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For two-dimensional (2D) arrays of metallic nanorods arranged perpendicular to a substrate several methods have been proposed to determine the electromagnetic near-field distribution and the surface plasmon resonances, but an analytical approach to explain all optical features on the nanometer length scale has been missing to date. To fill this gap, we demonstrate here that the field distribution in such arrays can be understood on the basis of surface plasmon polaritons (SPPs) that propagate along the nanorods and form standing waves. Notably, SPPs couple laterally through their optical near fields, giving rise to collective surface plasmon (CSP) effects. Using the dispersion relation of such CSPs, we deduce the condition of standing-wave formation, which enables us to successfully predict several features, such as eigenmodes and resonances. As one such property and potential application, we show both theoretically and in an experiment that CSP propagation allows for polarization conversion and optical filtering in 2D nanorod arrays. Hence, these arrays are promising candidates for manipulating the light polarization on the nanometer length scale.
©2008 Optical Society of America
1. Introduction
Nanostructures exhibiting strong plasmonic properties bear potential tunability in various areas of applications such as sensors [1–3], metamaterials [4–6], optoelectronics [7–9], and sub-wavelength photonics [10,11]. In the case of two-dimensional (2D) arrays of metallic nanorods arranged in the x-y plane as used here (see Fig. 1(a)), the large optical anisotropy makes the optical characteristics sensitively dependent on the angle of incidence and polarization state of the incoming light [12–15]. Highly localized fields inside these structures make them candidates for surface-enhanced Raman spectroscopy [16], for investigating exciton-plasmon coupling [17], or for tailoring the light transmission over short distances [7]. Similar to single nanorods, for which the surface plasmon resonance (SPR) is found to be strongly dependent on the geometry and dielectric constant of the surrounding mediumem [18,19], active tuning of the transmittance of such 2D nanorod arrays by varying em [20] becomes feasible as well.
While the optical properties of an isolated nanorod antenna seem to be well understood both theoretically and experimentally, nanorod arrays (see Fig. 1(a)) still need particular attention. The outcome of our treatment presented here is a better theoretical understanding of such 2D metallic nanorod arrays which directly allows optimizing them for specific applications as those mentioned above, but furthermore might also trigger novel approaches in nanooptics.
To date, the theoretical treatment of electromagnetic fields inside nanorod arrays has mostly been carried out by coupling localized surface plasmons (LSPs) [13]. The origin of this approach is the very popular description of an isolated nanorod [18], in which the nanorod is idealized as an elongated spheroid in the quasi-electrostatic approximation of the Mie theory [21]. In an array of nanorods, the LSPs of the individual nanorods can be viewed as harmonic oscillators that couple to each other, leading to a resonance shift as well as a changed near-field distribution. In order to describe the coupling, an adapted Maxwell-Garnett theory is normally used [13, 15]. Wurtz et al. [22] recently postulated that the dipolar-coupled LSPs may form an extended plasmonic mode propagating perpendicularly to the nanorod long axis, i.e., in the x-y plane.
Nevertheless, using this dipolar coupling of LSPs faces two problems. Firstly, Bryant et al. recently showed that the description of single nanorods using the quasi-static approximation of theMie theory is inaccurate in many respects [19], in particular for high aspect ratios. Since the Maxwell-Garnett theory is based on this approximation, it suffers from the same problematic. Secondly, the coupling of higher-order modes [14] is essential for explaining all optical properties in such metallic arrays. Since it is not possible to model these modes with the Maxwell-Garnett theory, it seems that the dipolar coupling of LSPs is inaccurate. Hence a novel approach to interpreting plasmons in nanorod arrays is needed.
Fig. 1. (a) Micrograph of a gold nanorod structure, and (b) model of the nanorod array combined with a depiction of the lateral field distribution of a collective surface plasmon (CSP).
Within this paper, we will introduce a novel analytical description of plasmons in nanorod arrays in order to improve the fundamental understanding. This will allow us to predict the surface plasmon resonances — including higher-order resonances — on an analytical basis for the first time without the need of fitting parameters. Furthermore, it is then possible to investigate the impact of the different geometrical parameters (arrangement, nanorod separation and length) on the optical properties without conducting time-consuming numerical calculations. To achieve that we will proceed as follows:
On the basis of surface plasmon polaritons (SPPs) propagating along the surface of an individual nanorod in z direction (see Fig. 1(b)), we firstly show that for nanorod arrays the SPPs couple laterally through near-field interaction, thereby forming collective surface plasmons (CSPs). Secondly, we find that CSPs propagating parallel to the nanorod long axis form standing waves when the nanorod length ℓ becomes finite. The resonance frequencies of these standing waves as calculated within this analytical approach are in good agreement with 3D numerical calculations using the method of multiple multipoles (MMP) [23, 24]. Thirdly, the excitation of CSPs is very phase sensitive and thus dependent on the angle of incidence, while it also depends on the polarization state of the incoming light. CSPs thus allow tuning the optical transmission across 2D nanorod arrays over a broad wavelength range, as will be demonstrated both experimentally and theoretically using ellipsometric measurements with crossed polarizers and the 3D MMP modeling, respectively.
