1. Introduction

Noble metal nanostructures have been explored in recent years to confine and enhance light over subwavelength volumes due to plasmonic resonances. The field enhancement takes place in gaps between nanostructures, near sharp corners, in film-coupled nanoparticles, and inside the metal for nanoantenna sizes of the same order as the skin depth. In bulk media, phase matching is required to maximize the nonlinear emission, whereas nanoscale field enhancement in the proximity of plasmonic nanoantennas can boost nonlinear optical processes with no phase matching requirement [1]. This has been exploited to enhance second harmonic generation (SHG) [2], third harmonic generation (THG) [3,4], four-wave mixing (FWM) [5], and difference frequency generation (DFG) [6].

Hybrid dielectric/plasmonic nanostructures, such as a metallic dipole nanoantenna with a nonlinear dielectric nanoparticle in the gap, have been proposed to enhance nonlinear emission. Though the hybrid dipole nanoantenna is well understood in the linear regime, in the nonlinear regime the origin of nonlinear emission is still under debate [7–10]. Since both metals and dielectrics exhibit nonlinearity, the nonlinear far-field contains contributions from each, and the impossibility of measuring these contributions separately has led to seemingly controversial interpretations about the origin of the nonlinear signal. For example, in [8] a dipole nanoantenna with an ITO disk in the gap was reported, and the enhanced THG emission was attributed to ITO due to the linear field enhancement in the gap. In [9] a similar experiment was carried out and a doubled THG intensity was observed. The conclusion there was that the THG signal comes primarily from gold, and the enhanced nonlinear radiation was attributed to the change of the linear properties of the nanoantenna due to the presence of the dielectric in the gap.

These results inspired us to use a full numerical approach to explore the design space of the hybrid dipole nanoantenna and illustrate regimes where THG is dominated by the gap dielectric, by the metal or by both. A detailed analysis is required to quantify how each part of the system contributes to the third harmonic emission. We consider THG because it requires less complex structures, e.g., a monopole nanoantenna, which is not suitable for SHG [11,12]. Furthermore, the third order nonlinear susceptibility can be derived from the linear properties of the medium based on Miller’s rule, whereas the application of this rule is not straightforward for SHG in nanostructures and metamaterials [13]. The design space of a metal monopole nanoantenna is also explored to show when this simpler structure should be preferred for THG.

