Dark-mode plasmon resonances can be excited by positioning a suitable nano-antenna above a nanostructure to couple a planar incident wave-front into a virtual point source. We explore this phenomenon using a prototypical nanostructure consisting of a silver nanotriangle into which a hole has been drilled and a rod-like nano-antenna of variable aspect ratio. Using numerical simulations, we establish the behavior of the basic drilled nanotriangle under plane wave illumination and electron beam irradiation to provide a baseline, and then add the nano-antenna to investigate the stimulation of additional dark-mode plasmon resonances. The introduction of a suitably tuned nano-antenna provides a new and general means of exciting dark-mode resonances using plane wave light. The resulting system exhibits a very rich variety of radiant and sub-radiant resonance modes.
© 2015 Optical Society of America
A diverse range of technological applications exploit the dipolar transverse and longitudinal plasmon resonances of nanoparticles. For example, chemical or biomedical analyses can be achieved by SERS , particle aggregation , shift in peak resonance , or resonant fluorescent enhancement ). In addition new medical treatments that exploit dipolar plasmon resonances are being trialed [5, 6] while applications in respect of medical imaging , transduction of intracellular processes , solar glazing [9, 10], solar thermal harvesting , spectrally and angularly selective coatings , dichroic filters , decorative applications  and photovoltaic cells  are under development. In contrast, the properties and possible applications of multipolar and dark mode plasmon resonances are less well studied. (‘Dark plasmon’ is a term applied to resonances that are only weakly radiative and which, for symmetry reasons, cannot be induced by a planar incident wave, at least not in the quasistatic limit ). It is possible that these higher-order resonances could also be exploited in selected technological applications, such as for coupling a complex electric field into a quantum emitter . We were interested, therefore, in the development of a platform on which the amplitude and quality of such complex resonances could be optimized. Here we show that a drilled nano-triangle, coupled with a suitable tuned nano-antenna, is a suitable candidate.
There are three reasons for selecting triangles for this exercise: first, due to their lower symmetry (relative to spheres or rods), even simple triangular shapes exhibit unusual and interesting resonances, with the quadrupolar and hexapolar resonances becoming discernible in optical spectra of sufficiently large nanoparticles [18–27]. Secondly, triangular shapes can be conveniently fabricated by wet chemistry and/or lithography [19, 21, 28, 29]. Thirdly, we envisaged that holes can be readily drilled through the face of such triangles using either electron beam etching or ion beam milling. In principle, the hole formed by the drilling operation could be used to house a fluorophore or quantum dot.
We use numerical simulations of plane wave and electron beam illumination to probe the higher-order resonances that can be excited on drilled nano-triangles. The base case is an equilateral silver triangle of 200 nm edge and 5 nm thickness into which a hole of variable size has been drilled mid-face. Having established the effects of planar wave fronts on such shapes, we then show what happens when a suitably tuned nano-rod antenna is introduced above the drilled triangle. We use the computationally intensive discrete dipole approximation to obtain quantitative, fully-retarded simulations and faster boundary element method calculations to obtain qualitative confirmation of the symmetry of the various resonances possible and on the effect of triangle thickness. Our focus throughout is on identifying the optimum conditions for exciting strong, multipolar plasmon resonances.
To avoid ambiguity we have defined some of the terms that we use. The term edges is used for the side flanks of the drilled triangular shape, whether in the hole or along the external perimeter of the triangle. Hole edge represents the edge around the hole while triangle edges designates the edges along the external perimeter of the shape. The case where the electric field lies in the plane of the triangle will be referred to as in-plane polarization. The far-field optical properties are not affected by the actual direction of polarization within the plane for triangles and other shapes with n-fold (n≥3) rotational symmetry . The term transverse polarization will be used to designate the case when the electric field is perpendicular to the plane of the triangle. If a cylindrical nano-antenna is positioned above the face of the triangle then transverse polarization has the appropriate orientation to activate longitudinal resonances in the nano-antenna.
