1. Introduction

Dielectric, semiconductor, and metallic nanowires have recently attracted significant attention due to their remarkable optical and electronic properties.^1–6 Among others, these include light guiding in subwavelength diameter metallic wires, negative permeability of pairs of closely spaced metallic rods, and optical plasmonic antenna.^5,6 All these effects rely on surface plasmon modes propagating along a metallic nanowire or nanorod. Progress in nanofabrication techniques has opened up the possibility to fabricate metallic nanorods in macroscopic size arrays with their long axes aligned perpendicular to the substrate.⁷ This geometry of oriented nanorods with a much reduced influence of the underlying substrate impacts both the spectral optical properties of and the spatial field distribution in the array that differ strongly from those of nanorods studied in colloidal solutions^8,9 or placed on substrates.^5,10 Maxwell-Garnet calculations of the optical properties of such arrays have shown that the strongly anisotropic optical behavior can be beneficial for spectroscopic, sensing, and imaging applications.^11–15 One original and fundamental property of nanorod arrays is in the possibility of wide range tuning of the geometrical parameters of the structure so as to control the electromagnetic interaction between the rods. This can be done for example by changing the inter-rod distance and is crucial for determining the reflection, transmission and extinction of the array. It is also important for the development of new applications such as guiding electromagnetic energy on the nanoscale along the chains of nanorods.

In this Letter we report on the experimental and numerical demonstration of the formation of an extended plasmonic mode in arrays of closely-spaced metallic nanorods and discuss the related optical properties. In contrast to widely studied plasmonic geometries involving nanorods on substrates or suspended in solutions, the nanorods studied here have a much higher aspect ratio, are standing on a substrate with their long axes oriented perpendicularly to it, and are placed in a quasi-regular lattice. We show that the optical response of the nanorods when forming such arrays is governed by a collective plasmonic mode resulting from the strong electromagnetic coupling between the dipolar longitudinal plasmon resonance supported by individual nanorods. Both the spectral position of this collective plasmonic resonance and the associated electromagnetic field distribution in the nanorod array strongly depend on the inter-rod coupling strength and is shown to differ from the dipolar response of isolated nanorods from which they originate. These unique properties therefore demonstrate strong potential for the use of oriented metallic nanorod assemblies in the manipulation of light in nanoscale waveguide applications, sensing and nonlinearity enhancement applications, as well as in subwavelength imaging.^16,17

2. Sample fabrication and characterization

Gold nanorod arrays were fabricated by electrodeposition into thin nanoporous anodized aluminum oxide (AAO) templates.⁷ The templates are formed by a self-organization process during the anodization of aluminum (Al) films sputter-deposited onto a multilayered conducting substrate made of a 5 nm thick Au film supporting the Al film, a 10 nm thick tantalum pentoxide base layer and a glass slide. The grown nanorods are embedded in the alumina template and placed in an arrangement determined by the geometry of the alumina template (Fig. 1(a)). Arrays formed by nanorods with a length in the range of 50 nm to 400 nm and diameters in the range of 10 nm to 40 nm were studied (aspect ratios 10 to 30) with inter-rod distances varying from about 100 nm to about 170 nm.

The optical properties of the arrays were investigated by measuring the transmittance T of the sample and plotting the quantity -ln(T) as a function of wavelength for different angles of incidence and polarizations of the incident light. Considering the length of the rods in the [300nm–400 nm] range both scattering and absorption are contributing to T and -ln(T) therefore represents the extinction of the sample.

 

Fig. 1. (a) SEM image of an array of Au nanorods in air. Inset: TEM cross-section of Au nanorods embedded in AAO (the scale bars are 100 nm), (b) zero-order optical extinction spectra for different of angles of incidence. The spectra have been taken for nanorods embedded in an AAO matrix. The nanorods length is 300 nm, diameter is 30 nm, and the inter-rod distance (center-to-center) is about 100 nm. (c) zero-order optical extinction spectra of arrays of Au nanorods in AAO as a function of rod aspect ratio. The nanorods length is 400 nm and the inter-rod distance (center-to-center) is about 100 nm. The curves are labeled according to the rod’s aspect ratio corresponding to rods with diameter ranging from about 15 nm to 30 nm.

