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The nanofocusing performance of hourglass plasmonic waveguides is studied analytically and numerically. Nonlocal effects in the linearly tapered metal-air-metal stack that makes up the device are taken into account within a hydrodynamical approach. Using this hourglass waveguide as a model structure, we show that spatial dispersion drastically modifies the propagation of surface plasmons in metal voids, such as those generated between touching particles. Specifically, we investigate how nonlocal corrections limit the enormous field enhancements predicted by local electromagnetic treatments of geometric singularities. Finally, our results also indicate the emergence of nonlocality assisted tunnelling of plasmonic modes across hourglass contacts as thick as 0.5 nm.
© 2013 Optical Society of America
1. Introduction
Nanofocusing devices [1, 2] are attracting great research attention due to their many potential applications in current on-chip photonic technology [3]. The ability of these plasmonic structures to concentrate light at the nanoscale has strong implications in areas as different as long-range waveguiding [4], quantum optics [5], or nanometrology [6]. Lately, this research field has been driven by enormous advances in nanofabrication, which make it possible to shape metallic structures with unprecedented precision [7–9]. The classical electrodynamics description of the optical modes supported by these devices predicts a strong reduction of the effective plasmonic wavelength as the metal size shrinks. Within this local approximation, electromagnetic (EM) fields vary significantly over length scales comparable to the characteristic Thomas-Fermi wavelength of conduction electrons (∼ 0.1 nm in noble metals such as gold or silver [10]). The occurrence of these rapid spatial EM oscillations means that the usual free electron description of the metal permittivity (Drude model) is no longer justified. Thus, in order to reflect the emergence of strong electronic interactions at this sub-nanometer length scale, the implementation of nonlocal (spatially dispersive) dielectric functions for metals is required.
Recently, we have investigated the impact of spatial nonlocality in the nanofocusing performance of metallic tips within the hydrodynamical model [11]. In this work, we consider the complementary structure, a linearly tapered metal-insulator-metal (MIM) waveguide, where spatial dispersion in the metal regions is again taken into account within the hydrodynamical approach [11–15]. First, we develop an analytical WKB treatment of the surface plasmon polaritons (SPPs) propagation along linearly tapered waveguides, valid only for small taper angles. Next, we present a finite element nonlocal solution of Maxwell’s equations for arbitrary MIM geometries. Then, we use these theoretical tools to extend our investigation to hourglass structures, in which two tapered waveguides with opposite orientations are merged. Our theoretical results show that spatial dispersion drastically modifies the propagation characteristics of SPPs along tapered waveguides and geometric singularities [11, 16]. Remarkably, our predictions indicate that nonlocal effects makes the tunnelling of surface plasmons through thin metal contacts (∼ 0.5 nm) possible.
2. Methods
First, we present an analytical WKB description [2, 11] of the SPP propagation along the hourglass waveguide schematically depicted in Figure 1(a). The hourglass parameters used in our study, defined in Figure 1(b), describe a tapered MIM stack of length 2L′ and angle α (measured wall to wall). The gap thickness is reduced (increased) linearly from D to W (from W to D) in the region −L′ < x < 0 (0 < x < L′). The length of the two waveguide arms along the x-direction, L = L′sec(α/2), is chosen to ensure that the SPP optical path is of identical length for all taper angles α. The resultant x-dependent modulation of the gap thickness d(x) can be expressed as d(x) = 2|x|tan(α/2).
Fig. 1 (a) 3D rendering of a linearly tapered metal-insulator-metal (MIM) hourglass waveguide with translational invariance in the direction normal to the page (y-direction). The arrow indicates the direction of SPP propagation. (b) Schematic xz-plane view of the structure depicted in panel (a), with input gap thickness D, hourglass waist thickness W, arm length L′, SPP path length L, and gap (metal) permittivity ε1 and (ε2). The hourglass angle α is measured wall to wall. Note that the x and z axes are displaced for clarity. The coordinate origin is located at the center of the hourglass waist.
