## Exact solution for velocity of plasmon-polariton in metallic nano-chain |

Optics Express, Vol. 22, Issue 16, pp. 18958-18965 (2014)

http://dx.doi.org/10.1364/OE.22.018958

Acrobat PDF (836 KB)

### Abstract

In equidistant infinite chain of metallic nanospheres the collective mode of surface plasmons propagates without radiative losses, i.e., the Lorentz friction losses in each nanosphere are compensated by energy income in near-, medium- and far-field from the rest of the chain. Within an approximate approach including numerical studies in Green function framework it has been indicated superluminal propagation of some plasmon-polariton modes. By the exact solution of the nonlinear dynamic equation we demonstrate that the superluminal modes were an artifact of the perturbation solution type and we show that the group velocities for both polarizations are limited by light velocity, though vary in large range depending on chain parameters and are typically one order lower than the light velocity.

© 2014 Optical Society of America

## 1. Introduction

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

3. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**, 10871–10875 (2004). [CrossRef] [PubMed]

4. J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. **82**, 2590–2593 (1999). [CrossRef]

*μ*m range propagation in gold nanoparticles with averaged radius

*a*= 50 nm aligned in the equidistant chain with separation between neighboring spheres

*d*= 200 nm. In series of papers [5

5. 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**, R16356 (2000). [CrossRef]

9. 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 inmetal nanoparticle plasmon waveguides,” Nat. Mater. **2**, 229–232 (2003). [CrossRef] [PubMed]

*μ*m and the energy transfer along the chain was directly confirmed. Such long range of plasmon-polariton propagation reaching visible light wave-lengths and undergoing with reduced group velocity below 0.1×

*c*[6

6. 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**, 193408 (2002). [CrossRef]

9. 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 inmetal nanoparticle plasmon waveguides,” Nat. Mater. **2**, 229–232 (2003). [CrossRef] [PubMed]

10. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. **98**, 011101 (2005). [CrossRef]

11. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. **38**, 4743–4746 (2013). [CrossRef] [PubMed]

14. V. A. Markel and A. K. Sarychev, “Comment on Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B **86**, 037401 (2012). [CrossRef]

## 2. Equation for dispersion of plasmon-polariton in the chain

**r**we deal with the dipole

**D**, then in the other place

**r**

_{0}(

**r**

_{0}is fixed to the end of

**r**) this dipole causes electric field in the form as follows (including relativistic retardation of electromagnetic signals) [15, 16]:

*r*

_{0}) contributions to dipole field. This allows for writing out the dynamical equation for plasmon oscillations at each nano-sphere of the chain, which can be numbered by integer

*l*(

*d*denotes the separation between nano-spheres in the linear chain; vectors

**r**and

**r**

_{0}are collinear if the origin is associated with one of nano-spheres in the chain). We assume

*d*> 3

*a*(

*a*–the nano-sphere radius), when multipole corrections beyond dipole interaction model are negligible [1

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

3. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**, 10871–10875 (2004). [CrossRef] [PubMed]

17. P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. **96**, 113002 (2006). [CrossRef] [PubMed]

*k*is a number for 1D case, though we will call it traditionally as wave vector). This corresponds to the discrete Fourier transform component with respect to the positions similarly as for phonons in 1D crystal. The dynamics equation for time-position Fourier picture of dipole self-oscillations in the chain attains the form (for particularities of the derivation cf. [18

18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics **8**, 1317–1333 (2013). [CrossRef] [PubMed]

*ω*

_{1}denotes the Mie dipole frequency of plasmons on single nanosphere. The damping ratio

5. 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**, R16356 (2000). [CrossRef]

*C*is a constant of unity order,

*v*is the Fermi velocity in the metal,

_{F}*λ*is the electron mean free path in bulk metal (including scattering of electrons on other electrons, on impurities and on phonons [5

_{b}5. 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**, R16356 (2000). [CrossRef]

*v*= 1.396 × 10

_{F}^{6}m/s and

*λ*≃ 53 nm (at room temperature); the latter term in the formula (5) accounts for scattering of electrons on the boundary of the nanoparticle, while the former one corresponds to scattering processes similar as in bulk. The other effects, as the so-called Landau damping (especially important in small clusters [19

_{b}19. C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, “Fragmentation of the photoabsorption strength in neutral and charged metal microclusters,” Phys. Rev. Lett. **63**, 255–258 (1989). [CrossRef] [PubMed]

20. W. Ekardt, “Anomalous inelastic electron scattering from small metal particles,” Phys. Rev. B **33**, 8803–8805 (1986). [CrossRef]

19. C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, “Fragmentation of the photoabsorption strength in neutral and charged metal microclusters,” Phys. Rev. Lett. **63**, 255–258 (1989). [CrossRef] [PubMed]

18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics **8**, 1317–1333 (2013). [CrossRef] [PubMed]

*ω*gives self-frequency of these oscillations.

