Edge-Plasmon Whispering-Gallery Modes in Nanoholes

Metallic apertures remain one of the most intensively investigated optical structures due to their rich behavior. This encompasses physical phenomena such as Bragg scattering, surface plasmons, or localized modes, offering great potential for many applications in the growing fields of nanophotonics and plasmonics. Here, we show that isolated metallic nanoholes can additionally support edge plasmons in the form of whispering-gallery modes and we demonstrate that, although high-order modes remain dark at normal incidence, they can be excited at oblique illumination. As revealed by simulations, the existence of these modes in nanohole arrays is fundamental to understanding complex resonant phenomena and to explaining previous unexpected experimental results found in metamaterials. Notably, the symmetries of the employed lattice determine the excited mode orders, which, together with a strong dependence on the film thickness, yields an easily and highly tailorable optical response. Furthermore, some modes display unusually narrow line widths (approximately 7 nm) and a high sensitivity to the environment, resulting in one of the highest sensing figures of merit (approximately 80) reported for a localized plasmon. These singular features may find application in the construction of special polarizers and filters, single-layer zero-transmittance absorbers, and high-performance nanosensors.

Figure 1

The transformation from straight to curved edges: from (a) a straight metallic edge to (b) a nanodisk following a convex bend and (c) a nanohole following a concave bend.

Figure 2

The optical response of isolated nanoholes. (a) The structure under consideration and the field polarization. The blue arrow represents the $\mathbf{E}$ field for $p$ polarization and the $\mathbf{H}$ field for $s$ polarization. (b),(c) The normalized scattering cross section for the $t=30$ nm case when (b) $r=160$ nm and $\theta$ varies between $0^{\circ }$ and ${60}^{\circ }$ and (c) $\theta ={45}^{\circ }$ and $r$ varies between 160 and 200 nm. (d)–(g) ${\mathbf{H}}_{\mathrm{scat}}$ at the input interface at a frequency of (d) 340 THz, (e) 333 THz, (f) 510 THz, and (g) 660 THz. The angle of incidence is as follows: (d) $\theta =0^{\circ }$; (e)–(g) $\theta ={45}^{\circ }$.

Figure 3

The optical response of periodic hole arrays. The geometrical dimensions are $t=30$ nm, $r=160$ nm, and $a=400$ nm. (a)–(c) Angular spectra for (a) $p$-polarization transmittance, (b) $p$-polarization absorption, and (c) $s$-polarization absorption. The first-order Wood’s anomaly is indicated with dashed lines. (d)–(f) Normalized $H_z$ at the hole center for $p$ polarization, $\theta =5^{\circ }$, and (d) $f=457.5$ THz, (e) $f=583$ THz, and (f) $f=661.4$ THz.

Figure 4

The influence of the lattice symmetries on the hole-array optical response. (a) The symmetry groups of the second- and third-order EP-WGMs (4- and 6-pole) and of the square and hexagonal lattices (those of a four-sided and a six-sided polygon, respectively). (b),(c) The $H_z$ distribution that may arise from the interactions between adjacent holes is shown for different EP-WGM orders in (b) a square lattice and (c) a hexagonal lattice. The red areas correspond to positive magnitudes while the blue areas indicate negative values. The appearance of a dipole moment is indicated with a green arrow. (d) The transmittance spectra of a hole array with $t=5$ nm, $r=160$ nm, and $a=400$ nm for different lattice and/or angle configurations. The number of poles is indicated next to each resonance.

Figure 5

The influence of the thickness and mode hybridization. (a) The absorption spectrum as a function of $t$ for $a=400$ nm and $r=160$ nm. The angle of incidence is $\theta =2^{\circ }$. (b) A schematic representation of a cut plane ($x=y,z$) passing through the center of the hole. (c),(d) The $H_z$ distribution through the cut plane shown in (b) for $t=300$ nm at (c) $f=579.3$ THz and (d) $f=592.7$ THz. The black lines indicate the location of the metal film.

Figure 6

The EP-WGM sensing performance in a square hole array with $t=50$ nm, $r=160$ nm, and $a=400$ nm. (a) The resonant wavelength, sensitivity, and FOM as a function of $n$ (each circle corresponds to a different simulation). (b) The variation in the transmittance as $n$ changes from 1 to 1.1. The solid and dashed lines correspond to the simulated response of an array of holes with sharp and blended edges, respectively. (c) The unit cell with a blended-edge hole considered in the simulations.

Figure 7

A quadrupolar EP-WGM supported by a square array (periodicity $a=400$ nm) of square holes (side $l=250$ nm) with round corners (bending radius $b=25$ nm) milled in a film with $t=35$ nm. (a) The simulated $H_z$ field. (b) The tetrahedral mesh employed in the simulation.