1. Introduction

Localized surface plasmon resonances (LSPRs), famous for their large optical extinction cross-sections, high field enhancements and sub-wavelength light confinement, can be considered as classical charge density oscillations at the nanoscale. The free electrons on the metal-dielectric boundaries in a metallic nanostructure can be driven into an oscillating motion by an external field with resonant frequencies typically in the visible and near-infrared [1]. Depending on the particle shape and material properties specific modes (oscillation patterns) may exist. As nanofabrication allows the production of increasingly complicated nanostructures and nanosystems, a fundamental understanding of LSPR modes and mode interactions in different geometries and arrangements is of considerable importance. Roughly, LSPR modes can be divided into dipole resonances and higher order resonances. The latter typically have lower radiative losses and higher quality factors (Q-factors). Applications such as plasmonic lasing [2, 3], plasmon enhanced luminescence [4], LSPR refractive index (RI) and biosensing [5] can substantially benefit from these sharp higher order plasmon modes. Some nanoparticles, for which multipolar resonances have been observed, are nanoshells [6, 7], nanorice [8], nanoprisms [9], nanodisks [10], nanorings [11], nanowires and nanorods [12, 13] and nanostars [14]. The unique plasmonic properties of branched nanoparticles such as chemically synthesized nanostars, hexapods [15] and octapods [16] enable high LSPR sensitivities [14] as well as zeptomol detection in SERS [17].

Here, we present an extensive experimental and theoretical study of a planar, multi-branched nanostructure supporting higher order plasmon modes: the nanocross. The cross geometry consisting of two intersecting nanobars was recently introduced by Cortie et al. [18] and us [19]. For a cross consisting of any number of nanobars (arms) k, each branch can be matched with exactly one lobe of the multipolar spherical harmonics $Y_l^m$ with l ≤ k. This allows a well defined charge separation into the harmonics’ lobes with each cross branch carrying exactly one charge type. We will show that, in addition to dipole modes, the branched concave shape enables a very efficient charge separation into quadrupolar (l = 2) and octupolar (l = 3) modes with high Q-factors. The spectral tunability of these modes will be discussed as well. The spectral position can easily be controlled by varying the arm length and arm angle. More interestingly, also the spectral separation of the plasmon modes can be tuned and optimized for specific applications by changing the cross arm angle α.

Coupling of a plane wave to higher order plasmon modes is only possible through either phase retardation of the incident radiation across the particle [11, 20] or by reducing the structural symmetry [21,22]. Symmetry breaking of a plasmonic system can enable hybridization of plasmon modes of different multipolar symmetry [23, 24]. Dark higher order modes can therefore mix with dipole plasmons rendering the hybridized mode dipole active. We will show that for different nanocross geometries the degree of rotational symmetry C_n critically influences the excitation of higher order modes with both phase retarded and non-phase retarded plane wave illumination. Furthermore, Fano interference can arise from the coherent interaction between spectrally overlapping sharp and broad resonances resulting in pronounced asymmetric line shapes and even antiresonances [25, 26]. Particularly important about nanocross geometries with reduced symmetry, is the fact that they allow the excitation of Fano resonances without the need for nanometer sized interparticle gaps. This subtle interplay between structural symmetry and plasmonic properties is expected to have an important influence on other symmetry dependent optical phenomena such as second harmonic generation (SHG) [27].

The well defined higher order resonances supported by a nanocross make this geometry an excellent building block for coherently coupled plasmonic nanocavities. In plasmonic dimers the individual particle’s dipole modes can hybridize into bonding and antibonding resonances with narrow subradiant and broad superradiant line widths. Additionally, symmetry breaking in heterodimers will strongly influence the allowed mode coupling leading to dipole active higher order modes and Fano resonances with complex spectral shapes [11, 28–30]. The benefits of this plasmon line width engineering in applications such as LSPR RI sensing have recently been demonstrated [19, 31–33]. For a heterodimer consisting of a cross with three arms and a nanobar we investigate in detail the generation of hybridized sub- and superradiant plasmon modes and how these can evolve into Fano antiresonances when increasing the particle size from the electrostatic regime to experimentally accessible dimensions. By changing the inter-particle distance the near-field coupling influence is studied. Other nanocross based plasmonic cavity designs with polarization independent response, improved line widths and multiple Fano antiresonances are demonstrated.

2. Experimental and simulation methods

The fabricated cavity structures are schematically shown in Fig. 1(a)–1(c) with the geometrical parameters indicated: cross arm length L_c and width W_c, cross arm angle α, bar length L_b and width W_b, and Au thickness T and SiO₂ pillar height H. The sample fabrication process is similar to the one described in [19]. The sample here consists of a standard microscope cover glass with a sputtered 50 nm Au / 3.5 nm Ag bi-layer. Using electron beam lithography on a negative tone hydrogen silsesquioxane (HSQ) resist coating and subsequent Xe ion milling the nanostructures are carved out of the metal layers.

 

Fig. 1 Structural parameters of the studied nanocross cavities. (a) two-arm cross: 2AX, (b) three-arm cross: 3AX and (c) three-arm cross and bar heterodimer: 3AX-I. Panel (d): SEM image taken under an angle of the three-arm cross in panel (c).

