From symmetric to asymmetric bowtie nanoantennas: electrostatic conformal mapping perspective

2.1 Conformal mapping: basics

A conformal map is an analytic transformation of the form z′=f(z), where z′=x′+iy′, which preserves oriented angles locally. Hence, the tangential component of the electric field E_|| and the normal component of the displacement field, D_⊥ are conserved under the transformation, implying that the material in the original and transformed spaces are identical. That is,

Furthermore, if a given function Φ(z) is a solution of Laplace’s equation for the z-plane, representing the quasi-static potential in such plane, then Φ′(z′)=Φ(f⁻¹(z′)) (the quasi-static potential in z′-plane) will be the Laplace solution for the z′-plane.

Let us now imagine that the geometry under study is a 2D bowtie geometry [i.e. a three-dimensional (3D) bowtie geometry with out-of-plane invariance] with a nanoemitter at (x′, y′)=(1 nm, 0 nm) modelled as a line dipole (Figure 1A). By using the natural logarithm mapping

such geometry is transformed into the periodic metal-insulator geometry shown in Figure 1B. In particular, the nanoemitter at (x′, y′)=(1 nm, 0 nm) is transposed to (x, y)=(0, 2πm), where m is an integer. If the nanoemitter were on-center in the original frame, it would be transposed to x=–∞. The fact that the bowtie has a gap at its center prevents the metal-insulator geometry to extend to –∞ in the x-direction.

If we restrict the bowtie to be at least one order of magnitude smaller than the wavelength, we can invoke the quasi-static approximation for the analytical analysis, whereby the radiation losses are neglected and the magnetic and electric fields are decoupled; the latter field can then be expressed via an electrostatic potential satisfying the Laplace equation. In this situation, the power dissipations in the original, Pabs(x′,y′), and transformed, Pabs(x,y), frames are identical. Hence, the former can be calculated using the latter, whereby the electric field is evaluated at the dipole position as

where P_nr is the nonradiative power emission; ω is the angular frequency; p_x and p_y are the x and y components of the dipole moment with magnitude |p|, respectively; and E1xS(0, 2 πm) and E1yS(0, 2 πm) are the x and y components of the electric field, respectively, at the position where the dipole is in the transformed frame. Normalizing P_nr to the power radiated by the line (2D) dipole

where μ₀ is the permeability in the free space, results in the nonradiative Purcell enhancement given by

provided an intrinsic quantum yield of 1 is used for the nanoemitter; under such condition, one can map the nonradiative decay experienced by the nanoemitter with the power absorbed by the bowtie nanoantenna [50].

Such nonradiative Purcell enhancement is modulated by the coupling between the array of line dipoles and the plasmonic eigen-modes of the system [notice the explicit field overlap in Eq. (3)] since the latter modes are the only effective nonradiative channels of the system. These plasmonic eigen-modes are nothing but the localised surface plasmons resulting from the interference of the surface plasmons triggered by line dipoles. These are reflected back and forth between the two ends of the periodic metal-insulator geometry (i.e. standing-wave plasmonic resonances) and have a wavenumber k=(nπ – Δφ)/(L₁+L₂), with n=1, 2, 3, …, and L₁+L₂ representing the order of the standing-wave plasmonic resonance and the total length of the periodic metal-insulator cavities, respectively, and Δφ is a reflection phase correction. This will be discussed in the following subsection. The reader is referred to Figure 1 to see the definition of geometrical parameters. The complete mathematical derivation describing this underlying mechanism can be found in Section 3.

2.2 Surface plasmon reflection phase

The surface plasmons acquire a nontrivial phase at both ends of the periodic metal-insulator geometry associated with the near-field energy storage at these end faces. This is included in our model through a semi-empirical phase correction Δφ, which is added to the reflection phase of an open boundary (i.e. π) [25]. An analytical solution to the surface plasmon reflection phase could be attempted, but only under some assumptions, such as ignoring the evanescent plasmonic modes [51], [52]. This approximation is valid for long enough periodic metal-insulator cavities (i.e. large enough bowtie), and results into an asymptotic value of the surface plasmon reflection phase at long wavelengths where metal has a large negative dielectric constant. However, for short periodic metal-insulator cavities, and thus, small bowties as those considered in our electrostatic conformal mapping works, this approximation is not valid.

