $t$ = 10 nm, and $h$ = 50 nm is shown (dashed line). As $d$ becomes comparable to the single-particle LSPR, two resonances emerge as a result of Fano resonance: the higher energy plasmon resonance (PR) and lower energy lattice resonance (LR). LRs were first reported in 2005 in nanosphere lattices [41] and are known to occur red-shifted of the lattice constant. Nanorings fabricated by NSL to date have not supported LRs because the LSPR wavelength has exceeded inter-particle spacing. Extinction intensity of the LR was smaller than the LSPR in nearly every case in contrast to spheroid lattices [7,8].

PRs arise from the collective LSPRs, but appear slightly shifted due to Fano interference. The PR appeared red- or blue-shifted from the single-particle LSPR when $d$ was below or above the single-particle LSPR, respectively. The PR appeared blue-shifted by 32 nm for $d$ = 900 nm, but returned near the single-particle LSPR as $d$ widened. At $d$ = 900 nm (green), the PR FWHM was 66% of the isolated ring LSPR as the two modes began coupling according to a Fano resonance. Lattice constants substantially smaller than the single-particle LSPR do not support Fano resonant interaction, which red-shifted the PR wavelength. For example, a PR was observed 10 nm red-shifted from the LSPR at $d$ = 800 nm. Other simulated array spacings of 700 nm and 1200 nm followed these described trends (data not shown).

Each resonance feature monotonically increased in wavelength as $d$ was tuned between 900 nm and 1000 nm, where Fano resonant diffractive coupling appeared to begin. This data is shown in Fig. 3(b) and 3(c). Increasing $d$ consistently red-shifted the PR (black circles) and LR (blue triangles) wavelengths, shown in Fig. 3(b). PR and LR features red-shifted approximately 0.7 nm and 0.4 nm, respectively, for every nm increase in lattice constant between 900 and 940 nm. This behavior changed when the lattice constant increased beyond 940 nm, but remained monotonic: the LR red-shifted identically with lattice constant while the PR maintained the single-particle LSPR location within 1%.

Extinction intensity of the PR changed non-monotonically across the same range of lattice constants while the LR magnitude decreased monotonically, as shown in Fig. 3(c). The magnitude of the PR reached a sharp maximum of 140.7 at 940 nm [shown in Fig. 3(c)]. This is a result of EM field enhancement on each nanostructure arising from Fano resonant in-phase constructive interference from scattering nanostructures within the lattice [14,15]. The PR magnitude dropped exponentially for $d\ge $ 940 nm [shown in Fig. 3(c)]. Meanwhile, intensity of the LR monotonically decreased to 78% of its value at $d$ = 900 nm as $d$ widened to 1000 nm.

Arrangement of nanorings into square lattices with $d$ = 940 nm resulted in the highest estimated extinction values, ostensibly due to particle polarizability ($\alpha $) [14]. This lattice configuration resulted in an extinction magnitude of 140.7 for the PR feature at 916 nm, and a LR with magnitude of 22.0 at 967 nm. This intense extinction occurs at a lattice constant 20 nm larger than the single-particle LSPR wavelength. The imaginary component of polarizability (Im($\alpha $), blue solid line in Fig. 4) is directly proportional to single-particle extinction [15]. This would be anticipated to provide maximal extinction. However, scattering contributions to extinction appear to modulate the proportional relationship between extinction magnitude and the Im($\alpha $) maxima at 924 nm. Positive and negative values of the real component of polarizability (Re($\alpha $), blue dashed line in Fig. 4) correspond to constructive and destructive interference, respectively, on neighboring nanoelements within a lattice [15]. A negative Re($\alpha $) at 924 nm appears to destructively interfere with the Im($\alpha $) maxima, before becoming positive after 925 nm to lend constructive interference to neighboring dipoles. Interestingly, summing Re($\alpha $) and Im($\alpha $) (red solid line in Fig. 4) yields a 944 nm maxima, which is close to the observed optimal lattice constant of 940 nm.

