Abstract

In this paper we study the spectral, localization and dispersion properties of dipolar modes in quasi-periodically modulated nanoparticle chains based on the Fibonacci sequence. By developing a transfer matrix approach for the calculation of resonant frequencies, oscillation eigenvectors and integrated density of states (IDS) of spatially-modulated dipole chains, we demonstrate the presence of large spectral gaps and calculate the pseudo-dispersion diagram of Fibonacci plasmonic chains. The presence of plasmonic band-gaps and localized states in metal nanoparticle chains based on quasi-periodic order can have a large impact in the design and fabrication of novel nanophotonics devices.

© 2007 Optical Society of America

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  1. M. Kohmoto, B. Sutherland, and C. Tang, "Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model," Phys. Rev. B 35, 1020-1033 (1987).
    [CrossRef]
  2. D. Levine and P. J. Steinhardt, "Quasicrystals: definition and structure," Phys. Rev. B 34, 596-616 (1986).
    [CrossRef]
  3. C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).
  4. T. Fujiwara and T. Ogawa, Quasicrystals (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  5. R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, "Gap states and localization properties of one-dimensional Fibonacci quasicrystals," Phys. Rev. B 42, 6402-6406 (1990).
    [CrossRef]
  6. M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic media," Phys. Rev. Lett.,  58, 2436-2438 (1987).
    [CrossRef] [PubMed]
  7. C. Benoit, G. Poussigue, and A. Azougarh, "Neutron scattering by phonons in quasi-crystals," J. Phys.: Condens. Matter 2, 2519-2536 (1990).
    [CrossRef]
  8. E. L. Albuquerque and M. G. Cottam, "Theory of elementary excitations in quasiperiodic structures," Phys. Rep. 376, 225-337 (2003).
    [CrossRef]
  9. A. Rudinger and F. Piechon, "On the multifractal spectrum of the Fibonacci chain," J. Phys. A.: Math. Gen. 31, 155-164 (1998).Q1
    [CrossRef]
  10. T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wavefunctions on a Fibonacci lattice," Phys. Rev. B 40, 7413-7416 (1989).
    [CrossRef]
  11. F. Igloi, L. Turban, and H. Rieger, "Anomalous diffusion in aperiodic environments," Phys. Rev. E. 59, 1465-1474 (1999).
    [CrossRef]
  12. W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633-636 (1994).
    [CrossRef] [PubMed]
  13. T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, "Photonic dispersion relation in a one-dimensional quasicrystal," Phys. Rev. B 50, 4220-4223 (1994).
    [CrossRef]
  14. L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, "Light transport through the band-edge states of Fibonacci quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
    [CrossRef] [PubMed]
  15. R. Zia, J. A. Schuller and M. L. Brongersma, "Plasmonics: The Next Chip-Scale Technology," Materials Today 9, 20-27 (2006).
    [CrossRef]
  16. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13, 1501 (2001).
    [CrossRef]
  17. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
    [CrossRef] [PubMed]
  18. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
  19. 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, 356-359 (2000).
    [CrossRef]
  20. S. Y. Park and D. Stroud, "Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation," Phys. Rev. B. 69, 125418 (2004).
    [CrossRef]
  21. C. Girard and R. Quidant, "Near-field optical transmittance of metal particle chain waveguides," Opt. Express,  12, 6141 (2004).
    [CrossRef] [PubMed]
  22. F. A. B. F. de Moura, L. P. Viana, A. C. Frery, "Vibrational modes in aperiodic one-dimensional harmonic chains," Phys. Rev. B. 73, 212302 (2006).
    [CrossRef]
  23. P. K. Datta and K. Kundu, "The absence of localization in one-dimensional disordered harmonic chains," J. Phys: Condens. Matter 6, 4465-4478 (1994).
    [CrossRef]
  24. M. Schroeder, Fractals, Chaos, Power Laws (Freeman, NY, 1991).
  25. R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, 2000).
  26. U. Frisch, Turbolence (Cambridge University Press, 2004).