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Although the structural dimensions still require expensive serial lithography steps, advances in nanostencil and nanoimprint lithography suggest that this cost issue can soon be circumvented [34]. In order to partially etch the substrate an additional Sulfur Hexafluoride + Oxygen (SF₆ + O₂) plasma etch is performed (Oxford instruments Plasmalab System 100 ICP 180). This etch provides a more homogeneous surrounding RI resulting in higher quality resonances and larger RI sensing sensitivity [19]. Figure 1(d) shows a scanning electron microscope (SEM) images of the structures taken under an angle of 60° after etching, clearly showing that the Au particles are elevated above the substrate. The height H of the SiO₂ support is about 170 nm. Prior to SEM imaging, a 20 nm thick layer of a conductive polymer is spun on the samples to avoid charging problems. The bright rim on top of the particles results from Au redeposition during the ion-milling. The cavities are arranged in 50 μm × 50 μm arrays with an interparticle distance of 2 μm in order to avoid near-field coupling and suppress diffraction related effects.

Transmission spectra were taken with a Fourier transform infrared (FTIR) microscope (Bruker Vertex 80v + Hyperion). The incident light from a tungsten lamp is focused on the sample via a 15 x magnification, NA = 0.4 reflective Cassegrain condenser and collected in transmission with an identical objective. This type of objective creates a light cone with angles between 9.8° and 23.6° impinging on the sample surface. The resulting k-vector spread leads to an experimental angle of incidence which is not uniquely defined. We will refer to the central k-vector when the angle of incidence is mentioned. The transmitted light is further polarized, spatially filtered with a metal knife edge aperture and detected with a liquid nitrogen cooled mercury-cadmium-telluride (MCT) and Si diode detector. All spectra are normalized to a reference spectrum taken on the bare substrate under identical conditions.

Calculated extinction spectra and surface charge distributions are obtained with a commercial FDTD solver using a mesh of 2 nm [35]. For the dielectric permittivity of gold we used a multicoefficient model fit, or Drude model fit where mentioned, to ellipsometry data of the Au films used in our fabrication process. This data matches well the values reported in literature [36]. The glass substrate was not taken into account, except in Fig. 9. All the charge plots are calculated at the middle cross-section of the Au structures.

3. Dark and bright plasmon modes in a two-arm nanocross

First we discuss the basic properties of a simple nanocross consisting of two crossed nanobars intersecting in their center points. This geometry, schematically shown in Fig. 1(a), will be referred to as a 2AX (two-arm cross). For comparison panels (a) and (c) in Fig. 2 show, respectively, the calculated and experimental extinction spectrum of a single cross arm. A broad resonance is observed with a dipole charge distribution as shown in the inset (black dot, D_I). Panels (b) and (d) show spectra for the corresponding 2AX. The structural dimensions of the cross are L_c = 370 nm, W_c = 70 nm, T = 50 nm and α = 45°. The red lines correspond to normal incident excitation for which a similar, slightly red-shifted, spectral response is observed as for the bar. The charge density distribution for this mode is shown in the insets and reveals an l = 1 dipole mode (D_2AX, red dot). The blue lines in panels (b) and (d) are spectra obtained with side illumination. Here, the sample was tilted relative to the optical axis such that the particles are illuminated at an angle of ∼ 20° to the sample surface in the measurement and at grazing incidence in the simulations.

 

Fig. 2 LSPR modes and spectral tunability in a two-arm nanocross. (a–b) Calculated extinction spectra of (a) a nanobar and (b) a 2AX. (c–d) Corresponding experimental spectra. Red lines: normal incident illumination, blue lines: side illumination. Polarization in vertical direction. L_b = L_c = 370 nm, W_b = W_c = 70 nm, T = 50 nm and α = 45°. Insets: calculated charge density distributions corresponding to the indicated resonances. (e–f) Spectral dependence of the D_2AX mode (red) and Q_2AX mode (blue) on (e) the arm length L_c (α = 45°) and (f) the arm angle α (L_c = 370 nm). Lines: simulation, symbols: experiment. Simulated and experimental data obtained with side illumination. The substrate was not taken into account in the simulations. SEM images corresponding to the measured particles in (f) are shown on top with arm angle α and associated direction of side illumination indicated. Dashed lines: comparison with the electrostatic model. The calculated extinction spectra for each α are displayed in the gray scale intensity map in (f).

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Since the in-plane particle dimensions are large enough for sufficient phase retardation, the resulting field gradient along the 2AX can couple to higher order modes that are dark for normal incidence. This results in the strong extra peak in the extinction spectrum (blue dot). From the corresponding charge distribution in the inset it is concluded that this resonance is an (l, m) = (2, 2) quadrupole mode. The narrow line width is a direct result of the mode’s zero dipole moment which strongly reduces radiative losses. Since the cross geometry has the clear distinct symmetry of a dipole and quadrupole charge distribution, an efficient charge separation into these modes is facilitated. For the 2AX in Fig. 1, the simulations result in a Q-factor of 4.4 for the D_2AX mode and 19 for the Q_2AX mode. The constituting nanobar’s dipole mode D_I in panel (a) has a Q-factor of only 4.3. Note that the measured quadrupole mode is not as intense and sharp as in the calculation. This is a common observation in this type of plasmonic systems and is usually attributed to extra losses in the metal due to grain boundaries and increased surface scattering [37]. Extra losses induced by the Au redeposition rim on top of the particles (see Fig. 1(d)) were found to be negligible. In our measurements, the absence of the substrate in the calculation and the non-zero angle of incidence and k-vector spread will add to the observed discrepancies in spectral width and position of the resonances [19].

