Abstract

Dispersion of the resonant properties exhibited by silver and gold nano-strips in a wide range of wavelengths is considered. The tunability and Q-factor of scattering resonances as well as the field enhancement achieved at strip terminations are analyzed in the wavelength range from visible to near infrared (400–1700 nm), confirming that the resonant behaviour is dominated by dispersion properties of short-range surface-plasmon polaritons (SR-SPPs) propagating along the strip. It is found that, while the Q-factor decreases for longer wavelengths due to the SR-SPP dispersion curve moving closer to the light line, the field enhancement depending also on the metal susceptibility magnitude remains largely unaffected. The results obtained are also used to estimate the phase change involved in the SR-SPP reflection by strip terminations.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  4. K. Imura, T. Nagahara, and H. Okamoto, "Near-field imaging of plasmon modes in gold nanorods," J. Chem. Phys. 122, 154701-1-5 (2005).
    [CrossRef] [PubMed]
  5. 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, 257403-1-4 (2005).
    [CrossRef] [PubMed]
  6. T. Laroche and C. Girard, "Near-field optical properties of single plasmonic nanowires," Appl. Phys. Lett. 89, 233119-1-3 (2006).
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  7. F. Neubrech,  et al., "Resonances of individual metal nanowires in the infrared," Appl. Phys. Lett. 89, 253104-1-3 (2006).
    [CrossRef]
  8. L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802-1-4 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, "Slow-plasmon resonant nano-strip antennas: Analysis and demonstration," Phys. Rev. B 77, 115420-1-5 (2008).
    [CrossRef]
  14. T. Søndergaard, "Modeling of plasmonic nanostructures: Green�??s function integral equation methods," Phys. Status Solidi(b) 244, 3448-3462 (2007).
    [CrossRef]
  15. D. W. Prather, M. S. Mirotznik, and J. N. Mait, "Boundary integral methods applied to the analysis of diffractive optical elements," J. Opt. Soc. Am. A 14, 34-43 (1997).
    [CrossRef]
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  17. E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
    [CrossRef]
  18. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
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2007 (4)

T. Søndergaard and S. I. Bozhevolnyi, "Metal nano-strip optical resonators," Opt. Express 15, 4198-4204 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi and T. Søndergaard, "General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators," Opt. Express 15, 10869-10877 (2007).
[CrossRef] [PubMed]

T. Søndergaard, "Modeling of plasmonic nanostructures: Green�??s function integral equation methods," Phys. Status Solidi(b) 244, 3448-3462 (2007).
[CrossRef]

S. Lal, S. Link, and N. J. Halas, "Nano-optics from sensing to waveguiding," Nat. Photonics 1, 641-648 (2007).
[CrossRef]

1997 (2)

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1969 (1)

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Aussenegg, F. R.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Beermann, J.

T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, "Slow-plasmon resonant nano-strip antennas: Analysis and demonstration," Phys. Rev. B 77, 115420-1-5 (2008).
[CrossRef]

Boltasseva, A.

T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, "Slow-plasmon resonant nano-strip antennas: Analysis and demonstration," Phys. Rev. B 77, 115420-1-5 (2008).
[CrossRef]

Bozhevolnyi, S. I.

T. Søndergaard and S. I. Bozhevolnyi, "Metal nano-strip optical resonators," Opt. Express 15, 4198-4204 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi and T. Søndergaard, "General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators," Opt. Express 15, 10869-10877 (2007).
[CrossRef] [PubMed]

T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, "Slow-plasmon resonant nano-strip antennas: Analysis and demonstration," Phys. Rev. B 77, 115420-1-5 (2008).
[CrossRef]

T. Søndergaard and S. I. Bozhevolnyi, "Slow-plasmon resonant nanostructures: Scattering and field enhancements," Phys. Rev. B 75, 073402-1-4 (2007).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Ditlbacher, H.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Economou, E. N.

