1. Introduction

Localized surface plasmons (LSP) are collective oscillations of conduction band electrons that are responsible for the unique optical properties of noble metal nanostructures [1–3]. Light incident on noble metal nanostructures excites LSP resonances (LSPRs) and the resulting coherent charge density oscillations generate intense oscillating E-fields in the vicinity of the nanoparticles (NPs). While individual NPs are already effective nanoscale antennas in the optical range of the electromagnetic spectrum, their performance and functionality can be optimized and extended by integrating multiple NPs into discrete structures or entire arrays [4–6]. Two coupling regimes determine the properties of these higher order noble metal NP structures. Quasistatic near-field interactions on short interparticle separations (< λ/2π) facilitate an efficient localization of the incident E-field into electromagnetic hot-spots located in the gap between two NPs. Consequently, nanoscale plasmonic antennas have large antenna apertures [6–8]. At the same time, the high density of local optical states (LDOS) associated with plasmon resonances with intrinsically small mode volumes boosts the radiative rate of quantum emitters [9–12]. NPs separated by longer distances interact via far-field diffractive coupling. Rayleigh anomalies that arise in regular NP arrays at the diffraction edge, when a diffraction order radiates at a grazing angle into the array plane, are of particular interest since under these conditions LSPRs and in-plane diffracted photonic modes can interact synergistically [13]. If the photonic mode overlaps with the LSPR, photonic-plasmonic hybrid modes with extraordinarily sharp lineshapes can be generated [13–16]. These collective array resonances are commonly referred to as surface lattice resonances (SLRs) [17,18]. The E-field experienced by an individual NP in an array is determined by the sum of the incident E-field and the E-field re-radiated by all other NPs in the array at the location of the NP. . Careful analyses of SLRs in one- and two-dimensional chains of NPs revealed that the narrow lineshapes are determined through wavelength dependent dipolar coupling [19] and that the observed narrow spectral features are the result of a positive singularity in the real part of the dipole sum [15,20,21]. Another beneficial aspect of the underlying delocalized modes, besides their sharp spectral linewidth, is an increase of E-field intensity and LDOS between the array defining metal NPs. This effect in particular has created interest in collective modes as a design strategy for more homogeneously “hot” plasmonic surfaces [22–27].

Most studies in the area of collective photonic-plasmonic arrays have so far focused on lattices comprising spherical NPs [10,15,28], nanodisks [26,29–31] or nanorods [17,18,23,32,33] made of gold or silver as building blocks. Some studies also considered nanoholes [34] and nanopillars [35]. Previous studies by Christ et al. have demonstrated in multilayer nanowire arrays that near-field coupling is an important additional control parameter in plasmonic lattices [36]. Recent advancements in the area of template guided self-assembly and NP manipulation techniques make it now possible to integrate clusters of electromagnetically strongly coupled NPs into regular arrays [10,37,38] or to combine noble metal NP clusters with high refractive index dielectric NPs into mixed metal-dielectric hybrid arrays [39]. The aim of this manuscript is to characterize the effect of this additional configurational variability on collective mode formation and to elucidate general design guidelines for photonic-plasmonic resonances with defined near- and far-field responses.

