Abstract

We compute the dispersion diagram of an infinite chain of silver nanospheres. The Drude model is used to define the permittivity of nanospheres, and the generalized multipole technique (GMT) is applied to solve the Maxwell’s equation and, thus, to analyze the plasmon excitation. The obtained dispersion diagram using the GMT is compared with the result of the dipolar interacting model as well as the quasistatic model. Results of the finite element method are also presented to verify the accuracy of our results. Finally, a finite chain of metal nanospheres is examined for its scattering and propagation length of the guided modes.

© 2011 Optical Society of America

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  1. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
    [CrossRef]
  2. Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
    [CrossRef] [PubMed]
  3. W. Nomuraa, M. Ohtsub, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. 86, 181108 (2005).
    [CrossRef]
  4. M. Conforti and M. Guasoni, “Dispersive properties of linear chains of lossy metal nanoparticles,” J. Opt. Soc. Am. B 27, 1576–1582 (2010).
    [CrossRef]
  5. S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69, 125418 (2004).
    [CrossRef]
  6. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
    [CrossRef]
  7. Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
    [CrossRef]
  8. A. Dhawan, S. J. Norton, M. D. Gerhold, and T. Vo-Dinh, “Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers,” Opt. Express 17, 9688–9703 (2009).
    [CrossRef] [PubMed]
  9. D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
    [CrossRef]
  10. 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]
  11. N. Talebi and M. Shahabadi, “Analysis of the propagation of light along an array of nanorods using the generalized multipole techniques,” J. Comput. Theor. Nanosci. 5, 711–716(2008).
    [CrossRef]
  12. E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
    [CrossRef]
  13. R. F. Harrington, Time-Harmonic Electromagnetic Fields(Wiley, 2001).
    [CrossRef]
  14. E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002).
    [CrossRef]

2010 (1)

2009 (1)

2008 (2)

N. Talebi and M. Shahabadi, “Analysis of the propagation of light along an array of nanorods using the generalized multipole techniques,” J. Comput. Theor. Nanosci. 5, 711–716(2008).
[CrossRef]

Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
[CrossRef] [PubMed]

2007 (1)

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

2005 (2)

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]

W. Nomuraa, M. Ohtsub, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. 86, 181108 (2005).
[CrossRef]

2004 (3)

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

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

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

2003 (1)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

2002 (2)

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Citrin, D. S.

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]

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

Conforti, M.

Dhawan, A.

Erni, D.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002).
[CrossRef]

Ford, G. W.

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

Gerhold, M. D.

Gosele, U.

Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
[CrossRef] [PubMed]

Guasoni, M.

Hafner, C.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002).
[CrossRef]

Hao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields(Wiley, 2001).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Knez, M.

Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
[CrossRef] [PubMed]

Liu, L.

Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
[CrossRef] [PubMed]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Moreno, E.

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002).
[CrossRef]

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Nomuraa, W.

W. Nomuraa, M. Ohtsub, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. 86, 181108 (2005).
[CrossRef]

Norton, S. J.

Ohtsub, M.

W. Nomuraa, M. Ohtsub, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. 86, 181108 (2005).
[CrossRef]

Park, S. Y.

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

Qin, Y.

Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
[CrossRef] [PubMed]

Shahabadi, M.

N. Talebi and M. Shahabadi, “Analysis of the propagation of light along an array of nanorods using the generalized multipole techniques,” J. Comput. Theor. Nanosci. 5, 711–716(2008).
[CrossRef]

Stroud, D.

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

Talebi, N.

N. Talebi and M. Shahabadi, “Analysis of the propagation of light along an array of nanorods using the generalized multipole techniques,” J. Comput. Theor. Nanosci. 5, 711–716(2008).
[CrossRef]

Vahldieck, R.

Vo-Dinh, T.

Weber, W. H.

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

Yang, R.

Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
[CrossRef] [PubMed]

Yatsui, T.

