1. Introduction

Plasmon photonics, typically relying on collective electron oscillations in metal surfaces exposed to open dielectric environments, holds great promise to become the natural link between current nanoelectronics and future integrated nanophotonics [1, 2]. The rapid decay of the electromagnetic field towards the interior of the metal is however accompanied by lateral oscillations extending over macroscopic distances with wavelength comparable to the free-space light wavelength. Metallic nanostructures have been devised to control plasmons over sub-wavelength scales via localized modes (e.g., in nanoparticles [3]) and propagating modes [4]. Several types of plasmon waveguides have been investigated, and in particular, thin metal layers have been shown to sustain long-range surface plasmons [5], both in symmetric [6] and asymmetric [7] environments. Channel plasmons [8], wedge plasmons [9], hybrid dielectric-metal plasmons [10], and gap plasmons [11] have been explored, with a view to optimizing the compromise between propagation distance and degree of confinement. Plasmon bands formed in periodically corrugated metal surfaces [12] have been also used for waveguiding [13], and so have plasmon hopping between neighboring nanoparticles arranged in strings [14, 15, 16, 17]. However, these types of plasmon waveguides suffer from the damaging effects of radiative losses, whereby plasmon scattering into propagating light modes causes depletion of the surface-bound signal, particularly at waveguide turns, splitters, and junctions.

We explore here a solution to eliminate radiative losses, consisting in fully burying the plasmon-supporting structures inside metal. Metal-insulator-metal (MIM) cavities, which belong to this category of structures, have actually attracted much interest in recent years due to their ability to produce extreme confinement [18] and effective negative index of refraction [19] in guided surface plasmons. We go one step forward by considering alternative structures that confine plasmons in two spatial directions and allow controlled propagation only in the remaining third direction. Our structures consist of linear arrays of coupled voids, in which hybridization between nearest-neighbor void plasmons is adjusted through void size and geometrical overlap. Propagation distances of tens of microns are predicted for these structures, which should be easy to fabricate by covering arrays of silica or latex beads with metal using electrochemistry techniques similar to those employed to manufacture porous metal surfaces [20]. Actually, linear chains of dielectric spheres have been produced upon infiltration in nanopores [21], and gold nanovoid linear arrays have been recently synthesized by burying in metal polystyrene spheres self-assembled in silicon V-grooves [22]. These waveguides are advantageous with respect to the complementary structures formed by particle arrays [23, 24], because the field is forced to be confined inside the cavities and radiative losses are automatically suppressed, thus allowing larger voids to be considered, and therefore also larger propagation distances and sharp turns with minimum losses.

Our work relies on rigorous solution of Maxwell’s equations using the boundary element method (BEM) [25], in which the electromagnetic field is expressed in terms of surface charges and currents that are self-consistently obtained upon application of the customary field boundary conditions. We describe axially-symmetric finite arrays, which greatly reduce the computational demand of the BEM [25], and this allows us to consider the effect of disorder by randomly varying the size and average position of the voids.

The modes and waveguiding properties of linear arrays of separated voids have been already discussed in the literature [26], and the advantages of the suppression of radiative losses identified. We present additional results by allowing the voids to overlap, so that plasmon propagation along the arrays occurs mainly through the openings between adjacent voids, thus decreasing signal absorption in the metal. Finite-size windows between nanovoids are typical in porous materials produced by electrochemically covering opal-like structures with metal [27, 22]. The overlap leads to strong void coupling, which has been studied in planar arrays [28] and in void dimers [29]. Large field enhancement can occur at the ring that defines the opening, similar to what happens in rim modes formed at the opening of a void that is partially buried by a planar metal surface [30], or in cusps between adjacent nanoapertures in a metal film [31].

2. Plasmon hybridization in coupled voids

Plasmons in strongly interacting systems exhibit interesting phenomena such as the singular transition between touching and non-touching regimes in particle dimers [32]. The lowest-energy dipole-dipole mode at large inter-particle distance evolves continuously towards longer wavelengths as the particles are brought into close proximity, eventually setting up a gap mode. This evolution is accompanied by participation of higher particle multipoles, which produce a huge pileup of induced charge in the gap region and numerous higher-energy resonances. Right after touching, a long-wavelength mode shows up in which (in contrast to what happens in separate particles) the induced charge in each particle does not present nodes, and charge neutrality is preserved by current flowing through the connecting neck.

