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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6450–6461
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Energy transport in metal nanoparticle chains via sub-radiant plasmon modes

Britain Willingham and Stephan Link  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6450-6461 (2011)
http://dx.doi.org/10.1364/OE.19.006450


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Abstract

We investigate the propagation of surface plasmon polaritons through coupling of light to sub-radiant dipole modes in finite chains of Ag nanoparticles. End excitation of collections of closely spaced particles reveals a band of sub-radiant modes whereby the decay of surface plasmon polaritons due to radiative losses is minimized. We show that excitation of any of these sub-radiant modes results in the most efficient energy transfer throughout the optical spectrum, with smaller interparticle separations resulting in the longest propagation.

© 2011 OSA

1. Introduction

Finite chains of metal nanoparticles have the potential to transport electromagnetic energy via the excitation of their localized surface plasmons [1

1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23(17), 1331–1333 (1998). [CrossRef]

,2

2. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

]. The efficient coupling of light to surface plasmons provides a pathway to distribute and direct energy among an ensemble of particles [3

3. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000). [CrossRef]

6

6. M. Sukharev and T. Seideman, “Phase and polarization control as a route to plasmonic nanodevices,” Nano Lett. 6(4), 715–719 (2006). [CrossRef] [PubMed]

]. In order to optimize energy propagation along linear chains, it is necessary to understand interparticle coupling as well as energy decay channels [7

7. C. Dahmen, B. Schmidt, and G. von Plessen, “Radiation damping in metal nanoparticle pairs,” Nano Lett. 7(2), 318–322 (2007). [CrossRef] [PubMed]

,8

8. A. O. Pinchuk and G. C. Schatz, “Nanoparticle optical properties: Far- and near-field electrodynamic coupling in a chain of silver spherical nanoparticles,” Mater. Sci. Eng. B 149(3), 251–258 (2008). [CrossRef]

]. For a particle assembly, optical spectra provide a means to analyze these important factors [9

9. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003). [CrossRef]

12

12. M. D. Arnold, M. G. Blaber, M. J. Ford, and N. Harris, “Universal scaling of local plasmons in chains of metal spheres,” Opt. Express 18(7), 7528–7542 (2010). [CrossRef] [PubMed]

]. In particular, the identification of super and sub-radiant modes provides essential information about propagating surface plasmon polaritons (SPPs) [13

13. P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]

,14

14. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]

].

In comparison to infinite chains, the use of finite nanoparticle chains as optical waveguides provides a more realistic system [21

21. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]

,23

23. S. Zou and G. C. Schatz, “Metal nanoparticle array waveguides: proposed structures for subwavelength devices,” Phys. Rev. B 74(12), 125111 (2006). [CrossRef]

25

25. D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: the role of chain length unravelled,” Nano Lett. 5(5), 985–989 (2005). [CrossRef] [PubMed]

], as practical applications would involve well defined finite structures [26

26. Q. H. Wei, K. H. Su, S. Durant, and X. Zhang, “Plasmon resonance of finite one-dimensional au nanoparticle chains,” Nano Lett. 4(6), 1067–1071 (2004). [CrossRef]

29

29. R. Quidant, C. Girard, J.-C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004). [CrossRef]

]. Here, non-degenerate modes appear as sub-radiant plasmon resonances that have suppressed radiative losses when compared to super-radiant plasmon modes [14

14. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]

,30

30. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87(16), 167401 (2001). [CrossRef] [PubMed]

32

32. M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009). [CrossRef] [PubMed]

]. Optical spectra of finite chains can be used to identify the spectral region where the sub-radiant modes occur, which are expected to support propagating SPPs with reduced losses. Thus, a theoretical investigation that considers the relationship between optical spectra and SPP damping in finite particle chains at close interparticle separations and hence includes multipolar plasmon interactions and retardation demands further attention.

In what follows, we investigate the effects of higher order plasmon modes of finite Ag nanoparticle chains on their collective optical properties. The exact solution to Maxwell's equations for spheres is used to demonstrate that the propagation of SPPs shows reduced decay with decreasing interparticle separations. More importantly, an excitation bandwidth formed from sub-radiant plasmon modes supporting low-loss SPPs is revealed in the optical spectra. The results are consistent with quasi-static solutions as the resonant energies of the sub-radiant modes follow calculated eigenvalue trends.