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Girard, C.

T. Laroche and C. Girard, "Near-field optical properties of single plasmonic nanowires," Appl. Phys. Lett. 89, 233119-1-3 (2006).
[CrossRef]

Halas, N. J.

S. Lal, S. Link, and N. J. Halas, "Nano-optics from sensing to waveguiding," Nat. Photonics 1, 641-648 (2007).
[CrossRef]

Hofer, F.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Hohenau, A.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Imura, K.

K. Imura, T. Nagahara, and H. Okamoto, "Near-field imaging of plasmon modes in gold nanorods," J. Chem. Phys. 122, 154701-1-5 (2005).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kobayashi, T.

Kreibig, U.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Krenn, J. R.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Lal, S.

S. Lal, S. Link, and N. J. Halas, "Nano-optics from sensing to waveguiding," Nat. Photonics 1, 641-648 (2007).
[CrossRef]

Laroche, T.

T. Laroche and C. Girard, "Near-field optical properties of single plasmonic nanowires," Appl. Phys. Lett. 89, 233119-1-3 (2006).
[CrossRef]

Link, S.

S. Lal, S. Link, and N. J. Halas, "Nano-optics from sensing to waveguiding," Nat. Photonics 1, 641-648 (2007).
[CrossRef]

Mait, J. N.

Mirotznik, M. S.

Morimoto, A.

Nagahara, T.

K. Imura, T. Nagahara, and H. Okamoto, "Near-field imaging of plasmon modes in gold nanorods," J. Chem. Phys. 122, 154701-1-5 (2005).
[CrossRef] [PubMed]

Neubrech, F.

F. Neubrech,  et al., "Resonances of individual metal nanowires in the infrared," Appl. Phys. Lett. 89, 253104-1-3 (2006).
[CrossRef]

Novotny, L.

L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802-1-4 (2007).
[CrossRef] [PubMed]

Okamoto, H.

K. Imura, T. Nagahara, and H. Okamoto, "Near-field imaging of plasmon modes in gold nanorods," J. Chem. Phys. 122, 154701-1-5 (2005).
[CrossRef] [PubMed]

Prather, D.W.

Rogers, M.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Shen, Y. R.

F. Wang, and Y. R. Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806-1-4 (2006).
[CrossRef] [PubMed]

Søndergaard, T.

T. Søndergaard and S. I. Bozhevolnyi, "Metal nano-strip optical resonators," Opt. Express 15, 4198-4204 (2007).
[CrossRef] [PubMed]

T. Søndergaard, "Modeling of plasmonic nanostructures: Green�??s function integral equation methods," Phys. Status Solidi(b) 244, 3448-3462 (2007).
[CrossRef]

S. I. Bozhevolnyi and T. Søndergaard, "General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators," Opt. Express 15, 10869-10877 (2007).
[CrossRef] [PubMed]

T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, "Slow-plasmon resonant nano-strip antennas: Analysis and demonstration," Phys. Rev. B 77, 115420-1-5 (2008).
[CrossRef]

T. Søndergaard and S. I. Bozhevolnyi, "Slow-plasmon resonant nanostructures: Scattering and field enhancements," Phys. Rev. B 75, 073402-1-4 (2007).
[CrossRef]

Takahara, J.

Taki, H.

Wagner, D.

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

Wang, F.

F. Wang, and Y. R. Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806-1-4 (2006).
[CrossRef] [PubMed]

Yamagishi, S.