## 3. Exact solution for group velocity of plasmon-polariton in the chain

*ω*of Eq. (3)) with respect to the wave vector

*k*defines the group velocity of particular modes. Because of logarithmic singular term in the far-field transversal contribution to dipole interaction in the chain (taking advantage of the formula,

*ω*with

*ω*

_{1}(Mie frequency) in functions

*F*in r.h.s. of Eq. (3). This produces, however, the hyperbolic singularity in transversal group velocity resulted by virtue of Eq. (6). Moreover, within the perturbation approach the logarithmic singularity occurs for both polarizations, which is noticeable if one takes the derivative with respect to

*k*from perturbation terms to the real parts of frequencies given by real parts of Eqs. (4) taken at

*ω*=

*ω*

_{1}. All this singularities occur at isolated points for which

*kd*±

*ω*

_{1}

*d/v*=

*lπ*(

*l*–integer). Both hyperbolic and logarithmic divergences for perturbation formula for group velocities in this points would result in local exceeding

*c*by corresponding group velocities. To resolve the problem of this unphysical divergence the exact solution of Eq. (3) must be found, because of divergence of the expression (6) the corresponding contribution cannot be treated still as perturbation. To avoid the perturbative method of solution of Eq. (3) resolving itself to substitution of

*ω*

_{1}instead of

*ω*in the r.h.s. of this equation [18

18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics **8**, 1317–1333 (2013). [CrossRef] [PubMed]

*k*∈ [0, 2

*π/d*) (in 1000 points). Such a procedure can be treated as the accurate solution of Eq. (3) without deficiency caused by the perturbative approach. The exact solution of Eq. (3), found numerically, is plotted in Fig. 1 for both polarizations of plasmon-polaritons.

**8**, 1317–1333 (2013). [CrossRef] [PubMed]

*c*). This quenched logarithmic singularity into small local minimum is presented in Fig. 2.

*F*and thus perturbative contribution to

_{z}*ω*(

_{z}*k*) (Eq. (4)) is given by a continuous function, the corresponding group velocity will have logarithmic singularity as the derivative of

*ω*(

_{z}*k*) with respect to

*k*still will contain the sum of (cos(

*m*(

*kd*+

*ω*

_{1}

*d/v*)) −

*cos*(

*m*(

*kd*−

*ω*

_{1}

*d/v*)))/

*m*, arising from the term with factor

*ω*

_{x(y)}(

*k*) in corresponding function given by Eq. (4). The origin of these terms for both polarizations is the medium-field contribution to dipole interaction in the chain. In points

*kd*±

*ω*

_{1}

*d/v*=

*l*2

*π*,

*l*–integer, the logarithmic singularity produces an artifact in the group velocities–exceeding

*c*. This again precludes applicability of the perturbation approach, at least close to singularity points.

*ω*can be determined in the whole region

*kd*∈ [0, 2

*π*). For the longitudinal polarization of plasmon-polariton its group velocity found by the exact solution is presented in Fig. 3. This solution for

*v*does not exhibit any singularities–the logarithmic singularity of perturbation term is quenched to only small local extrema limited by

_{z}*c*, similarly as it was demonstrated above for the transversal polarization.

*c*visibly due to relativistic constraints imposed on the dynamic equation and manifesting themselves in the form of its solution. The retardation of electric signals prohibits the collective excitation group velocity to exceed the light velocity. This quenching concerns infinite singularities which occur in perturbation expressions for self-energy and in perturbation formulae for group velocities. The relativistic invariance of dynamic equation for collective dipole plasmon oscillations in the chain prevents, however, exceeding the light velocity by the group velocity of all plasmon-polariton modes. The exact solutions of the dynamical Eq. (3) inherently posses this property in opposition to perturbative its solutions. Exact self-energies have suitably regularized their dependence with respect to

*k*, that their derivatives do not exceed

*c*.

11. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. **38**, 4743–4746 (2013). [CrossRef] [PubMed]

14. V. A. Markel and A. K. Sarychev, “Comment on Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B **86**, 037401 (2012). [CrossRef]

*k*-modes in singular regions. In the simulations [11

11. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. **38**, 4743–4746 (2013). [CrossRef] [PubMed]

12. 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**, 035403 (2008). [CrossRef]

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

*k*modes including also those in singular regions. The different character of truncated singularities, originally infinite divergences in perturbation series, for longitudinal and transversal polarizations will result in different features of the corresponding modes (especially in relative narrowing of truncated singularities, stronger for longitudinal polarization). Though truncated at

*c*level, the singular modes give the same long distance propagation for both polarizations (ca. two orders longer than for modes with other values of wave-vector

*k*).

## 4. Damping of plasmon-polariton in the chain

*k*for which the exact cancellation of the Lorentz friction takes place in the chain. They are still inconvenient and are of similar magnitude as in separate nanospheres (for the Au nanosphere with radius

*a*= 10 nm,

23. W. Jacak, J. Krasnyj, J. Jacak, R. Gonczarek, A. Chepok, L. Jacak, D. Hu, and D. Schaadt, “Radius dependent shift in surface plasmon frequency in large metallic nanospheres: Theory and experiment,” J. Appl. Phys. **107**, 124317 (2010). [CrossRef]

24. T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. **80**, 4249–4252 (1998). [CrossRef]

25. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B **85**, 165419 (2012). [CrossRef]

## 5. Conclusions

*k*domain (diminishing with

*d/a*growth). Both the low (only Ohmic) attenuation of self-oscillations and the possibility to excite appropriately formed (by excitation of some part of the chain) wave packet of plasmon-polaritons modes in the chain propagating with suitably selected group velocity, may be responsible for experimentally observed of practically undamped propagation of collective plasmon signal over relatively large distances in several

*μ*m length nano-chains.