2. Optical characteristics of finite chains of Ag nanoparticles

First, as shown in Fig. 1
Fig. 1 Optical properties of finite chains with varying particle numbers N and center-to-center separations d(σ) = σa, calculated by generalized Mie theory for Ag spheres with radius a = 25 nm. (a) Energy splitting, ∆E, between the longitudinal L (red) and transverse T (blue) dipole modes as a function of N for σ = 2.1, and σ = 3.0. (b) Optical extinction spectra of linear chains for L and T incident polarization and varying N at σ = 2.1. (c) Dependence of the optical absorption and scattering spectra of chains with N = 10 on σ for L-polarization. Note the different scales for Qabs and Qscat.
, we demonstrate that the energy splitting, ∆E, between the longitudinal and transverse polarized dipole resonances is highly dependent on both particle separations and numbers, N. Using the partial scattering solution to Mie's theory for multiple spherical particles as described by Gẽrardy and Ausloos [33

33. J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. II. optical properties of aggregated metal spheres,” Phys. Rev. B 25(6), 4204–4229 (1982). [CrossRef]

], the optical spectra of finite linear chains are obtained to gain significant physical insight into the characteristics of propagating SPPs. Multipoles up to l = 20 were included in the calculations and all results have been independently tested for smaller chains of particles using the finite-element method.

In Fig. 1(a), we compare ∆E as a function of N for two interparticle separations σ = 2.1 and σ = 3.0, where σ is defined as the ratio of center-to-center distance, d, and nanoparticle radius, a, i.e. σ = d/a. The shift of the dipole resonance for longitudinal and transverse modes converges for particle chains of roughly N = 10, independent of σ [25

25. D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: the role of chain length unravelled,” Nano Lett. 5(5), 985–989 (2005). [CrossRef] [PubMed]

]. The structure analyzed is shown in the lower corner of Fig. 1. The constituent particles have a = 25 nm and are surrounded by vacuum. In all cases, the optical cross-sections are normalized by N, while the incident wave vector, k, is fixed perpendicular to the chain of particles situated along the z-axis. The material response neglects small particle effects, and is described by a Drude fit to the dielectric function of Ag, where the bulk resonance frequency is ωb = 9.5 eV, the non-radiative decay is given by Γ = 0.1 eV, and the high-frequency limit is ε = 5.0 [34

34. B. Willingham, D. Brandl, and P. Nordlander, “Plasmon hybridization in nanorod dimers,” Appl. Phys. B 93(1), 209–216 (2008). [CrossRef]

]. For all calculations the surface-to-surface separations are ≥ 1 nm so that non-local dielectric effects can be neglected [35

35. J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. 9(2), 887–891 (2009). [CrossRef] [PubMed]

].

To determine the presence of higher order plasmon modes, we turn to the extinction spectra calculated for varying N at constant interparticle separation, σ = 2.1 (Fig. 1(b)). We first consider the longitudinal mode, which shows a drastic red-shift from the single particle response at 3.46 eV to the eventual saturation at 2.31 eV as N increases. The red-shift, due to increasing N and hybridization of higher order plasmon modes between particles at close separations, is accompanied by a broadening of the dipole peak. Radiative damping as well as additional dipole-like modes, seen as ripples to the blue of the main peak, contributes substantially to the width for N ≥ 5.

For comparison, Fig. 1(b) also shows the extinction for transverse polarization, resulting from the formation of anti-bonding hybridized modes which slightly blue-shift to 3.64 eV at N = 50. This much smaller shift in resonance energy is due to reduced plasmonic interactions of the anti-bonding modes compared to the bonding modes for longitudinal polarized excitation [34

34. B. Willingham, D. Brandl, and P. Nordlander, “Plasmon hybridization in nanorod dimers,” Appl. Phys. B 93(1), 209–216 (2008). [CrossRef]

].

By studying the effect of interparticle separation on the scattering and absorption spectra separately, we can determine the contributions of super and sub-radiant plasmon modes to the collective optical properties of finite particle chains. This is shown in Fig. 1(c), where scattering and absorption cross-sections, Qscat and Qabs, of chains with N = 10 are compared for longitudinal polarization while varying σ. Important differences are seen for different σ in the scattering and absorption contributions to the overall extinction. Excluding the single particle Mie spectra at σ = ∞, it is clear that there are multiple peaks at increasingly distributed energies for smaller σ. Regardless of σ, however, the super-radiant dipole mode in the Qscat plots, having all particle dipoles aligned parallel to the chain, is the most efficient at losing energy through far-field scattering.