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3. Results and discussion

The experimental extinction spectra, measured for an array of Au nanorods embedded in an AAO matrix (n≈1.6) are shown in Fig. 1(b) for different angles of incidence. At normal incidence or also with s-polarised light (the incident electric field vibrates perpendicular to the nanorod long axis), the spectra reveal one single peak at around 520 nm wavelength. This resonance is associated with the transverse (T) plasmonic excitation in the direction normal to the nanorod long axes.¹¹ At oblique incidence and with p-polarized light (incident electric field has a component both along and perpendicular to the nanorod long axes) the spectra contain two peaks: the above mentioned T-mode as well as a longer wavelength resonance, the L-mode associated with a plasmonic excitation polarized along the nanorod long axes. This long-wavelength peak becomes more pronounced at larger angles of incidence for which this longitudinal plasmon resonance is excited more effectively. The angular sensitivity of the spectra of Fig. 1(b) reflects the strong anisotropy of the structure defined by the orientation of the nanorods in the assembly. The general behavior of these two eigenmodes is consistent with the response of the dipolar modes supported by isolated or weakly interacting rods.¹⁸

As can be seen form Fig. 1(b), in addition to its amplitude, the spectral position of the L-mode in nanorod arrays depends on the angle of incidence of the probe light and, with an increase in the angle of incidence, shifts towards shorter wavelengths. This angular dispersion of the mode is shown in Fig. 2(a) for the spectra of Fig. 1(b) and contrasts with the dispersionless behavior of the localized plasmonic modes of isolated plasmonic modes. The angular dispersion of the L-mode was shown to depend on the inter-rod coupling strength as it increases for a decreasing effective inter-rod distance n̄d, where n̄ is the average refractive index of the medium embedding the rods and d is the distance between neighboring nanorods. This property is plotted in Fig. 2(b) and was observed in the spectral dependency of nanorod arrays embedded in alumina with an additional air shell around the nanorods.¹⁷

 

Fig. 2. (a) Angular dispersion of the L-mode as measured from Fig. 1(b) for incidence angles varying from 10° to 50°. The momentum is the projection of the incident photon momentum perpendicular to the rods long axis, (b) dispersion ħΔω/Δk _‖ for the L-mode measured as a function of effective inter-rod distance between the incidence angles of 10° and 50°. The lines in (a) and (b) are a guide to the eye.

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The L-mode resonance wavelength is strongly dependent on both the rod aspect ratio and the distance between the rods in the array. In accordance with the dipolar plasmonic response of nanorods, an increase in the nanorod aspect ratio leads the two resonances to split further apart spectrally with the T-mode undergoing a blue-shift while the L-mode moves towards longer wavelengths. This behavior is illustrated in Fig. 1(c) in the particular case of a varying rod diameter but is also observed when the rod length is varied. The effective inter-rod distance n̄d can also be varied in order to tune the spectral position of the resonances in the array throughout the visible spectrum.^16,17 The spectral position of this resonance for a given nanorod aspect ratio is not, however, consistent with that of the localized surface plasmon resonances of an isolated rod or even for geometries involving dimers, for which the longitudinal resonance is expected to appear at a much longer wavelength.^19–22 For example, an isolated nanorod with a diameter of 30 nm and an aspect ratio of 10 embedded in AAO exhibits a longitudinal dipolar resonance at around 2 µm. The resonance demonstrating similar polarization properties in the array of nanorods of Fig. 1 is observed in the visible region of the spectrum at around 650 nm, therefore strongly blue shifted compared to the expected position of the dipolar resonance. The direction of the L-mode shift as a function of aspect ratio is qualitatively identical to the behavior of the longitudinal dipolar resonance of an isolated nanorod, the value of the shift, being however much smaller than one would expect for an isolated dipolar resonance.¹⁹ Figure 3(a) illustrates the spectral dependency of both the dipolar longitudinal plasmonic resonance supported by an isolated ellipsoid and the L-mode of a nanorod assembly as a function of particle aspect ratio. It is clear from Fig. 3(a) that both the spectral position of the resonances for any given aspect ratio and their qualitative variation as a function of aspect ratio are specific to the resonance considered. First, as mentioned earlier, the L-mode is strongly blue-shifted compared to the dipolar resonance of the isolated particle with a saturation in the red-shift occuring for rods with larger aspect ratios. The most compelling observation made in Fig. 3(a), however, appears at smaller aspect ratios for which the L-mode is located around 550 nm, overlapping the T-mode in the assembly while an isolated particle of similar aspect ratio would show two very distinct dipolar resonances spectrally separated by a few hundred nanometers.¹⁹ This spectral overlap is clearly observed in the spectrum obtained for nanorods with an aspect ratio of 12 in Fig. 1(c) and agrees with the observation of strong electromagnetic coupling.^20,22–24