2006 (2)

R. Zia, J. A. Schuller and M. L. Brongersma, "Plasmonics: The Next Chip-Scale Technology," Materials Today 9, 20-27 (2006).
[CrossRef]

F. A. B. F. de Moura, L. P. Viana, A. C. Frery, "Vibrational modes in aperiodic one-dimensional harmonic chains," Phys. Rev. B. 73, 212302 (2006).
[CrossRef]

2004 (2)

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

C. Girard and R. Quidant, "Near-field optical transmittance of metal particle chain waveguides," Opt. Express,  12, 6141 (2004).
[CrossRef] [PubMed]

2003 (3)

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

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, "Light transport through the band-edge states of Fibonacci quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

E. L. Albuquerque and M. G. Cottam, "Theory of elementary excitations in quasiperiodic structures," Phys. Rep. 376, 225-337 (2003).
[CrossRef]

2001 (1)

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13, 1501 (2001).
[CrossRef]

2000 (1)

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, 356-359 (2000).
[CrossRef]

1999 (1)

F. Igloi, L. Turban, and H. Rieger, "Anomalous diffusion in aperiodic environments," Phys. Rev. E. 59, 1465-1474 (1999).
[CrossRef]

1998 (1)

A. Rudinger and F. Piechon, "On the multifractal spectrum of the Fibonacci chain," J. Phys. A.: Math. Gen. 31, 155-164 (1998).Q1
[CrossRef]

1994 (3)

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633-636 (1994).
[CrossRef] [PubMed]

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, "Photonic dispersion relation in a one-dimensional quasicrystal," Phys. Rev. B 50, 4220-4223 (1994).
[CrossRef]

P. K. Datta and K. Kundu, "The absence of localization in one-dimensional disordered harmonic chains," J. Phys: Condens. Matter 6, 4465-4478 (1994).
[CrossRef]

1990 (2)

C. Benoit, G. Poussigue, and A. Azougarh, "Neutron scattering by phonons in quasi-crystals," J. Phys.: Condens. Matter 2, 2519-2536 (1990).
[CrossRef]

R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, "Gap states and localization properties of one-dimensional Fibonacci quasicrystals," Phys. Rev. B 42, 6402-6406 (1990).
[CrossRef]

1989 (1)

T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wavefunctions on a Fibonacci lattice," Phys. Rev. B 40, 7413-7416 (1989).
[CrossRef]

1987 (2)

M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic media," Phys. Rev. Lett.,  58, 2436-2438 (1987).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, and C. Tang, "Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model," Phys. Rev. B 35, 1020-1033 (1987).
[CrossRef]

1986 (1)

D. Levine and P. J. Steinhardt, "Quasicrystals: definition and structure," Phys. Rev. B 34, 596-616 (1986).
[CrossRef]

Adv. Mater. (1)

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13, 1501 (2001).
[CrossRef]

J. Phys. A.: Math. Gen. (1)

A. Rudinger and F. Piechon, "On the multifractal spectrum of the Fibonacci chain," J. Phys. A.: Math. Gen. 31, 155-164 (1998).Q1
[CrossRef]

J. Phys.: Condens. Matter (1)

C. Benoit, G. Poussigue, and A. Azougarh, "Neutron scattering by phonons in quasi-crystals," J. Phys.: Condens. Matter 2, 2519-2536 (1990).
[CrossRef]

J. Phys: Condens. Matter (1)

P. K. Datta and K. Kundu, "The absence of localization in one-dimensional disordered harmonic chains," J. Phys: Condens. Matter 6, 4465-4478 (1994).
[CrossRef]

Materials Today (1)

R. Zia, J. A. Schuller and M. L. Brongersma, "Plasmonics: The Next Chip-Scale Technology," Materials Today 9, 20-27 (2006).
[CrossRef]

Nat. Mater. (1)

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

Opt. Express (1)

Phys. Rep. (1)