W. Nomuraa, M. Ohtsub, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. 86, 181108 (2005).
[CrossRef]

Zhao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

W. Nomuraa, M. Ohtsub, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. 86, 181108 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

J. Comput. Theor. Nanosci. (1)

N. Talebi and M. Shahabadi, “Analysis of the propagation of light along an array of nanorods using the generalized multipole techniques,” J. Comput. Theor. Nanosci. 5, 711–716(2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nano Lett. (3)

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

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]

Y. Qin, L. Liu, R. Yang, U. Gosele, and M. Knez, “General assembly method for linear metal nanoparticle chains embedded in nanotubes,” Nano Lett. 8, 3221–3225 (2008).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. B (4)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

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

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

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Other (1)

R. F. Harrington, Time-Harmonic Electromagnetic Fields(Wiley, 2001).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Multipole setting for analyzing a periodic structure.

Fig. 2
Fig. 2

Multipole setting that automatically satisfies the periodic boundary condition.

Fig. 3
Fig. 3

Dispersion diagram of infinite chain of MNPs ( ω 0 = ω p / 3 ). Open circles show higher order modes of the structure obtained using the GMT method.

Fig. 4
Fig. 4

Distribution of the magnetic field ( H y ) on the x z plane for K L = π at t = 0 .

Fig. 5
Fig. 5

Distribution of the magnetic field for higher order modes on the x z plane for K L = π at t = 0 .

Fig. 6
Fig. 6

Computed transverse and longitudinal dispersion diagrams using the GMT, FEM, dipolar model, and quasistatic approximation. The results of the dipolar model and quasistatic approximation are from [6].

Fig. 7
Fig. 7

Scattering from a finite chain of nanospheres for λ = 350 nm . As can be seen, the electromagnetic field is decaying.

Fig. 8
Fig. 8

Plasmon propagation along a finite chain of nanospheres at λ = 390 nm . (a) Intensity of the normalized H. (b)–(e) Corresponding values of H x for λ = 390 nm at t = 0 , T / 2 , T, and 3 T / 2 , respectively.

Equations (15)

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E r n m = n ( n + 1 ) j ω ϵ r 2 P n m ( cos θ ) H ^ n ( 2 ) ( k r ) e j m ϕ n m n ,
H r n m = n ( n + 1 ) j ω μ r 2 P n m ( cos θ ) H ^ n ( 2 ) ( k r ) e j m ϕ n m n .
E r = n ( n + 1 ) j ω ϵ r 2 P n m ( cos θ ) H ^ n ( 2 ) ( k r ) e j m ϕ ,
E s D 2 = k = 1 M n 2 m n ( C n m E T E r n m ( r r k ) + D n m E T M r n m ( r r k ) ) ,
E t D 1 = k = 1 M n 1 m n ( C n m E T E r n m ( r r k ) + D n m E T M r n m ( r r k ) ) ,
E ( r , θ , ϕ ) = p = e j p K L n m , k ( C n m E T E r n m ( r r k p L z ^ ) + D n m E T M r n m ( r r k p L z ^ ) ) ,
E ( r , θ , ϕ ) = p = e j p K L n m , k ( C n m e j K L E T E r n m ( r r k p L z ^ ) + D n m e j K L E T M r n m ( r r k p L z ^ ) ) = e j K L E ( r , θ , ϕ ) .
E ( r , θ , ϕ ) = p = e j p K L ( E T E r 1 , 0 ( r r S p L z ^ ) + E T M r 1 , 0 ( r r S p L z ^ ) ) ,
... + [ A ] 2 [ C ] e 2 j K L + [ A ] 1 [ C ] e j K L + [ A ] 0 [ C ] + [ A ] 1 [ C ] e j K L + [ A ] 2 [ C ] e 2 j K L + ... = [ A ] out [ C ] out + [ B ] exc ,
p = e j p K L [ A ] p [ C ] = [ A ] out [ C ] out + [ B ] exc .
A ( ω ) C ( ω ) = B ( ω ) ,
A PEC ( ω ) C PEC ( ω ) + B exc ( ω ) = 0 ,
error = A ( ω ) C ( ω ) B ( ω ) C ( ω )
v ϵ / 2 ( | E ( n + 1 ) | 2 | E n | 2 ) + μ / 2 ( | H ( n + 1 ) | 2 | H n | 2 ) d v v ϵ / 2 | E n | 2 + μ / 2 | H n | 2 d v < 0.05
ϵ ( ω ) = 1 ω p 2 ω ( ω + j ν )

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