## Abstract

A new surface plasmon resonator, in the form of a metal cylinder shaped like a teardrop, is proposed and modeled using hypersingular integral equations. Illuminating the apex of the teardrop along its axis of symmetry with TM-polarised light excites equal-magnitude surface plasmons counter-propagating around its periphery. Interference of these counter-propagating surface plasmons leads to resonant modes having very sharp line-widths (~0.6 nm) and large quality factors (~3400). In water, a Au teardrop supports resonant modes having large bulk sensitivities (~1900 nm/RIU) and remarkably large figures-of-merit (~1700), about 1000 × larger than the figure-of-merit of, *e.g.*, isolated Au nano-spheres. The scattering cross-section of the structure near resonant modes exhibits a Fano-like line-shape. The teardrop should find applications as a high-quality surface plasmon resonator or as a biosensing element.

© 2012 OSA

## 1. Introduction

Metal nano-particles (NPs) [1, 2] are of considerable interest due to their ability to support strongly confined surface plasmon-polariton (SPP) resonances. They have found numerous applications, particularly to biosensors [3, 4]. Various metal geometries, ranging from curved-surface structures like disks [5] and cylinders [6], to sharp-angled structures like cubes [7] and wedges [8], have been investigated. SPP resonances on such structures depend strongly on their shape, size and composition, and on the surrounding medium. Fano resonances involving SPPs in metal nano-structures have recently been observed [9–13]. In general, such resonances arise from the coupling of a narrow discrete resonance with a broad one [12]. SPP Fano resonances are strongly dispersive and they are sharp so they have a high quality factor (*Q*) [9–12], and thus are also of strong interest for applications.

In this paper we propose a new surface plasmon resonator, in the form of a metal cylinder shaped like a teardrop, as sketched in Fig. 1
. The structure is invariant along *z* (out of the page) and obtained from a wedge of angle *β* by closing its two flat surfaces of length *a* with a circular surface of radius *r* = *a* × tan(*β*/2). The normal to the circular surface coincides with the normal of the two flat surfaces at their ends. The permittivity of the metal is denoted *ε _{m}* and that of the dielectric background by

*ε*.

_{d}The resonator is assumed illuminated by a transverse-magnetic (TM) polarised plane wave of magnitude *E _{0}* in the plane of incidence perpendicular to the

*z*axis, at an angle of incidence

*ϕ*= 90°; the source is thus an

_{inc}*x*-polarised plane wave incident on the apex. The apex is taken as being perfectly sharp, although in practice a rounded apex of radius ~10 nm would behave similarly as far as the resonant modes are concerned.

Illuminating the teardrop directly onto its apex generates single-interface SPPs of equal magnitude that propagate along the flat surfaces, as sketched in Fig. 1 (the apex acts as a free-space to SPP coupler). The SPPs then propagate in opposite directions around the curved section (possibly with radiation), and back to the apex. Resonant modes are thus formed by the interference of oppositely propagating SPPs propagating along the teardrop perimeter.

## 2. Analysis in vacuum

The teardrop (*ε _{m}*) is assumed formed of Au [14] and, initially, surrounded by vacuum (

*ε*=

_{d}*ε*). The geometry is set to

_{0}*β*= 30° and

*a*= 1 μm, yielding

*r*= 268 nm. We arrived at these dimensions via trials to produce resonant modes within the free-space wavelength range of

*λ*= 600 to 2400 nm (altering the perimeter alters the resonant wavelengths). We investigate the scattering and resonant properties of this structure using single hypersingular integral equations (HIEs) [15] (the teardrop shape is inspired from [15]). HIEs can model accurately electromagnetic fields to distances as small as ~10

_{0}^{−4}

*λ*from the structure, including the apex.

_{0}Figure 2 shows the scattering cross-section of the teardrop bounded by vacuum (thick red curve), computed over the range of interest, as [1]:

*R*= 60 µm is the radius of the circle on which the electric field is sampled (far from the structure),

*dδ*is the elemental angle, and |

*E*(

^{scat}*λ*)| and |

_{0}*E*(

^{inc}*λ*)| are the scattered and incident electric field magnitudes at

_{0}*λ*, respectively. From Fig. 2, the structure is observed to resonate at four wavelengths (sharp peaks) with the longest resonant wavelength belonging to the lowest order resonance. It is noted that the resonances are asymmetric, especially at high-order, with line-shapes characteristic of Fano resonances [12].

_{0}Table 1
lists the resonant free-space wavelength *λ _{r}*, full-width at half-maximum (FWHM) and quality factor (

*Q*) of the resonant modes (

*Q*~

*λ*/FWHM which holds at high

_{r}*Q*). The FWHM increases and the

*Q*decreases as the mode order increases. This is consistent with the increasing attenuation of SPPs as

*λ*decreases (with mode order). The highest

_{r}*Q*of 3400 is observed for the first resonance (mode 1), which also has the narrowest FWHM of 0.60 nm.

Figure 3
shows the distribution of |*E ^{scat}*| and of its

*y*-component |

*E*|, both normalised to

_{y}^{scat}*E*and plotted on log and linear scales, for mode 1 (top row) and mode 2 (bottom row). The log plots (left) illustrate the rapid field decay in the metal, characteristic of SPPs. The electric field of mode 1 is localised around the curved section of the teardrop (Fig. 3(d)), while for mode 2 the field is distributed around the curved section and the two flat surfaces (Fig. 3(h)).