Figure 2 plots Γnr¯ along with the corresponding phase correction Δφ for θ′=20° bowtie nanoantennas of varying size l′=2(L′₁+L′_2′)+g′ (see definition of geometrical parameters in Figure 1) and gap size g′ where the nanoemitter is vertical or horizontal. For the horizontal case, all plasmonic eigen-modes emerge at a similar wavelength regardless of l′ and g′. Specifically, the wavelength of the fundamental mode is around 575 nm, whereas higher order plasmonics modes are agglomerated close to the surface plasmon wavelength. This is confirmed by plotting the cut-off wavelength of the first three eigen-modes (i.e. computed via the divergence of the coefficient of the scattered potential in the region where the dipole is present, B_±, see Figure 2F). The vertical case shows a completely different picture. The fundamental plasmonic eigen-mode dramatically red-shifts significantly with l′ (see Figure 2A and E) with decreasing g′ for a fixed l′ (Figure 2A). In addition, the first higher order eigen-mode is not agglomerated close to the surface plasmon wavelength as the rest of higher order modes, but it appears around 525 nm.

Figure 2:

Influence of the bowtie nanoantenna’s length (l′).

The analytical (symbols) and numerical (solid lines) results of the Γnr¯ for θ′=20° bowtie nanoantenna of varying length l′ and gap size g′ illuminated by a nanoemitter with vertical (A) and horizontal (B) polarizations placed at (x′, y′)=(1 nm, 0 nm). Notice that the symbols and solid lines overlap and are largely indistinguishable. Phase correction Δφ as a function of l′ for the vertical (C) and horizontal nanoemitters (D) for the three gap sizes considered in A and B. The cut-off wavelength for the first three localized surface plasmon modes with a vertical (E) and horizontal nanoemitter (F) when g′=1 nm. Notice that the cut-off wavelength for all modes is rather stable against g′ for the horizontal nanoemitter case.

From Figure 2C and D, one can observe how the phase correction tends rapidly to an asymptotic value as the size of the bowtie (i.e. length of the periodic metal-insulator cavities) increases for a vertical nanoemitter (panel C), whereas the phase correction undergoes a larger excursion and drops more monotonically down to 0 rad for a horizontal nanoemitter (panel D). This polarization-dependent response can be understood from the transmission line representation of the classical Sommerfeld half-space problem [53], [54] and based on the strong influence of evanescent plasmonic modes in the surface plasmon reflection phase [51]. A vertical Hertzian dipole contributes a voltage source on a transverse-magnetic (TM) transmission line analog, whereas a horizontal dipole contributes current sources on both the TM and transverse-electric (TE) transmission lines [53]. As such, the latter deals with a larger number of evanescent plasmonic modes that can only be neglected for a larger bowties than for vertical dipoles. The fact that Δφ for g′=0.2 nm and a vertical nanoemitter is rather flat compared to the other gap sizes is due to the diminishing influence of the evanescent plasmonic modes in the corresponding long metal-insulator cavity.

2.3 Asymmetric nanoantennas

Manufactured nanoantennas suffer from inevitable imperfections due to the nanofabrication process. Bowtie nanoantennas are not free of such issues. Thus, it would be useful to understand the impact of these imperfections in the response of bowtie nanoantennas. The analytical frame described here cannot describe precisely all types of imperfections, but it can at least assist with simple deformations, such as asymmetries between the monomers composing the dimer. We study in this section the effect of asymmetric circular sectors comprising the bowtie nanoantennas. A general illustration of this nanoantenna is shown in Figure 1A, where the angles of the two arms (made of gold) are different θ′1≠θ′2. Notice that without loss of generality, the nanoemitter remains at (x′, y′)=(1 nm, 0 nm) in the original frame.