2. Results and discussion

2.1 Effect of NP clustering on SLRs

We simulated two-dimensional arrays of 60 nm diameter NPs embedded in a homogeneous ambient medium of refractive index n _r = 1.50 through finite difference time domain (FDTD) simulations and using periodic boundary conditions [40]. We used the gold dielectric function by Johnson and Christy [41]. To quantify the effect of NP clustering we replaced the individual NPs with NP dimers, which represent the simplest possible model to capture the essential physics of the electromagnetic coupling in NP clusters. We chose an interparticle separation of s = 4 nm as a conservative estimate of the interparticle spacing in NP cluster arrays (NCAs) [10,37,38,42,43] obtained through template guided self-assembly [44]. One class of material of interest and our simulation model is presented in Fig. 1.We assumed a normal incidence of the light on the sample and we used linearly polarized light. In the case of the dimers, the E-vector was oriented along the x-axis and pointed parallel to the long dimer axis. We systematically varied the lattice spacing, Λ, in the array and calculated extinction spectra and E-field intensity enhancements for different values of Λ. Figure 2(a) and b contains the extinction spectra as function of the modulus of the lattice vector k = 2π/Λ, k = 0.79 – 1.79 × 10⁷ rad/m (corresponding to Λ = 800 – 350 nm), for square arrays of 60 nm gold NP monomers and dimers, respectively. The plots also contain the diffraction edges, k’(m, n), of the square array as solid lines, which are given by the Bragg diffraction condition as:

 

Fig. 1 (a) Schematic illustration of amixed metal-dielectric NP hybrid array. (b) SEM of fabricated array with Λ = 1100nm. Scale bar = 500nm.

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Fig. 2 Extinction (a, c, e) and E-field intensity enhancement (b, d, f) as function of the lattice vector k in gold NP square monomer (a, b), square dimer (c, d), and centered square dimer (e, f) arrays. Spectra of isolated monomers and dimers are added as separate column next to the respective dispersion diagram in (a-d). The isolated monomer spectra in (a) and (c) were scaled by a factor of 1000 and 500, respectively. The extinction was calculated as (1-T)A, where T is the transmission and A is the unit cell area (m²). The E-field intensity for monomers is evaluated 2 nm away from the NP surface along the E-field polarization axis. For the dimer it is evaluated in the center of the gap. The diffraction edges of the (0, 1), (1, 1), (0, 2), (1, 2) modes are included as red, yellow, white, and green lines, respectively. The unit cell geometries of corresponding arrays are shown on the right side of each row.

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The LSPR energy of 2.19 eV for the monomer and 1.90 eV for the dimer are indicated in Fig. 2. In the investigated k-range the monomer LSPR (Fig. 2(a)) interacts with the (0, 2), (1, 1), and (0, 1) Rayleigh anomalies. Once an in-plane diffracted order approaches the LSPR, increased extinction along the diffraction edge (below the peak LSPR energy) marks the onset of photonic-plasmonic coupling. The coupling enhances with increasing k, but finally the resonance detaches from the diffraction edge to converge against the LSPR wavelength. In case of the (0, 1) SLR, the E-field of the incident plane wave and the dipolar LSP mode responsible for the in-plane radiation have a parallel alignment, which results in a particularly efficient SLR excitation. We note in passing that the SLRs exhibit a spectrally sharp character for energies that are red-shifted relative to the LSPR and that the spectra gradually broaden as their energy approaches the LSPR. For energies higher than the LSPR energy, photonic and plasmonic modes cease to couple since for energetic reasons as well as for phase-match considerations (the phase of the plasmon in the individual NP monomer and dimers shifts by π when the frequency is scanned across the resonance), the plasmons do not excite SLRs above the LSPR energy [14].

The extinction spectra of the dimer square array (Fig. 2(c)) show overall the same trends as the monomer but the longitudinal plasmon resonance is red-shifted to 1.89 eV due to plasmon hybridization [46–50]. A noteworthy beneficial aspect of the resonance red-shift is that it is accompanied by a decrease in the imaginary part of the gold dielectric function, which results in a reduction of the dissipative losses in the metal [5]. A second anticipated result of the NP dimerization is a strong increase in the provided E-field enhancement due to plasmon coupling between the two NPs of the dimer [51]. The calculated peak E-field intensity enhancement spectra for the rectangular monomer (Fig. 2(b)) and dimer (Fig. 2(d)) arrays confirm that the E-field intensities in the dimer array are 2 orders of magnitude higher than in the monomer array. The simulations in Fig. 2 include the far- and near-field spectra of isolated NPs and dimers. A comparison of the near-field spectra of the isolated building blocks and the resulting peak E-field intensity enhancement in the arrays reveals that both collective mode formation and direct near-field coupling contribute to the E-field intensity enhancement in the dimer array. The array effect is, however, weaker than the clustering effect, which underlines the value of NP clusters as building block for plasmonic metasurfaces and arrays in which phase, amplitude, and powerflow can be varied through rational design of the array morphology [7,37,38,42,52,53].