2. Simulations and discussion

We simulate gold nanostructures on a semi-infinite SiO₂ substrate (Fig. 1(a)) by using the finite-difference time-domain (FDTD) method in 3D [14]. The thickness of all nanostructures was t = 50 nm, the width was chosen among the set w = {10, 20, 30} nm, and the gap of the dipole nanoantennas among the set g = {30, 50, 70} nm filled by a dielectric nanoparticle of length g and width w. We simulate nine dipole nanoantennas (each combination of w and g) and three monopole nanoantennas (each w value), one at a time, without periodic boundary conditions. The nanoantennas are oriented along the z-direction on an xz-plane, and the simulation domain is discretized with 1 nm resolution. A z-polarized and y-propagating plane wave excitation at ω₀ (λ₀ = 900 nm), implemented as a sinusoidal signal modulated by a Gaussian pulse, is introduced from air. Plane wave excitation of a single nanoantenna is a good approximation to experiments performed with loosely focused beams. The lengths of the dipole nanoantennas (L_d) and the monopole nanoantennas (L_m) were selected such that each nanoantenna is resonant at λ₀, resulting in 255 ≤ L_d ≤ 324 nm and 174 ≤ L_m ≤ 198 nm. This was determined by linear extrapolation because the resonance wavelength shifts linearly with the nanoantenna length [15]. To monitor the nonlinear far-field, we use a large simulation domain of (800 nm)³. The analytical Kirchhoff near-to-far field transformation cannot be applied because the nanoantenna is not embedded in a homogeneous medium. The corners and edges of the metal nanostructures are rounded by r = 10 nm curvature radius to reduce numerical artifacts and to neglect nonlocal effects [16]. Linear dispersion is modelled by the Drude+2CP, Lorentz, and Drude models for gold (Johnson and Christy data) [17], SiO₂ [18], and ITO [19], respectively. We use an isotropic third order nonlinearity (instantaneous Kerr effect) as in [20]. The simulation of the linear and nonlinear responses run at the same time, naturally taking into account pump depletion. We use ${\chi }_{\mathit{Au}}^{(3)}=7.6\cdot {10}^{-19}\hspace{0.17em}{m}^2/V^2$ [21] and neglect nonlinearity in the substrate. The gap dielectric is a fictitious material with the linear properties of ITO and variable ${\chi }_g^{(3)}$ to explore the design space. We seek the value of ${\chi }_g^{(3)}$ for which the emission from the gap material and gold is the same. The excitation at λ₀ = 900 nm allows us to exploit plasmonic field confinement in gold and to use a real valued ${\chi }_{\mathit{Au}}^{(3)}$ (complex values are reported in [21] for λ < 630 nm). At 3ω₀, gold behaves as a lossy dielectric, such that the radiation from the nonlinear source is unaffected by plasmonic resonances [20]. Furthermore, working at λ₀ allows us to keep the study general by avoiding the ITO absorption resonance (∼ 250 nm) and its epsilon-near-zero (ENZ) region (∼1400 nm) where the transition to the metallic regime occurs. This regime has been recently exploited to enhance the nonlinear emission in hybrid nanostructures due to two nested plasmonic resonances [22].

 

Fig. 1 (a) Sketch of dipole and monopole nanoantennas. (b) η_THG for a dipole nanoantenna (w = 30 nm, g = 10 nm). (c) η_THG for the nine dipole nanoantennas (•, ▪) and the three monopole nanoantennas (dashed lines): w = 30 nm (red), w = 50 nm (green), and w = 70 nm (blue), g = 10 nm (dark shade), g = 20 nm (medium shade), g = 30 nm (light shade).

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To isolate the contributions to the overall THG emission, we consider the dipole nanoantennas in the following cases: (I) emission only from the gap material ( ${\chi }_g^{(3)}\ne 0$, ${\chi }_{\mathit{Au}}^{(3)}=0$), (II) emission only from gold ( ${\chi }_g^{(3)}=0$, ${\chi }_{\mathit{Au}}^{(3)}\ne 0$), (III) emission from the gap material and gold ( ${\chi }_g^{(3)}\ne 0$, ${\chi }_{\mathit{Au}}^{(3)}\ne 0$). We also consider (IV) emission from gold in monopole nanoantennas ( ${\chi }_{\mathit{Au}}^{(3)}\ne 0$). We quantify the nonlinear emission by calculating the THG scattered power at 3ω₀ as the integral of the normal component of the Poynting vector through a closed surface in the far-field. We report the relative THG efficiency η_THG by normalizing with respect to the nanostructure with the largest THG emission, which is the monopole nanoantenna with w = 30 nm.

