From Near-Field to Far-Field Coupling in the Third Dimension

LETTER pubs.acs.org/NanoLett

From Near-Field to Far-Field Coupling in the Third Dimension: Retarded Interaction of Particle Plasmons Richard Taubert,† Ralf Ameling,† Thomas Weiss,†,‡ Andr
e Christ,† and Harald Giessen*,† † ‡

4th Physics Institute and Research Center SCoPE, University of Stuttgart, 70550 Stuttgart, Germany LASMEA, University Blaise Pascal, 24 Avenue des Landais, 63177 Aubiere Cedex, France

bS Supporting Information ABSTRACT: We study the transition from the near-field to the far-field coupling regime of particle plasmons in a threedimensional geometry. In the far-field regime, retardation plays the dominant role and the plasmonic resonances are radiatively coupled. When the spatial arrangement of the oscillators is matched to their resonance wavelength, superradiant-like effects are observed. KEYWORDS: Gold nanowires, plasmonic dimers, hybridization, Fabry-P
erot resonance, superradiant mode, retardation

T

he optical response of metallic nanoparticles is governed by particle plasmon resonances (PPRs). These PPRs can focus optical energy to subwavelength domains and allow for tailoring their spectral response. Hence they are used in several scientific fields, for example, as metamaterials, plasmonic sensors, and optical nanoantennas. In particular, coupling between two and more particles has received strong attention during the past decade. Commonly, near-field coupling with particle spacings well below their resonance wavelength has been investigated.1
8 This regime can be understood very well within the plasmon hybridization model.6 In this approach, radiative effects are neglected and interaction is mediated by quasi-electrostatic forces between the particles. However, one of the most striking features of the PPR is its strong coupling to the radiation field, making it a promising candidate for optical antennas. Therefore, neglecting the radiation fields of the oscillators gives only a very limited description of the system. As an example, it is not possible to describe far-field coupling of optical antennas9,10 in the hybridization model. Far-field coupling in plasmonic structures has been investigated theoretically11 before. Experimental investigations have focused on lateral coupling in planar structures.12
19 In vertically stacked three-dimensional structures, coupling should be stronger for two reasons. First, the oscillators are arranged along the direction of the incoming excitation light and are excited with a certain phase retardation. Spatial forward
backward symmetry is broken due to this effect. Eigenmodes that are dark in a planar geometry, such as the antisymmetric hybridized plasmon mode in a lateral dimer structure,5 can become bright in a stacked geometry.3,4,20 Second, in a stacked geometry the dipole emission patterns are oriented along the axis which connects both dipoles. Therefore, a large fraction of the radiation field of one dipole is directed toward the other dipole, which is not the case in a planar arrangement. r 2011 American Chemical Society

Figure 1. (a) Schematic of the structure. The individual wires have a width of w = 200 nm and a height of h = 20 nm. Their lateral period is 400 nm. The vertical distance dz is varied. The spacer layer is a spin-ondielectric. (b
d) Scanning electron microscope images of the structure with dz = 80, 270, and 650 nm.

Here, we investigate the coupling behavior of stacked plasmonic dimers from the near-field regime to the far-field regime. The structure consists of two stacked gold nanowires, schematically depicted in Figure 1a. We vary the vertical distance dz over Received: July 29, 2011 Revised: August 29, 2011 Published: August 31, 2011 4421

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Nano Letters

Figure 2. Schematic of the near-field plasmon hybridization scheme (left). The modes of the single wires split into the two new modes of the coupled structure that are characterized by the symmetry of the charge oscillations. In the far-field case (right), retardation plays a crucial role and the system exhibits coupled FP-PPR modes. Here, not only the symmetry of the charge oscillations in the wires but also the symmetry of the field distribution between the wires has to be taken into account: symmetric modes are characterized by a symmetric field distribution (odd FP mode) and charge oscillation, antisymmetric modes by antisymmetric field distribution (even FP mode) and charge oscillations. The mode index N is a natural number.

a very large range, from very small distances, where the limit of near-field coupling holds, up to large distances with a spacing larger than the PPR wavelength. To excite the PPR in the wire, incoming light with wavelength λ has to be polarized perpendicular to the wires. In this case, the single layer structure exhibits a PPR at λPPR = 826 nm. The refractive index of the spacer layer is nSp = 1.46. For the description of near-field coupling, the incoming plane wave is usually approximated as a homogeneous electric field across the structure. Then the system can be described in the electrostatic limit using the plasmon hybridization scheme,6 which is shown in the left part of Figure 2. The coupled system exhibits two new resonances: the antisymmetric and the symmetric mode. The splitting between the two modes is governed by the coupling strength that decreases upon increasing dz. If one takes only near-field coupling into account, the coupling should vanish for large dz, and both modes should eventually converge toward the resonance of the isolated particle. Instead, large spectral effects can be observed at larger dz. The quasi-electrostatic model alone does not fully describe the system, and coupling via the radiation fields of the particle plasmons plays a key role. For the transition to far-field coupling, retardation has to be taken into account, as the structure dimensions are on the order of the wavelength. Additionally, the system supports FabryP
erot (FP) modes for large dz. Hence, the structure can be described in terms of a FP cavity with strongly dispersive resonant mirrors. In contrast to plasmonic modes, FP mirror cavity modes are characterized by transmittance maxima, and their symmetry corresponds to the field distribution inside the resonator rather than to charge oscillations at the boundaries. The schematic in the right part of Figure 2 shows that these symmetry considerations can be transferred to the nanowire dimer system: symmetric charge oscillations in the wires occur when dz = (2N
1)(λ/2) with a positive integer number N. In this case, the field between