In Fig. 3(a) we plot the E-field intensity enhancement map for the unit cell of the Λ = 623 nm (k = 1.01 × 10⁷ rad/m) square array at the energy of the highest peak E-field enhancement (1.83eV). The E-field was evaluated in the equatorial plane of the metal NPs. For E-field intensity enhancement maps at additional energies please refer to Fig. 6. The Bragg grating vector for the (1, 1) SLR in the square array is oriented along the xy-diagonal direction. The maps show that a resonant excitation of this mode creates E-field intensity enhancement not only at the location of the NP dimers but also weaker enhancement in the center of the square array unit cell. In the next step, we determined the impact of placing an additional dimer in this central position. Figure 2(e) and f show the extinction and E-field intensity enhancement spectra for the resulting centered square (rhombic) array. The geometric transformation of the square array into a centered square array changes the geometric conditions for in-plane diffraction and the position of the diffraction edges (Fig. 7). In the k-range of interest the LSPR couples primarily to the (0, 1) Rayleigh anomaly (solid red line) of the centered square array. The integration of the additional NP dimer achieves an additional enhancement of the peak E-field intensity associated with the SLR by a factor of approximately 3. A map of the E-field intensity enhancement in the unit cell of the centered square array at 1.83 eV is included in Fig. 3(b). The addition of the new dimer does not only increase the peak E-field intensity enhancement in the gap junction between the NPs, but - consistent with the delocalized nature of the SLR - also enhances the E-field intensity in the lattice space between the dimers. We will analyze this effect in more details in the subsequent section.

 

Fig. 3 Electric field intensity enhancement map for the square array (a) and centered square array (b) of gold NP dimers. Normalized Poynting vectors are included as cyan arrows. All maps are evaluated with Λ = 623nm (k = 1.01 × 10⁷ rad/m) and at an energy of 1.83eV.

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2.2 Incorporation of dielectric NPs

So far, we have limited our analysis to homogeneous arrays in which metal NPs or metal NP clusters provide both LSPRs as well as delocalized photonic modes due to diffractive coupling. The latter is, however, not unique to metal NP arrays. In fact, light trapping in dielectric NPs has less rigid requirements with regard to particle size and geometric array conditions [54,55]. Furthermore, dielectric NPs make it possible to avoid high dissipative losses in metal NPs at optical frequencies. However, dielectric NPs do not achieve the same level of light localization as is possible with metallic nanostructures, in particular, noble metal NP clusters. One approach to overcome this dilemma is to synergistically combine dielectric NPs and noble metallic NPs in one array in which the dielectric NPs define the grating and noble metal NP clusters provide efficient E-field localization [39,56].

We investigated the electromagnetic interactions in a hybrid array formed by a square array of 250 nm diameter dielectric NPs containing dimers of 60 nm gold NPs in the center of the unit cell (Fig. 4(c), (d)).The refractive index of the dielectric NPs was chosen as n_r = 2.40 to resemble TiO₂ NPs and the interparticle separation of the gold NPs was again 4 nm. We emphasize that similar geometries have already been experimentally realized through template guided self-assembly strategies [39].

 

Fig. 4 Extinction (a, c) and E-field intensity enhancement (b, d) spectra of square arrays of dielectric (n_r = 2.40) NPs (a, b) and metal-dielectric (c, d) hybrid array comprising both gold NP dimers and dielectric NPs. In the hybrid structure the E-field intensity was evaluated in the center gold NP dimer gap. In the dielectric array, the E-field intensity was evaluated in the center of the unit cell at the same z-coordinate as in the hybrid array. The diffraction edges of the (0, 1), (1, 1), (0, 2), (1, 2) modes are included as red, yellow, white, and green lines, respectively. The unit cell geometries of corresponding arrays are shown on the right side of each row.

