Abstract

In this paper we investigate for the first time the near-field optical behavior of two-dimensional Fibonacci plasmonic lattices fabricated by electron-beam lithography on transparent quartz substrates. In particular, by performing near-field optical microscopy measurements and three dimensional Finite Difference Time Domain simulations we demonstrate that near-field coupling of nanoparticle dimers in Fibonacci arrays results in a quasi-periodic lattice of localized nanoparticle plasmons. The possibility to accurately predict the spatial distribution of enhanced localized plasmon modes in quasi-periodic Fibonacci arrays can have a significant impact for the design and fabrication of novel nano-plasmonics devices.

© 2008 Optical Society of America

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  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. Maters. 2, 229-232 (2003).
    [CrossRef]
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    [CrossRef]
  3. . 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]
  4. . E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. VanDuyne, L. Gunnarson, T. Rindzevicius, B. Kasemo, and M. Kall, "Controlling Plasmon Line Shapes through Diffractive Coupling in Linear Arrays of Cylindrical Nanoparticles Fabricated by Electron Beam Lithography," Nano Lett. 5, 1065 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. . 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, 668 (2003).
    [CrossRef]
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  8. . C. Girard and R. Quidant, "Near-field optical transmittance of metal particle chain waveguides," Opt. Express 12, 6141 (2004).
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  14. . G. Veronis and S. Fan, "Bands and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
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  26. . M. Ghulinyan, C. J. Oton, L. Dal Negro, L. Pavesi, R. Sapienza, M. Colocci, and D. Wiersma, "Light pulse propagation in Fibonacci quasicrystals," Phys. Rev. B 71, 094204 (2005).
    [CrossRef]
  27. . R. Lifshitz, "The square fibonacci tiling," J. Alloys and Compounds 342, 186 (2002).
    [CrossRef]
  28. . X. Fu,Y. Liu, B. Cheng, and D. Zheng, "Spectral structure of two-dimensional Fibonacci quasilattices," Phys. Rev. B 43, 10808 (1991).
    [CrossRef]
  29. . N. Ferralis, A. W. Szmodis, and R. D. Diehl, "Diffraction from one and twoo dimensional quasicrystalline gratings," Am. J. Phys. 72, 1241 (2004).
    [CrossRef]
  30. . L. Dal Negro, N. N. Feng, and A. Gopinath, "Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays," J. Opt. A, in print.
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    [CrossRef]
  32. . C. L. Hynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, N. Gunnarsson, J. Prikulis, B. Kasemo, and M. Kall, "Nanoparticle Optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B. 107, 7337 (2003).
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    [CrossRef]
  35. . B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, "Facts and artifacts in near-field optical microscopy," J. Appl. Phys. 81, 2492 (1997).
    [CrossRef]
  36. . A. Taflove, Computational Electrodynamics:The Finite-Difference Time-Domain Method (Artech House, 1995).
  37. . OmniSim software by Photon Design, Oxford, UK.
  38. . A. C.  Cangellaris and D. B.  Wright, "Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena," IEEE Trans. Antennas Propag.  39, 1518 (1991).
    [CrossRef]
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    [CrossRef]

2007 (3)

2006 (3)

. A. Boltasseva and S. I. Bozhevolnyi, "Directional couplers using long-range surface plasmon polariton waveguides," IEEE JSTQE. 12, 1233 (2006).

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

. E. Macia, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
[CrossRef]

2005 (3)

. E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. VanDuyne, L. Gunnarson, T. Rindzevicius, B. Kasemo, and M. Kall, "Controlling Plasmon Line Shapes through Diffractive Coupling in Linear Arrays of Cylindrical Nanoparticles Fabricated by Electron Beam Lithography," Nano Lett. 5, 1065 (2005).
[CrossRef] [PubMed]

. M. Ghulinyan, C. J. Oton, L. Dal Negro, L. Pavesi, R. Sapienza, M. Colocci, and D. Wiersma, "Light pulse propagation in Fibonacci quasicrystals," Phys. Rev. B 71, 094204 (2005).
[CrossRef]

. G. Veronis and S. Fan, "Bands and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

2004 (3)

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

. N. Ferralis, A. W. Szmodis, and R. D. Diehl, "Diffraction from one and twoo dimensional quasicrystalline gratings," Am. J. Phys. 72, 1241 (2004).
[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]

