## Energy transport in metal nanoparticle chains via sub-radiant plasmon modes |

Optics Express, Vol. 19, Issue 7, pp. 6450-6461 (2011)

http://dx.doi.org/10.1364/OE.19.006450

Acrobat PDF (1429 KB)

### 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

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]

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]

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]

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

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]

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. 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. 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]

## 2. Optical characteristics of finite chains of Ag nanoparticles

*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]

*l*= 20 were included in the calculations and all results have been independently tested for smaller chains of particles using the finite-element method.

*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]

*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 ω

_{b}= 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]

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]

*E*, which does not increase further upon addition of more particles, is a direct measure of the particle-particle interaction strength and alludes to an upper limit of the group velocity of SPPs [10

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]

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]

*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.

*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]

_{scat}and Q

_{abs}, 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 Q

_{scat}plots, having all particle dipoles aligned parallel to the chain, is the most efficient at losing energy through far-field scattering.

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

_{scat}, but always remain weaker than the super-radiant mode. Absorptive losses, as inferred from the Q

_{abs}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]

_{abs}compared to the main dipole peak. For small σ, the Q

_{scat}and Q

_{abs}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.

_{pol}], for the super-radiant and two lowest energy, bright sub-radiant modes of the

*N*= 10 particle chain at σ = 2.1, corresponding to Q

_{abs}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.

_{0}, is equivalent to twice the length of the nanoparticle chain.

_{0}/

*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

*N*= 50 Ag nanoparticles with

*a*= 25 nm for different σ, and supports the notion that sub-radiant modes efficiently absorb incident radiation, sustain suppressed radiative losses, and thereby minimize SPP decay. The near-field intensity

*I*= |

**E**|

_{s}^{2}

**E**is the total scattered electric field including near and far-field contributions, and ε

_{s}_{0}and μ

_{0}are the vacuum permittivity and permeability, respectively. The chain is end-excited with multiple wavelengths of longitudinally polarized plane waves, exciting both bright and dark sub-radiant modes [13

**2**(7), 387–388 (2008). [CrossRef]

*I*is then fit to an exponential decay,

*I*=

*I*

_{0}exp

*(-b*z

*)*with decay constant,

*b*. Figure 3 shows the intensity profiles for the excitation energies that yield maximum SPP propagation at each σ. For σ = 3.0, the decay follows previous calculations showing maximum SPP propagation for excitation energies near the single particle resonance [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]

*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[ρ

_{pol}], 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

**2**(7), 387–388 (2008). [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]

*μ*m. 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.

## 4. Origin of sub-radiant modes

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

**P**of each particle and insisting that no bound charge is induced except within an infinitesimal volume at the surface

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

*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]. By solving for

**P**in spherical coordinates and neglecting the contribution of the energy contained within the magnetic fields, the multipolar moments

*q*are obtained by solvingwhere the

_{lmi}*ω*'s are the resonant modes of the system of spheres and

_{l}*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]

*ith*sphere with unit vector,

**n**

_{i}, normal to its surface is then given bywhere

*r*

_{i}is the radial distance from the center of the particle and

*Y*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, ρ

_{lm}_{pol,i}=

**P**

_{i}•

**n**

_{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.

*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]

*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

**2**(7), 387–388 (2008). [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]

**P**•

**n**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]

**2**(7), 387–388 (2008). [CrossRef]

*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

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]

*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

**E**•

_{s}**z**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.

*v*[22

_{g}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]

*v*/

_{g}*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]

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. 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]

## 6. Conclusion

## Acknowledgments

## 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. |

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. |

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 |

4. | S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Photonics |

5. | A. V. Malyshev, V. A. Malyshev, and J. Knoester, “Frequency-controlled localization of optical signals in graded plasmonic chains,” Nano Lett. |

6. | M. Sukharev and T. Seideman, “Phase and polarization control as a route to plasmonic nanodevices,” Nano Lett. |

7. | C. Dahmen, B. Schmidt, and G. von Plessen, “Radiation damping in metal nanoparticle pairs,” Nano Lett. |

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 |

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 |

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. |

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 |

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 |

13. | P. Nordlander, “Plasmonics: Subwavelength imaging in colour,” Nat. Photonics |

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. |

15. | K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. |

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 |

17. | A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B |

18. | A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B |

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. |

20. | D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. |

21. | W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B |

22. | S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B |

23. | S. Zou and G. C. Schatz, “Metal nanoparticle array waveguides: proposed structures for subwavelength devices,” Phys. Rev. B |

24. | V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B |

25. | D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: the role of chain length unravelled,” Nano Lett. |

26. | Q. H. Wei, K. H. Su, S. Durant, and X. Zhang, “Plasmon resonance of finite one-dimensional au nanoparticle chains,” Nano Lett. |

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 |

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 |

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 |

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. |

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 |

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. |

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 |

34. | B. Willingham, D. Brandl, and P. Nordlander, “Plasmon hybridization in nanorod dimers,” Appl. Phys. B |

35. | J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. |

36. | M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. |

37. | R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B |

38. | R. Brako, “Optical properties of composite media,” J. Phys. C Solid State Phys. |

39. | J. D. Jackson, |

40. | D. J. Bergman, “Dielectric constant of a two-component granular composite: a practical scheme for calculating the pole spectrum,” Phys. Rev. B |

41. | F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B |

42. | K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B |

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 |

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. |

45. | R. M. Dickson and L. A. Lyon, “Unidirectional plasmon propagation in metallic nanowires,” J. Phys. Chem. B |

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. |

**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

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