In Figs. 1(b) and 1(c), we demonstrate that there are three regimes in hybrid plasmonic/dielectric nanoantennas for THG emission: a gold dominated regime, a gap dominated regime, and a regime where the emission from gold and gap are comparable. In Fig. 1(b), we plot η_THG vs ${\chi }_g^{(3)}/{\chi }_{\mathit{Au}}^{(3)}$ for cases (I)–(III) for the dipole nanoantenna (w = 30 nm, g = 10 nm) with the highest η_THG. The dashed line is for emission from the gap material only – case study (I) – and shows a quadratic behavior with ${\chi }_g^{(3)}/{\chi }_{\mathit{Au}}^{(3)}$. The solid line is for emission from gold only – case study (II) – which is horizontal since we are considering a single value for ${\chi }_{\mathit{Au}}^{(3)}$. The cross point ( ${\chi }_g^{(3)}/{\chi }_{\mathit{Au}}^{(3)}=x_c$) between dashed and solid lines – red circle ( ) – identifies the value such that the nonlinear emission from gold equals that from the gap material. The white squares are for emission from gold and gap – case study (III) – and they show an asymptotic behavior, approaching the “gold only” regime on the left and the “gap only” regime on the right. The red square ( ) identifies the case study (III) for ${\chi }_g^{(3)}=x_c\cdot {\chi }_{\mathit{Au}}^{(3)}$. For ${\chi }_g^{(3)}/{\chi }_{\mathit{Au}}^{(3)}>>x_c$ emission from the gap material dominates, while for ${\chi }_g^{(3)}/{\chi }_{\mathit{Au}}^{(3)}<<x_c$ the emission from gold dominates.

The red circle and square in Fig. 1(b) are sufficient to identify the boundary between gold and gap dominated regimes, as well as η_THG at the cross point x_c. Thus, only these two points are shown in Fig. 1(c) for each of the nine dipole nanoantennas. For comparison, dashed lines are used to indicate η_THG of the three monopoles – case study (IV). For the dipole nanoantenna, by changing w and g we observe the variation of four parameters: η_THG, x_c, the horizontal spacing between circles (d_h), and the vertical spacing between circle and square (d_v). Increasing w produces a strong decrease in η_THG, a decrease in x_c and a decrease in d_h. Increasing g produces a slight decrease in η_THG, an increase in x_c, and a decrease in d_v. A high η_THG means that the baseline emission from gold is high. A high x_c means that the emission from gold is more prone to dominate over the emission from the gap material. The lower d_h for the largest width indicates that a larger gap volume compensates the lower field enhancement. The smaller d_v for the largest gap size means that the gap material’s contribution to η_THG is small. The dipole nanoantennas have a lower nonlinear emission from gold than a monopole nanoantenna with the same w, since a continuous structure supports a less damped linear harmonic oscillation, which is a cause of strong nonlinear emission [23]. Our simulations show that nonlinear emission from gold depends strongly on w, preferring threadlike rather than bulky shapes, as reported in [24]. Considering ITO ( ${\chi }_g^{(3)}=2.16\cdot {10}^{-18}{m}^2/V^2$ [25]) for the dipole nanoantenna in Fig. 1(b), we obtain η_THG ∼ 4 ·10³ by quadratic extrapolation, falling within the gap dominated regime. Nonlocal effects due to surface roughness can increase the effective ${\chi }_{\mathit{Au}}^{(3)}$ and make gold more prone to dominate over the gap material [26].

 

Fig. 2 Absolute value (top) and phase (bottom) of E_z(3ω₀): emission from (a) gap only ( Visualization 1), (b) gold only ( Visualization 2), (c) gap material and gold ( Visualization 3) in a dipole nanoantenna; (d) gold in a monopole nanoantenna ( Visualization 4).

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In the linear regime, the spectral peak of the field enhancement in the gap is slightly red-shifted with respect to the peak of the extinction spectrum [27]. Since the nonlinear emission by the gap material is driven by the linear field enhancement in the gap, and the nonlinear emission from gold is linked to the linear extinction, our simulations show that the nonlinear spectra for the emission from the gap and for the emission from gold exhibit a shift in wavelength, as confirmed in [28]. We use the linear extinction spectrum to determine the nanoantenna lengths required for resonance at λ₀. This corresponds to maximizing the nonlinear emission from gold. Thus, the nonlinear emission from the gap is not optimal. This shift for our dipole nanoantennas in the linear regime ranges from ∼ 10 nm (w = 30 nm, g = 10 nm) to ∼ 35 nm (w = 70 nm, g = 30 nm).