2. Dispersion relation of nanowire arrays
We start our calculations by considering a single infinitely long (ℓ→∞) gold nanowire of diameter 2r, embedded in a homogeneous medium with dielectric constant εm, say anodized aluminum oxide (AAO). Using literature values for εAu [25] and a simple 2D MMP model, we are able to compute the dispersion relation for such an isolated nanowire. Our results show that SPPs propagate along the wire in z direction with their electric field being bound to the nanowire. For larger radii r, the dispersion relation approaches that of a planar gold-air interface, while for small r the magnitude of the k⃗ vector increases significantly for a given frequency. Fig. 2 illustrates the case of a Au nanowire of 25 nm in diameter (marked with +). A clear separation from the light line (×) is visible, which exhibits no crossing with the calculated dispersion curve.
Fig. 2. Dispersion relation of a SPP/CSP on infinitely long Au nanowires (diameter 25 nm, embedded in nAAO=1.6, single nanowire or array of nanowires).
In a next step, we arrange such infinitely long nanowires in a square lattice with a variable center-to-center wire separation d ranging between 60 and 200 nm (see Fig. 2). The SPPs localized on the individual wires will now couple and thus become delocalized. It is appropriate to call these changed SPPs collective surface plasmons (CSPs). In general, plasmons in these structures may propagate both parallel and normal to the nanowire long axis similar to guided modes [26] or as reported by [22]. However, we found that restricting ourselves to the parallel-propagating plasmons is sufficient for obtaining accurate values for the resonance frequencies, as shown later. Furthermore, calculations of the energy flux with the 3D MMP model suggest that the flux in the x-y plane vanishes at the first resonance. The full investigation of the energy flux for the different modes will be addressed in a further publication. Hence, for the sake of simplicity, we set kx and ky to zero, so that the CSPs possess a wave vector component kz only. Note that, for kx and ky unequal to zero, the numerical eigenvalue search needed to obtain the dispersion relation within the 2D MMP model becomes very challenging and error-prone.
For d≥200 nm, the dispersion shows no changes with respect to the aforementioned isolated gold nanowire; hence, the nanowire array can be described by the solution of the isolated wire alone as shown in Fig. 2. For a reduced d, however, significant changes become evident; the slope of the dispersion curve drastically decreases for lower kz, becoming quasi-linear for d=60 nm. This means that the group velocity of the CSPs decreases with decreasing d. We find the same behavior when arranging the nanowires in a hexagonal fashion rather than using the square arrangement from above. As seen in Fig. 2, the quasi-linear behavior for d=60 nm is maintained, while the dispersion curve shifts to even higher energies or frequencies, as the fractional packing density is larger in the hexagonal arrangement.
3. Coupled surface plasmons: standing waves in nanorod arrays
Let us consider now an array of nanorods of finite length ℓ. The CSPs will be excited e.g. at the top of the nanorods by a p-polarized incident plane wave (k⃗inc). This is in fact possible, because for finite ℓ, the translational symmetry along the z axis is broken — hence no crossing with the light line is required — and momentum can be transferred. Moreover, optical plane-wave excitation may be accomplished also at angles α varying between 0° and 90° with respect to the z axis. The excited CSP propagates down the nanorods, experiencing partial reflection and transmission at the bottom face. While the reflected portion propagates back along the rods, the transmitted part of the CSP decays to light and constitutes a transmitted plane wave with wave vector k⃗trans (see Fig. 4). Note that the transmitted wave leaves the array structure at exactly the same angle α at which the exciting wave was incident, since all CSPs have the same group velocity when traveling through the array of uniform thickness ℓ. Nevertheless, note also that the phase lag Δφ of the transmitted with respect to the incident wave depends on several factors, such as the nanorod length ℓ, the angle of incidence α, as well as the wavelength. That part of the CSP that is reflected propagates up the nanorods, is partly reflected at the top face as well, propagates down the nanorods again, and is superposed on the first CSP. This effect is similar to multiple reflections in a dielectric slab. Hence, the nanorod array acts as a resonator promoting the formation of a standing-wave pattern. The standing-wave condition for such a resonator reads
where m is an odd integer, θ a specific phase jump that occurs upon reflection of the CSPs at the extremities of the nanorods, and ℓ the length of the nanorods. Inserting kz into the dispersion relation of Fig. 2 then directly yields the resonance wavelengths for a given interrod separation d.