E. L. Albuquerque and M. G. Cottam, "Theory of elementary excitations in quasiperiodic structures," Phys. Rep. 376, 225-337 (2003).
[CrossRef]

Phys. Rev. B (5)

R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, "Gap states and localization properties of one-dimensional Fibonacci quasicrystals," Phys. Rev. B 42, 6402-6406 (1990).
[CrossRef]

M. Kohmoto, B. Sutherland, and C. Tang, "Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model," Phys. Rev. B 35, 1020-1033 (1987).
[CrossRef]

D. Levine and P. J. Steinhardt, "Quasicrystals: definition and structure," Phys. Rev. B 34, 596-616 (1986).
[CrossRef]

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, "Photonic dispersion relation in a one-dimensional quasicrystal," Phys. Rev. B 50, 4220-4223 (1994).
[CrossRef]

T. Fujiwara, M. Kohmoto, and T. Tokihiro, "Multifractal wavefunctions on a Fibonacci lattice," Phys. Rev. B 40, 7413-7416 (1989).
[CrossRef]

Phys. Rev. B. (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, 356-359 (2000).
[CrossRef]

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

F. A. B. F. de Moura, L. P. Viana, A. C. Frery, "Vibrational modes in aperiodic one-dimensional harmonic chains," Phys. Rev. B. 73, 212302 (2006).
[CrossRef]

Phys. Rev. E. (1)

F. Igloi, L. Turban, and H. Rieger, "Anomalous diffusion in aperiodic environments," Phys. Rev. E. 59, 1465-1474 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633-636 (1994).
[CrossRef] [PubMed]

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, "Light transport through the band-edge states of Fibonacci quasicrystals," Phys. Rev. Lett. 90, 055501 (2003).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic media," Phys. Rev. Lett.,  58, 2436-2438 (1987).
[CrossRef] [PubMed]

Other (6)

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

T. Fujiwara and T. Ogawa, Quasicrystals (Springer-Verlag, Berlin, 1990).
[CrossRef]

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

M. Schroeder, Fractals, Chaos, Power Laws (Freeman, NY, 1991).

R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, 2000).

U. Frisch, Turbolence (Cambridge University Press, 2004).

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Figures (4)

Fig. 1.
Fig. 1.

(a). Schematics of the aperiodic nanoparticle chain, and real parts of the transfer matrix element Q11 for (b) periodic and (c) aperiodic Fibonacci chains.

Fig. 2.
Fig. 2.

Integrated density of states (IDS) for the (a) periodic and (b) quasi-periodic Fibonacci chains.

Fig. 3.
Fig. 3.

Dipole moments of the (a) extended state of a periodic chain, (b) critical state (ω/ω0 = 0.986) and (c) the most localized states (ω/ω0 = 0.984) of a quasi-periodic Fibonacci chain. The participation ratios for cases (b) and (c) are 0.201 and 0.044.

Fig. 4.
Fig. 4.

Dispersion diagrams for (a) periodic chain: (b) quasi-periodic Fibonacci chain, transverse polarization; (c) quasi-periodic Fibonacci chain, longitudinal polarization.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

i , n = ω 0 2 p i , n Γ e p ˙ i , n + Γ R ω 0 2 p i , n γ i α n 2 ( p i , n 1 + p i , n + 1 )
u n + 1 = [ ω 2 ω 0 2 i ( Γ e ω Γ R ω 3 ω 0 2 ) γ i α n 2 ] u n u n 1 u n + 1
u n + 1 u n = T n u n u n 1
T n = ω 2 ω 0 2 i ( Γ e ω Γ R ω 3 ω 0 2 ) γ i α n 2 1 1 0
u N + 1 u N = T N T N 1 T N 1 T 2 T 1 u 1 u 0 = Q u 1 u 0
Q 11 ( ω ) = 0
g ( ω ) = k δ ( ω ω k )
P ( ω k ) = n e n ( ω k ) 2 N n e n ( ω k ) 4
n e n ( ω k ) 2 = 1

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