Our WKB description assumes that the reduction of the slab thickness along the direction of surface mode propagation in this hourglass waveguide takes place in a length scale much longer than the SPP effective wavelength. Within the WKB approximation, the electric field associated with the lowest plasmonic mode supported by the metal void can be expressed as [2]
where is a line integral corresponding to the phase accumulated by the SPPs along their optical path and A(x) is a slowly varying amplitude factor ensuring power flow conservation. Within the WKB approximation, the propagation wavevector k(x′) experiences only a small variation over a distance dx′ (i.e., |dk(x′)−1/dx′| ≤ 1) [17]. Therefore, for each position x, k(x) can be obtained from the dispersion relation of an infinite, spatially dispersive, plasmonic MIM geometry with gap thickness d(x). Treating nonlocality within the hydrodynamical Drude model we can calculate the dispersion relation of the lowest SPP mode supported by MIM waveguides, obtaining [18] where qi(x) (i = 1, 2, 3) are the various normal wavevectors associated to the SPP mode: q1(x) = [k(x)2 − ε1(ω/c)2]1/2 and q2(x) = [k(x)2 − ε2(ω/c)2]1/2 correspond to the transverse wavevector components into free space and metal, respectively, and is the longitudinal plasmon wavevector into the metal. The permittivities of air and metal regions are respectively given by ε1 = 1 and ε2 = εD(ω), where denotes the local permittivity of the Drude metal with bulk plasma frequency ωP and damping constant γ. c is the velocity of light. Note that in the limit β = 0, Eq. (2) recovers the local dispersion relation [19]The amplitude factor in Eq. (1) is fixed to satisfy energy conservation along the propagation direction. Thus, we have , where the time-averaged Poynting vector within the metal is given by
Similarly, the time-averaged Poynting vector outside the metal is given by The WKB description presented here does not take into account reflection and scattering effects experienced by the SPPs in their propagation along the surfaces of the metal void. This makes our description valid only for hourglass angles, α, which are consistent with the geometric assumptions presented above.To overcome the limitations of the WKB approach we employ a finite element method (FEM) solving strategy, based in the same hydrodynamical model for the metal permittivity. Within this nonlocal FEM numerical approach, the time harmonic electric field E = E(ω, r) associated with the SPP modes supported by an arbitrary metal nanostructure is described by the following set of coupled equations
where the first and second source terms on the right hand side of the inhomogeneous Maxwell wave equation (Eq. (6)) describe the positive background charge due to the ion cores and the nonlocal displacement current Jd = Jd(ω, r), respectively. We solve this wave equation simultaneously with Eq. (7), the transport equation that governs the dynamics of the electron density in the metal. In this hydrodynamical approach, the first term of Eq. (7) describes the electron-electron interactions and the remaining ones are the usual Drude terms for a noninteracting electron gas. The nonlocal parameter β, which measures the impact of nonlocality, is proportional to the Fermi velocity in the metal [20]. It is closely related to the decay length δLP of the longitudinal plasmons excited in the metal structure due to spatial dispersion. Specifically, at frequencies well below the plasma frequency, ωP, we can write β ≈ δLPωP [16]. Eq. (7) can be used to derive nonlocal expressions for the longitudinal and transverse components of the dielectric function used in the derivation of Eq. (2) [21]. Note that in the limit β = 0 Eq. (7) recovers the local description, which provides us with the fitting parameters for the usual Drude permittivity.Our FEM solving strategy for Eqs. (6) and (7) follows the approach outlined in [22]. Utilising highly adaptive meshes resolving geometrical features as small as 0.1 nm, we solve these equations using Comsol Multiphysics, a state-of-the-art commercial FEM solver. In order to fully specify the coupled set of equations, the usual Maxwell boundary conditions for the continuity of the surface parallel EM fields have to be augmented by an additional boundary condition. Since the lattice constant (interionic distance) is similar to the Coulomb screening length (interelectronic distance) for noble metals such as Ag or Au, the appropriate additional condition for our problem is the continuity of the normal component of the electric field.
Note that both our WKB solution and our FEM approach omit electron tunnelling effects. Recent theoretical [23–25] and experimental [7, 26] studies have reported that plasmonic enhancement is significantly affected by electron tunnelling effects in the close encounter between metallic nanoparticles at optical frequencies.