At σ = 2.1, the secondary dipole peaks between the main super-radiant mode at 2.35 eV and the quadrupole resonance at 3.47 eV are sub-radiant in nature. As shown below (Fig. 2
Fig. 2 Surface charge density, Re[ρpol], along a chain of N = 10 Ag nanoparticles of radii a = 25 nm at interparticle separations of σ = 2.1. Polarization induced charge density waves are highlighted by the magenta lines and have standing wavelengths of λ0/n, which is further illustrated by a line segment (black line) of the surface charge density taken along the surface of each particle through the chain axis. Collective excitation at energies corresponding to Qabs, shown on the left, induces optically active plasmon modes of both super- (n = 1) and sub-radiant (n = 3, 5) nature.
), for sub-radiant modes, the surface charge density due to the induced polarization forms domains of dipole nature which alternate along the chain in standing wave configurations [13

13. P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]

]. These optically active, or “bright”, sub-radiant plasmon modes have non-zero dipole strength at close interparticle separations and hence contribute to Qscat, but always remain weaker than the super-radiant mode. Absorptive losses, as inferred from the Qabs plots, exhibit a different trend. The super-radiant dipole resonance broadens and decreases in intensity due to dynamic depolarization [36

36. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8(11), 581–583 (1983). [CrossRef] [PubMed]

], while the bright sub-radiant modes, like the one at 2.74 eV for σ = 2.1, display enhanced Qabs compared to the main dipole peak. For small σ, the Qscat and Qabs plots illustrate that the most efficient coupling of light to the chain occurs at energies corresponding to sub-radiant and higher order modes, where losses due to far-field scattering are minimized.

We can identify super and sub-radiant plasmon modes by their distinctive surface charge densities. Analysis of the induced polarization charge density waves can characterize the resonant nature of optical features, like those discussed in Fig. 1. Shown in Fig. 2 are the real surface charge densities, Re[ρpol], for the super-radiant and two lowest energy, bright sub-radiant modes of the N = 10 particle chain at σ = 2.1, corresponding to Qabs of Fig. 1(c). From this vantage point, the incident wave vector, k, is directed out of the page and is longitudinally polarized along the length of the chain.

The induced polarization charge density wave at 2.35 eV is characteristic of a super-radiant plasmon mode. A line segment (black line) taken along the surface of each particle through the chain axis, highlights the opposing concentrations of positive and negative charge. A standing wave pattern is visible in the surface charge density and is indicated by the magenta line, which identifies the plasmon mode as a collective dipole antenna, whose wavelength, λ0, is equivalent to twice the length of the nanoparticle chain.

At 2.74 and 3.04 eV, the surface charge density also displays standing wave patterns due to their induced polarization (magenta lines). These standing waves have wavelengths that are equal to integer fractions of the wavelength for the super-radiant mode, i.e. λ0/n. The overall dipole moment for these collective resonances is diminished as localized charge distributions form dipole-like domains which alternate along the chain. Bright sub-radiant modes, like those shown in Fig. 2 are identified by odd fractions of the fundamental charge density wave, n = 3, 5, …, and possess net dipole moments when excited in a collective manner. Dark sub-radiant modes, characterized by even integer fractions n = 2, 4, …, possess no net dipole moment when excited collectively, and therefore cannot be seen in the optical spectra in Fig. 1. Both bright and dark sub-radiant plasmon modes give rise to suppressed scattering, and can increase coupling to the incident field when excited asymmetrically (see section 4), as the surface charge density wave now forms nodal patterns which decay in magnitude along its length. Therefore it is expected that intrinsic sub-radiant eigenmodes should support propagation of low-loss SPPs along finite chains of metal nanoparticles [13

13. P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]

].

3. SPP Propagation in finite chains of Ag nanoparticles

The inset to Fig. 3 relates the energies at which maximum propagation takes place to each interparticle separation, σ. With decreasing σ, an increasingly broad band of energies is observed where exponential decay of I takes place, coinciding with the sub-radiant modes described in section 2 and section 4. Here the loss is mainly dominated by absorption as the intensity of the sub-radiant modes increases upon end-excitation (see section 4). The SPP amplitudes decay exponentially like a system of damped harmonic oscillators, consistent with a free electron response. Shown on top of Fig. 3 is the surface charge density, Re[ρpol], at an excitation energy of 2.62 eV and an interparticle spacing σ = 2.1. The induced polarization charge density alternates along the chain, and displays even symmetry with respect to the center of the nanoparticle chain. This is characteristic of exciting a dark sub-radiant plasmon mode [13

13. P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]

], which supports maximum SPP propagation with minimal radiative decay [32

32. M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009). [CrossRef] [PubMed]

].