We will use the term multipolar resonance to mean a localized surface plasmon resonance with a spatial distribution displaying more than two poles of surface charge. Examples are quadrupolar (4 poles), hexapolar (6 poles) and octupolar (8 poles). These are to be distinguished from a dipolar resonance which is characterized by having only two poles, obviously of opposite charge. In addition complex resonances may display a changing number of poles of surface charge as a function of phase, i.e. when plotted against elapsed time. This may be due to such a resonance being a superposition of two simpler resonance modes which peak at different times in the cycle (i.e have different phases with respect to the incoming light). For example, a visualization of surface charge against time can show the presence of first one resonance component (e.g. dipolar) and then, later in the cycle, another (e.g. quadrupolar). Therefore, counting poles of surface charge is not an unambiguous way to characterize a resonance. Superimposition of the electric field’s phasor (phase and amplitude) onto the real-space depiction of the resonance resolves this difficulty. We define the term multimodal here to mean a resonance that exhibits differing numbers of peaks as a function of time.
The major portion of the simulations were conducted using the Discrete Dipole Approximation (DDA), as implemented in the DDSCAT code of Draine and Flatau . In this technique the target material (silver in our case) is simulated by a set of closely-spaced dipoles. We used the dielectric data of Johnson and Christie . The DDA technique has been found to give accurate results provided that the dipole spacing is sufficiently small (of the order of 1 nm or less for most metals) [18, 33–36]. The extinction spectra were calculated over the visible and near-infrared regions of the spectrum. The near-fields were calculated in a plane located in vacuum 1 nm above the faces of the triangles, at wavelengths corresponding to the resonance peaks of interest. The out-of-plane component of the electric field in this plane (Ey in our geometry) was considered to provide a good approximation of the surface charge. Furthermore, since electrodynamic resonances have complex nature, we also present the near-field electric field intensities as phasor plots which have been color mapped to the HSV color wheel, so that intensity and phase can be simultaneously depicted. Finally, it has previously been shown that large triangles of a higher aspect ratio have stronger and better separated multipolar modes than small shapes of low aspect ratio [20, 26]. Therefore, for the DDA calculations we chose to consider an equiangular triangle 5 nm thick with a side length, L, of 200 nm, with a hole located in the geometric center of the triangle (ie. mid-face) in order to separate and amplify the higher-order resonances in the DDA calculations.
Although accurate, DDA calculations are slow. Therefore, we also made use of code based on the boundary elements method (BEM) to produce alternative depictions of the various resonances that might be possible in these shapes. These calculations were qualitative only, because the shapes used had well-rounded edges in order to achieve numerical stability and convergence in the calculations. Therefore, the energies at which resonances occurred in the BEM and electron energy loss spectroscopy (EELS) calculations are not identical to those of the DDA calculations. Nevertheless, the symmetry and sequence of the resonances should be similar and hence comparable. The speed of the BEM methods allowed for the effect of different triangle thicknesses to be sampled. BEM calculations were performed using a slightly modified version of the MNPBEM v13eels toolbox [37, 38]. A drilled triangle was created with similar size parameters to those used in DDA, but the corners (r = 5 nm) and edge profile (r = 2.5 nm) had to be rounded to obtain convergence. The triangle was subjected to light with in-plane polarization (E along z and k along y, Fig. 1(a)) or to a 200 keV electron beam parallel to y passing either down the axis of three-fold rotational symmetry, or rastered over the whole triangle. Convergence against mesh-size was tested, with excellent convergence for static calculations and satisfactory convergence for retarded calculations. The final maximum patch edge was 2.5 nm, with finer meshing around curves and points. There was strong qualitative agreement between static and retarded calculations, and moderate quantitative agreement in resonance energy but the dipole resonance was significantly damped by retardation. The complex surface charge was colored using the same scheme as other plots, and compared to the charge of the static eigenmodes.
Figure 1(a)-(c) shows a selection of the drilled triangles generated with a 200 nm edge length, L, and a 5 nm thickness. The baseline behavior of the drilled triangle motif was first established by systematically varying the hole diameter from 0 to 120 nm and simulating the optical properties for the in-plane polarization. Note that placing holes of diameter equal to or larger than ~60% of L will breach the sides of the triangle when the hole is positioned mid-face. In the second phase of the work a standard hole diameter of 40 nm was chosen for the DDA calculations and orthogonal cylindrical nano-antennas of systematically varying lengths introduced into the simulations. In these cases E was transverse polarized (relative to the triangles) so that the coupling effect of the nano-antenna could be activated.