With this configuration, let us first consider the asymmetric l′=20 nm and g′=1 nm bowtie nanoantennas with θ′1=20° and varying θ′2. The schematic representations, including dipole position and polarization, are shown in Figure 3A and B above their corresponding results. The results of the nonradiative Purcell enhancement (Γnr¯) spectra as a function of θ′2 ranging from 10° to 40° are shown in Figure 3E and F considering a vertical and horizontal polarization of the nanoemitter, respectively. The main observation from these results is that the localized surface plasmon modes are blue-shifted when changing θ′2. This performance is in agreement with our previous works considering symmetric bowtie and tripod nanoantennas and nanocavities [13], [25], [26], [27]. However, there is an interesting difference: the blue-shift is smaller for the asymmetric nanoantennas compared to the symmetric ones. For instance, for the vertical nanoemitter shown in Figure 3A, the localized surface plasmon mode of order n=1 is blue-shifted from λ=697 nm to λ=600 nm, i.e. a Δλ=97 nm shift. However, for the symmetric case [25], considering gold bowtie nanoantennas (not shown), the blue-shift goes from λ=731 nm to λ=566 nm (Δλ=165 nm) when changing θ′1=θ′2 from 10° to 40°. Interestingly, a similar response is obtained for the orthogonal polarization (i.e. horizontal nanoemitter), shown in Figure 3F, with the localized surface plasmon of order n=1 at the same location as the vertical polarization, λ=697 nm for θ′1=10° and λ=600 nm for θ′1=40°. For symmetric bowtie nanoantennas, however, the fundamental localized surface plasmon mode is blue-shifted from λ=638 nm to λ=517 nm (Δλ=121 nm) within the same range of θ′1=θ′2. These results demonstrate that the overall Γnr¯ spectra will be changed when introducing asymmetries into the nanoantennas, an aspect that is crucial in real-world scenarios where it is likely that fabrication errors and tolerances may produce asymmetric instead of symmetric bowtie nanoantennas.

Figure 3:

Non-radiative and radiative Purcell enhancement for asymmetric nanoantennas.

The schematic representation of the l′=20 nm and g′=1 nm bowtie nanoantenna with θ′1=20° (A, B) and θ′1=30° (C, D) illuminated by a vertical (A, C) and horizontal (B, D) nanoemitters; the nanoemitters are depicted by the green arrow. The corresponding Γnr¯ (second row) and Γr¯ (third row) 2D bowties, and simulated Γnr¯ for a 1 nm-thick bowtie (bottom row). Insets in (E) and (G) depict the geometry of the bowtie for different θ′2.

Another noteworthy information is that the blue-shift of the Γnr¯ spectra observed in Figure 3E and F result from the hybridization between the spectrum produced by each of the arms of the bowtie nanoantennas (θ′1 and θ′2). Hence, the arm with smaller θ′ will dominate the spectrum. This can be corroborated by the results in Figure 4 where it is shown how for θ′2<(θ′1=20°), the shift of the localized surface plasmon modes is larger compared to the case when θ′2>θ′1. This is due to the fact that, in the former scenario, the Γnr¯ spectrum is dominated by the arm with θ′2. When θ′2>θ′1, the spectrum is dominated by the top arm with θ′1=20° and then the Γnr¯ is saturated when increasing θ′2. For the sake of completeness, the results of the radiated Purcell enhancement Γr¯ spectra, which is calculated as the ratio between the power scattered by the bowtie nanoantenna and that radiated by the nanoemitter alone (Γr¯=Pr/P0), as a function of θ′2 are shown in Figure 3I and J for both vertical and horizontal nanoemitters, respectively. As observed, the Γr¯ is almost two orders of magnitude lower than the Γnr¯, thus confirming the assumption that radiation losses are negligible because l′≪λ₀ (see Methods section for more details). We have also calculated the spectral response considering a different value of θ′1=30° and the results of the Γnr¯ are shown in Figure 3G and H for the vertical and horizontal polarization of the nanoemitter, respectively, along with the results of the Γr¯ for both polarizations (Figure 3K and L). Note that a similar performance as those previously discussed when considering θ′1=20° is obtained with this case where the localized surface plasmon modes are blue-shifted when changing θ′2 from 10° to 40°.