In Fig. 4
Fig. 4 Correlation between optical spectra and SPP decay for N = 50 chains of Ag nanoparticles with a radius of a = 25 nm for varying σ = 2.1, 2.4, and 3.0. Upper left panel schematically illustrates the asymmetric end-excitation used in the calculations of SPP propagation, where intensity values are sampled at the nanoparticle gaps. (a) Plot of the 1/e decay length for each σ showing an energy band where maximum SPP propagation takes place. Accompanying optical extinction spectra confirm the assignment of this energy band to sub-radiant plasmon modes. (b), (c), and (d) show intensity profiles where a maximum decay length is obtained at each separation σ (color plots), compared to excitation at energies just outside the bands highlighted in (a), which leads to strongly non-exponential intensity profiles.
we further verify the sub-radiant nature of SPP propagation in finite nanoparticle chains by correlating energies at which maximum propagation takes place to the optical extinction when the chain is excited collectively. Shown in the lower part of Fig. 4(a) are the 1/e SPP decay lengths as a function of excitation energy for the three interparticle separations highlighted in Figs. 1 and 3. Comparison to the corresponding optical extinction spectra (top part Fig. 4(a)) illustrates that maximum SPP propagation takes place at energies which correspond to sub-radiant plasmon modes for all σ considered. At energies which lie outside this energy band, the intensity fails to decay exponentially, uncharacteristic of pure propagating SPPs. This can be seen from the intensity profiles for the three analyzed separations at different excitation energies, as shown in Figs. 4(b)4(d). In particular, at σ = 2.1, (Fig. 4(b)), an excitation energy of 2.62 eV results in a maximum 1/e decay length of 0.81 μm. Excitation at energies outside the band corresponding to sub-radiant modes (and highlighted in Fig. 4(a)), e.g. at 3.14 and 2.2 eV, drastically reduces SPP propagation. The same trend is also observed for separations of σ = 2.4 and σ = 3.0, shown in Fig. 4(c) and Fig. 4(d). These results suggest that low-loss SPP propagation occurs via excitation of sub-radiant dipole modes.

Interestingly, for the smallest interparticle separations, the excitation band for maximum SPP propagation partially overlaps with the super-radiant plasmon mode, possibly due to enhanced near-field coupling at close distances. Although an exponential intensity decay is observed (which is the definition of the energy band in Fig. 4(a) and the inset of Fig. 3), the SPP decay length is significantly shorter for excitation at energies corresponding to the super-radiant mode.

4. Origin of sub-radiant modes

Assuming a quasi-static response of the polarization P of each particle and insisting that no bound charge is induced except within an infinitesimal volume at the surfaceP=0, the macroscopic fields are described by the solutions to Laplace's equation [38

38. R. Brako, “Optical properties of composite media,” J. Phys. C Solid State Phys. 11(15), 3345–3355 (1978). [CrossRef]

]. Because of the time-harmonic nature of the material response, a polarization current, J pol = ∂P/∂t with constant charge carrier density is included. Using the Hamilton relations, this leads to the equations of motion for the multipolar moments of each particle [39

39. J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley and Sons, 1975).

]. By solving for P in spherical coordinates and neglecting the contribution of the energy contained within the magnetic fields, the multipolar moments qlmi are obtained by solving
l2l+1qlmi+jiNpkHpkjlmiqpkj=ωl2ωb2qlmi,
(1)
where the ωl's are the resonant modes of the system of spheres and H is a Hermitian matrix representing the interaction between multiple particles [40

40. D. J. Bergman, “Dielectric constant of a two-component granular composite: a practical scheme for calculating the pole spectrum,” Phys. Rev. B 19(4), 2359–2368 (1979). [CrossRef]

]. The dipole per unit volume of the ith sphere with unit vector, n i, normal to its surface is then given by
Pi=i(lmqlmilai2l+1rilYlm(θi,ϕi)),
(2)
where r i is the radial distance from the center of the particle and Ylm are the spherical harmonics. By adding to the Hamiltonian the energy loss of a system of dielectric spheres placed in an alternating uniform field in vacuum, we can solve for the induced surface charge density, ρpol,i = P in i, and the fields attributed to it. In Fig. 5(a) we show the k = 0 quasi-static results from collectively exciting a small chain with N = 5 particles at a separation of σ = 2.1.