The spectral positions of the cross modes strongly depend first of all on the arm length L_c. Figure 2(e) illustrates this effect for the 2AX with α = 45°. The red dots correspond to the measured dipole mode extinction resonance with a linear slope of 284 nm spectral red shift per 100 nm increase in L_c. For the quadrupole, indicated by the blue dots, this is slightly lower. A linear fit results in a 257 nm red shift per 100 nm L_c increase. Simulations (full lines) show the same behavior, although blue shifted since the substrate was not taken into account. Secondly, the spectral position and, more interestingly, the spectral separation of the plasmon modes can be tuned by changing the cross arm angle α. The insets in Fig. 2 show SEM images of crosses with fixed L_c = 370 nm and increasing α in steps of 5°. In Fig. 2(f), the experimental spectral positions in function of α of the D_2AX (red dots) and Q_2AX (blue dots) modes are shown. The α dependence is nicely reproduced in the simulations (full lines) with a substrate-induced red shift. For the dipole mode we see a saturating red shift with increasing angle caused by the increasing effective length of the resonator cavity. The position of the quadrupole, on the other hand, is symmetric around α = 45° or, equivalently, the quadrupole plasmon energy is independent of the direction of incidence (left or front side). This distinct different modal behavior makes it possible to control the spectral separation with the arm angle. Indeed, for decreasing α the dipole and quadrupole resonances approach, starting from a large spectral separation to finally merge together for the smallest angles. A simple calculation of the electrostatic energy E_st of a system of point charges localized at the extremities of the cross arms subject to an external field E_y further elucidates the α dependence. For the dipole and quadrupole mode we find E_st ∝ – sinα and E_st ∝ − cosα· sinα, respectively. If we overlay E_st with the results of Fig. 2(f) (dashed lines), it becomes clear that the angle dependence can be understood from simple static considerations where the symmetric charge distribution for α = 45° is the lowest energy configuration for the Q_2AX mode. Although the general trend is retained, for angles close to 0° and 90° the finite width of the cross arms can not be neglected and our simplified point charge description starts to deviate. Especially the D_2AX mode for α between ∼ 70° and 90° shows a blue-shift with increasing α instead of a red shift. The spectral behavior here is similar to concave hyperbolic particles where a blue-shift is expected for particles with smaller base size (here larger α) and follows the rule that increased interactions between the opposite sign surface charges leads to a blue-shift of the resonance [38].

Since a cross has a finite rotational symmetry a strong anisotropy can occur in the excitation efficiency of the different plasmon modes. Figure 3 illustrates this for a two-arm cross illuminated from different in-plane directions θ, as defined in panel (a). The bottom graphs in panel (b) and (c) contain simulated extinction spectra where the black spectrum corresponds to θ = 0°, the gray spectrum to θ = 90° and the cyan spectrum to the indicated angle. In the top right is shown a polar plot of the different modes’ peak intensity obtained from a multi-peak Lorentz fit to the calculated spectra. The different plasmon modes in the polar plot are indicated by the dashed lines in the spectra. All previously shown data with side illumination was obtained for θ = 0°. For the 2AX with α = 45° in panel (b) the lower energy D_2AX mode is nearly isotropically excited for all incident angles. When the cross arm angle is changed to, e.g., α = 60°, panel (c), the degeneration for θ = 0° and θ = 90° is lifted and a mode splitting occurs. A first red shifted dipole mode, located at 1280 nm (red line in the polar plot), is concentrated around θ = 0°, i.e. when the incident electric field is along the long axis of the cross. A second blue shifted dipole resonance at 1050 nm (orange line in the polar plot) can be excited when there’s a field component along the short cross axis (θ = 90°). This behavior is similar to what is expected for ellipsoidal particles [39]. The distinct behavior of the dipole modes in Fig. 3(b) and 3(c) clearly reflects, resp., the C ₄ (4-fold) and C ₂ (2-fold) rotational symmetry of the crosses.

 

Fig. 3 Rotational anisotropy of mode excitation in two-arm nanocrosses. (a) Schematic drawing defining the plane wave angle of incidence θ. The light is incident from the side in the k-vector direction and polarized along the E-field vector. (b) An α = 45° 2AX with C ₄ rotational symmetry. (c) An α = 60° 2AX with C ₂ rotational symmetry. (b) and (c) top left graph: experimental data, bottom graph: corresponding simulated spectra without substrate, top right: polar plot of the different modes’ peak intensity. Black spectra correspond to θ = 0°, gray spectra to θ = 90°, and cyan spectra to the indicated θ. Dashed lines indicate the plasmon modes in the polar plots (red and orange: dipolar, blue: quadrupolar). Scale bar in SEM image: 100 nm.

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The quadrupole mode (blue dashed lines) is most efficiently excited when θ = 0° and 90°. For θ = 45° the resonance disappears from the extinction spectrum as the incident field polarization can not be coupled. An experimental study of the full θ dependence is hard to realize. The experimental data, shown here in the top left graph of each panel, corresponds only to θ = 0° (black lines) and θ = 90° (gray lines) for the nanocrosses shown in the SEM insets.