3.1 Resonances in an isolated drilled triangle with in-plane polarization
Figure 2 shows the extinction efficiency (Qext) of an isolated, drilled triangle as a function of the hole size. Due to its high aspect ratio, the solid triangle shows a dipolar resonance in the infrared at 1.1 eV and this is further red-shifted to 0.7 eV as the diameter of the hole is increased towards 80 nm. This resonance is labelled ‘A’ in Fig. 2. In the case of a drilled triangle, mode ‘A’ can be considered to correspond to the lower energy ‘bonding’ mode formed by hybridization between the dipole resonance of the triangle and its hole. There are also several higher order resonances at higher energies. For convenience, some of these have been labelled as ‘B’, ‘C’, ‘D’, ‘E’ and ‘F’ on Fig. 2. These, higher-order, resonances have appeared due to the breaking of symmetry caused by retardation of the light as it travels the relatively large distance (200 nm) across the triangle. They would be absent in the case of a very much smaller shape.
Mode A is both a dipolar and a diphase resonance, Fig. 3(a). The surface charge on the two ends of the shape (base and tip here) simply oscillates back and forth between negative and positive. Furthermore, the introduction of a hole has little effect on distribution of the near-field although it does lead to a progressive red-shifting of the mode
Mode B is associated with a solid triangle and is extinguished when a hole of sufficient diameter is inserted into the face of the triangle. As shown in Fig. 3(b), it has a complex hexapolar distribution of surface charge. Note the two reddish lobes in the phasor map on the sides of the triangle. These are about π/2 out of phase with the maxima on the sides or the tips and base of the triangle. Hence both quadrupole and hexapole modes are present in this shape at the frequency of the simulations.
Mode C, Fig. 3(c), begins as a stable decaphase resonance for the solid triangle, acquires an octuphase component to become multimodal for hole diameters between about 20 and 90 nm, and finally ends as just a stable octuphase resonance for holes of diameters 100 nm or greater.
Modes D, E and F are higher order resonances that are strongly developed at a hole diameter of about 40 nm. Their distribution of spatial charge is complex, Fig. 3(d)-(f). Mode E (at 2.70 eV) is interesting because it has the greatest intensity of electric field located on the hole edge. This could be useful if a quantum emitter was located in the hole. The component of the overall field located in the region of the hole edge obviously corresponds to the higher energy of the two hybridized modes and is clearly a 'hole mode', that, in this instance, has hybridized with a complex higher-order mode of the basic triangle.
Further insight into the above higher-order resonances was gained by running BEM calculations of drilled nano-triangles. These calculations reveal the eigenmodes possible in a particular geometry irrespective of the nature of illumination, i.e. resonances of both bright or dark characteristic. The resulting optical extinction spectra of the triangle with a 40 nm hole are shown in Fig. 4(a). For comparative purposes we also show the spectrum calculated for this geometry using the DDA. The latter is slightly red-shifted relative to the BEM results due to the BEM calculation having been applied to a shape with rounded corners and edges. The dipole mode A of Figs. 2 and 3, clearly matches the dipole mode at 1.14 eV in Fig. 4(a). The quadrupole mode B of Figs. 2 and 3 is matched in Fig. 4(a) by a mode at 1.85 eV. The complex modes C through to E of Figs. 2 and 3 are respectively matched by modes in Fig. 4 at 2.22, 2.54, and 2.74 eV. The electric charge distribution of Mode F of Fig. 3 is not, however, matched exactly by that of the corresponding mode of the BEM calculation due probably to divergence in the numerical results at these high energies. An advantage of a BEM calculation is that it can identify all the vibrational modes, not only those excitable by an optical source. The most prominent of the lower energy eigenmodes are depicted in Fig. 3(b). The symmetric eigenmodes at 1.18 eV, 1.88 eV, 2.25 eV and 2.56 eV match the resonances A, B, C and D respectively. As we will see shortly, some of the other eigenmodes match the dark modes predicted for electron beam excitation.