These experimental observations: the spectral position of the L-mode as a function of rod aspect ratio, nanorod concentration, and inter-rod distance, as well as the dispersive behavior of the resonance, all indicate that although the L-mode in the extinction spectra is related to the plasma oscillations along the long axis of the nanorods, the nature of this mode is more complex. The observed behavior of the L-mode can be explained by the formation of an spatially extended (collective) plasmonic resonance in the array caused by the interaction between the plasmonic modes supported by the nanorods. Particularly, in a simple consideration, this extended mode can be treated as a linear combination of the plasmonic modes of each individual nanorod in the assembly experiencing electromagnetic interaction from neighboring nanorods if k_pl<2π, where k_p is the field extension of the plasmonic mode outside metal and l is the distance between nanorods in the array. Considering the interaction between two nanorods only, the coherent interaction between the longitudinal dipolar modes supported by these rods would result in the formation of two eigenstates representing the symmetric and antisymmetric configurations of the plasmonic dipole moments of the coupled rod system.²⁰ Including more interacting nanorods in this consideration will result in the formation of the band of plasmonic states of the nanorod assembly.

This interaction is schematically shown in the energy diagram of Fig. 3(b) where strong coupling between the nanorods shifts the symmetric state (parallel orientation of the dipole moments of the nanorods) in the short-wavelength spectral range compared to the longitudinal dipolar mode of the isolated nanorods. The mode associated with this latter state is the L-mode studied above. The antisymmetric state corresponds to the antiparallel arrangement of the dipole moments. The dependence of the spectral position of the L-mode on the angle of incidence further emphasizes the collective behavior of this state and its delocalized character in particular.

 

Fig. 3. (a) L-mode wavelength dependence on the aspect ratio for nanorods in the array (circles) and isolated ellipsoids (squares). The L-mode data are from Fig. 1(c). (b) Schematic energy diagram of the plasmonic resonances in the array of nanorods with regards to the isolated modes. Strong electromagnetic coupling of the plasmonic resonances supported by individual nanorods leads to the formation of a collective plasmonic mode. J measures the inter-rod coupling strength. See text for details. (b) Spectra of Re(ε_zz) calculated in the modified Maxwell-Garnet model for assemblies of Au ellipsoids with different concentrations: (squares) 0.08 rods/µm², (circles) 0.3 rods/µm², (triangles) 0.8 rods/µm², and (crosses) 48 rods/µm². The ellipsoids are embedded in AAO, have a major and minor axis length of 300 nm and 25 nm, respectively.

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The spectral dependencies of the effective anisotropic permittivity of the nanorod array calculated in the Maxwell-Garnet model are presented in Fig. 3(c) for different separation between nanorods in the array.¹¹ For clarity of the figure, only the real part of the dielectric constant ε_L seen by light polarized along the long axis of the nanorods, Re(ε_zz), is presented. The behavior of the ε_xx component is described by a single Lorentzian resonance related to the plasmonic mode associated with a short axis of nanorods. This resonance red-shifts with decreased inter-rod distance but more importantly, its localized nature is independent of nanorod separation. From Fig. 3(c) it can be seen that for larger inter-rod distances, when near-field interactions between the rods are negligible (k_pl>2π), a Lorentzian resonance dominates the spectral behavior of the permittivity, which is associated with the longitudinal plasma resonance of the nanorods. In this regime the rods are weakly interacting and the nature of the resonance is that of an isolated nanorod embedded in an effective medium made of AAO and neighboring nanorods. With an increase in the nanorod concentration, the real part of the effective permittivity becomes negative indicating a metallic behavior of the nanorod array as a whole for light polarized along the nanorods’ long axes. For the orthogonal polarization (along the short axis of the rods), however, the real part of the permittivity (not shown) remains positive as for a transparent dielectric. The frequency at which Re(ε_zz) changes sign can be considered as the effective plasma frequency $\tilde{\omega }$_p of the metallic nanorod assembly. It corresponds to the excitation of the symmetric modes supported by the nanorod assembly with dipoles in adjacent nanorods excited in phase.