Fig. 3. Three electric-field plots showing the phase-averaged electric field (a) in a nanorod array (3D MMP calculations, α=25 and λ=690 nm), (b) on an isolated nanorod and (c) of two SPPs forming a standing wave on an isolated cylinder with θ=0.
The phase jump θ depends on the area density, i.e., the absolute number of nanorods per µm2: for small densities (large d) θ is close to zero, while for high densities (small d) θ→π. This can clearly be seen from the plots of the electric near field within the nanorod arrays; for large d or isolated nanorods, high electric field intensities occur at the extremities (Fig. 3(b) and 3(c)), whilst for small d the fields at the ends are minimal (see Fig. 3(a)). Nullifying the field at the ends of the nanorods is possible only through destructive interference of CSPs (⇒θ≈π). Unfortunately, it is not possible to give exact numbers for θ due to “end effects” similar to those found for individual nanorods [19,27]. A systematic analysis of these effects, however, is beyond the scope of this paper and will be left for the future.
Nevertheless, by assuming a phase jump of π for small d, we are able to quantitatively predict the plasmon resonance wavelengths for any CSP mode in such an array structure. As an example, we display in table 1 the data for a square gold nanorod array with ℓ=300 nm, r=12.5 nm, and d=60 nm embedded in an AAO matrix with refractive index nAAO=1.6. We obtain the following values for the first three modes:
Table 1. The first three plasmon modes calculated for CSPs in a nanorod array using the analytic approach. Settings are: ℓ=300 nm, r=12.5 nm, and d=60 nm.
Note that, as 3D MMP calculations show, the upward- and downward-propagating plasmons interfere asymmetrically for modes with m=2, 4, 6, …. Also, we observe relatively high transmittance instead of resonances for even m.
4. Comparison to MMP calculations and experiment
We then compared our analytical approach reported above to a full 3D calculation of the optical near and far fields in our nanorod arrays, using the 3D MMP model as reported in [14] and the same literature data for εAu [25]. In contrast to [14], we compute here phase-resolved electric-field plots (see Fig. 4) rather than time-averaged values. As depicted, a p-polarized plane wave (k⃗inc) is incident on a nanorod array at an angle a of 28.6°, and exits the structure at the bottom face at the same angle (k⃗trans). The field distribution changes dramatically within the array with the excitation phase. As the excitation is a continuous wave, a true plasmon propagation cannot be resolved. However, because of damping and off-resonant excitation, the reflected CSP will be weak or even negligible. Hence, only little interference is present, and the CSP may propagate quasi-unaffected along the nanorods in Fig. 4.
For comparison, we performed our 3D MMP calculations for exactly the same lattice structure as used for the CSP modeling above, i.e., a square lattice of nanorods having hemispherical ends, d=60 nm, r=12.5 nm, ℓ=300 nm, and nAAO=1.6. When calculating both the phase difference Δ between the s and p components and the amplitude of the transmitted wave (k⃗trans) as functions of the incident angle α, we find that the different resonance frequencies (modes) appear at different α (see Fig. 5 or [14]). This indicates that, despite the above-mentioned symmetry breaking, kz depends on the incident angle of the excitation light, as the different resonances do not appear simultaneously. Nevertheless, the peak positions are well reproduced by both the 3D MMP and CSP calculations. The first three resonances at
| mode | 1 | 2 | 3 |
|---|---|---|---|
| λ[nm] | 690 | 650 | 610 |
are in good agreement with the values in table 1 obtained using the CSP approach. Note that for s-polarized light these resonances do not appear; hence, no CSPs are excited and the array simply acts as an effective medium.
Fig. 4. Calculated electric-field distribution for a p-polarized plane wave hitting the array, which is embedded in AAO, at an angle α=28.6° (external angle 50°). The free-space wavelength is λ=760 nm and the plot is phase-resolved. Δφ (here ~90° - see Fig. 5) is the phase delay of the p component due to the array.
Furthermore, additional support for the CSP hypothesis is provided by the phase of the transmitted wave, in particular the phase difference between its s- and p-polarized components. According to the CSP dispersion curve we expect the p component to propagate significantly more slowly through the array than the s component, leading to a phase difference that increases with ℓ. In Fig. 5, results from our 3D MMP calculations are plotted. They confirm the CSP theory, showing drastic phase changes in the plasmonically active regime. Most interestingly, we observe an oscillatory behavior of the phase for angles smaller than the angle of the first resonance (mode 1). However, for larger angles a phase jump of 2π occurs. This holds also for the next resonance (mode 2), leading to an overall phase jump of 4π, and so forth.