It is instructive to consider the resonances that would be excited in the drilled triangle using an electron beam, Fig. 5(a)-(c). The EELS simulations were performed two ways. In Fig. 5(b) the whole triangle was rastered with the electron beam to produce a qualitative EELS-type ‘heat map’ whilst in Fig. 5(a) and (c) the electron beam was fixed at the hole center and the resulting transient electric field rendered as a phasor color map. It is clear in Fig. 5(a) that the primary effect of making the triangle flatter is to red-shift and spread out all of the vibrational modes. The resonances identified in the spectra of Fig. 5(a) have been imaged in Figs. 5(b) and (c), arranged in ascending order of energy to facilitate correlation with one another. They are all characterized, as expected, by perfect three-fold rotational symmetry. In the raster maps of Fig. 5(b), mode Z1 is obviously excited most prominently when the electron beam passes close to the corners of the triangle and is the dipole mode. Modes Z3 and Z4 are excited most strongly when the electron beam passes at the triangle edges and are higher order edge modes. Modes Z2 and Z6 are most strongly associated with an electron beam through the hole and they respectively match the hexapole and breathing modes identified by Schmidt et al.  for triangular geometries. Mode Z2 can be excited by either a center-fire electron beam or by a beam at the middle of an edge. Mode Z5 is complex in nature and can be excited by a variety of beam positions. Z6 is the standard center mode which is less intense due to the presence of the hole. In Fig. 5(c) we show the six distinct resonances, labelled X1 to X6, that were identified for the case of the electron beam passing through the center of a 40 nm diameter hole in a 200 x 200 x 200 x 5 nm triangle. The electric field distributions of the phasor maps for resonances X1 and X2 respectively match the Z2 and Z5 of the raster maps, and the symmetric eigenomodes at 1.78 eV and 2.44 eV of Fig. 4(b), while X3 matches Z6. An animation of the evolution of the resonance modes in the standard drilled triangle, as the frequency of the light is increased, is available in the Supplementary Materials as Visualization 7. The animation shows clearly how each resonance, Xn, or A to E, is associated with a marked change in phase of its electric field.
3.2 Resonances in a hybrid system of drilled triangle and coupled nano-antenna
Next we examine whether the introduction of a cylindrical nano-antenna positioned orthogonally over the hole of the triangle could be used as a virtual point source to excite additional higher-order or ‘dark modes’ beyond those already discussed. In this case the drilled triangle is oriented transverse to the light but the nano-antenna, which is perpendicular to the plane of the triangle, obviously undergoes longitudinal excitation. The light propagates from left to right across the electric field maps that follow. The optimum length of the nano-antenna for coupling to the dark modes was established by systematically varying the aspect ratio (L/D) the nano-antenna from 0.25 to 4, but the hole diameter was fixed at 40 nm.
The addition of the nano-antenna to the drilled triangle resulted in dramatically changed behavior, Fig. 6(a). The introduction of the electric field radiating outwards from the end of the nano-antenna excites previously unseen resonance modes in the triangle. The longitudinal dipolar resonance of the nano-antenna itself is designated P in the figure. This is red-shifted slightly relative to the situation for an isolated nano-antenna due to the interaction with the triangle but is mundane and not discussed further here. More importantly, there are several new resonances in the triangle, the most prominent of which have been denoted as Q, R, S, U, V, Y, T1 and T2. Each of the resonances Q, R, S, U, V or Y can be tuned to their maximum amplitude by suitable adjustment of the length of the nano-antenna. The effect of nano-antenna aspect ratio on the strength of the strong resonances Q and S is shown in Fig. 6(b) as an example.