LETTER

Figure 3. Calculated dz-dependent transmittance spectra. The colorcoded transmittance is shown as a function of the wavelength. The single layer PPR is denoted by the orange line. Unperturbed FP modes are indicated by the dashed white lines, and the modes calculated by the analytical model (eqs 1 and 3) are shown as solid white lines. The white arrows at the upper border indicate vertical distances for which samples were experimentally realized. The left panel displays a vertical cross section for dz = 30 nm.

the wires is also symmetric with respect to the structure geometry (see upper right situation of Figure 2). In the same way, antisymmetric charge oscillations correspond to dz = (2N)(λ/2) and exhibit an antisymmetric field distribution. However, this model is only correct for frequencies far away from the PPR. Around the PPR, the particle plasmon phase varies strongly and additional considerations have to be taken into account, which we will discuss further below. In order to investigate the coupling properties of the dimer structure systematically, calculations have been carried out using the scattering matrix method.21,22 In the calculations, the wire layers are placed on a quartz substrate (nSub = 1.46), and each layer is covered by a spacer layer (nSp = 1.46). For the dielectric materials, dispersion and losses are neglected, and the optical properties of gold are taken from the experimental data of Johnson and Christy.23 The incoming field is a plane wave at normal incidence polarized perpendicular to the wires. Figure 3 shows the color-coded transmittance over the wavelength for a continuous variation of dz. The area up to dz ≈ 150 nm is governed by near-field coupling. Symmetric and antisymmetric plasmon modes are visible as dips in transmittance. The splitting between the symmetric mode at around 600 nm and the antisymmetric mode at around 1200 nm is fairly large for small dz. Upon increasing distance, the splitting becomes smaller due to a decreasing coupling strength. For larger distances, the modes of the system show a strong spectral dependence on dz and strongly resemble FP modes. However, the wavelength position of FP modes in a normal cavity should depend linearly on dz. Here, this is obviously not the case. Strong coupling of the PPR to the FP modes leads to a modification of spectral position and line width of the modes, as will be elaborated in the following: To describe the modified spectral position of the modes, one has to include the spectral response of the plasmonic oscillators into the cavity model. A FP mode is characterized by a total phase shift of Δjtot = 2πN. Assuming empty cavities and neglecting the 4422

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Nano Letters

LETTER

extension of the boundary layers, only the retardation phase shift Δjret = 2π(2nSpdz/λ) has to be considered. In this case the FP resonance wavelength depends linearly on dz. These unperturbed FP modes are shown in Figure 3 as white dashed lines. For our plasmonic system however, an additional phase contribution due to the phase behavior of the resonant mirrors, that is, the nanowire layers, has to be taken into account24,25 and the resonance condition reads 2nSp dz þ 2jres ðλÞ ð1Þ λ To calculate jres(λ) for a single plasmonic wire array layer, we apply an analytical oscillator model. The plasmonic wire array is modeled as a thin homogeneous layer with a model dielectric function 2πN ¼ 2π

εPPR ðωÞ ¼ n2Sp


f ω
ωPPR þ iγ

ð2Þ

Here, ωPPR denotes the PPR frequency, γ is the damping, and f is the oscillator strength. The reflection phase of this layer then can be approximated as jres ðλÞ≈

π λλPPR þ arctan 2 Γðλ
λPPR Þ

ð3Þ

with Γ as damping parameter. Inserting this into eq 1 accurately reproduces the resonance positions for dz larger than 150 nm as shown by the white solid lines in Figure 3. It is furthermore notable that the line width of the coupled PPR-FP-mode which is spectrally close to the single PPR decreases drastically for certain vertical distances dz. This happens at the intersections of the FP modes (indicated by white lines in Figure 3) with the single layer PPR resonance (orange line in Figure 3), for example around dz = 260, 540, and 820 nm. These positions are characterized by the equation dz = N(λPPR/2nSp); therefore, the Bragg criterion at the PPR wavelength is fulfilled. This can be regarded as a matching of the spatial arrangement of the oscillators to their emission wavelength. Around the PPR the resonant part of eq 3 varies strongly. At λ = λPPR the mode becomes close to optically inactive and the spectra are dominated by the neighboring modes that have opposite symmetry. Therefore, at Bragg distance, plasmon oscillation in the symmetric branches become antisymmetric and vice versa. As a consequence, the PPR far-fields interfere destructively in forward direction and constructively in backward direction. Only one broad feature is observed in the transmittance spectrum. Its width is strongly enhanced compared to the single PPR width, which can be attributed to the tailored spatial arrangement of the oscillators. The coupled system acts like a single oscillator with enhanced dipole moment and oscillator strength. This is similar to a superradiant mode in a system of coupled quantum emitters, where all optical modes but one become dark (subradiant) and all the energy is contained within one superradiant mode.27 While our system exhibits also one vanishing subradiant mode at Bragg distance, the broad spectral feature originates in contrast to quantum emitters from two neighboring branches with opposite symmetry compared to the subradiant mode. As the term superradiance is usually connected to quantum systems, it has to be pointed out that a broadening of the radiative width, which is usually associated with superradiance, can also be observed in a system of classical oscillators.28 26