## Acknowledgments

## References and links

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

2. | L. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: Influence of array structure on plasmon resonance wavelength and width,” J. Phys. Chem. B |

3. | S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. |

4. | J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. |

5. | 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 |

6. | 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 |

7. | S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B |

8. | S. A. Maier, P. G. Kik, L. A. Sweatlock, H. A. Atwater, J. J. Penninkhof, A. Polman, S. Meltzer, E. Harel, A. Requicha, and B. E. Koel, “Energy transport in metal nanoparticle plasmon waveguides,” Mat. Res. Soc. Symp. Proc.777, T7.1.1 (2003). |

9. | 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 inmetal nanoparticle plasmon waveguides,” Nat. Mater. |

10. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

11. | I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. |

12. | 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 |

13. | Y. Hadad and B. Z. Steinberg, “Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B |

14. | V. A. Markel and A. K. Sarychev, “Comment on Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B |

15. | L. D. Landau and E. M. Lifshitz, |

16. | J. D. Jackson, |

17. | P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. |

18. | W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics |

19. | C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, “Fragmentation of the photoabsorption strength in neutral and charged metal microclusters,” Phys. Rev. Lett. |

20. | W. Ekardt, “Anomalous inelastic electron scattering from small metal particles,” Phys. Rev. B |

21. | I. S. Gradshteyn and I. M. Ryzhik, |

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

23. | W. Jacak, J. Krasnyj, J. Jacak, R. Gonczarek, A. Chepok, L. Jacak, D. Hu, and D. Schaadt, “Radius dependent shift in surface plasmon frequency in large metallic nanospheres: Theory and experiment,” J. Appl. Phys. |

24. | T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. |

25. | E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B |

**OCIS Codes**

(240.5420) Optics at surfaces : Polaritons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Plasmonics

**History**

Original Manuscript: May 26, 2014

Revised Manuscript: July 6, 2014

Manuscript Accepted: July 8, 2014

Published: July 29, 2014

**Citation**

Witold A. Jacak, "Exact solution for velocity of plasmon-polariton in metallic nano-chain," Opt. Express **22**, 18958-18965 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-18958

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### References

- D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: The role of chain length unravelled,” Nano Lett.5, 985–989 (2005). [CrossRef] [PubMed]
- L. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: Influence of array structure on plasmon resonance wavelength and width,” J. Phys. Chem. B107, 7343–7350 (2003). [CrossRef]
- S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120, 10871–10875 (2004). [CrossRef] [PubMed]
- J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett.82, 2590–2593 (1999). [CrossRef]
- 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. B62, R16356 (2000). [CrossRef]
- 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. B65, 193408 (2002). [CrossRef]
- S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67, 205402 (2003). [CrossRef]
- S. A. Maier, P. G. Kik, L. A. Sweatlock, H. A. Atwater, J. J. Penninkhof, A. Polman, S. Meltzer, E. Harel, A. Requicha, and B. E. Koel, “Energy transport in metal nanoparticle plasmon waveguides,” Mat. Res. Soc. Symp. Proc.777, T7.1.1 (2003).
- 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 inmetal nanoparticle plasmon waveguides,” Nat. Mater.2, 229–232 (2003). [CrossRef] [PubMed]
- S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005). [CrossRef]
- I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett.38, 4743–4746 (2013). [CrossRef] [PubMed]
- 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. B78, 035403 (2008). [CrossRef]
- Y. Hadad and B. Z. Steinberg, “Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B84, 125402 (2011). [CrossRef]
- V. A. Markel and A. K. Sarychev, “Comment on Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B86, 037401 (2012). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Field Theory (Nauka, 1973).
- J. D. Jackson, Classical Electrodynamics (John Willey and Sons Inc., 1998).
- P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett.96, 113002 (2006). [CrossRef] [PubMed]
- W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics8, 1317–1333 (2013). [CrossRef] [PubMed]
- C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, “Fragmentation of the photoabsorption strength in neutral and charged metal microclusters,” Phys. Rev. Lett.63, 255–258 (1989). [CrossRef] [PubMed]
- W. Ekardt, “Anomalous inelastic electron scattering from small metal particles,” Phys. Rev. B33, 8803–8805 (1986). [CrossRef]
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic Press, Inc., 1994).
- V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B75, 085426 (2007). [CrossRef]
- W. Jacak, J. Krasnyj, J. Jacak, R. Gonczarek, A. Chepok, L. Jacak, D. Hu, and D. Schaadt, “Radius dependent shift in surface plasmon frequency in large metallic nanospheres: Theory and experiment,” J. Appl. Phys.107, 124317 (2010). [CrossRef]
- T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett.80, 4249–4252 (1998). [CrossRef]
- E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B85, 165419 (2012). [CrossRef]

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