4. Dark, bright and Fano resonances in a three-arm nanocross

As a next step we can add an extra arm to the cross. The resulting three-arm cross, further referred to as 3AX, has typical extinction spectra as shown in Fig. 4(a). We see that increasing the number of cross arms introduces additional modes. Two resonances are observed for normal incidence (red lines) and three for side illumination (blue lines). The lower energy mode is a dipole resonance (orange dot) while the second 3AX mode (cyan dot) is very similar to the 2AX quadrupole mode. In the latter, only the diagonal arms are in resonance and induce small mirror charges on the center bar as can be seen in the charge plot in the inset. For the higher energy extinction peak the charge density plot (green dot) reveals an (l, m) = (3, 3) octupole mode. Here, the dipole moment in the diagonal arms oscillates out-of-phase with the dipole moment in the center bar. We will further refer to the 3AX dipole, quadrupole and octupole modes as D_3AX, Q_3AX and O_3AX, respectively. The sharp simulated O_3AX extinction resonance has a very high Q-factor of 23, compared to only 3.7 for the D_3AX and 15 for the Q_3AX mode.

 

Fig. 4 LSPR modes and spectral tunability in a three-arm nanocross. (a) Calculated and (b) experimental extinction spectra of a 3AX. Red lines: normal incident illumination, blue lines: side illumination. Polarization in vertical direction. L_{c 1} = L_{c 2} = 370 nm, W_c = 70 nm, T = 50 nm and α = 45°. Insets: calculated charge density distributions corresponding to the indicated resonances. (c) Simulated normal incidence extinction spectra of the 3AX in (a) for varying α between 20° and 40°. The displayed spectral window corresponds to the dashed box in (a) and shows the O_3AX dipole activation for reduced rotational symmetry and spectral shift for varying α. (d–e) Extinction (red), scattering (black) and absorption (green) cross-sections showing Fano interference in the 3AX. (d) A zoom-in of the 3AX side illumination spectrum indicated by the dashed box in (a). (e) Same as (d) but for normal incidence.

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Interestingly, the higher energy octupolar mode is also excited with non-phase retarded excitation, i.e. normal incidence (red spectra). In fact, the rotational symmetry of the 3AX with α = 45° shown in panels (a) and (b) is C ₂ which is lower than the C ₆ symmetry required for a pure (l,m) = (3,3) octupole with zero dipole moment. This reduced structural symmetry allows the l = 3 mode to mix with l = 1 dipole components and consequently dipole activate the octupolar resonance. Although not a pure l = 3 resonance, the major contribution is octupolar nevertheless and for simplicity we keep the reference O_3AX octupole mode.

A simple static calculation of the net dipole moment p = Σ_jq_jd_j, assuming equal point charges in each cross branch as we did for the 2AX, results in p proportional to 2sinα − 1. This means that for a 3AX with C ₆ rotational symmetry, i.e. α = 30°, p vanishes and the O_3AX mode should become dark. This is indeed the case as shown in Fig. 4(c). This figure shows extinction spectra of three-arm crosses with different arm angles between 20° and 40° for normal incidence. The spectral window corresponds to the dashed box in panel (a). It is seen that the octupolar resonance reduces its intensity as α approaches 30° and finally becomes completely dark when α = 30°. The l = 2 quadrupole mode remains dark regardless of α since the centrosymmetry of the particle is maintained [23, 40]. Experimentally, all three 3AX modes are observed for side illumination of a 3AX with α = 45°. For normal incidence the quadrupole mode disappears while the octupole mode remains, although with a reduced extinction cross-section, in agreement with the simulated spectra. Note that also for normal incidence a small bump is seen at the quadrupole mode. This can be explained by the k-vector spread of the incident light introduced by the Cassegrain objective. Fig. 4(c) also illustrates the spectral tunablity of the octupolar mode with α. A maximum resonant wavelength is obtained for α = 30° while the octupole mode further blue-shifts for increasing or decreasing arm angle, similar to the behavior of the Q_2AX mode in Fig. 2(f).

Upon close inspection of the 3AX extinction spectrum in Fig. 4(a), the O_3AX resonance line shape is clearly asymmetric. A zoom-in on the octupole and quadrupole modes is shown in panels (d) for side illumination and (e) for normal incidence. When comparing absorption (green lines) and scattering (black lines) cross-sections, it is seen that the octupole absorption resonance (green dot) lies within the lower energy tail of the scattering resonance. This is a key indication of Fano interference [26, 41, 42]. The interference, here, occurs between the spectrally overlapping broad superradiant tail of the D_3AX mode and the narrow subradiant O_3AX mode. Their coherent coupling results in a destructive interference in the scattering on the lower energy side of the octupole absorption peak and in a constructive interference on the higher energy side. For the l = 2 quadrupole (cyan dot) no Fano interference is observed since the centrosymmetry of the particle does not allow coupling to either of the l = 1 and l = 3 modes.

The influence of the finite rotational symmetry of the 3AX on the excitation efficiency of the different plasmon modes is shown in Fig. 5. Panel (a) shows that by increasing the structural symmetry from C ₄, for the α = 45° 2AX, to C ₆, for the α = 30° 3AX, the dipole and quadrupole mode extinction cross-sections become more isotropic for varying angle of side incidence θ. While the dipole mode is fully isotropic, the Q_3AX mode has local extinction maxima for θ = 30° and 90°. Although this cross arrangement supports an octupolar resonance, it only appears as a weak shoulder to the quadrupole mode when θ is exactly 0° or 60°. This results in six lobes in the polar plot, indicating again a strong relationship between plasmonic properties and the particle’s geometrical symmetry.