The largest resonant peaks in the extinction spectrum occur at the super-radiant dipole and quadrupole modes at 2.86 and 3.5 eV, respectively, as shown in the lower plot of Fig. 5(a). However, secondary peaks due to bright sub-radiant dipole modes are also visible in the optical cross-section at 3.19 and 3.53 eV. Although the energies are blue-shifted compared to generalized Mie theory calculations due to the lack of phase retardation, the energy trend for the different plasmon modes is in excellent agreement with the results discussed in section 2. The inset plots the energy of the N non-degenerate eigenvalues [21

21. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]

] as a function of σ. At large σ, all modes converge to the l = 1 mode of an isolated sphere. Only the super-radiant and the two bright sub-radiant modes marked by the colored boxes can be excited optically by a symmetric field, while the two dark sub-radiant modes are optically inactive as a result of the mirror symmetry of their charge distribution [13

13. P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]

,41

41. F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30(9), 4989–4999 (1984). [CrossRef]

,42

42. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005). [CrossRef]

]. As σ decreases, the eigenvalues follow the same trend as the excitation energies that yield minimum SPP decay (Figs. 3 and 4).

The physical mechanism behind the sub-radiant mode at 3.18 eV is illustrated by plotting Pn in the upper part of Fig. 5(a). The charge density varies asymmetrically with respect to the center particle, which leads to near-fields that have odd parity upon negation of their coordinates [42

42. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005). [CrossRef]

]. Hence, they possess a finite dipole moment and can thus be optically excited. In contrast, dark sub-radiant modes, which lie close in energy to their bright counterparts (black lines in the eigenvalue plot in the inset of Fig. 5(a)) become optically active upon end-excitation [13

13. P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]

]. The charge density obtained by quasi-static calculations shown in Fig. 5(a) agrees well with the results from generalized Mie theory (section 2).

Shown in Fig. 5(b) are the results of exciting the leading particle within chains of increasing N at a separation of σ = 2.1, which demonstrate that end-excitation creates a sub-radiant band of energies at which radiative decay is minimized and efficient coupling to particle chains occurs. The cumulative dipole moments along the z-axis are plotted for chains with varying N. For N = 5, the sub-radiant mode at 3.18 eV gains magnitude with respect to the lowest energy super-radiant dipole at 2.86 eV. This trend continues for larger N and is accompanied by a gradual red-shift as indicated by the red arrow. We can further conclude that for chains with larger N and at small σ, a broad band emerges consisting of low energy sub-radiant modes, in agreement with Fig. 4 and the inset of Fig. 3. This is evident from the appearance of additional peaks associated with both bright and dark sub-radiant modes, which at N = 50 become a broad featureless shoulder at the high energy side of the super-radiant dipole mode. The results of these quasi-static calculations should also apply to larger metallic nanoparticles with local dielectric properties that are governed by a free electron response.

5. Group velocity of SPPs in finite chains of Ag nanoparticles

The results presented so far have shown that SPP damping is strongly suppressed for sub-radiant modes. However, it is also necessary to examine the group velocity of the propagating SPPs, which can be obtained from the SPP dispersion [3

3. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000). [CrossRef]

]. Shown in Fig. 6
Fig. 6 Dispersion relation for SPPs supported by an Ag nanoparticle chain with a = 25 nm and N = 50 at σ = 2.1. The red colored region denotes the energy band formed by the sub-radiant plasmon modes at which minimum SPP decay is found. The inset illustrates the method used to extract the propagating wave-number k|| within the sub-radiant band and compares collective (2.62 eV) to localized excitation (3.45 eV).
is a dispersion curve at energies corresponding to the sub-radiant plasmon modes for an N = 50 chain of Ag nanoparticles with a = 25 nm separated by σ = 2.1. The association between SPP propagation and sub-radiant plasmon modes in this finite chain is illustrated by the red colored region, corresponding to the same colored part in the optical extinction spectrum plotted in Fig. 4(a). The wave-number, k||, for a particular incident excitation energy with longitudinal polarization was obtained by fitting the Esz components along the particle chain to the equation of a damped harmonic oscillator (shown in inset). For excitation at 2.62 eV, corresponding to a sub-radiant mode, the decay constant, b, of the damped harmonic oscillator equation is taken to be the exponential decay parameter in Fig. 4(a). For comparison, near the single particle plasmon resonance at 3.45 eV, excitation of localized plasmons leads to large radiative and intrinsic damping and therefore inefficient coupling along the Ag nanoparticle chain.

The slope of the dispersion curve determines the SPP group velocity vg [22

22. S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004). [CrossRef]

]. In the region of minimum SPP decay, ~2.62 eV, we find that the group velocity is vg / c ~0.3, where c is the speed of light in vacuum. In addition to the small damping and large group velocity for SPPs excited via sub-radiant plasmon modes, it is important to point out that the dispersion curve in Fig. 6 lies below the light line within the error of the fitting analysis. This is consistent with low losses due to minimized radiative decay as observed from the optical scattering spectra. Furthermore, it should be pointed out that the dispersion relation for the sub-radiant plasmon modes in these Ag particle chains resembles the behavior of continuous plasmonic nanowires [43

43. G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher, A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu, B. Monacelli, and G. Boreman, “Plasmon dispersion relation of Au and Ag nanowires,” Phys. Rev. B 68(15), 155427 (2003). [CrossRef]

], which support long propagation distances due to reduced radiative decay [44

44. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

46

46. Y. Fang, Z. Li, Y. Huang, S. Zhang, P. Nordlander, N. J. Halas, and H. Xu, “Branched silver nanowires as controllable plasmon routers,” Nano Lett. 10(5), 1950–1954 (2010). [CrossRef] [PubMed]

].