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Figure 4 contains the simulated extinction spectra for both the dielectric NP (Fig. 4(a)) and the metal-dielectric hybrid array (Fig. 4(c)) as function of k. The general structure of the dispersion diagram of the hybrid lattice in terms of the number and position of the diffraction edges resembles that of the regular square array. However, while the dielectric NP array exhibit extinction only along the edges of the various diffraction orders, the extinction spectrum of the hybrid array shows the formation of SLRs. Strikingly, the hybrid array contains electromagnetic features from two interdigitated square arrays with two sets of distinct lattice sites. Unlike for the metal NP arrays, the hybrid arrays shows extinction (albeit weaker) along the diffraction edge even for energies that lie above the LSPR energy due to contributions from the dielectric NP sub-array. Several features of the spectra indicate electromagnetic interactions between the metal and dielectric sub-arrays. For instance, in the hybrid array the extinction of the hybrid mode associated with the (1, 1) Rayleigh anomaly has gained in intensity compared to that associated with the (0, 1) Rayleigh anomaly, while in both the square array metal NP dimer (Fig. 2(c)) and the dielectric NP (Fig. 4(a)) array the (0, 1) mode is clearly the dominating feature. The propagation direction of the (1, 1) SLR in the hybrid array points along the dielectric – gold NP dimer diagonal, so that the light scattered into the array plane from high refractive index dielectric NPs achieves an effective excitation of the metal NP clusters. Interestingly, the extinction associated with the (0, 1) diffraction edge at energies above the gold LSPR energy is also noticeably higher in the hybrid array than in the corresponding all-dielectric NP array due to increased absorption in the array plane in the presence of the metal NPs.

In Fig. 4(b) and d we present the E-field intensity enhancement spectra as function of k for the dielectric NP square array and for the hybrid array. The photonic modes of the dielectric NP arrays generate only modest E-field intensity enhancements. This behavior changes dramatically upon integration of a NP dimer into the unit cell of the dielectric NP square array. The E-field intensity enhancement tracks the SLRs and the relative intensities are consistent with the far-field extinction data, in particular, the peak E-field intensity enhancement is overall higher for the (1, 1) hybrid mode than for the (0, 1) mode. Interestingly, the peak E-field intensity enhancement in the hybrid array is even higher than that of the centered square array of noble metal NP dimers in Fig. 2(f). The gain in peak E-field enhancement in the hybrid array confirms a synergistic interplay between light trapping in the array plane through the high refractive index dielectric component and E-field enhancement provided by the plasmonic clusters.

Photonic-plasmonic coupling in hybrid arrays has benefits other than boosting the peak E-field intensity enhancement. Figure 5(a) contains a map of the E-field intensity enhancement in the hybrid array unit cell with Λ = 623nm (k = 1.01 × 10⁷ rad/m) at 1.83 eV (maps for additional energies are provided in Fig. 8). While the peak E-field intensity enhancement is confined to the immediate vicinity of the gold NP dimer, the map shows increased enhancement of up to one order of magnitude in the array plane at locations remote from the metal NPs. Maximizing the E-field penetration in a large fraction of the ambient medium is highly desirable for many practical applications of plasmonic arrays and metasurfaces. In Fig. 5(b) we compare the cumulative distribution plots of the E-field intensity enhancement for the unit cells of the hybrid array, the NP dimer square array and the dimer centered square array for Λ = 623nm (k = 1.01 × 10⁷ rad/m) at 1.83 eV. In all three cases, we did not include the immediate vicinity of the metal NPs since our focus is here not on the hot-spots provided by the metal NPs but on the average E-field enhancement in the remaining unit cell. The distribution plots reveal that the average E-field enhancement throughout the entire unit cell is systematically higher in the hybrid array than in both gold NP dimer arrays. Considering that the hybrid arrays achieve higher E-field enhancement with half as many gold NPs as the centered square array, we conclude that the hybrid array is an effective design approach for boosting the E-field intensity in the ambient medium surrounding the integrated metal NPs.