 

Fig. 5 Rotational anisotropy of mode excitation in three-arm nanocrosses. (a) An α = 30° 3AX with C ₆ rotational symmetry. (b) An α = 45° 2AX with C ₂ rotational symmetry. (c) An α = 60° 2AX with C ₂ rotational symmetry. (a–c) top left graph: experimental data, bottom graph: corresponding simulated spectra without substrate, top right: polar plot of the different modes’ peak intensity. Black spectra correspond to θ = 0°, gray spectra to θ = 90°, and cyan spectra to the indicated θ. Plasmon modes indicated with the dashed lines are the dipolar (red and orange), quadrupolar (blue and purple) and octupolar modes (green). Insets: calculated charge density distributions corresponding to the indicated resonances. Scale bar in SEM image: 100 nm.

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For the 3AX with α = 45° in panel (b) the rotational symmetry is reduced to C ₂ and introduces an additional dipolar and quadrupolar resonance. The dipole D_3AX and quadrupole Q_3AX modes already introduced in Fig. 4 are indicated with the red and blue dashed lines, respectively. Their θ dependence is similar to that of the α = 60° 2AX in Fig. 3(c). A second dipole mode is observed at 1080 nm (orange dashed line) once θ ≠ 0°. The corresponding charge distribution is shown in the inset (orange dot). One diagonal arm together with the vertical cross arm forms an α = 22.5° 2AX, indicated with the cyan and yellow dashed contour in the charge plot (purple dot). These two arms can support another quadrupole mode. From Fig. 2(f) we know that the smaller arm angle will result in a blue-shifted resonance compared to the quadrupole in the two diagonal arms that form an α = 45° 2AX (blue dashed line). The extinction spectrum for θ = 45° in Fig. 5(b) (cyan line) indeed reveals an additional resonance at 850 nm (purple dashed line). From the charge density plot in the inset (gray dot), we see that this mode is in fact a superposition of two quadrupole modes. One mode in each α = 22.5° 2AX built up from a diagonal arm and the common vertical bar. The reduced structural symmetry further has an important effect on the excitation of the octupole mode (green dashed line). This mode now only appears for incident angles close to 0°, giving rise to two lobes in the polar plot. Increasing the arm angle from α = 45° to 60° leads to comparable results. However, it is seen that bringing the arms closer together further splits the two dipole modes. The longer wavelength dipole mode (red dashed lines) slightly red-shifts, while the lower wavelength mode (orange dashed lines) strongly blue-shifts leading to a spectral overlap with the lower energy quadrupole mode (blue dashed line). This spectral overlap was also observed for a 2AX with small α in Fig. 2(f). The octupole and higher energy quadrupole mode both experience a blue shift.

5. Broken inversion symmetry in a nanocross: the scissor cross

As already mentioned above, the centrosymmetry of the 2AX and 3AX in Fig. 2 and Fig. 4 do not allow the l = 2 quadrupole modes to mix with l = 1 components. Therefore, they are dark for non-phase retarded excitation (i.e. normal incidence). The dashed red line in Fig. 6(a) shows again the normal incidence extinction of the centrosymmetric 2AX for which only the dipole resonance can be excited. When, however, the two arms are translated 20 nm up and down, respectively, the structural centrosymmetry is broken, rendering the quadrupole modes dipole active. Consequently, an additional resonance appears in the extinction cross-section (red full line). The lower energy mode at 1220 nm is the dipole resonance as can be seen in the charge distribution (red dot). In this mode the main dipole moment and absorption contribution comes from the longer, right half of the cross. At the higher energy resonance, indicated by the blue dot, the clear quadrupolar charge distribution resembles the Q_2AX mode in the symmetric cross. Here, the dipole moment and absorption is strongest in the left, smaller cross half. It is this mismatch in spectral resonance of both sides of the cross that provides the necessary phase difference to generate the out-of-phase quadrupolar charge oscillation under non-phase retarded illumination. The asymmetric cross will further be referred to as the scissor cross.

 

Fig. 6 Symmetry breaking in a two-arm cross: scissor cross. (a) Calculated extinction (red), scattering (black) and absorption (green) spectra of a 2AX with α = 45°, L_c = 370 nm, W_c = 70 nm and center offset of 40 nm under vertically polarized normal incidence showing the excitation of a quadrupole (blue dot) and dipole (red dot) resonance and resulting Fano line shape. Dashed line: symmetric 2AX (offset = 0 nm). Insets show corresponding charge density maps. (b) Evolution of the resonances as the offset (and hence asymmetry) of the scissor cross is increased. The calculated extinction spectra for each offset value, normalized to maximum extinction, are displayed in the gray scale intensity map in (b).

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As for the 3AX, in this configuration no nanometer sized gaps are involved to excite a Fano resonance. In the 3AX of Fig. 4(d) the quadrupole mode (cyan dot) has a clear symmetric scattering resonance that spectrally overlaps with the absorption resonance. For the scissor cross in Fig. 6(a) this is clearly not the case. Instead, an asymmetric Fano interference, similar to the O_3AX Fano line shape, is observed. The Fano resonance now results from the spectral overlap of the narrow quadrupole with the broader dipole mode.

Panel (b) shows the calculated spectral evolution of the scissor cross resonances as the offset between the arms is varied from 0 nm (symmetric case) to 120 nm. Since the dipole mode (red dots) is dominated by the longer cross half, which becomes longer with increasing offset, it red-shifts accordingly. At the same time a shorter left part induces a linear blue shift for the antiparallel mode (blue triangles).