In Resonance Q, which is the strongest of the new modes induced by the nano-antenna, the end of the nano-antenna excites the perimeter of the drilled hole uniformly, and the hole acts like a virtual point source in turn, Fig. 7(a). The rest of the triangle undergoes a complex resonance that had both octupolar and decapolar characteristics, depending on the phase angle at which it is sampled. The strength of this phenomenon is greatest for a nano-antenna with L/D = 1.625 and for this geometry the resonance peaks at 2.43 eV. This would normally be a dark mode because of its radial symmetry. The interplay of the hole and triangle edge modes generates a complex pattern of charge with time. Resonance R is strongest for the nano-antenna with L/D = 1.125 and light of 2.60 eV. It too would normally be a dark mode. The charge around the hole edge alternates from unipolar to dipolar to quadrupolar as phase sweeps form 0 to π, while the triangle edges display a strong hexapolar resonance. Resonance S is the second strongest of the induced modes and, like Q, is also a dark mode, but is even more complex, with the perimeter of the hole undergoing a tripolar resonance, while the triangle edges alternate through one oscillation between charge distributions with 12 and 24 poles, Fig. 7(c). The interplay of the hole and triangle edge modes generates a complex pattern of charge. The tips of the triangle hardly participate in this resonance, as also shown by a plot of the norm. The peak of this resonance is greatest when L/D = 2.5 and the light is at 2.85 eV. Resonances U and V and similarly complex and have the charge distribution characteristic of a dark mode.
No direct match could be made between modes Q, R, S, U and V and the modes Z2 to Z6 of Fig. 5(a)-(c). Evidently, the combined effects of hybridization between modes, the phase-shifting effect of retardation and, possibly, the scattering of light off the nano-antenna have combined to yield more complex resonances than are possible by electron beam stimulation of a simple drilled triangle. Finally, the relatively weak resonances T1 and T2 of Fig. 6(a) are simply edge resonances, and arise irrespective of whether the nano-triangle is drilled or not, Fig. 7(f). T2, in particular, take the form of SPPs propagating along the triangle edges. Resonance T2 is extinguished when a sufficiently long nano-antenna is introduced, but resonance T1 is essentially insensitive to the presence of any nano-antenna. Modes T1 and T2 are examples of the phenomenon reported by Schmidt et al. , in terms of which a flattened plasmonic particle develops the edge modes of a thin film.
3.3 Magnetic modes
Due to the coupled nature of electric and magnetic fields, each of the above resonances is also associated with an oscillation in the magnetic near-field as well. Evlyukhin et al.  have shown how a far-field parameter like Qext can be decomposed into contributions from electric and magnetic oscillations, and the results provide considerable insight into the resonances. Here we characterize the magnetic oscillations of the drilled triangle motif using maps of their near-field distributions. The transverse resonances T1 and T2 had negligible magnetic components and are not shown. As with the electric fields, the strongest magnetic fields were associated with the dipolar resonances A and P, as expected . The remaining resonances displayed interesting patterns of magnetic near-field, Fig. 8. In particular, resonances V and Y were strongly magnetic. Some of the resonances contain locations (red hues) where the amplitude of the magnetic near-field is reduced to below that of the incident field. The best example of this is Resonance C.
The drilled triangle motif is very flexible, and our simulations predict that it could be used for the controlled excitation of a variety of higher-order plasmon resonances, either with, or without, an associated nano-antenna. Some of these resonances are just multipolar, others are multimodal (i.e. resonances that are the sum of more than one component, for example quadrupole/octupole hybrids). Visualization and analysis of these complex resonances is facilitated by plotting the special distribution of the electric field phasor.
In broad outline, five broad classes of resonant behavior were found in the basic drilled triangle, exposed to light without the benefit of a coupled nano-antenna. The simplest of these (Resonance A) is just a dipolar mode, which is also diphase in nature. At a slightly higher energy there is an octuphase resonance (Resonance B). On close inspection this was actually comprised of a quadrupolar component and a hexapolar component, 90° out-of-phase with one another. As the radius of the hole gets larger, this resonance is attenuated and eventually extinguished. At higher energy, there is a decapolar mode (Resonance C) which, as the radius of the hole is increased, morphs smoothly into a purely octupolar resonance. At even higher irradiation energies there are resonances unique to the drilled triangle motif. We analyzed the most prominent of these. Resonance D has a complex charge and phase distribution which is strongly expressed at two positions along the base of the triangle. Resonance E is most strongly expressed around the hole in the triangle. It arises from the hybridization of the hole plasmon with a complex higher-order triangle resonance. Resonance F consists of a dipole resonance around the hole hybridized with multimodal sexapolar and octupolar resonances along the sides of the triangle.