 

Fig. 5 (a) Electric field intensity enhancement map and Poynting vector map for hybrid array. (b) Cumulative distribution plots of the E-field intensity in one unit cell for 3 types of array. All Figs. are evaluated with Λ = 623nm (k = 1.01 × 10⁷ rad/m) and energy = 1.83eV.

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Fig. 6 Electric field intensity enhancement map for the gold dimer square array evaluated with Λ = 623nm (k = 1.01 × 10⁷ rad/m) at energy = (a)2.25eV, (b) 2.07eV, (c) 1.91eV, (d) 1.77eV, (e) 1.65eV, (e) 1.55eV.

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2.3 Analysis of the energy flow in hybrid arrays

In conventional metal NP arrays the control over spatial and spectral light intensity relies on interference, which is governed by the relative phase relationships between the different emitters in the array. In a grating of identical metal NPs, the array geometry, which determines the separation between individual particles, is the only parameter available to control phase relationships. Template-guided self-assembly strategies provide now the ability to include individual NPs or NP clusters with red-shifted resonance wavelengths, as well as NPs of different chemical composition (e.g. metal vs. dielectric) at pre-defined locations in one array [39]. The association level and chemical composition of the NPs influence the local resonance condition and, thus, the phase. This is important as the time-averaged Poynting vector S = ½ Re[E × H*], which describes the optical powerflow density, is proportional to the phase gradient [57,58]. The impact of a change in the chemical composition on a subset of lattice sites on the optical powerflow is illustrated by comparing the Poynting vector maps for the centered rectangular gold NP dimer array (Fig. 3(b)) and the hybrid array (Fig. 5(a)). The Figs contain the Poynting vectors in the xy-plane of the unit cell for one selected grating period (Λ = 623nm, k = 1.01 × 10⁷ rad/m) and energy (1.83 eV) corresponding to the peak E-field intensity in the hybrid array. The replacement of the gold NP dimers on the edges of the unit cells with high refractive index dielectric NPs in the hybrid array has substantial effects on the Poynting vector map and leads to a much higher degree of topological complexity than observed in the gold NP dimer array. In particular, the hybrid array contains optical vortices around locations of destructive interference where the field intensity is zero and the phase, consequently, undetermined [59,60]. Analogous locations of two-dimensional circulating optical powerflow are absent in the Poynting vector map of the metal dimer arrays in Fig. 3.

Our simulations show that hybrid NP arrays provide a viable approach for generating a high density of optical vortices with lateral dimensions of < 300 nm. The fact that some of these optical vortices exist outside of the NPs where they can interact with the ambient medium, makes this strategy very appealing for enhancing light-matter interactions through the circulating optical power in the vortex as well as for controlling the spatial distribution of amplitude and phase in the entire plane of the array. This gain in functionality could prove useful in future mixed metal-dielectric (“optoplasmonic”) metasurfaces.