Also in the 3AX geometry, the quadrupole mode is expected to become dipole active for normal incidence once the centrosymmetry is broken. These additional spectrally sharp Fano features for normal incident illumination make the asymmetric variants of the cross geometries very appealing for sensing applications. Note that reducing the cross symmetry from C ₄ to C ₂ by choosing unequal arm lengths is not sufficient to make the l = 2 mode dipole active.

6. Mode coupling in nanocross heterodimers

When two plasmonic resonators are brought into each others near-field, their plasmon modes will couple. In this section we first investigate the spectral response of a 3AX-I nanocross heterodimer consisting of a three-arm cross in close proximity to a nanobar I. As was the case for the asymmetric scissor cross, the reduced structural symmetry enables higher order modes in the cross to take part in the light-matter interaction. In the second part and in the next section, several other nanocross heterodimer and -trimer configurations with enhanced functionalities are presented.

The relevant geometrical parameters for the 3AX-I configuration are indicated in Fig. 1(c). We limit our discussion to a 3AX with α = 45°. The left panels in Fig. 7 show a hybridization diagram illustrating the nature of the plasmon resonances in a 3AX-I cavity structure with dimensions much smaller than the incident wavelength. In this electrostatic limit the extinction spectrum is scale invariant and size dependent phase retardation is absent [19,43]. In the FDTD simulations no substrate was introduced and in order to further facilitate resolving the involved modes the permittivity of Au was modeled with a Drude fit to the experimental data: $\varepsilon (f)={\varepsilon }_{\infty }-{\omega }_P^2/[2\pi \cdot f(i{\nu }_c+2\pi \cdot f)]$, with ε _∞ = 10.094, ω_P = 1.36422 × 10¹⁶ rad/s and ν_c = 1.09289 × 10¹⁴ rad/s. Figure 7(a) shows the scattering (blue), absorption (red) and extinction (black) cross-sections of the cross under grazing incidence. For these small particles in the electrostatic limit the extinction is almost completely determined by absorption. Since we are in the quasistatic regime there is no phase retardation along the particle and the cross quadrupole mode is not resolved in the extinction spectrum. When calculating the quadrupole moment (green dashed line), however, the mode and its spectral position can still be observed [44]. Cross-sections for an isolated bar are presented in panel (c).

 

Fig. 7 Hybridization diagrams for a 3AX-I heterodimer. Calculated extinction (black), scattering (blue) and absorption (red) cross-sections. Left panels: (b) 3AX-I and corresponding individual cross (a) and bar (c) in the electrostatic limit (L_{c 1 , c 2} = 30 nm, W_c = 7 nm, α = 45°, L _b = 17 nm, W _b = 9 nm, T = 5 nm and G = 2 nm). Green dashed line: cross quadrupole moment. Black dashed lines: relevant mode interactions. Mode color code corresponds to Fig. 4. (e) Scaling of the 3AX-I from 0.1 (same as panel (b)) to 1 times the experimental dimensions (same as panel (g)). Left axis: absorption, right axis: scattering. Dashed lines indicate the spectral shifts of the hybridized modes: HD_I (black), BOD (green), BQD (cyan) and BDD (orange). Right panels: the fabricated unscaled 3AX-I structure (L_{c 1 ,c 2} = 300 nm, W_c = 70 nm, α = 45°, L_b = 170 nm, W_b = 90 nm, T = 50 nm and G = 20 nm). (d, i): phase difference between the cross and bar dipole moment Δϕ(p_bar-p_cross) in the 3AX-I. Au was modeled with a Drude fit and no substrate was used. Spectra calculated for vertically polarized normal incidence, except for (a) and (f).

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In panel (b), coupling of the cross and bar resonances in the 3AX-I structure gives rise to four distinct hybridized modes. The black dashed lines indicate the main interactions. The nature of the modes can be derived from their respective charge density distributions shown in the top left insets. The first lower energy mode is an out-of-phase oscillation of the cross D_3AX and bar D_I dipole modes (orange dot: bonding dipole-dipole, BDD) [45]. To further support the assignment of an out-of-phase hybridized BDD mode, panel (d) shows the calculated phase difference between the dipole moments in the cross and the bar Δϕ(p_bar-p_cross). At the BDD mode Δϕ reaches −0.92π rad. The reduced structural symmetry in the heterodimer and strong near-field coupling due to the small interparticle separation enables the bar D_I mode to couple easily to higher order modes in the cross rendering these modes dipole active. This results in three other hybridized modes: a bonding combination of the Q_3AX and D_I mode (bonding quadrupole-dipole - BQD) indicated by the cyan dot, a bonding combination of the O_3AX and D_I mode (bonding octupole-dipole - BOD) indicated by the green dot, and the higher energy mode which is dominated by the bar D_I mode weakly coupled to the dipole and higher order modes in the 3AX which are irrelevant for the further discussion and we’ll refer to this hybridized mode simply as HD_I (black dot). Note that for vertical polarization the plasmon coupling is weaker than for horizontal polarization. Therefore, the hybridized HD_I, BOD, BQD and BDD modes are spectrally not much shifted compared to the constituting cross and bar resonances. Since the BQD mode has a substantial overlap with the BOD, the latter is mixed in the charge density distribution of the BQD mode. Therefore, the charge density in the 3AX at the BQD mode (cyan dot) deviates slightly from the antisymmetric distribution in Fig. 4 (cyan dot). Nevertheless, an overall quadrupolar charge distribution is seen in the cross.