In Table 1

Table 1. Decay Constants b(σ) as Function of Nanoparticle Radius a

table-icon
View This Table
, we present SPP propagation decay constants b [μm−1] for linear chains of N = 50 Ag nanoparticles with sizes ranging from the quasi-static limit to a = 50 nm. For increasing σ and decreasing a, the decay constant increases, denoting smaller propagation lengths. Smaller σ results in reduced decay constants as multipolar contributions lead to suppressed radiative losses when excited at energies corresponding to the sub-radiant modes. For a = 50 nm and σ = 2.1 the 1/e propagation length is ~3 μm when excited at 2.47 eV, nearly twice as large when compared to interparticle separations of σ = 3.0. For each particle size, the excitation energies that yield minimum SPP decay are easily correlated to the optical spectra, and lie between the main dipole resonance and the quadrupole mode. The width of this band broadens for close interparticle separations, providing a unique optical window for excitation of sub-radiant modes within which efficient energy transfer takes place. Particles with sizes near the quasi-static limit displayed decay profiles that were in very good agreement with electrostatic calculations. The energy propagation in chains of such small sizes is interpreted to be supported by the optical near-field and its dependence on higher order multipole modes, especially for small separations σ.

6. Conclusion

In conclusion, using generalized Mie theory we have shown that propagating SPPs along linear chains of Ag nanoparticles exhibit maximum propagation distances at small separations when exciting sub-radiant plasmon modes. As the number of particles increases, these discrete modes form an excitation band at which radiative losses are minimized, becoming increasingly broad at small interparticle separations. Electromagnetic energy transport by optically induced SPPs in collections of closely spaced nanoparticles offers distinct advantages for creating optical interconnects between other waveguides because SPPs can be transported around sharp corners or can be split, for example, into two SPPs in a ‘T’ – like nanoparticle geometry. The results presented here also point to the use of chemical methods to arrange nanoparticles in close proximity to each other, thereby maximizing SPP propagation as well as the excitation bandwidth formed by the sub-radiant modes.

Acknowledgments

This work was supported by the Robert A. Welch Foundation (C-1664), a 3M Non-Tenured Faculty Grant, and the NSF (CHE-0955286). We acknowledge Wei-Shun Chang, Christy Landes, Liane Slaughter, Peter Nordlander, and Mark Stockman for fruitful discussions.

References and links

1.

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23(17), 1331–1333 (1998). [CrossRef]

2.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

3.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000). [CrossRef]

4.

S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Photonics 2(7), 438–442 (2008). [CrossRef]

5.

A. V. Malyshev, V. A. Malyshev, and J. Knoester, “Frequency-controlled localization of optical signals in graded plasmonic chains,” Nano Lett. 8(8), 2369–2372 (2008). [CrossRef] [PubMed]

6.

M. Sukharev and T. Seideman, “Phase and polarization control as a route to plasmonic nanodevices,” Nano Lett. 6(4), 715–719 (2006). [CrossRef] [PubMed]

7.

C. Dahmen, B. Schmidt, and G. von Plessen, “Radiation damping in metal nanoparticle pairs,” Nano Lett. 7(2), 318–322 (2007). [CrossRef] [PubMed]

8.

A. O. Pinchuk and G. C. Schatz, “Nanoparticle optical properties: Far- and near-field electrodynamic coupling in a chain of silver spherical nanoparticles,” Mater. Sci. Eng. B 149(3), 251–258 (2008). [CrossRef]

9.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003). [CrossRef]

10.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002). [CrossRef]

11.

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002). [CrossRef]

12.

M. D. Arnold, M. G. Blaber, M. J. Ford, and N. Harris, “Universal scaling of local plasmons in chains of metal spheres,” Opt. Express 18(7), 7528–7542 (2010). [CrossRef] [PubMed]

13.

P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]

14.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]

15.

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007). [CrossRef] [PubMed]

16.

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008). [CrossRef]

17.

A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006). [CrossRef]

18.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74(3), 033402 (2006). [CrossRef]

19.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(6), 066606 (2005). [CrossRef]

20.