To assess the spatial distribution of the electromagnetic field associated with the L-mode resonance of the array, we performed full 3D numerical calculations using a Finite Element Method. The geometry used is shown in Fig. 4. The model consists of a periodic arrangement of Au rods placed in a square array and supported by a glass substrate. Considering that the spacing between the rods in the sample is smaller than k ^-1 ₀ (the wave vector of the incident light), the periodic geometry greatly diminishes the amount of memory required to perform the calculations without affecting the generality of the results. The Au rods were modeled using a cylindrical shape with rounded extremities. Their diameter and length were set to 30 nm and 300 nm, respectively, with a dielectric response described by a Drude-Lorentz law fitting ellipsometric measurements on a sputtered Au film.²⁵ The eigenmodes of the nanorods arrays were identified by computing the zero-order transmission spectra of the structures for a TM-polarized illumination field incident on the sample through the substrate with an angle in air varying between 0° (normal incidence) and 80°. The influence of the inter-rod coupling strength was investigated by changing the period of the array from 500 nm to 100 nm. The results are shown in Fig. 4 in which both the magnitude and direction of the power flow, represented by the Pointing vector, is plotted alongside the spatial distribution of the norm of the electric field $\sqrt{{\overrightarrow{E}}^{*}.\overrightarrow{E}}$ for the L-mode. For the larger inter-rod distance of 500 nm, the field distribution around the rods is dominated by the dipolar mode of the longitudinal plasmon resonance occurring at a wavelength of 1948 nm. In this case, the field is localized at the rods extremities with a small braking in symmetry due to the presence of the substrate and illumination conditions. The effect of a reduction in the inter-rod distance leads to a blue-shift of the L-resonance, consistent with both the results from the effective medium calculations and experimental observations. Note that in the conditions of the calculations, strong coupling is responsible for the optical response of the assemblies in Fig. 4(b–d). The most surprising result comes from a dramatic change in both the spatial distribution of the electric field and the power flow in the array. In fact, the electric field concentrates within the assembly, in the middle part of the rods as the inter-rod separation decreases. This evolution corresponds to a simultaneous decrease of the field amplitude at the rods’ extremities. This reconfiguration of the spatial distribution of the field in the array is supported by experimental observations that showed the lack of sensitivity in the spectral position of the array’s extinction, and the L-mode position in particular, when the index of refraction of the superstrate is modified, i.e. when only the rods extremities are subjected to a change in refractive index.¹⁷

 

Fig. 4. L-mode’s electric field distribution in the primitive cell of the nanorod array for different inter-rod distances: (a) 500 nm, (b) 200 nm, (c) 150 nm, and (d) 100 nm calculated using the finite element method. The nanorod length and diameter is 300 nm and 30 nm, respectively. The nanorods are embedded in an AAO matrix (n=6) and supported by a glass substrate (n=1.5). The superstrate is air (n=1). The angle of incidence of TM-polarized probe light is 45°. The arrows in (a)–(d) show the direction of the Poynting vector.

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It is instructive to examine the behavior of the Pointing vector within the assembly as a function of inter-rod coupling strength (Fig. 4 (a–d)). Going from the larger inter-rod distances to the strongly coupled regime, the electromagnetic energy flow around the rods evolves from being localized at the rod extremities to a net energy flux in the layer containing the nanorods in a direction perpendicular to their long axes. Thus, the electromagnetic energy coupled to the array, can effectively propagate from nanorod to nanorod within the assembly. This can be used in applications aimed at guiding light at the nanoscale. ^26–30 In the geometry considered in the calculations, the maximum bandwidth of the L-mode defined as B_L=2|ω ₀-ω_L|, where ω ₀ and ω_L are the resonant frequencies of the dipolar longitudinal mode of the isolated nanorodsand the L-resonance, respectively, is obtained for the smallest inter-rod distance and is estimated to be about 1.4 eV for an inter-rod distance of 100 nm. This corresponds to a maximum group velocity of about 1.7×10⁷ m/s if a linear dependence of the group velocity of the L-mode with bandwidth is assumed. The rate at which energy is lost in the assembly would then correspond to 1.2×10¹³ Hz, considering the 100 meV homogeneous broadening of the L-mode.³¹ The damping rate follows a monotonous and decreasing function of the inter-rod coupling strength, supporting the observation that the formation of the L-mode enables electromagnetic energy to propagate in the layer defined by the nanorod array.