By gradually increasing the structural size, a clear evolution of the individual hybridized modes is obtained and the appearance of an asymmetric Fano antiresonance is revealed [41–43]. Figure 7(e) shows the evolution of the 3AX-I scattering and absorption spectra as all dimensions are scaled up from 10% (i.e. panel (b)) to 100% (i.e. panel (g)) of the experimental size. For clarity, absorption and scattering spectra have different axes. Increasing the structural dimensions results in a red shift and broadening of all modes due to dynamic depolarization (phase retardation) effects and radiative losses [46]. However, the HD_I mode experiences a faster shift and broadening than the BOD and BQD modes. As a consequence, the HD_I mode increasingly spectrally overlaps with the BOD, as is seen in the absorption spectra in panel (e). For a scale factor of 0.5, both modes are clearly separated, while for 0.8 times the experimental size, the HD_I is only seen as a shoulder. Finally, at scale factor 1.0, both modes are indistinguishable. Since we now have a spectrally overlapping dipole active broad (HD_I) and dipole active narrow (BOD) resonance, their interaction is allowed and the requirements for Fano interference are satisfied [26]. The result is a destructive interference with a typical antiresonance spectral dip seen in the scattering spectra for scale factors 0.8 and 1.0 [19, 21, 41–43].

The hybridization diagram of the full size (scale factor 1.0) 3AX-I heterodimer can be found in panels (f–h). Again, panel (i) shows the phase difference Δϕ(p_bar-p_cross). At the BDD mode, Δϕ reaches −0.93π rad, demonstrating the out-of-phase oscillation of the dipole moments at this mode. This results in a reduced net dipole moment, making this mode subradiant with an obvious line width narrowing when compared to the D_3AX mode in panel (f). The longer wavelength tail of the HD_I mode extends weakly under the subradiant BDD absorption peak, inducing also here Fano interference with a slightly asymmetric extinction line shape as a consequence.

The corresponding experimental spectrum and SEM image of the 3AX-I heterodimer in Fig. 7(g) is shown in Fig. 8(a) (red line). The dashed arrows point out, from short to long wavelengths, the BOD Fano antiresonance, BQD and BDD resonances. Arrays with different interparticle distance G were fabricated in order to investigate the influence of the near-field coupling strength. As the bar and cross are brought closer together from G = 50 nm (green line) to 40 nm (blue line) and 20 nm (red line), first of all a stronger plasmon hybridization red-shifts the BQD and BDD modes and reduces the line width of the BDD. Secondly, the stronger near-field interaction at smaller gap sizes leads to an increased cross-section for the BQD mode and more pronounced BOD Fano dip due to more efficient excitation of the Q_3AX and O_3AX modes [29]. The agreement with the corresponding simulated spectra in panel (b) is striking. The BOD Fano resonance is somewhat deviating. This is most likely caused by fabrication imperfections in the 3AX geometry which become more important at higher order resonances.

 

Fig. 8 Influence of interparticle distance G in a 3AX-I heterodimer cavity. (a) Measured and (b) calculated extinction spectra without substrate. Calculated gap sizes: G = 20 nm (red), G = 40 nm (blue), G = 50 nm (green). Spectra are offset on the y-axis for clarity. The thin black lines in (a) and (b) are four oscillator Fano model fits to the red spectra (G = 20 nm). Arrows indicate from short to long wavelength the BOD Fano antiresonance, BQD and BDD modes. Insets: SEM images of the corresponding fabricated cavities.

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Fig. 9 Nanocross based heterodimers and -trimers. (a, e) 2AX-I dimer. (b, f) Xx dimer. (c, g) 2AX-I trimer. (d, h) 2AX-I-2AX trimer. (a–d) Normal incidence calculated spectra taking the substrate and substrate etching into account. (e–h) Experimental data with an SEM image of the corresponding geometry (scale bar: 100 nm). Red lines: vertical polarization, blue lines: horizontal polarization, black lines: unpolarized. Top panels show charge density distributions of the indicated resonances.

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It has been shown that the amplitude of transmission or reflection from plasmonic lattices as well as the absolute value of the polarizability of disk, ring and disk-in-ring particles can be well described by a Fano-type line shape [47–49]. We recently demonstrated that also far field extinction spectra E(ω) = 1-T(ω) = |e(ω)|² of a 2AX-I nanocross heterodimer can be expressed with an analytical Fano interference model [19]:

The 2AX-I configuration consisting of a 2AX and a bar is a simplified variation of the 3AX-I heterodimer and proved to be very valuable for high sensitivity LSPR RI sensing due to its sharp spectral features. The multiple sharp resonances even enable multiwavelength spectroscopic sensing. In Fig. 9(a) the spectral response of a 2AX-I with α = 45° is shown. The main features here are again a subradiant bonding dipole-dipole mode (red dot), and a pronounced Fano antiresonance dip resulting from destructive interference between a broad superradiant background (parallel alignment of cross and bar dipole moments) and a narrow coupled Q_2AX -D_I mode (blue dot) [19]. Note that, due to spectral overlap with the superradiant tail, also here Fano interference occurs at the subradiant BDD mode resulting in a slightly asymmetric line shape.