3. Results and discussion
Figure 2(a) plots the dispersion relation for the plasmonic modes supported by an infinite silver-air-silver MIM geometry of gap thickness d schematically depicted in Figure 2(b). ω(k) only for the lowest SPP modes is shown. Note that the modal frequency and wavevector are normalized to the surface plasmon frequency, defined as Re(εD(ωSPP)) = −1. Results for various gap thicknesses ranging from 1 nm to 50 nm are rendered. SPP bands obtained within the local (β = 0) and the nonlocal (β = 0.0036c) descriptions are plotted in black solid and red dashed lines, respectively. In all calculations, the local Drude constants were taken from the fitting to Ag experimental data [27] used in [11], and a realistic value of the parameter β for silver was considered in the nonlocal treatment [28]. Figure 2 shows that the reduction of the waveguide size red-shifts the SPP bands. At a fixed frequency, this band lowering with decreasing d can be interpreted as an increase of the mode binding to the structure surfaces [29]. This fact makes hourglass waveguides generated by the adiabatic reduction of d very suitable for light localisation purposes. Note that the insets of Figure 2 demonstrate the long propagation lengths (Λ = (2Im{k})−1) of these plasmonic modes over a wide spectral window, which reach many SPP wavelengths even for very small waveguide dimensions.
Fig. 2 (a) Dispersion relation for the lowest SPP mode supported by a silver-air-silver MIM waveguide with gap thickness d, ranging from 1 nm to 50 nm. Black dashed (red solid) lines show local (nonlocal, β = 0.0036c) results. The inset renders the modal propagation length as a function of frequency. (b) Schematic of the infinite MIM geometry used in panel (a), fully characterised by the gap thickness d, gap permittivity ε1 and metal permittivity ε2.
The comparison between local and nonlocal results in Figure 2 indicates that spatial dispersion blue-shifts the SPP frequencies in hourglass waveguides. This is a well-known nonlocal feature reported in the past for simple plasmonic waveguides [30] and nanoparticle geometries [13]. Note that the increase in ω(k) due to spatial dispersion is more pronounced for smaller d, where the thickness of the gap approaches the longitudinal plasmon decay length (δLP ≈ 0.1 nm for realistic Ag parameters).
Figure 3 renders the real part of the z-component of the electric field, Ez/E0, along the surface of the waveguide defined in Figure 1(b), with L = 1000 nm, D = 60 nm, W = 0 nm, and α = 3.44°. Note that these parameters imply a geometrical singularity at the point (x, y) = (0, 0). The fields have been normalized to the incident amplitude E0. Both local (black) and nonlocal (red) results are rendered at 0.7ωSPP, where the modes depicted in Figure 2(a) exhibit a convenient balance between confinement and propagation length. This balance arises because, whilst the binding of modes to the metal surfaces is tighter at high frequencies, the propagation length increases towards lower mode energies due to the reduced absorption damping (see Figure 2(a)). Excellent agreement between WKB analytical (lines) and FEM numerical (dots) results is obtained. Remarkably, we observe that in the nonlocal description SPPs are able to tunnel through the geometrical singularity located at x = 0 nm, penetrating into the non-illuminated side of the waveguide in the region x > 0. This is in contrast to the local limit, where SPPs are stopped adiabatically as they approach this singularity. Similar effects have been already reported theoretically in other singular MIM geometries that support EM surface modes. For example, as we have demonstrated in [16], the nonlocality enabled tunneling of SPPs has important consequences for kissing nanowire geometries, where it leads to a set of discrete nonlocal resonances that emerge below the surface plasma frequency. This tunnelling effect can be understood as a result of the smearing of the charge density in the nonlocal approximation, which prevents localizations of fields at very small distances (of the order of the Thomas-Fermi wavelength) [11].
Fig. 3 Real part of the electric z-field component (Ez/E0) evaluated at 0.7ωSPP, plotted along the surface of a silver-air-silver plasmonic hourglass waveguide with L = 1000 nm, D = 60 nm and W = 0 nm. Dots and solid lines render FEM and analytical WKB calculations, respectively. Local (nonlocal) results are plotted in black (red).
In accordance with the band blue-shifting identified in Figure 2, Figure 3 reveals an increase in the effective wavelength of plasmonic modes in the nonlocal description, which is accompanied by a reduction in the field amplification that can be achieved. Both these nonlocal effects are strongest around x = 0 nm, and vanish in regions where the metal surfaces are far apart.
Up to this point we have restricted our study to waveguides at a fixed operating frequency. In Figure 4(a) we lift this restriction by plotting the frequency dependent z-field amplitude |Ez/E0| at the waist of the hourglass waveguide as a function of frequency, for waist diameters ranging from 0.1 nm to 5 nm (α = 3.44°). Note that we plot the component in the direction perpendicular to the metal surfaces (z-direction) as it is the one that undergoes the highest amplification. Local (black) and nonlocal (red) results are rendered, and good agreement between WKB analytical (lines) and FEM numerical (dots) results is observed. Both descriptions yield a reduction of the field amplification at the hourglass waist at larger frequencies, which comes as a result of the increased metal absorption damping. The difference between local and nonlocal results is most pronounced for W = 0.1 nm, the narrowest hourglass waist thickness considered in Figure 4. In such diminutive junctions, the real part of the nonlocal wavevector in the SPP propagation direction decreases significantly due to spatial dispersion, which results in lower field enhancements due to the weaker SPP binding to the metal surfaces.