Figure 4:

Comparison between symmetric and asymmetric bowtie nanoantennas.

The Γnr¯ values for different sets of θ′1 and θ′2 corresponding to a vertical (A) and horizontal nanoemitter (E); the solid lines are the corresponding results obtained with COMSOL Multiphysics. Snapshot of the E′φ′ field (B–D) and the E′ρ′ field (F–H) at the fundamental peaks in (A) and (E), respectively, computed using COMSOL Multiphysics.

As we have shown in our previous works, the Γnr¯ is determined by the coupling between the nanoemitters and the field profile of each localized surface plasmon mode supported by the bowtie nanoantennas [25]. As shown in Figure 3, the Γnr¯ is increased when changing θ′2 to larger values, and this coupling can also be modified by changing the position of the nanoemitter along the x′ or y′ axis [13]. In this realm, there will either be a strong or low coupling of the radiated field from the nanoemitter to each LSP mode depending on its spatial location within the field distribution of each mode. This performance can also be observed in Figure 3, and it is more evident for the horizontal polarization of the nanoemitter (Figure 3F and H). The figures show how there is an LSP mode that is only excited when θ′2≠θ′1, which is a result of the hybridization between the different modes excited in the asymmetric bowtie nanoantenna.

To further study this case, let us first consider the case of a vertical polarized nanoemitter illuminating an asymmetric bowtie nanoantenna with θ′1=20° and θ′2=12°. The Γnr¯ spectrum for this case, extracted from Figure 3E, is shown in Figure 4A as a black curve along with the Γnr¯ spectra for symmetric bowtie nanoantennas with θ′1=θ′2=12° (dark blue) and θ′1=θ′2=20° (light blue). From these results, as expected, the peak of Γnr¯ related to the LSP mode of order n=1 for the asymmetric case lies in between those obtained using symmetric nanoantennas. For completeness, the field distribution for this LSP mode considering the two symmetric and asymmetric nanoantennas are shown in Figure 4B and C, respectively, which indicate how the same field distribution is obtained in all cases. For the orthogonal polarization, the results of the Γnr¯ spectra are shown in Figure 4E considering the asymmetric and the two symmetric nanoantennas with angles of the arms as in Figure 4A. From these results, one can observe how the first peak of Γnr¯ for the asymmetric case is no longer in between the two peaks of the symmetric nanoantennas and is now red-shifted. Moreover, note that its field distribution is completely different (Figure 4H) to those of the symmetric nanoantennas (Figure 4F and G) due to the hybridization between the modes.

2.4 Three-dimensional nanoantennas

Conformal mapping enables us to provide an analytical solution only for 2D bowties (i.e. 3D bowties with infinite height). However, the physical insight gained can be transferred to realistic 3D bowties. We demonstrate this by using COMSOL Multiphysics to reproduce some of the previous results for 1-nm thick and 10-nm thick bowties with point dipoles at 0.5 nm and 5 nm height, respectively, modelling a 3D nanoemitter.

The bottom row in Figure 3 shows the Γnr¯ for the 1-nm thick bowtie. These color maps show a more complex electromagnetic response than the 2D counterpart, but one can appreciate that the trends remain: the localized surface plasmon modes are blue-shifted when changing θ′2 and one can find a θ′2 such that Γnr¯ vanishes. In fact, θ′2 is the same for the 2D and 3D cases.

The 1-nm thick bowtie is the most extreme case compared to the 2D scenario. Thicker bowties will not only retain the trends, but they will also have a similar spectra to the 2D scenario qualitatively and quantitatively in terms of wavelength. To demonstrate this, we refer the reader to compare Figure 5 with Figure 4A and E.

Figure 5:

3D asymmetric bowtie nanoantennas excited by a localized emitter.