As a next example of a nanocross based heterodimer we discuss an Xx side-by-side arrangement of two 2AX particles with α = 60° and different arm length L_c = 300 nm and 360 nm. The results are shown in Fig. 9(b) and 9(f). For vertical polarization, we have essentially the same situation as for a 2AX-I dimer. The dipole modes of the crosses hybridize into a lower energy subradiant bonding dipole-dipole mode with slight asymmetric Fano line shape (red dot) and a higher energy antibonding dipole-dipole mode which gives the broad superradiant envelop centered around 1200 nm. The quadrupole mode of each cross overlaps with this superradiant envelop and result in a destructive Fano dip. Since L_c is different, the quadrupole modes are spectrally well separated and two distinct Fano dips arise: one for the small cross quadrupole (blue dot) and one for the bigger cross quadrupole (green dot). For horizontal (transverse) polarization (blue lines), in the 2AX-I configuration, only a broad dipole response was observed. For the Xx heterodimer, however, something interesting happens. Since also for horizontal polarization both crosses have a strong dipole resonance (see e.g. Fig. 2(f)), an hybridization into a lower energy bonding (in-phase oscillation of dipole moments) and higher energy antibonding (out-of-phase oscillation of dipole moments) mode takes place. The reduced symmetry, introduced by the unequal arm lengths, will render the antibonding mode dipole active [45, 50]. This mode is now a narrow subradiant mode, due to its small net dipole moment, and overlaps spectrally with the lower energy bonding mode resulting in a Fano interference around 910 nm, as is confirmed by absorption and scattering data and charge density plots (not shown).

7. Mode coupling in nanocross heterotrimers

For applications, such as LSPR RI sensing, signal throughput is often an important factor in determining the signal-to-noise ratio. Working with polarized light typically reduces the available light intensity. Polarization anisotropy is to be considered as well when studying the emission from local dipole emitters like fluorescent dyes or quantum dots coupled to a plasmonic cavity [51]. Figure 9(c) shows the extinction from a trimer consisting of the 2AX-I dimer in panel (a) and an extra horizontal bar. The spectra of the dimer and trimer present the same spectral features for vertical polarization (red lines). For horizontal polarization (blue line), however, the dimer only has a broad dipole response while the structural symmetry of the trimer dictates an identical response for both polarizations. Corresponding measurements are shown in panels (e) and (g). The black lines are spectra recorded with unpolarized illumination. In the trimer configuration both the subradiant mode (red dot) and Fano resonance (blue dot) retain its spectral shape when going from vertically polarized to unpolarized excitation. In the dimer on the other hand, the subradiant mode is reduced to a small bump, hence making it useless for RI sensing. It has to be noted, however, that the line width of the Fano resonance in the trimer is considerably increased compared to the dimer with vertically polarized excitation.

A line width improvement of the Fano and subradiant resonances can be obtained with a 2AX-I-2AX trimer shown in the SEM image of panel (h) for α = 60°. Here, a sharp subradiant mode results from the out-of-phase oscillation of the bar dipole moment and the cross dipole moments as shown in the charge density distribution in the insets (red dot). Note that the line shape acquires an even more asymmetric Fano character. At the shorter wavelength Fano antiresonance (blue dot) the charge plots clearly reveal the simultaneous excitation of the Q_2AX mode in both crosses, which enhances the Fano dip. From a three oscillator Fano model fit a Q-factor of 10.8 and 9.7 are found for the simulated trimer Fano antiresonance and subradiant mode. For the dimer with only one cross the corresponding values are lower: 8.5 and 6.1, respectively. The improved line widths can be explained by a more symmetric suppression of the net dipole moment in the C ₂ symmetric trimer configuration. Experimentally, a line width of 0.086 eV and 0.079 eV is found for the Fano antiresonance and subradiant mode, respectively. A similar dimer with only one cross resulted in line widths of ∼0.15 eV and ∼0.13 eV, resp. [19]. The RI spectral shift sensitivity for the specific fabricated samples is known to be approximately 0.605 eV/RIU at 1.10 eV and 0.41 eV/RIU at 0.73 eV. Taking into account the line width using the LSPR sensitivity Figure of Merit, FoM = (δλ/δn)/FWHM [(eV/RIU)/(eV)], an estimated experimental FoM of 7.0 and 5.2, resp., results, compared to ∼4 in the corresponding dimer [19]. Hence, the reduced line width in the 2AX-I-2AX trimer immediately results in a substantial increase of the LSPR RI sensitivity.

It would be interesting to investigate these dimer and trimer modes from a group theory point of view [41, 52]. This goes, however, beyond the scope of this paper.

8. Conclusion

LSPR modes in several nanocross geometries were studied experimentally and by means of FDTD calculations. The ability to posses discrete degrees of rotational symmetry situated between rotational invariance (C _∞), like a disk, and no rotational symmetry at all (C ₁), like a non-equilateral triangle, makes the nanocross a fundamentally interesting plasmonic structure. The branched concave cross shape was shown to support high Q-factor quadrupolar and octupolar resonances for which the spectral position and separation can be tuned by changing the arm length and arm angle. Accessing the multipoles with non-phase retarded illumination, which is of interest for several applications such as LPSR RI sensing, is possible when reducing the rotational symmetry of the particles. Not only Lorentzian shaped but also asymmetric Fano resonances were identified in individual crosses, indicating that Fano effects in plasmonic nanostructures are rather common and no exceptions [26, 42]. We demonstrated that the nanocross is a versatile building block for coherently coupled plasmonic nanocavities. Hybridized bonding and antibonding combinations of the primitive dipole and multipole resonances as well as multiple Fano interferences were observed in several nanocross based heterodimers and heterotrimers. This complex mode coupling gives one a tool to control radiative losses and do plasmonic line shaping. Our findings are of considerable importance in the understanding and design of more complex branched nanoparticle systems where the nanocross can serve as a model system.