Introduction of a nano-antenna converted a portion of the incoming planar incident wave into a virtual point source located above the hole of the triangle. Strong coupling occurred between antenna and triangle when the resonance frequency of the nano-antenna was of the same order as that of the primary dark mode of the drilled triangle. The point excitation provided by the end of the nano-antenna was able to excite a range of additional multipolar resonances in the triangle, all with a quite different symmetry to those occurring on the basic drilled shape. The most prominent of these new resonances were labelled Q, R, S, U, V and Y. The interesting symmetries of some of these new modes suggests possible directions for future application. For example, the modes could be used to generate multipolar fields for the excitation of other nanoparticles or nanostructures. In particular, location of an object such as a quantum dot, nano-diamond or other fluorophore at strategic locations on such triangles may permit the plasmonically-enhanced generation of otherwise forbidden quadrupolar or higher order emission modes.
The authors thank the Australian Research Council for support under grants DP120102545 and LE140100002. Computing resources were provided by the University of Technology Sydney and the NCI National Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government.
References and links
1. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]
2. R. Elghanian, J. J. Storhoff, R. C. Mucic, R. L. Letsinger, and C. A. Mirkin, “Selective colorimetric detection of polynucleotides based on the distance-dependent optical properties of gold nanoparticles,” Science 277(5329), 1078–1081 (1997). [CrossRef] [PubMed]
3. A. D. McFarland and R. P. Van Duyne, “Single silver nanoparticles as real-time optical sensors with zeptomole sensitivity,” Nano Lett. 3(8), 1057–1062 (2003). [CrossRef]
4. W. Deng, F. Xie, H. T. M. C. M. Baltar, and E. M. Goldys, “Metal-enhanced fluorescence in the life sciences: Here, now and beyond,” Phys. Chem. Chem. Phys. 15(38), 15695–15708 (2013). [CrossRef] [PubMed]
5. L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U.S.A. 100(23), 13549–13554 (2003). [CrossRef] [PubMed]
7. C. Xu, J. Xie, D. Ho, C. Wang, N. Kohler, E. G. Walsh, J. R. Morgan, Y. E. Chin, and S. Sun, “Au-Fe3O4 dumbbell nanoparticles as dual-functional probes,” Angew. Chem. Int. Ed. Engl. 47(1), 173–176 (2008). [CrossRef] [PubMed]
9. S. Schelm and G. B. Smith, “Dilute LaB6 nanoparticles in polymer as optimized clear solar control glazing,” Appl. Phys. Lett. 82(24), 4346–4348 (2003). [CrossRef]
10. N. L. Stokes, J. A. Edgar, A. M. McDonagh, and M. B. Cortie, “Spectrally selective coatings of gold nanorods on architectural glass,” J. Nanopart. Res. 12(8), 2821–2830 (2010). [CrossRef]
11. O. Neumann, C. Feronti, A. D. Neumann, A. Dong, K. Schell, B. Lu, E. Kim, M. Quinn, S. Thompson, N. Grady, P. Nordlander, M. Oden, and N. J. Halas, “Compact solar autoclave based on steam generation using broadband light-harvesting nanoparticles,” Proc. Natl. Acad. Sci. U.S.A. 110(29), 11677–11681 (2013). [CrossRef] [PubMed]
12. J. Liu, B. Cankurtaran, L. Wieczorek, M. J. Ford, and M. B. Cortie, “Anisotropic optical properties of semitransparent coatings of gold nanocaps,” Adv. Funct. Mater. 16(11), 1457–1461 (2006). [CrossRef]
13. M. B. Cortie, X. Xu, and M. J. Ford, “Effect of composition and packing configuration on the dichroic optical properties of coinage metal nanorods,” Phys. Chem. Chem. Phys. 8(30), 3520–3527 (2006). [CrossRef] [PubMed]
14. A. Iwakoshi, T. Nanke, and T. Kobayashi, “Coating materials containing gold nanoparticles,” Gold Bull. 38(3), 107–112 (2005). [CrossRef]