These properties were also investigated experimentally by gradually tuning the inter-rod coupling strength varying the inter-rod distance from about 170nm to about 100nm. This results in a shift of the L-Mode resonance position from 1.74 eV to 2.25 eV.¹⁷ Estimating the longitudinal resonance frequency of the isolated rod to be located around 0.54 eV, the effect of decreasing the inter-rod distance from 170nm to 100nm is to increase the bandwidth from 2.4 eV to 3.4 eV, respectively. Accounting for the bandwidth, the inter-rod distance, as well as the L-mode linewidth, we can estimate the group velocity and propagation length of electromagnetic energy in the nanorod assembly. These quantities are plotted in Fig. 5 as a function of inter-rod distance. The behavior of the group velocity is quite interesting as it undergoes a non-monotonous variation as a function of inter-rod coupling strength with a maximum obtained for an inter-rod distance of about 155nm. In this case, a group velocity of about 5×10⁷m/s is reached, a value that is comparable to group velocities reported for other plasmonic guiding geometries.^32,33 This observation suggests that the guiding properties of the structure are not obtained for maximum inter-rod coupling strength but rather for an optimal distribution of the electromagnetic field of the L-mode. Turning back to Fig. 5, the propagation length of electromagnetic energy in the rod assembly is shown to decrease from a value of about 2.7 µm estimated for coupling strengths varying between 1.2eV and 1.4eV, to a value of 1.66 µm for the maximum coupling strength of 1.7eV. We attribute this increased damping to losses in the L-mode due to an increased overlap between the L-mode resonance and the inter-band band edge of gold. This behavior was not observed in the numerical calculations for the array parameters for which the L-mode is always situated well in the infra-red spectral range.

 

Fig. 5. Group velocity (dotted labels) and propagation length (square labels) of electromagnetic energy of the L-Mode as a function of inter-rod distance, a decreasing distance corresponds to an increasing coupling strength.

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Finally, defining the inter-rod coupling strength for the L-mode as 0 J=ħ|ω ₀-ω_L|, coupling strengths on the order of 1eV are typically observed in assemblies of nanorods but more importantly, can be easily varied in order to control the degree of localization of electromagnetic energy within the assembly. For comparison, coupling strength on the order of 60 meV to 200 meV, depending on polarization, have been measured for other plasmonic structures,^20,23 while calculations show that coupling strength comparable to our observations can be reached in more complex plasmonic systems made of chains of interacting plasmonic nanoparticles.³⁴

4. Conclusion

In conclusion, we studied the optical properties of closely spaced and oriented Au nanorods standing on a transparent substrate. The optical response of such nanorod assemblies exhibits two plasmonic resonances in the visible spectral range with perpendicular polarizations: the T-mode and the L-mode with an electric field polarized perpendicular and parallel to the long axis of the rods, respectively. The L-mode was shown to exhibit peculiar spectral and spatial properties reflecting electromagnetic interactions between the nanorods in the array. In particular, strong electromagnetic coupling between the dipolar longitudinal resonance supported by the nanorods when isolated, leads to the formation of hybrid plasmonic modes of similar polarization. The high-frequency L-mode was shown to be coupled to photons and demonstrated a dramatic modification of the associated spatial field distribution compared to the dipolar plasmonic response with an electric field concentrated in the middle part of the rod layer. Both the field distribution and the dispersion of this mode are indicative of the spatial delocalization of electromagnetic energy coupled to this state and numerical calculations were used to further estimate the guiding characteristics of the L-mode for potential applications in electromagnetic guiding and manipulation applications on the nanometric scale.^35,36 These applications are particularly relevant since the recent demonstrations of the possibility to produce laterally confined assemblies of nanorods with single nanorod resolution. Such a structure also represents also a strongly anisotropic material with either positive or negative values of the permittivity for different polarization directions. It may therefore find numerous applications in the design of metamaterials with specific sensing and imaging capabilities providing a route for the development of materials with indefinite dielectric tensor, highly phase-matched surfaces as well as near-zero refractive index materials.