D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4(9), 1561–1565 (2004). [CrossRef]

21.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]

22.

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004). [CrossRef]

23.

S. Zou and G. C. Schatz, “Metal nanoparticle array waveguides: proposed structures for subwavelength devices,” Phys. Rev. B 74(12), 125111 (2006). [CrossRef]

24.

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B 75(8), 085426 (2007). [CrossRef]

25.

D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: the role of chain length unravelled,” Nano Lett. 5(5), 985–989 (2005). [CrossRef] [PubMed]

26.

Q. H. Wei, K. H. Su, S. Durant, and X. Zhang, “Plasmon resonance of finite one-dimensional au nanoparticle chains,” Nano Lett. 4(6), 1067–1071 (2004). [CrossRef]

27.

K. B. Crozier, E. Togan, E. Simsek, and T. Yang, “Experimental measurement of the dispersion relations of the surface plasmon modes of metal nanoparticle chains,” Opt. Express 15(26), 17482–17493 (2007). [CrossRef] [PubMed]

28.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76(20), 201403 (2007). [CrossRef]

29.

R. Quidant, C. Girard, J.-C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004). [CrossRef]

30.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87(16), 167401 (2001). [CrossRef] [PubMed]

31.

J. J. Choquette, K.-P. Marzlin, and B. C. Sanders, “Superradiance, subradiance, and suppressed superradiance of dipoles near a metal interface,” Phys. Rev. A 82(2), 023827 (2010). [CrossRef]

32.

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009). [CrossRef] [PubMed]

33.

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. II. optical properties of aggregated metal spheres,” Phys. Rev. B 25(6), 4204–4229 (1982). [CrossRef]

34.

B. Willingham, D. Brandl, and P. Nordlander, “Plasmon hybridization in nanorod dimers,” Appl. Phys. B 93(1), 209–216 (2008). [CrossRef]

35.

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. 9(2), 887–891 (2009). [CrossRef] [PubMed]

36.

M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8(11), 581–583 (1983). [CrossRef] [PubMed]

37.

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975). [CrossRef]

38.

R. Brako, “Optical properties of composite media,” J. Phys. C Solid State Phys. 11(15), 3345–3355 (1978). [CrossRef]

39.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley and Sons, 1975).

40.

D. J. Bergman, “Dielectric constant of a two-component granular composite: a practical scheme for calculating the pole spectrum,” Phys. Rev. B 19(4), 2359–2368 (1979). [CrossRef]

41.

F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30(9), 4989–4999 (1984). [CrossRef]

42.

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005). [CrossRef]

43.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher, A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu, B. Monacelli, and G. Boreman, “Plasmon dispersion relation of Au and Ag nanowires,” Phys. Rev. B 68(15), 155427 (2003). [CrossRef]

44.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

45.

R. M. Dickson and L. A. Lyon, “Unidirectional plasmon propagation in metallic nanowires,” J. Phys. Chem. B 104(26), 6095–6098 (2000). [CrossRef]

46.

Y. Fang, Z. Li, Y. Huang, S. Zhang, P. Nordlander, N. J. Halas, and H. Xu, “Branched silver nanowires as controllable plasmon routers,” Nano Lett. 10(5), 1950–1954 (2010). [CrossRef] [PubMed]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(130.2790) Integrated optics : Guided waves
(260.3910) Physical optics : Metal optics
(350.3950) Other areas of optics : Micro-optics
(350.4990) Other areas of optics : Particles
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Optics at Surfaces

History
Original Manuscript: November 1, 2010
Revised Manuscript: February 28, 2011
Manuscript Accepted: March 14, 2011
Published: March 22, 2011

Citation
Britain Willingham and Stephan Link, "Energy transport in metal nanoparticle chains via sub-radiant plasmon modes," Opt. Express 19, 6450-6461 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6450