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

Nat. Photonics (1)

S. Lal, S. Link, and N. J. Halas, "Nano-optics from sensing to waveguiding," Nat. Photonics 1, 641-648 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. (1)

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Phys. Status Solidi (1)

T. Søndergaard, "Modeling of plasmonic nanostructures: Green�??s function integral equation methods," Phys. Status Solidi(b) 244, 3448-3462 (2007).
[CrossRef]

Other (11)

H. Rather, Surface Plasmons (Springer, 1988).

J. Jin, The Finite Element Method in Electromagnetics (John Wiley & Sons, New York 2002).

O. Svelto, Principles of Lasers (Springer, 4th ed., 1998).

T. Søndergaard and S. I. Bozhevolnyi, "Slow-plasmon resonant nanostructures: Scattering and field enhancements," Phys. Rev. B 75, 073402-1-4 (2007).
[CrossRef]

T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, "Slow-plasmon resonant nano-strip antennas: Analysis and demonstration," Phys. Rev. B 77, 115420-1-5 (2008).
[CrossRef]

F. Wang, and Y. R. Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806-1-4 (2006).
[CrossRef] [PubMed]

K. Imura, T. Nagahara, and H. Okamoto, "Near-field imaging of plasmon modes in gold nanorods," J. Chem. Phys. 122, 154701-1-5 (2005).
[CrossRef] [PubMed]

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, 257403-1-4 (2005).
[CrossRef] [PubMed]

T. Laroche and C. Girard, "Near-field optical properties of single plasmonic nanowires," Appl. Phys. Lett. 89, 233119-1-3 (2006).
[CrossRef]

F. Neubrech,  et al., "Resonances of individual metal nanowires in the infrared," Appl. Phys. Lett. 89, 253104-1-3 (2006).
[CrossRef]

L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802-1-4 (2007).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1.
Fig. 1.

(a) Effective index and propagation length of SR-SPP modes sustained by silver films of different thickness t. Lines: exact numerical computation. Points: values from approximate analytical formula given by Eq. (1) [12]. (b) Same as (a) for gold films. Peaks in the propagation length derive from discontinuities in the slope of the metal refractive index curve taken from experimental data [18].

Fig. 2.
Fig. 2.

(a) Schematic of the metal (silver or gold) nanostrip. The strip is illuminated with a p-polarized plane wave propagating at an angle of 45° with respect to the x axis. (b) Scattering cross section (normalized to the strip width) as a function of wavelength for 10 nm thick silver strips of three different widths w. Black and white arrows indicate first and second order (harmonic) resonances, respectively.

Fig. 3.
Fig. 3.

Strip width w as a function of the desired (first or second order) resonance wavelength for silver strips of different thickness t. Lines represent linear fitting of the data.

Fig. 4.
Fig. 4.

Strip width w as a function of the desired (first or second order) resonance wavelength for gold strips of different thickness t. Lines represent linear fitting of the data.

Fig. 5.
Fig. 5.

Q-factor of the first and second order resonances for silver (black) and gold (red) strips of 10 nm thickness.

Fig. 6.
Fig. 6.

(a) Field enhancement as a function of the first order resonance wavelength for silver (black) and gold (red) strips of 10 nm thickness, computed at the maximum of the field inside the metal (point M in the inset) and in the dielectric just outside the strip edge (point D in the inset). Inset reproduces a typical cross section of the electric field magnitude along the x-axis through the center of the strip. (b) Same as (a) for second order resonance. Inset shows field enhancement under detuning condition with respect to the resonant wavelength for 10-nm thick, 300 nm wide silver (black squares) and gold (red triangles) nanostrips.

Fig. 7.
Fig. 7.

(a) Strip width w as a function of the desired (first, second and third order) resonance wavelength for 10 nm thick silver strips. Points: exact numerically computed data. Dashed lines: linear fitting of exact data. Solid curves: prediction according to Eq.(2) and assuming a wavelength independent phase. (b) Estimated phase change due to reflection at strip terminations. Points: phase estimation from exact data. Curves: phase estimation from linear fitting data.

Equations (3)

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k srsp = k 0 1 + ( 1 ε m ) ε m 2 · tanh 2 ( 0.5 k 0 t 1 ε m )
w 2 π λ n eff = m π ϕ
ϕ m ( λ ) = m π 2 π λ n eff ( λ ) w m ( λ )

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