 

Fig. 1. Evolution of void-dimer plasmon energies with overlap distance d. The plasmons are trapped in silica-filled gold-void dimers. The modes are probed through the photonic local density of states (LDOS) at the center of one of the voids, projected along a direction perpendicular to the dimer axis. The void radius is a=240nm. The electric dipole plasmon of the spherical cavity near 900 nm is recovered in the single-void (full overlap at d=-2a) and separated-voids (d=30nm) limits. The LDOS is normalized to its value in an infinite silica environment, ${\epsilon }_{{\mathrm{SiO}}_2}^{1⁄2}{\omega }^2⁄{\pi }^2{c}^3$. A vertical offset has been introduced in consecutive spectra for clarity. The intensity of the induced field produced by a dipole at the center of the left void is shown in the insets for selected separations.

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When studying open structures such as particles, it is natural to examine absorption and scattering properties, which relate to far-field experimental characterization techniques. However, with voids buried in metal, we must rely on near-field properties. Here, we use the photonic local density of states (LDOS), which represents the weight of all normalized electromagnetic modes at a certain point of space for a certain light frequency. In practice, the LDOS is proportional to the imaginary part of the electric field induced by a dipole on itself [33], which we calculate using the noted BEM technique. Furthermore, the LDOS is proportional to the decay rate of an excited atom, and therefore, it can be measured in voids by investigating the decay of dopants in a regime of weak coupling between the cavities and free space through a thinned metal layer separating them. Alternatively, the LDOS is accessible through electron energy-loss spectroscopy, with the advantage of nanometer spatial control over the sampled region [34].

Unlike metallic particles, the complementary system consisting of two dielectric inclusions buried in metal does not present a singular transition between touching and non-touching [29]. The interaction of separate voids dies off when the surface-to-surface separation is large compared to the metal skin-depth. This is illustrated in Fig. 1, which shows LDOS spectra at the center of one of the voids for separations d ranging from overlapping geometries (d<0, solid curves) to separate voids (d>0, broken curves). We concentrate in Fig. 1 on the case of transverse polarization, perpendicular to the dimer axis of revolution (i.e., modes with m=±1 azimuthal symmetry). (We have also explored the m=0 case, with polarization along the axis, and found similar behavior for the modes [29].) The degenerate dipole mode of separate identical voids (d=30nm) gradually splits into two hybridized plasmons as the cavities approach each other (d=1nm) and continue splitting smoothly with increasing overlap (d=-10nm). No signature of singularities near touching are observed. Interestingly, the lowest-energy mode displays large values of the field intensity near the neck and also a redshift that is largest when the degree of overlap becomes significant (d~-a). This mode evolves towards the single-cavity dipole for complete overlap. The other, higher-energy mode loses intensity with increasing overlap and approaches the quadrupole energy at full overlap.

The different behavior in void and particle dimers near touching can be explained by analyzing the electrostatic limit, in which the plasmon frequencies of a system formed by two materials of permittivities ε ₁ and ε ₂ satisfy the relation [35]

where δ takes real values in the (-1,1) range [36], determined by the geometry of the boundaries between both materials, but independent of ε ₁ and ε ₂. In particular, for a separate-particle dimer near touching, the lowest-energy mode corresponds to 1-δ~√d when ε ₂ represents gold and ε ₁ a dielectric surrounding it [32]. This leads to large negative values of ε _gold/ε _diel~-d ^-1/2, which can be reached by the gold permittivity in the near-infrared region. However, the mode of the same symmetry in the complementary void dimer must satisfy ε _gold/ε _diel~-d ^1/2 [i.e., by interchanging ε ₁ and ε ₂ in Eq. (1), while maintaining the same value of δ], so that it is located near the bulk plasmon frequency (Re{ε _gold}=0) for vanishing spacing d, but the Re{ε _gold}=0 condition is not satisfied by gold within the spectral range considered here [37]. Therefore, the singular-transition behavior of gold-particle dimers is averted in the complementary void structures. Metals in general tend to have significant absorption (i.e., large Im{ε}) at the bulk plasmon frequency, so it is doubtful that a singular transition can be observed in void dimers. Although we base these ideas upon an electrostatic analysis, we expect them to apply to the dimer discussed in Fig. 1, even though the voids are large enough to produce important retardation effects (the wavelength of the single-void dipole plasmon inside the silica is only 30% larger than the void diameter). However, the small d limit is governed by the junction region, which becomes small compared to the wavelength for d≈0.