In Fig. 2, we plot the near-fields for the dipole nanoantenna and the monopole nanoantenna with the highest η_THG. We show the absolute value and phase of E_z(3ω₀) in the xz plane for cases (I)–(IV) in Figs. 2(a)–2(d), respectively, assuming ${\chi }_g^{(3)}=x_c\cdot {\chi }_{\mathit{Au}}^{(3)}$. The electric field hot spots show nearly uniform phase and thus can be considered as idealized dipole sources. Visualization 1– Visualization 4 show the evolution of the third harmonic electric field in the yz plane for cases (I)–(IV), respectively. In the case of THG emission from the gap, as expected, the dipole source is localized in the gap (Fig. 2(a)). More interesting is the emission from gold in the dipole nanoantenna, where we observe one dipole source in each branch of the dipole nanoantenna, and an enhancement of the nonlinear field in the gap (Fig. 2(b)). The electric field distribution when both the gap and gold are emitting (Fig. 2(c)) shows constructive interference in the gap and at the extremes of the dipole nanoantenna, and destructive interference along the dipole nanoantenna branches. The monopole shows a dipole source in the middle (Fig. 2(d)), as also reported in [29], which can be used as an ideal source in metasurfaces for nonlinear beam shaping [20].

In Fig. 3 we show the far-field radiation patterns at 3ω₀ corresponding to the nanoantennas in Fig. 2. We consider xy and yz cut planes, where the angle is measured starting from the y-axis. The radiation patterns were calculated by considering a polar coordinate system with the scattering center in the middle of the nanoantenna, and evaluating the normal projection of the Poynting vector through a circle in the far-field. The radiation patterns are normalized with respect to the monopole with highest η_THG. The radiation can be seen as produced by a collinear array of phased emitters aligned along z with subwavelength spacing. This produces radiation patterns in the xy plane with the same shape in the four cases in Fig. 3. In all cases, the radiation is observed to be primarily directed through the substrate because it has a larger refractive index than the air above the antennas. The different electric field distributions shown in Fig. 2 play a role only for the radiation patterns in the yz plane. The emission from the gap material only produces a broad radiation pattern (Fig. 3(a)), which can be understood as diffraction by a small aperture; in fact, the broadening decreases as the gap size increases (not shown). In the case of emission from gold only in the dipole nanoantenna (Fig. 3(b)), the two nonlinear dipole sources produce a directive lobe. The simultaneous emission from gold and the gap material produces a hybrid radiation pattern (Fig. 3(c)). When the THG emission from gold and the gap material are comparable (x_c region), complex ${\chi }_{\mathit{Au}}^{(3)}$ and/or ${\chi }_g^{(3)}$ values could lower the η_THG value due to destructive interference in the far-field. The radiation pattern produced by the monopole nanoantenna (Fig. 3(d)) has the same shape as the one in Fig. 3(b), but is less directive. This is because the central emitter dominates over the sources localized at the extremes of the monopole nanoantenna. Furthermore, we have observed that the forward-to-backward ratio increases with w when the emission comes from the gap, and decreases with w when the emission comes from gold. These radiation properties help to identify the position of the nonlinear emitters, and to understand which emission, whether from the gap material or gold, is dominating in an experiment.

 

Fig. 3 Radiation pattern at 3ω₀: emission from (a) gap material only, (b) gold only, and (c) gap material and gold in a dipole nanoantenna; (d) gold in a monopole nanoantenna.

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3. Conclusion

In conclusion, we investigated third harmonic generation (THG) in hybrid dielectric/metal nanoantennas and found that both nanoantenna design and gap material nonlinear susceptibility determine whether the THG emission is primarily from the gap material, or from gold. In terms of design, nanoantenna width is a primary factor. In addition, THG emission from gold in a dipole is always much lower than that in a monopole of the same width. Due to its fabrication simplicity, the monopole nanoantenna should be preferred, unless the nonlinear permittivity of the gap material is strong enough to make its nonlinear emission much larger than that from gold.