Figure 4. Transmittance (dark red) and reflectance (light blue) spectra for plasmonic dimers at different vertical distances dz. The left panel shows the measured spectra. The right panel shows the calculated transmittance and reflectance for the corresponding spacer thicknesses.

To experimentally verify our results, we fabricated a set of 11 samples with different spacer thicknesses dz which were varied from values as low as 30 nm up to more than 1 μm. Every fabricated sample should exhibit a spectrum according to a vertical cross section at the corresponding dz in Figure 3, where the exact positions for each sample are indicated by white triangles at the top of the graph. The structures were fabricated using an electron-beam lithography multilayer process.29 Before patterning the structure, several reference gold markers were processed on the substrate. Then, the structure of the first layer was defined in a negative tone resist (Allresist AR-N 7500), which was spin coated on top of a 20 nm gold film. The pattern was transferred into the gold layer using Ar ion beam etching. After removal of the resist, a planarization layer was applied by spin coating of a spin-on dielectric (Futurrex IC1-200). Finally, the second layer was exposed in the same way, aligned with respect to the common coordinate system. The varying spacer thickness was controlled by using different dilutions of the spin-on-dielectric and multiple coating runs. Transmittance as well as reflectance measurements are performed in a Fourier-transform infrared spectrometer (Bruker Vertex 80) with an attached microscope (Bruker Hyperion) using a Si detector for the wavelength range below 950 nm and an InGaAs detector for the range above 950 nm. The spectra are subsequently combined. Transmittance is measured with respect to the substrate, reflectance with respect to a gold mirror. The left panel of Figure 4 shows the experimentally acquired transmittance and reflectance spectra for the different samples. The calculated results are shown in the right panel for comparison. No free parameters are used in the calculations. The spectral features observed in the simulations are well reproduced by our 4423

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Nano Letters experimental results. The differences can be mainly attributed to inhomogeneous broadening, imperfections in the structure, and a high numerical aperture of our microscope objective. For the samples with low distances, a distance-dependent near-field splitting is clearly observable. It is present for distances up to 150 nm, which corresponds to roughly λPPR/5nSp. For larger distances, the spectra are dominated by far-field effects. The spectrum for dz = 270 nm shows only one broad feature, indicating the first Bragg distance. This situation is repeated at dz = 550 nm for the second Bragg case. For higher distances, more and more modes are present leading to an increased number of resonant features in the spectrum. In conclusion, we have demonstrated the transition from nearfield to far-field coupling in a plasmonic dimer system in a threedimensionally stacked arrangement. Our findings are consistently supported experimentally and by simulations using S-matrix methods as well as a coupled oscillator model that takes retardation and the phase behavior of the plasmonic oscillators into account. In analogy to the plasmon hybridization scheme, which holds in the near-field regime, our system can be considered as a coupled ensemble of plasmonic oscillators and Fabry-P
erot modes, which gives an intuitive picture for this coupling regime. Special attention has to be paid to stacking distances corresponding to a multiple of half the particle plasmon resonance wavelength, where matching of the spatial arrangement of the plasmonic oscillators to their resonance wavelength leads to a superradiant-like coupling. These results are of importance as they elucidate far-field coupling schemes for plasmonic systems. This could be useful for tailoring unusual optical properties of plasmonic nanostructures. Especially a Bragg-spaced arrangement of more than two oscillators, of which one or more could be a quantum emitter, will lead to a strong collective resonance with enhanced radiative decay rate without the common problems of near-field quenching.

’ ASSOCIATED CONTENT

bS

Supporting Information. The Supporting Information contains details on the derivation of eq 3 as well as graphs of the electric field distributions for different vertical distances dz. This material is available free of charge via the Internet at http:// pubs.acs.org.

LETTER

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’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors would like to thank S. G. Tikhodeev, N. A. Gippius, L. Langguth, and J. Dorfm€uller for stimulating discussions as well as H. Gr€abeldinger and R. Kr€uger for technical support. We gratefully acknowledge funding by BMBF (13N10146), DFG (SPP1391, FOR557/730, and DFG GI 269/11-1) and DFH/UFA. ’ REFERENCES (1) Halas, N. J.; Lal, S.; Chang, W.-S.; Link, S.; Nordlander, P. Chem. Rev. 2011, 111, 3913–61. (2) Liu, N.; Giessen, H. Angew. Chem., Int. Ed. 2010, 49, 9838–9852. (3) Liu, N.; Guo, H.; Fu, L.; Kaiser, S.; Schweizer, H.; Giessen, H. Adv. Mater. 2007, 19, 3628–3632. 4424

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