When more voids are added to the structure, new hybridized modes show up in the single-void dipole region (one per void), as shown in Fig. 2. The LDOS at the center of the leftmost sphere in Fig. 2 shows a number of spectral features equal to the number of coupled voids, N. Furthermore, these modes are ordered in energy according to the number of intensity maxima, from 1 (lowest energy) to N (highest energy). This is an indication that the modes are standing waves established by reflection at the ends of the arrays, and with the energy determined by the condition of constructive interference between waves traveling from left to right and viceversa.

Incidentally, the near-field contour plots of Figs. 1 and 2 represent the induced-field intensity for a source dipole placed at the center of the leftmost sphere (see double arrows). Interestingly, the field in different voids of the same structure has similar strength for dimers and trimers, and the intensity distributions are nearly symmetric with respect to the center of the structure, since we are preferentially exciting a single mode, from which the induced field is constructed. In contrast, absorption broadening causes mode overlap in larger (and spectrally denser) structures, so that the intensity distribution is no longer associated to a single mode (see below).

The LDOS at a point other than the center of a single spherical void shows a l=2 mode near 1.9eV (Fig. 2, upper blue curve). This mode leads to hybridized states of dominant l=2 character in void arrays (see weak feature near the same energy in all curves of Fig. 2), and we show below that this is relevant in reducing plasmon propagation.

 

Fig. 2. Plasmon mixing and splitting with increasing number of gold voids in a linear chain of overlapping silica-filled cavities of radius a=240nm and overlap distance d=-a. Black curves represent the LDOS projected along a direction perpendicular to the chain axis at the center of the leftmost void, while blue curves correspond to an off-axis position. The LDOS is normalized as explained in Fig. 1. A vertical offset has been introduced in consecutive spectra for clarity. The near-field intensity associated to different spectral features is shown as created by dipoles with the orientation and position shown by double arrows in the color-plot insets.

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3. Plasmon bands in void arrays

The standing waves observed in Fig. 2 develop into propagating Bloch waves in larger arrays, as those considered in Fig. 3. Namely, N=16 voids with d=-30nm overlap in the left plots and N=30 voids with d=-240nmoverlap in the right plots. For each chain, we plot the field along the axis produced by a dipole placed at the center of the leftmost sphere. The field is represented as a function of photon energy for two different orientations of the dipoles: perpendicular (m=1 azimuthal symmetry) and parallel (m=0) with respect to the void-chain axis. The number of modes equals the number of voids N, as we showed in Sec. 2. Consequently, for large N, there is significant spectral overlap, which results in continuous propagation bands in the N→∞ limit.

For perpendicular polarization (m=1), a depletion in transmission is clearly observed at ~1.9eV for both types of chains [Figs. 3(c),(g)]. This gap originates in absorption produced by the l=2 modes noted at the end of Sec. 2. The topmost plots [Figs. 3(a),(e)] illustrate the strong attenuation at this energy, which limits propagation to just 2–3 voids. At lower energies, there is a wide region of propagation bands, which emerge as standing waves in these finite geometries, with the induced field covering the entire chain cavity. This is illustrated by the induced-field plots of Figs. 3(b),(f). The propagation in Fig. 3(f) is so good that there is little evidence in the plot as of where the exciting dipole is sitting.