2003 (6)

. C. L. Hynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, N. Gunnarsson, J. Prikulis, B. Kasemo, and M. Kall, "Nanoparticle Optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B. 107, 7337 (2003).
[CrossRef]

.S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 20540 (2003).
[CrossRef]

. 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. Maters. 2, 229-232 (2003).
[CrossRef]

. 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]

. L. L. Zhao, K. L. Kelly, and G. C. Schatz, "The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonant wavelength and width," J. Phys. Chem. B 107, 7343 (2003)
[CrossRef]

. 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, 668 (2003).
[CrossRef]

2002 (2)

. R. Lifshitz, "The square fibonacci tiling," J. Alloys and Compounds 342, 186 (2002).
[CrossRef]

. H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, "Two-dimensional optics with surface plasmon polaritons," Appl. Phys. Lett. 81, 1762 (2002).
[CrossRef]

2001 (2)

. J. C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, "Near-field observation of surface plasmon polariton propagation on thin metal stripes," Phys. Rev. B. 64, 045411 (2001).
[CrossRef]

. 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]

1998 (1)

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

1997 (1)

. B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, "Facts and artifacts in near-field optical microscopy," J. Appl. Phys. 81, 2492 (1997).
[CrossRef]

1994 (2)

. 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]

1993 (1)

1991 (2)

. A. C.  Cangellaris and D. B.  Wright, "Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena," IEEE Trans. Antennas Propag.  39, 1518 (1991).
[CrossRef]

. X. Fu,Y. Liu, B. Cheng, and D. Zheng, "Spectral structure of two-dimensional Fibonacci quasilattices," Phys. Rev. B 43, 10808 (1991).
[CrossRef]

1987 (2)

. 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]

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

1986 (1)

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

1976 (1)

.P. Clippe, R. Evrard, and A. A. Lucas, "Aggregation effect on the infrared absorption spectrum of small ionic crystals," Phys. Rev. B 14, 1715 (1976).
[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]

Am. J. Phys. (1)

. N. Ferralis, A. W. Szmodis, and R. D. Diehl, "Diffraction from one and twoo dimensional quasicrystalline gratings," Am. J. Phys. 72, 1241 (2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

. H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, "Two-dimensional optics with surface plasmon polaritons," Appl. Phys. Lett. 81, 1762 (2002).
[CrossRef]

. G. Veronis and S. Fan, "Bands and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

IEEE JSTQE. (1)

. A. Boltasseva and S. I. Bozhevolnyi, "Directional couplers using long-range surface plasmon polariton waveguides," IEEE JSTQE. 12, 1233 (2006).

IEEE Trans. Antennas Propag. (1)

. A. C.  Cangellaris and D. B.  Wright, "Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena," IEEE Trans. Antennas Propag.  39, 1518 (1991).
[CrossRef]

J. Alloys and Compounds (1)

. R. Lifshitz, "The square fibonacci tiling," J. Alloys and Compounds 342, 186 (2002).
[CrossRef]

J. Appl. Phys. (1)

. B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, "Facts and artifacts in near-field optical microscopy," J. Appl. Phys. 81, 2492 (1997).
[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).
[CrossRef]

J. Phys. Chem. B (2)

. L. L. Zhao, K. L. Kelly, and G. C. Schatz, "The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonant wavelength and width," J. Phys. Chem. B 107, 7343 (2003)
[CrossRef]

. 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, 668 (2003).
[CrossRef]

J. Phys. Chem. B. (1)

. C. L. Hynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, N. Gunnarsson, J. Prikulis, B. Kasemo, and M. Kall, "Nanoparticle Optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B. 107, 7337 (2003).
[CrossRef]

Materials Today (1)

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

Nano Lett. (1)

. E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. VanDuyne, L. Gunnarson, T. Rindzevicius, B. Kasemo, and M. Kall, "Controlling Plasmon Line Shapes through Diffractive Coupling in Linear Arrays of Cylindrical Nanoparticles Fabricated by Electron Beam Lithography," Nano Lett. 5, 1065 (2005).
[CrossRef] [PubMed]

Nat. Maters. (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. Maters. 2, 229-232 (2003).
[CrossRef]