3. Conclusion

The coupling of in-plane diffracted photonic modes and LSPRs in regular arrays of noble metal NPs achieves the formation of photonic-plasmonic SLRs that generate a cascaded E-field enhancement in the vicinity of the NPs and – due to the contribution from delocalized photonic modes – an overall enhanced E-field intensity throughout the entire array. Different from cascaded E-field enhancement in discrete NP clusters [61], the arrays of NP clusters investigated in this manuscript sustain electromagnetic coupling interactions simultaneously on multiple length scales. With the continuing perfection of nanofabrication approaches, the structural complexity of electromagnetic materials continues to increase. It is now possible to integrate NPs of different association level and chemical composition into regular arrays [10,37,38]. In this manuscript we have systematically investigated the impact of this compositional variability on the SLRs in two dimensional NP arrays assembled i.) from individual 60 nm diameter gold NPs, ii.) from dimers of electromagnetically strongly coupled NPs, and iii.) from both high refractive index dielectric NPs and gold NP dimers at separate lattice locations. Our simulations provide a vivid demonstration of the advantage of NP clusters (plasmonic metamolecules) over individual NPs as building blocks for arrays that maximize both peak and average E-field intensity. Interestingly, the combination of quasi loss-less dielectric NPs and gold NP dimers in one array can further enhance the E-field intensity at the plasmonic hot-spots and throughout the array. Furthermore, we have demonstrated that the ability to pattern the phase landscape through both the array morphology and the chemical composition at defined lattice sites provides important additional degrees of freedom for controlling the optical energy flow in two-dimensional arrays, which are absent in conventional arrays. We anticipate that the ability to pattern the phase and amplitude of optical fields in mixed metal-dielectric NP arrays will lead to new applications in photonics and plasmonics.

Appendix materials

1. Electric field intensity enhancement map for the gold dimer square array

2. Calculation for the shift of diffraction edges

An in-plane propagation wave vector k can be written as _{$k=k_{\left|\right|}+G$}, where k_|| is the in-plane component of the incident wave vector (_{$k_{\left|\right|}=k_0\sin (\theta )$}; θ: incident polar angle). G is the Bragg grating vector. Because we choose the incident normal to the array plane, we have_{$k=G=i b_1 +j b_2$}, where i, j are integers and b₁ and b₂ are reciprocal lattice vectors.

The square lattice array has primitive lattice vectors a₁ and a₂ that can be expressed through unit vectors n_x and n_y oriented along the x- and y- axis, respectively.

We obtain the reciprocal lattice vectors b₁ and b₂ using the relationship a_i•b_j = 2πδ_ij:As a result, when mode i = 0, j = 1 or (0, 1) is excited, we get_{$G=\frac{2\text{π}}{\text{Λ}} n_y$}, which means that k points along the y-axis. If mode (1, 1) is excited, we get_{$G=\frac{2\pi }{\Lambda } n_x+\frac{2\pi }{\Lambda } n_y$}, which means k points along the xy diagonal.

For the rhombic array in Fig. 7(b), a similar treatment leads to the following expressions:As shown in Fig. 7(b), when mode (0, 1) is excited k points along the xy direction with_{$k=G=\frac{\sqrt{2}\text{π}}{\text{Λ}} (n_x+n_y)$}. When mode (1, 1) is excited, k points along the x axis with_{$k=G=\frac{2\sqrt{2}\text{π}}{\text{Λ}} n_x$}.

 

Fig. 7 Schematic real space of (a) square lattice and (b) rhombic lattice.

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3. Electric field intensity enhancement map for the hybrid array

 

Fig. 8 Electric field intensity enhancement map for the hybrid array evaluated with Λ = 623nm (k = 1.01 × 10⁷ rad/m) at energy = (a)2.25eV, (b) 2.07eV, (c) 1.91eV, (d) 1.77eV, (e) 1.65eV, (e) 1.55eV.

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4. FDTD simulation

The numerical modeling of all structures was performed with Lumerical [41]. The gold NPs were spheres with diameters of 60 nm. We used the gold dielectric function of Johnson Christy [42]. The edge-to-edge separation in NP dimers was kept constant at 4nm. The dielectric nanoparticles were spheres of a diameter of 250 nm with a refractive index = 2.40. Structures were embedded in a homogeneous media (n_r = 1.50). All structures were illuminated by a plane wave with a k vector normal to the array plane. The incident E-field vector was chosen to point along the x-axis, which coincided with the long axis in NP dimers. Periodic boundary conditions were applied along x- and y- axes and a perfect metal layer boundary condition was applied for the z-axis. The mesh grid size was chosen as 1nm in the surrounding of the gold NPs.

Acknowledgments

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DOE DE-SC0010679.

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