_{0}The scattered *x*-directed electric field component |*E _{x}^{scat}*| (not shown) is distributed similarly to |

*E*|. A different number of extrema was observed in the field distribution of all modes of Table 1 (

^{scat}*cf*. Figures 3(d) and 3(h)). Given the

*x*-polarized excitation and the symmetry of the teardrop about the

*y*-axis, |

*E*| = 0 along this axis for all modes. There is no field localisation near the apex and negligible field enhancement throughout. Other resonant modes may exist and may be revealed for a non-symmetric excitation (

_{y}^{scat}*i.e.*, for excitation at other angles of incidence

*ϕ*).

_{inc}Contrary to metal nano-particles, where the “cloud” of conduction electrons becomes polarised on resonance, the resonant modes of the teardrop originate from SPPs propagating in opposite directions around the periphery of the structure and interfering constructively at a set of discrete wavelengths (Fig. 3). Very narrow FWHM result because the resonances originate from wave (SPP) interference.

## 3. Analysis in water

The high *Q* of the teardrop suggests its suitability for biosensing. To examine this possibility we assume the same teardrop but with water [16] replacing vacuum (an aqueous solution is generally used as a carrier fluid for analyte [17]). The scattering cross-section of the teardrop computed in this case is plotted as the thin blue curve in Fig. 2. The four resonances are red-shifted relative to vacuum. This is due to an increase in the effective index of the SPPs propagating on the teardrop commensurate with the increase in surrounding refractive index (from *n _{d}* = 1 to ~1.30).

From the experimental perspective, a resonance and its FWHM could be interrogated by measuring the scattered intensity at a location far from the teardrop due to excitation by an incident beam. Thus, the radar cross-section is representative, and in 2D is given by [1]:

The mono-static radar cross-section corresponds to the case where the transmitter and receiver are co-located. We assume co-location at *y* = 60 µm (*R* = 60 µm in Eq. (2)) and we compute the responses near the resonances on the thin blue curve of Fig. 2. The results are plotted in Fig. 4
, from which *λ _{r}*, FWHM and

*Q*are determined, and listed in Table 2 .

The FWHM of the teardrop in water (Table 2) does not follow the mode order, as for the teardrop in vacuum (Table 1). In water, the FWHM of mode 1 is broader than the second and third modes. This is due to the absorption of water which increases rapidly with increasing wavelength in this range [16]. The narrowest FWHM (0.64 nm), obtained for mode 2, is much narrower than that of metal nano-structures such as ring-disk nano-cavities (FWHM > 130 nm) [9], nano-cubes (FWHM ~18 nm) [7], arrays of nano-rods (FWHM ~8.37 nm) [18], and nano-wires of rectangular to triangular cross-section (FWHM > 14 nm) [19]. The largest *Q* (2019), also obtained for mode 2, is higher than in Ag-coated large diameter toroid micro-resonators (Q ~1000) [20] and in arrays of nano-rods (Q < 900) [18], but comparable to a micro-resonator based on a channel waveguide [21], and 10 × smaller than long-range surface plasmon waveguide resonators [22].

The bulk sensitivity of the resonant modes ∂*λ _{r}*/∂

*n*was computed by perturbing the surrounding refractive index about the nominal value (

_{d}*i.e. n*± 10

_{d}^{−4}) and approximating the partial derivative with a central difference formula [17].The sensitivity of the four modes are listed in Table 2, and observed to decrease with mode order. The largest bulk sensitivity (1930 nm/RIU), obtained for mode 1, is much higher than that of isolated metal nano-particles, including nano-spheres and nano-branches [4], nano-rod chains [3], and ring-disk nano-cavities [9]. It also exceeds that of ring resonators in Si-on-insulator waveguides (SOI) [23].

Finally, we compute the figure of merit (FoM) of the four resonant modes via [7]:

and list them in Table. The largest FoM (1702), observed for mode 2, greatly exceeds that of standalone metal nano-structures, such as nano-spheres (FoM ~0.6) [4], nano-bipyramids (FoM ~4.5) [4], and nano-cubes (FoM ~8 to 20) [11]. It also greatly exceeds that of arrangements of metal nano-structures, such as clusters of nano-disks (FoM ~5.7) [24], arrays of nano-rods (FoM ~2 to 90) [18], and aggregates of nano-spheres (FoM ~8 to 10) [10]. Finally, it also exceeds the FoM of an SPP interferometer in SOI (FoM ~240) [25].## 4. Summary and concluding remarks

We proposed a metal teardrop-shaped cylindrical resonator and demonstrated that it can find applications as a high *Q* resonator and as a high-performance biosensor element. Resonant modes involve SPPs circulating and interfering around the periphery of the structure, leading to narrow resonant line-widths. The resonant modes of a teardrop in water have a bulk sensitivity in the range of 538 to 1930 nm/RIU, a FWHM in the range of 0.64 to 1.9 nm, a *Q* in the range of 374 to 2019, and a FoM in the range of 282 to 1702. In vacuum, the smallest FWHM observed is 0.6 nm and the largest *Q* observed is 3400. These performance characteristics should be attainable in carefully fabricated structures having smooth sidewalls.

## Acknowledgments

The High Performance Computational Virtual Laboratory (Canada) and the Research Institute for ICT (Iran) are gratefully acknowledged.

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