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References

  1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23(17), 1331–1333 (1998). [CrossRef]
  2. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]
  3. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000). [CrossRef]
  4. S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Photonics 2(7), 438–442 (2008). [CrossRef]
  5. A. V. Malyshev, V. A. Malyshev, and J. Knoester, “Frequency-controlled localization of optical signals in graded plasmonic chains,” Nano Lett. 8(8), 2369–2372 (2008). [CrossRef] [PubMed]
  6. M. Sukharev and T. Seideman, “Phase and polarization control as a route to plasmonic nanodevices,” Nano Lett. 6(4), 715–719 (2006). [CrossRef] [PubMed]
  7. C. Dahmen, B. Schmidt, and G. von Plessen, “Radiation damping in metal nanoparticle pairs,” Nano Lett. 7(2), 318–322 (2007). [CrossRef] [PubMed]
  8. A. O. Pinchuk and G. C. Schatz, “Nanoparticle optical properties: Far- and near-field electrodynamic coupling in a chain of silver spherical nanoparticles,” Mater. Sci. Eng. B 149(3), 251–258 (2008). [CrossRef]
  9. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003). [CrossRef]
  10. S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002). [CrossRef]
  11. S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002). [CrossRef]
  12. M. D. Arnold, M. G. Blaber, M. J. Ford, and N. Harris, “Universal scaling of local plasmons in chains of metal spheres,” Opt. Express 18(7), 7528–7542 (2010). [CrossRef] [PubMed]
  13. P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics 2(7), 387–388 (2008). [CrossRef]
  14. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]
  15. K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007). [CrossRef] [PubMed]
  16. A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78(3), 035403 (2008). [CrossRef]
  17. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006). [CrossRef]
  18. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74(3), 033402 (2006). [CrossRef]
  19. C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver spheres and its possible application,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(6), 066606 (2005). [CrossRef]
  20. D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4(9), 1561–1565 (2004). [CrossRef]
  21. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]
  22. S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004). [CrossRef]
  23. S. Zou and G. C. Schatz, “Metal nanoparticle array waveguides: proposed structures for subwavelength devices,” Phys. Rev. B 74(12), 125111 (2006). [CrossRef]
  24. V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B 75(8), 085426 (2007). [CrossRef]
  25. D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: the role of chain length unravelled,” Nano Lett. 5(5), 985–989 (2005). [CrossRef] [PubMed]
  26. Q. H. Wei, K. H. Su, S. Durant, and X. Zhang, “Plasmon resonance of finite one-dimensional au nanoparticle chains,” Nano Lett. 4(6), 1067–1071 (2004). [CrossRef]
  27. K. B. Crozier, E. Togan, E. Simsek, and T. Yang, “Experimental measurement of the dispersion relations of the surface plasmon modes of metal nanoparticle chains,” Opt. Express 15(26), 17482–17493 (2007). [CrossRef] [PubMed]
  28. A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76(20), 201403 (2007). [CrossRef]
  29. R. Quidant, C. Girard, J.-C. Weeber, and A. Dereux, “Tailoring the transmittance of integrated optical waveguides with short metallic nanoparticle chains,” Phys. Rev. B 69(8), 085407 (2004). [CrossRef]
  30. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87(16), 167401 (2001). [CrossRef] [PubMed]
  31. J. J. Choquette, K.-P. Marzlin, and B. C. Sanders, “Superradiance, subradiance, and suppressed superradiance of dipoles near a metal interface,” Phys. Rev. A 82(2), 023827 (2010). [CrossRef]
  32. M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009). [CrossRef] [PubMed]
  33. J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. II. optical properties of aggregated metal spheres,” Phys. Rev. B 25(6), 4204–4229 (1982). [CrossRef]
  34. B. Willingham, D. Brandl, and P. Nordlander, “Plasmon hybridization in nanorod dimers,” Appl. Phys. B 93(1), 209–216 (2008). [CrossRef]
  35. J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. 9(2), 887–891 (2009). [CrossRef] [PubMed]
  36. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8(11), 581–583 (1983). [CrossRef] [PubMed]
  37. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975). [CrossRef]
  38. R. Brako, “Optical properties of composite media,” J. Phys. C Solid State Phys. 11(15), 3345–3355 (1978). [CrossRef]
  39. J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley and Sons, 1975).
  40. D. J. Bergman, “Dielectric constant of a two-component granular composite: a practical scheme for calculating the pole spectrum,” Phys. Rev. B 19(4), 2359–2368 (1979). [CrossRef]
  41. F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30(9), 4989–4999 (1984). [CrossRef]
  42. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005). [CrossRef]
  43. G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher, A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu, B. Monacelli, and G. Boreman, “Plasmon dispersion relation of Au and Ag nanowires,” Phys. Rev. B 68(15), 155427 (2003). [CrossRef]
  44. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]
  45. R. M. Dickson and L. A. Lyon, “Unidirectional plasmon propagation in metallic nanowires,” J. Phys. Chem. B 104(26), 6095–6098 (2000). [CrossRef]
  46. Y. Fang, Z. Li, Y. Huang, S. Zhang, P. Nordlander, N. J. Halas, and H. Xu, “Branched silver nanowires as controllable plasmon routers,” Nano Lett. 10(5), 1950–1954 (2010). [CrossRef] [PubMed]

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