Fig. 4 (a) Frequency dependent electric z-field amplitude enhancement |Ez/E0| evaluated at the waist of the waveguide, for different hourglass waist thicknesses W, for fixed L = 1000 nm and α = 3.44°. Good agreement between WKB analytical (lines) and FEM numerical (dots) results is observed. Nonlocal and local predictions are shown in red and black, respectively (b) |Ez/E0| for ω = 0.7ωSPP, plotted as a function of the hourglass angle α. The hourglass waist is fixed to W = 0.1 nm, with all other parameters taking the same values as in panel (a).
Next, we investigate the sensitivity of the device performance to variations in the hourglass angle α. In Figure 4(b) we plot the α-dependent electric field, evaluated at 0.7ωSPP, at one of the walls of the hourglass waist for W = 0.1 nm (the smallest waist considered in Figure 4(a)). Local (nonlocal) results are rendered in black (red). As α increases, a trend towards larger field enhancements is observed due to the significantly increased propagation length Λ of modes sustained by straight MIM geometries with large gap thicknesses (see the inset of Figure 2(a)).
Note that we have fixed the SPP optical path, L, to 1000 nm for all α. Only this way, the effect of the tapering angle on the losses experienced by the SPP modes can be investigated. The comparison between WKB analytical results (lines) and FEM numerical results (dots) reveals good agreement between the two approaches for hourglass angles as large as α ∼ 15°. At larger angles, the WKB assumption of a reflection-free SPP propagation towards the hourglass waist breaks down, with significant reflection taking place as the modes accommodate to the rapidly changing geometry.
The dependence of the field enhancement at the hourglass waist on the taper angle is governed by two competing effects. On the one hand, increasing angles lead to larger propagation lengths, resulting in more efficient energy transfer and light localization. On the other hand, as α increases, the back scattering of SPPs caused by the metal walls becomes more pronounced, yielding lower field enhancements at the hourglass waist. Note that the latter effect is not described in our analytical WKB treatment of the system. For the geometric parameters considered in Figure 4(b) the balance of these two effects leads to a maximum field enhancement at an angle α ∼ 28° for both local and nonlocal predictions..
Finally, we consider the influence that the hourglass waist geometry has on the ability of SPPs to tunnel into the non-illuminated side of the waveguide (x > 0). Figs. 5(a)–5(c) show results for open hourglass geometries (W > 0). Panel (a) plots the local (black) and nonlocal (red) z-field enhancement |Ez/E0| on the surface of the hourglass waveguide defined in Figs. 1(a)–1(b), but for hourglass waist thicknesses of W = 0.15 nm and W = 5 nm. The fields are evaluated at 0.7ωSPP using the FEM numerical approach. Nonlocal electric field maps for W = 0.15 nm and W = 5 nm are reproduced in panels (b) and (c), respectively. The comparison between local and nonlocal results in Figure 5(a) reveals that the two models yield extremely similar results for W = 5 nm. This result can be understood from the dispersion bands in Figure 2(a), which reveal that, for 5 nm layer width, local and nonlocal predictions for both k and Λ differ by only 0.2% at ω = 0.7ωSPP.
Fig. 5 (a) |Ez/E0| evaluated at 0.7ωSPP, along the surface of a silver-air-silver plasmonic hourglass waveguide with length L = 1000 nm and input gap thickness D = 60 nm. Nonlocal (local) FEM numerical results are plotted in red (black). Results for hourglass waist thicknesses of both W = 0.15 nm and W = 5 nm are rendered. (b) and (c) Nonlocal |Ez/E0| map for the W = 0.15 nm and W = 5 nm cases in panel (a). (d) Same as panel (a), but for W < 0 (g = W /(tan(α/2)). Results for separation metal thicknesses of g = 0.5 nm and g = 5 nm are rendered. (e) and (f) Nonlocal |Ez/E0| map for the g = 0.5 nm and g = 5 nm cases in panel (d).