Inter-related effects of geometry and lattice constant remain to be evaluated for nanoring lattices which do not feature particle centrosymmetry or regular inter-particle spacing. The nanoring in Fig. 1 appears comprised of fused 30 nm grain elements. It is reported that similar self-assembled structures exhibit LSPR extinction profiles nearly identical to those modeled from centrosymmetric structures, but with a higher absorption to scattering ratio because of the highly absorptive spheroidal grains [42]. Effects of lattice spacing disorder in diffractive coupling have been investigated for spheres [43], but not complex ring structures. Modeling of these structural and spacing disorders within Fano resonant lattices is the subject of ongoing work.

#### 3.3 Refractive Index Sensitivity Analysis

Sensitivities of 760 nm RIU^{−1} and 1075 nm RIU^{−1} were observed for the PR and LR peaks, respectively, from a nanoring lattice with ${r}_{in}$ = 80 nm, $t$ = 10 nm, $h$ = 50 nm, and $d$ = 940 nm. Figure 5 shows red-shifted extinction spectra of this lattice (solid lines) when surrounding RI is changed from air (green) and water (blue), with corresponding sensitivity labels. Spectra of the isolated nanoring are also shown in Fig. 5 as dashed lines. Plasmonic behavior is dependent on the local RI environment, causing spectral peak shifts for refractive index sensors [44]. Sensitivity, $S$, to local RI is calculated as the change in peak location ($\Delta \lambda $) per change in RI unit ($\Delta RI$), i.e. $S=\Delta \lambda \Delta R{I}^{-1}$. Nanorings exhibit heightened sensitivity to RI changes than other nanostructures (e.g., spheres, rods, polyhedra) due to their relatively high surface area to volume ratio [26]. Sensitivity of non-coupling nanorings has been reported in the range of 350 to 700 nm RIU^{−1} for vacuum LSPR in the 800 nm to 1300 nm range [9,10,31]. Comparatively, spheres are reported at ~60 nm RIU^{−1} for 530 nm vacuum LSPR [45]. Higher RI sensitivity accrues from increased surface area where plasmons interact with the analyte, which increases in nanorings as $t$ decreases. However, Tsai *et al.* showed sensitivity increases in proportion to thickness [10]. Work by Larsson *et al.* further showed sensitivity increases in proportion to outer radius [31].

Nanorings configured into Fano resonant lattices exhibited enhanced sensitivity in the near infrared (NIR) of 1% and 43%, respectively for the PR and LR, over the isolated nanoring LSPR. NIR operation supports biosensing because visible light cannot probe biological analytes due to interactions with hemoglobin, water, and lipids. Detection of DNA and proteins using nanorings has been demonstrated [9,28]. The isolated nanoring LSPR at 919 nm in air shifted to 1170 nm in water (dashed lines in Fig. 5) for a sensitivity of 754 nm RIU^{−1}. Values of 740 and 880 nm RIU^{−1} have been measured for nanorings of 45 nm and 60 nm inner diameters with $t\approx $ 15 nm [31]. Configuring this nanoring into a square lattice (solid lines) with $d$ = 940 nm resulted in two distinct peaks arising from Fano resonance (PR and LR), as shown in Fig. 5. The PR of the lattice at 916 nm in air shifted to 1169 nm in water. This 760 nm RIU^{−1} sensitivity was a 1% increase over the single-particle LSPR. The LR at 967 nm in air shifted to 1325 nm in water. This 1075 nm RIU^{−1} sensitivity was a 43% increase over the single-particle LSPR sensitivity.

However, the PR appears more detectable than the higher sensitivity LR after consideration of peak breadth within a non-vacuum RI environment. FWHM of the LR in water is almost doubled relative air, consistent with observed broadening in non-coupling nanorings [10]. The LR redshifts linearly with RI due to wavelength contraction while the PR redshifts sub-linearly due to medium polarization. Mismatched shifting of the two resonances causes decoupling of the Fano resonance. This results in decreased peak amplitude and shape distortion, as observed in Ref [7], for lattices of nanospheres. Adjusting sensitivity by the average FWHM in both refractive index environments, to give the sensor figure of merit (FoM), is commonly used to account for this effect. FoM is calculated as $S/{(FWH{M}_{1}{}^{2}+FWH{M}_{2}{}^{2})}^{1/2}$ where subscripts correspond to RI mediums, e.g. air and water in this case. The sensor FoM of the three peaks in Fig. 5 are 4.12, 7.52, and 6.48 nm RIU^{−1} FWHM^{−1} for the single-particle LSPR, PR of the lattice, and LR peaks, respectively. The FoM for the PR is almost double its corresponding LSPR random array (equal to single-particle LSPR). In this work, particle spacing was chosen based on vacuum resonances. As illustrated in Ref [7], for nanospheres, lattice geometry and ring morphology can be identified computationally to maximize coupling at a specific working RI for a higher FoM. Otherwise, the FoM may become smaller with higher RI analytes. The sensor FoM of non-coupling nanorings in the literature is generally higher than nanospheres. For example, a 3.1 nm RIU^{−1} FWHM^{−1} FoM for rings has been reported [9], whereas spheres are within 1.0-1.5 nm RIU^{−1} FWHM^{−1}. Other published nanorings exhibit a FoM near 1.5 nm RIU^{−1} FWHM^{−1} despite high sensitivity due to LSPR broadening in non-vacuum RI environments [10,31].