For long chains, it is possible to describe these bands in the reduced Brillouin zone defined for the corresponding infinite chains. We have derived information on the bands by following two different procedures: (1) We have Fourier-transformed the field amplitude with respect to the z coordinate (along the axis) in order to obtain a field amplitude in wavevector space q along the chain, with each wavevector referring to a field component of spatial dependence e^iqz. This produces Fourier-space maxima that should correlate with the band structure of the infinite chain [Figs. 3(d),(h)]. (2) We have obtained the infinite-chain dispersion relation from the assumption that the field near the center of the chain is just a combination of Bloch waves, Ae^iqz+Be^-iqz, where A and B do not change from cell to cell, far from the chain ends. By examining three equivalent points in three contiguous voids, we derive values of A, B, and q. The real part of q is represented in Figs. 3(d),(h) versus energy for m=1 (solid blue curves). We obtain good agreement between these two procedures for the lowest-energy m=1 bands. At high energies, bands can be degenerate, so that more propagation terms with different values of q need to be included in the above Bloch expansion.

We have compared these results to the analytical dispersion relation of a flat cylindrical cavity (i.e., a silica wire covered with gold) for a radius such that its volume coincides with the volume inside the chain of voids (broken curves in Fig. 3). The similarity with our previous findings is quite reasonable in both polarizations for the N=30 chain [Figs. 3(h),(p)], which has large void overlap, and therefore, relatively shallow corrugations. In contrast, the N=16 chain, with only 30nm overlap, strongly deviates from the silica wire: the lowest-energy band for m=1 polarization has smaller curvature and is blue-shifted as a result of the sharp corrugations; the effect is even more dramatic for m=0 polarization, which leads to weaker inter-void interaction [29] and produces a nearly-flat low-energy band. The latter resembles the tight-binding regime, in which propagation occurs via hopping between contiguous voids. In contrast, the parabolic bands of the N=30 chain (with larger overlap) are reminiscent of the nearly-free particle regime, in which a two-band model is generally good to explain the small gap opening at the Brillouin zone boundary [38]. In particular, the first gap of Fig. 3(h) is ~0.15eV.

We have represented in Fig. 4 the propagation distance obtained by comparing N=20 and N=30 arrays, as explained in the figure caption. This should coincide with L=1/2Im{q}, although the values of Im{q} retrieved from the analytical Bloch expansions contain an important relative error, because Im{q}≪|q|. Propagation distances of tens of microns are predicted over the transmission bands for both light polarizations. The 1.9 eV gap is clearly resolved for m=1.

 

Fig. 3. Plasmon propagation in long silica-filled void chains buried in gold. Two different chains are considered: 16 voids and d=-30nm overlap (left); 30 voids and d=-240nm (right). All voids have the same radius a=240nm. The upper plots (a-h) correspond to excitation by a dipole located at the center of the leftmost cavity and oriented perpendicularly with respect to that axis (m=1 azimuthal symmetry relative to the chain axis). The lower plots (i-p) correspond to excitation by a dipole oriented along the axis (m=0 symmetry). For each chain and polarization, the plot on the left (c,g,k,o) shows the induced electric field intensity along the axis of the array, whereas the dispersion plot on the right (d,h,l,p) is the intensity of the Fourier transform of the field along the axis, represented for wavevectors within the first Brillouin zone of the infinite chain (-π/P < q <π/P, where P=2a+d is the period). Near-field plots are shown for photon energies corresponding to non-propagating modes (a,e,i,m) and near the maximum of a propagating mode (b,f,j,n). The dashed curves in the dispersion plots stand for the plasmons of an infinite cylindrical cavity with the same volume as the void-chain cavity. The blue curves in the m=1 case are obtained from the Bloch-wave expansion explained in the text.

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Fig. 4. Propagation distance in void chains. The field intensity at the center of the rightmost void in the 30-void waveguide of Fig. 3 (a=-d=240nm) is represented by thin red curves for m=1 (upper plots) and m=0 (lower plots). The same quantity is represented by thin black curves for a 20-void waveguide. The source dipole is placed at the center of the leftmost sphere. The smooth thick curves are made to intersect the average of consecutive maxima and minima in each oscillation of the thin curves. A propagation length (right scale) is determined from the ratio of the average curves for the two void lengths considered [ratio=exp(-L/10a)].