To verify these results experimentally, we performed spectroscopic measurements of the phase difference Δ between p- and s-polarized light occurring upon transmission through a nanorod array, using the method of rotating-analyzer ellipsometry. The array parameters were similar to the ones used in our CSP and MMP calculations, i.e., ℓ=300 nm, r=12.5 nm, and d=60 nm; however, the lattice was quasi-hexagonal (see Fig. 1(a)). The results are plotted in Fig. 6 and confirm the calculated properties. The differences in amplitude and peak position between theory and experiment can be explained easily (see [14]). Note that it is not possible to measure phase changes ≥2π by ellipsometry, since two waves with a phase difference of 2π are indistinguishable, at least when continuous-wave excitation is used, like in our experiment. Hence, it is legitimate to add a phase increment of 2π to the left-hand resonance branch, in order to make the data match the calculations. Note also that, as the extinction in the vicinity of the resonance peak is very pronounced, it would be very difficult and erroneous to assign absolute phase values to the measured data. However, the characteristics of our ellipsometric measurements agree very well with the calculations, showing that the CSP model not only fits with other (fully numerical) calculation methods, but also is in agreement with the experimental observations.
Fig. 5. Calculated extinction and phase behavior of a gold nanorod array for various angles of incidence (ℓ=300 nm, r=12.5 nm, d=60 nm, nAAO=1.6, square symmetry). (a) Extinction for p-polarized light, and (b) phase between the s and p components. Note, in MMP phases of plane waves are in the range (-180°,180°] which leads to unphysical jumps in the phase behavior. Hence, multiples of 2π were added to obtain continuous curves. The peak around 520 nm corresponds to the short-axis resonance of the structure, which is not discussed in this paper.
Fig. 6. Measured extinction and phase behavior of a gold nanorod array structure for various angles of incidence (ℓ=300 nm, r=12.5 nm, d=60 nm, in AAO, quasi-hexagonal symmetry). (a) Extinction for p-polarized light and (b) phase between the s and p components. For determining the phase, arcsin(sin(D)) was used up to 30°, while for larger angles arccos(+cos(Δ)) (solid line) and 360°-arccos(-cos(Δ)) (dashed line) were used, respectively. Note, the peak around 520 nm corresponds to the short-axis resonance of the structure, which is not discussed in this paper.
5. Polarization converter
One promising application of these findings in nanorod arrays is polarization conversion. When illuminating a 2D nanorod array with light polarized linearly at 45° with respect to the plane of incidence, we generate, in general, elliptically polarized light, due to the phase lag Δ between the p and s components. To prove this prediction, we arranged two polarization filters orthogonal to each other after the light source such that light transmission was prevented. We then inserted a nanorod structure in-between the two filters and scanned the angle of incidence. For α=0°, the structure behaves isotropically and, hence, no light is transmitted. For increasing α, though, CSPs become excited and a resonance appears for the p component. Then, light passes through the array structure with an angle-dependent spectrum as shown in Fig. 7. By quantitatively comparing the optical densities (OD) between 640 and 780 nm, we find that the transmission increases from OD>5 to OD≈1.2 when a is swept from 0° to 40°. With the OD at normal incidence measuring around 0.7 and the device not beeing optimized yet, the performance of this polarization converter is very remarkable. Hence, metallic nanorod arrays are promising candidates for novel applications in micro- and nanooptical devices.
Fig. 7. Extinction of a gold nanorod array structure for the normal (p-polarized) and X-pol (cross-polarized) case for various incident angles. The two different setups are sketched as insets. Note that the two polarization filters work reliably only between 450 and 780 nm.
6. Conclusion
In conclusion, we have introduced a new way of understanding the field distribution in nanorod arrays, using collective surfaces plasmons (CSPs). The CSPs, which propagate along the nanorods, allowed us to predict the resonances of the structure, which were confirmed by numerical calculations and ellipsometric measurements. Finally, as an outcome of this improved understanding, we were able to predict how the structures convert the polarization. These properties were also shown experimentally. Furthermore, by studying the phase relation in more detail, it should be possible in the future to determine the geometric length of the nanorods by optical means.
Acknowledgments
This work was supported by the German DFG Research Training Group 1401/1, the European Network of Excellence Plasmo-Nano-Devices, as well as the Specific Target Research Project PLEAS in the European Union Framework Program 6.
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