The schematic representation of the 3D l′=20 nm and g′=1 nm bowties (A, D). The Γnr¯ values for different sets of θ′1 and θ′2 corresponding to the vertical (second column) and horizontal nanoemitters (third column) for the 10-nm thick (top row) and 1-nm thick bowtie (bottom row).

3.1 Analytical formulation

In this section, we show the analytical solution of the periodic metal-insulator geometry described in Figure 1B. As observed, this geometry is similar to the one shown in our previous work [25]. However, our aim here is to evaluate a more general case when d₃ ≠ d₄ (i.e. the angles θ′1 and θ′2 of the two arms of the bowtie nanoantenna are different). To work within the quasi-static approximation, we assumed that bowtie nanoantennas are smaller than the wavelength of the illuminating source (l′≪λ₀). In this realm, the electric and magnetic fields are decoupled and the former can be fully described by an electrostatic potential satisfying Laplace’s equation. Moreover, since L1+L2≫θ′1, θ′2, the modes with phase variation along the y axis (longitudinal modes) can be neglected. Hence, it can be assumed that the excited localized surface plasmon will be those with phase variation along the x axis (transversal modes). With these considerations, the potentials inside and outside the metals of the multislab geometry in Figure 1B can be defined as follows:

where ϒ=[1−e2ik(L1+L2+2iΔφ)]−1,k=(nπ – Δφ)/(L₁+L₂) is the wave number of the localized surface plasmon modes with order n=1, 2, 3 …, Δφ is the phase correction added to the system to consider the nonperfect reflections at the end faces of the periodic metal-insulator cavities, and (B₊, B₋) and (E₊, E₋) are the constants corresponding to the potential in those regions where the nanoemitter is present or absent, respectively. In addition, (C₊, C₋) and (D₊, D₋) are the constants inside the metal strips d₄ and d₃, respectively. Finally, the coefficients for the incident potential (A₊, A₋) can be calculated by expanding the potential of the dipole nanoemitter using a Fourier transform along the x axis as A_±=(±p_y – ip_xsgn(k))/(2ε₀), with p_x,y as the components of the dipole moment along the x and y axis, respectively, and ε₀ as the permittivity in free space. The other coefficients (B_±, C_±, D_± and E_±) can be calculated by simply introducing two boundary conditions: the continuity of the parallel component of the electric field (E_x) along with the conservation of the perpendicular component of the displacement current (D_y=ε_AuE_y) at the boundaries d₁, d₂, d₂+d₃ and d₁+2d₂+d₃+d₄:

with ε_Au as the permittivity of the metal used in this work for the bowtie nanoantennas (gold). As in our previous works, the solutions of these equations can be calculated straightforwardly either manually or via a mathematical software. For the sake of brevity, however, we will not show the whole expressions for the constants in this paper. After applying an inverse transform to the induced potentials, the solutions for the potentials where the dipole is present (Φ1s) and absent (Φ2s) as well as inside the metal slabs, (Φ1m) and (Φ2m), are as follows:

with Ω=[2ε₀(L₁+L₂)]^−1/2. Once the potentials are calculated, the final step consists of simply differentiating these expressions to calculate the x and y components of the electric field in each region of periodic metal-insulator geometry:

3.2 Numerical simulations

All simulations in this manuscript were carried out using the frequency domain solver of the commercial software COMSOL Multiphysics following the same setup as in our previous studies [25], [26], [27]. Gold permittivity was modeled using an analytical polynomial equation. This function fits Palik’s experimental data [55]. The bowties were immersed within a box of 600 nm side filled with vacuum. Scattering boundary conditions were applied around this box to avoid undesirable reflections. The nanoemitter was modeled using two anti-parallel, in-plane magnetic currents with a distance of 5 pm between them. A refined mesh with a minimum and maximum size of 3 pm and 2 nm was implemented for the vacuum box to ensure accurate results. The mesh used for the bowtie nanoantennas was refined to be two times smaller than that of the 600-nm box.