15. S. Pillai, K. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl. Phys. 101(9), 093105 (2007). [CrossRef]
16. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004). [CrossRef]
17. A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun. 4, 1750 (2013). [CrossRef] [PubMed]
18. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003). [CrossRef]
19. D. Aherne, D. M. Ledwith, M. Gara, and J. M. Kelly, “Optical properties and growth aspects of silver nanoprisms produced by a highly reproducible and rapid synthesis at room temperature,” Adv. Funct. Mater. 18(14), 2005–2016 (2008). [CrossRef]
20. C. Awada, T. Popescu, L. Douillard, F. Charra, A. Perron, H. Yockell-Lelièvre, A. L. Baudrion, P. M. Adam, and R. Bachelot, “Selective excitation of plasmon resonances of single Au triangles by polarization-dependent light excitation,” J. Phys. Chem. C 116(27), 14591–14598 (2012). [CrossRef]
22. Y. He and G. Shi, “Surface plasmon resonances of silver triangle nanoplates: graphic assignments of resonance modes and linear fittings of resonance peaks,” J. Phys. Chem. B 109(37), 17503–17511 (2005). [CrossRef] [PubMed]
23. A. L. Koh, A. I. Fernández-Domínguez, D. W. McComb, S. A. Maier, and J. K. W. Yang, “High-resolution mapping of electron-beam-excited plasmon modes in lithographically defined gold nanostructures,” Nano Lett. 11(3), 1323–1330 (2011). [CrossRef] [PubMed]
24. R. Morarescu, H. Shen, R. A. L. Vallée, B. Maes, B. Kolaric, and P. Damman, “Exploiting the localized surface plasmon modes in gold triangular nanoparticles for sensing applications,” J. Mater. Chem. 22(23), 11537–11542 (2012). [CrossRef]
25. S. Viarbitskaya, A. Teulle, R. Marty, J. Sharma, C. Girard, A. Arbouet, and E. Dujardin, “Tailoring and imaging the plasmonic local density of states in crystalline nanoprisms,” Nat. Mater. 12(5), 426–432 (2013). [CrossRef] [PubMed]
26. P. Yang, H. Portalès, and M.-P. Pileni, “Identification of multipolar surface plasmon resonances in triangular silver nanoprisms with very high aspect ratios using the DDA method,” J. Phys. Chem. C 113(27), 11597–11604 (2009). [CrossRef]
28. S. Banerjee, K. Loza, W. Meyer-Zaika, O. Prymak, and M. Epple, “Structural evolution of silver nanoparticles during wet-chemical synthesis,” Chem. Mater. 26(2), 951–957 (2014). [CrossRef]
30. B. Hopkins, W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Optically isotropic responses induced by discrete rotational symmetry of nanoparticle clusters,” Nanoscale 5(14), 6395–6403 (2013). [CrossRef] [PubMed]
31. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering csalculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]
32. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
34. V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37(9), 1792–1805 (2008). [CrossRef] [PubMed]
35. B. M. Nebeker, J. L. de la Peña, and E. D. Hirleman, “Comparisons of the discrete-dipole approximation and modified double interaction model methods to predict light scattering from small features on surfaces,” J. Quant. Spectrosc. Radiat. Transf. 70(4-6), 749–759 (2001). [CrossRef]
36. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007). [CrossRef]
37. U. Hohenester and A. Trügler, “MNPBEM - A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Commun. 183(2), 370–381 (2012). [CrossRef]
38. U. Hohenester, “Simulating electron energy loss spectroscopy with the MNPBEM toolbox,” Comput. Phys. Commun. 185(3), 1177–1187 (2014). [CrossRef]
40. F.-P. Schmidt, H. Ditlbacher, U. Hohenester, A. Hohenau, F. Hofer, and J. R. Krenn, “Universal dispersion of surface plasmons in flat nanostructures,” Nat. Commun. 5, 3604 (2014). [CrossRef] [PubMed]
41. A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation,” Phys. Rev. B 84(23), 235429 (2011). [CrossRef]