Large hourglass waist thicknesses enable efficient transfer of EM energy into the x > 0 region of the structure. This energy transfer is less efficient for smaller W, which in turn yield much larger field enhancement values. Figure 5(a) shows these two effects for W = 0.15 nm, where the fields at the point x = 10 nm, located in the non-illuminated side of the waveguide, have dropped to less than half of the amplitude when compared to the W = 5 nm case. Interestingly, the nonlocal field amplitudes are larger than the local ones for most of the x > 0 region. This inversion results from absorption losses, which are increased in the local case due to the significantly larger local SPP wavevectors (shorter effective wavelengths λSPP = 2π/k). At x = 0 nm the effective wavelength of local and nonlocal SPPs are 5.3 and = 9.0 nm, respectively, which is in agreement with the periodicity observed in the reflection induced beating of |Ez/E0| apparent in Figure 5(a).
Next, in Figs. 5(d)–5(f), we consider the case of overlapping hourglass walls, characterised by W < 0. As W becomes negative, a metallic separation layer of thickness g = W /tan(α/2) emerges in between the two arms of the waveguide, occupying the region −g/2 < x < g/2. Figure 5(d) plots |Ez/E0| at 0.7ωSPP along the hourglass walls of the same geometry considered in Figure 5(a), but for negative values of W.
Structures with two metallic contact thicknesses, g = 0.5 nm and 5 nm, are considered. For of g = 5 nm the thick stopping layer is seen to be effective in terminating the SPP propagation within both the local (black lines) and nonlocal (red lines) descriptions. Interestingly, the thin metallic stopping layer (g = 0.5 nm) results in a behaviour which is both quantitatively and qualitatively different. As in the case of the singular geometry studied in Figure 3, we observe here that, whereas local SPPs are adiabatically stopped as they approach the hourglass waist, SPPs treated within the nonlocal model are able to tunnel accross this metallic contact, as evidenced by the non-zero nonlocal field amplitude for x > 0 in Figs. 5(d) and 5(f). This result can be understood by considering the SPP group velocity, which in the local limit always approaches zero at the singularity. Thus, even for metallic stopping layers whose thickness is thinner than the skin depth, SPPs are unable to tunnel accross adjacent metallic contacts, irrespective of their thickness. On the other hand, within the nonlocal description, the electronic charge distribution is smeared out accross the metal boundaries, which increases the effective thickness of the dielectric region and, in turn, leads to a non-zero SPP group velocity even at geometrical singularities. This nonlocal smearing of surface charges is measured by the longitudinal plasmon decay length, which for the parameters considered in our calculations is δLP ∼ 0.1 nm. This value sets the metal thicknesses g, for which nonlocal tunnelling of SPPs in tapered MIM waveguides takes place.
4. Conclusion
To briefly conclude, we have introduced a nonlocal WKB solution for the propagation of SPPs along linearly tapered plasmonic hourglass waveguides. Effects of nonlocality were included using the hydrodynamical Drude model, and good agreement of these WKB analytical results with a fully retarded nonlocal FEM numerical solution of the underlying equations was obtained. This formalism has enabled us to study the nanofocusing performance of plasmonic hourglass waveguides both analytically and numerically. Our results indicate that for a waist thickness of 0.1 nm spatial dispersion limits the field enhancements that can be achieved at the waist of metallic hourglass waveguides by up to a factor of two. Therefore, we can conclude that in terms of their nanofocusing performance, these devices are less sensitive to nonlocal effects than three dimensional metallic tips. [11] Interestingly, we have found that spatial dispersion enables SPPs to tunnel across metallic contacts as thick as 0.5 nm, which contradicts the usual vanishing surface plasmon group velocity predicted at sharp metallic contacts obtained within the local approximation. This effect is general and has ramifications for all singular metallic geometries that support SPPs, including kissing metallic nanowires [16], wedges [31] and nanocrescents. [32]
Our nonlocal description of EM fields propagation provides a powerful tool for the determination of the parameters which optimize the light-concentration performance of tapered and hourglass plasmonic waveguides.
Acknowledgments
This work was supported by the Centre for Doctoral Training on Theory and Simulation of Materials (TSM-CDT), the EPSRC Active Plasmonics Programme, the ESF Plasmon-Bionanosense Network, and the Leverhulme Trust. A.I.F.-D. and J.B.P. would like to acknowledge the funding provided by the Gordon and Betty Moore Foundation.
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