RI sensitivity has been evaluated in complex ring structures such as dimers, trimers, concentric ring-disks, and overlapping rings. Modulation of nanoring dimer sensitivity by widening the inter-ring gap from 10 nm to 250 nm increased sensitivity from 547 nm RIU^{−1} to 1084 nm RIU^{−1} [37]. Arranging nanorings into trimers was found to increase sensitivity relative to nanodisks of the same geometric dimensions [26]. Placing disks inside of rings has resulted in sensitivities ranging between 500 and 1200 nm RIU^{−1} [46,47]. Liu *et al.* found that a peak arising from Fano interference between bright and dark plasmon modes (i.e., plasmon-plasmon coupling) in overlapping rings had 36% higher sensitivity relative to the LSPR [38]. Protein sensing has been performed with ellipsoidal rings to investigate bonding modes between the major and minor axes, but no sensitivity figure was reported [28]. Whether configuring more complex arrangements of rings into diffractive coupling lattices would augment their refractive index sensitivity and FoM is a topic for future consideration.

## 4. Conclusion

Wavelength shifting, extinction intensity, and bandwidth of resonance features from lattices of nanorings were attributed to changes in radius, wall thickness, lattice constant, and dielectric environment. Combined use of the DDA and rsa-CDA allowed rapid attribution of spectra features to geometric influences over a large parametric range, improving intuition as well as computation time relative to higher order methods such as FEM. Increasing inner radius red-shifted the LSPR, and generally increased its bandwidth and extinction. Widening the wall thickness decreased absorption relative to scattering, as well as expanded (tuned) the LSPR bandwidth (wavelength). Lattice resonances (LR) arose from Fano resonant diffractive coupling and primarily influenced the extinction intensity of the plasmon resonance (PR) of the lattice. LR intensity remained near or less than that of single-particle LSPR. Adjusting the lattice constant in a square lattice of nanorings whose geometry maximized the product of extinction and absorption enhanced resonant extinction magnitude 4.3-fold relative to an isolated nanoring at a particular lattice constant. This particular lattice constant appeared to result from interactions between complex components of particle polarizability. A refractive index sensitivity of 1075 nm RIU^{−1} was observed in simulation for the LR peak of an optimized lattice. This was a 43% increase relative to the LSPR of an isolated nanoring. A sensor figure of merit based on amplitude to peakwidth ratio of spectral features of a lattice indicated sensitivity may be enhanced 4- to 5-fold relative to isolated nanorings. This *in silico* approach supports rational design of Fano resonant nanoring lattices for optoelectronic applications, such as sustainable energy and medical diagnostics.

## Acknowledgments

This work was supported in part by NSF CBET-1134222, NSF ECCS-1006927, NSF GRF awarded to G.T. Forcherio, the University of Arkansas Foundation, and the Walton Family Charitable Foundation. DDA simulations were performed on the supercomputers of the Arkansas High Performance Computing Center, supported by NSF-918970 and NSF-959124. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. G.T. Forcherio performed numerical modeling and prepared text and figures for the manuscript. P. Blake developed codes for numerical modeling, aided in text preparation, and initially evaluated isolated nanorings for this work. D. DeJarnette performed numerical modeling, and aided in text revision. D.K. Roper directed the work and refined compilation of the final text. The authors would like to thank Morgan E. Ware for helpful conversations and refinement of the draft text.

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