Opt. Express (4)

Phys. Rev. B (7)

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

.S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 20540 (2003).
[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]

. X. Fu,Y. Liu, B. Cheng, and D. Zheng, "Spectral structure of two-dimensional Fibonacci quasilattices," Phys. Rev. B 43, 10808 (1991).
[CrossRef]

.P. Clippe, R. Evrard, and A. A. Lucas, "Aggregation effect on the infrared absorption spectrum of small ionic crystals," Phys. Rev. B 14, 1715 (1976).
[CrossRef]

. M. Ghulinyan, C. J. Oton, L. Dal Negro, L. Pavesi, R. Sapienza, M. Colocci, and D. Wiersma, "Light pulse propagation in Fibonacci quasicrystals," Phys. Rev. B 71, 094204 (2005).
[CrossRef]

Phys. Rev. B. (3)

. J. C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, "Near-field observation of surface plasmon polariton propagation on thin metal stripes," Phys. Rev. B. 64, 045411 (2001).
[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. 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]

Phys. Rev. Lett. (3)

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

. 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]

Rep. Prog. Phys. (1)

. E. Macia, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
[CrossRef]

Other (6)

. L. Dal Negro, N. N. Feng, and A. Gopinath, "Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays," J. Opt. A, in print.

. A. Apostolico and V. E. Brimkov, "Fibonacci arrays and their two-dimensional properties," Theret. Comp. Sci. 237, 263 (200).
[CrossRef]

. A. Taflove, Computational Electrodynamics:The Finite-Difference Time-Domain Method (Artech House, 1995).

. OmniSim software by Photon Design, Oxford, UK.

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

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

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Figures (6)

Fig. 1.
Fig. 1.

(a). First two generations of 2D Fibonacci sequence shown along with the inflation rules, (b) Generation 7 of the 2D Fibonacci lattice, (c) Calculated diffraction spectra of the 2D Fibonacci lattice shown in (b), (d) Cut along the vertical arrow shown in (c) showing the preservation of 1D Fibonacci Fourier characteristics.

Fig. 2.
Fig. 2.

SEM images of (a) Periodic and (b) Fibonacci Au nanoparticle array. The insets show the dimension of the particles (200nm) and the inter-particle separation (50nm); (c) Normalized extinction of periodic arrays with 100 nm particle spacing. The inset shows the true-color image of the array under white light illumination. The reddish edge region of the array, which is responsible for the near-infrared background scattering, arises from structural imperfections due to partial lift-off at the array edge; (d) Normalized extinction of the Fibonacci array (100 nm particle spacing). The inset shows the true-color image of the Fibonacci array under white light illumination.

Fig. 3.
Fig. 3.

(a) and (b) show topographical AFM images on the Fibonacci and Periodic arrays respectively, (c) and (d) are NSOM images obtained from Fibonacci and Periodic nanoparticle arrays obtained under identical conditions.

Fig. 4.
Fig. 4.

(a) and (b) are NSOM images obtained from Fibonacci and Periodic nanoparticle arrays obtained under identical conditions, (c) shows the intensity profile along the horizontal cut in the NSOM images along the white lines in (a) and (b).

Fig. 5.
Fig. 5.

(a) and (c) are NSOM images obtained from Fibonacci and Periodic nanoparticle arrays obtained under identical conditions. (b) and (d) are FDTD simulations of periodic and Fibonacci Au nanoparticle arrays respectively, the particles were 200nm and min. interparticle separation was 50nm. Te number of particles for the Fibonacci array is 80 while it is 64 for the periodic lattice. The Au was modeled using the Drude parameters collision frequency =250 THz, and Plasma Frequency =6790 THz. The illumination was with a CW excitation polarized in the plane of the particle at 520nm.

Fig. 6.
Fig. 6.

(a). Calculated hot-spot locations are shown (red) along with the Fibonacci nanoparticle array (black). (b) Fourier spectrum of the hot-spots sequence along the vertical direction. (c) Fourier spectrum of the hot-spots sequence along the horizontal direction. This spectrum coincides with the one of a one-dimensional Fibonacci sequence.

Equations (2)

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g n + 2 = g n + 1 ϕ g n
ϕ = { 1 if n even 0 if n odd

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