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Fig. 5. Effect of void disorder on plasmon propagation: real-space analysis. The induced electric field intensity is represented in 10-void chains for an exciting dipole situated in the center of the leftmost void and oriented perpendicular (m=1) or parallel (m=0) to the chain axis. The average void radius is a=240nm in all cases and the overlap is d=-a and d=-60nm, as indicated by labels. The upper row of near-field plots corresponds to perfect chains (Δ=0). Lower rows represent the field in chains with different degrees of disordered in the radius a and overlap d (the maximum random displacement is ±Δ; see labels on the left). Two different chain realizations are considered for each non-vanishing value of Δ. The photon energy is taken to match a propagating mode of the perfect chain: 1.45 eV for d=-60nm and 1.48 eV for d=-240nm.

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4. The effect of disorder

Fabrication imperfections in actual structures involve certain degree of disorder. We study in particular the effect of disorder produced by a finite distribution of void diameters and overlap distances in the void arrays. We represent in Fig. 5 the induced field set up by a dipole placed at the center of the leftmost sphere of a N=10 chain, with the dipole oriented either perpendicular (m=1) or parallel (m=0) with respect to the chain axis. Two different average values of the void overlap are considered (240 nm and 60 nm). The degree of disorder is quantified by a random variation of the overlap and void radius, ±Δ.

For m=1, Fig. 5 (left) shows that the waveguides are tolerant to a disorder of ±3nm. Larger values of Δ produce significant depletion of the plasmon signal away from the position of the dipole, and this effect is more pronounced for 60 nm average overlap, because the relative random displacement is larger in this case. For m=0 (Fig. 5, right), the interaction between void modes is weaker, and the case of large overlap is only affected by large degrees of disorder (above ~±10nm), whereas the low-overlap chain shows already short propagation in the perfect chain, that can be facilitated by disorder due to random widening of the windows connecting contiguous voids.

 

Fig. 6. Effect of void disorder on plasmon propagation: reciprocal-space analysis. The dispersion relation of 10-void chains with average radius a=240nm and overlap d=-60nm is represented for different degrees of disorder in a and d (the maximum random displacement is ±Δ; see upper labels). The plots show the squared modulus of the Fourier transform of the on-axis induced electric-field amplitude for m=1 azimuthal symmetry.

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The disorder in real space translates into disorder in the dispersion relation, as shown in Fig. 6 for N=10 chains with 60nm overlap and m=1 polarization. The dispersion diagrams in the figure have been obtained from the Fourier transform of the on-axis electric field. The perfect chain (Δ=0) shows a smooth low-energy band that is broadened by the finite size of the chain cavity. This band is still recognizable with a disorder Δ=±12nm, but larger degrees of disorder produce devastating effects and tend to cause further broadening and flattening of the band, thus revealing the presence of localization at certain defects of the randomized structure. This localization is clearly observable in the near-field plots of Fig. 5 for d=-60nm, m=1, and large degree of disorder.

5. Concluding remarks

The buried plasmonic waveguides that we explore in this work can prevent crosstalk between adjacent structures if they are separated by more than a few times the skin depth (~20 nm in gold over the visible and near infrared), thus suggesting a plausible approach towards highly-integrated plasmon photonics, with a huge range of possible topologies to be addressed by three-dimensional packaging. Furthermore, buried structures automatically suppress radiative losses, which are conspicuous in open structures other than straight waveguides. The addition of periodic corrugations along the waveguides is a natural way of tailoring the propagation bands.

We find that bands of m=±1 symmetry involve large coupling between neighboring voids compared to bands with m=0 symmetry. As a consequence, the latter results in flatter, narrower bands, which are more sensitive to absorption and exhibit shorter propagation distances. These type of symmetries are easily excited by an on-axis dipole oriented either perpendicular or parallel to the chain axis, respectively.

The effects of disorder, which we have quantified by introducing random displacements in the position and radius of the voids, are weaker for chains with larger degree of geometrical overlap between neighboring voids. Such chains can tolerate ±3nm random variations in these geometrical parameters. Interestingly, large disorder leads to flattening of the plasmon bands, encompassing localization of the plasmon signal at specific locations of the waveguides.

Our results provide a characterization of some basic buried-plasmon structures that constitute an excellent playground for versatile optical circuits, compact interferometers, sharp turns, and other nano-optical elements to be integrated in sub-micron dimensions with no radiation losses and minimum crosstalk. Further research into the performance and design of these devices is still needed.