Abstract

We investigate the dispersion relation and extinction properties of surface plasmons in an array of gold nanoparticle chains under s-polarized plane wave excitations, through experiment and simulation. Our results reveal that the dispersion and extinction properties of gold nanoparticle chains at an air/glass interface are significantly different from those in a uniform medium. Under total internal reflection, the dispersion is much larger than that above total internal reflection and 100% extinction can be reached. We show that the large dispersion under total internal reflection can be explained by dipole fields and coupling at the air/glass interface.

© 2008 Optical Society of America

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  3. 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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
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  4. N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
    [Crossref]
  5. J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
    [Crossref]
  6. B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
    [Crossref] [PubMed]
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    [Crossref]
  9. Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004).
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  11. 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).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  18. P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
    [Crossref]
  19. K. B. Crozier, E. Togan, E. Simsek, and T. Yang, “Experimental measurement of the dispersion relations of the surface plasmon modes of metal nanoparticle chains,” Opt. Express 15, 17482–17493 (2007).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. S. J. Radzeviciusa, C.-C. Chenb, L. Peters Jr., and J. J. Daniels, “Near-field dipole radiation dynamics through FDTD modeling,” J. Appl. Geophys. 52, 75–91 (2003).
    [Crossref]
  28. L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” New J. Phys. 8, 264 (2006).
    [Crossref]
  29. L. Novotny L. and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Ch. 10.

2007 (4)

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403(R) (2007).
[Crossref]

J. Sung, E. M. Hicks, R. P. Van Duyne, and K. G. Spears, “Nanoparticle Spectroscopy: Dipole Coupling in Two-Dimensional Arrays of L-shaped Silver Nanoparticles,” J. Phys. Chem. C 111, 10368–10376 (2007).
[Crossref]

P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
[Crossref]

K. B. Crozier, E. Togan, E. Simsek, and T. Yang, “Experimental measurement of the dispersion relations of the surface plasmon modes of metal nanoparticle chains,” Opt. Express 15, 17482–17493 (2007).
[Crossref] [PubMed]

2006 (2)

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[Crossref]

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” New J. Phys. 8, 264 (2006).
[Crossref]

2005 (1)

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

2004 (4)

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]

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]

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]

Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004).
[Crossref]

2003 (4)

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[Crossref]

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

S. J. Radzeviciusa, C.-C. Chenb, L. Peters Jr., and J. J. Daniels, “Near-field dipole radiation dynamics through FDTD modeling,” J. Appl. Geophys. 52, 75–91 (2003).
[Crossref]

2002 (2)

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]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[Crossref]

2000 (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, R16356–16359 (2000).
[Crossref]

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

J. Mertz, “Radiative absorption, fluorescence, and scattering of a classical dipole near a lossless interface: a unified description,” J. Opt. Soc. Am. B 17, 1906–1913 (2000).
[Crossref]

1999 (1)

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

1998 (2)

1979 (1)

1977 (2)

Atwater, H. A.

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[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. B 65, 193408 (2002).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[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, R16356–16359 (2000).
[Crossref]

Aubard, J.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

Aussenegg, F. R.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998).
[Crossref]

Badenes, G.

P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
[Crossref]

Bourillot, E.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

Brongersma, M. L.

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]

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, R16356–16359 (2000).
[Crossref]

Chenb, C.-C.

S. J. Radzeviciusa, C.-C. Chenb, L. Peters Jr., and J. J. Daniels, “Near-field dipole radiation dynamics through FDTD modeling,” J. Appl. Geophys. 52, 75–91 (2003).
[Crossref]

Conway, J.

J. Lu, C. Petre, J. Conway, and E. Yablonovitch, “Numerical optimization of a grating coupler for the efficient excitation of surface plasmons at an Ag-SiO2 interface,” arXiv:physics/0703036v1 [physics.optics] (2007).

Cormack, I. G.

P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
[Crossref]

Crozier, K. B.

Daniels, J. J.

S. J. Radzeviciusa, C.-C. Chenb, L. Peters Jr., and J. J. Daniels, “Near-field dipole radiation dynamics through FDTD modeling,” J. Appl. Geophys. 52, 75–91 (2003).
[Crossref]

de Waele, R.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403(R) (2007).
[Crossref]

Dereux, A.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

Ditlbacher, H.

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

Djurisic, A. B.

Durant, S.

Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004).
[Crossref]

Duyne, R. P. V.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

Elazar, J. M.

Félidj, N.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[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]

Ghenuche, P.

P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
[Crossref]

Goudonnet, J. P.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

Gunnarsson, L.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Ch. 7 (Artech House Antennas and Propagation Library, 2000).

Harel, E.

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[Crossref]

Hartman, J. W.

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, R16356–16359 (2000).
[Crossref]

Haynes, C. L.

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Hecht, B.

L. Novotny L. and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Ch. 10.

Hicks, E. M.

J. Sung, E. M. Hicks, R. P. Van Duyne, and K. G. Spears, “Nanoparticle Spectroscopy: Dipole Coupling in Two-Dimensional Arrays of L-shaped Silver Nanoparticles,” J. Phys. Chem. C 111, 10368–10376 (2007).
[Crossref]

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

Hohenau, A.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

Janel, N.

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]

Kall, M.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

Kall, M. J.

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Kasemo, B.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Ketterson, J. B.

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” New J. Phys. 8, 264 (2006).
[Crossref]

Kik, P. G.

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[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. B 65, 193408 (2002).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[Crossref]

Koel, B. E.

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[Crossref]

Koenderink, A. F.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403(R) (2007).
[Crossref]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[Crossref]

Krenn, J. R.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998).
[Crossref]

Kunz, R. E.

L., L. Novotny

L. Novotny L. and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Ch. 10.

Lacroute, Y.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

Lamprecht, B.

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

Lechner, R. T.

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

Leitner, A.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998).
[Crossref]

Lévi, G.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

Loza-Alvarez, P.

P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
[Crossref]

Lu, J.

J. Lu, C. Petre, J. Conway, and E. Yablonovitch, “Numerical optimization of a grating coupler for the efficient excitation of surface plasmons at an Ag-SiO2 interface,” arXiv:physics/0703036v1 [physics.optics] (2007).

Luan, L.

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” New J. Phys. 8, 264 (2006).
[Crossref]

Lukosz, W.

Maier, S. A.

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[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. B 65, 193408 (2002).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[Crossref]

Majewski, M. L.

McFarland, A. D.

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Meltzer, S.

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[Crossref]

Mertz, J.

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]

Peters Jr., L.

S. J. Radzeviciusa, C.-C. Chenb, L. Peters Jr., and J. J. Daniels, “Near-field dipole radiation dynamics through FDTD modeling,” J. Appl. Geophys. 52, 75–91 (2003).
[Crossref]

Petre, C.

J. Lu, C. Petre, J. Conway, and E. Yablonovitch, “Numerical optimization of a grating coupler for the efficient excitation of surface plasmons at an Ag-SiO2 interface,” arXiv:physics/0703036v1 [physics.optics] (2007).

Polman, A.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403(R) (2007).
[Crossref]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[Crossref]

Prangsma, J. C.

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403(R) (2007).
[Crossref]

Prikulis, J.

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Quidant, R.

P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
[Crossref]

Quinten, M.

Radzeviciusa, S. J.

S. J. Radzeviciusa, C.-C. Chenb, L. Peters Jr., and J. J. Daniels, “Near-field dipole radiation dynamics through FDTD modeling,” J. Appl. Geophys. 52, 75–91 (2003).
[Crossref]

Rakic, A. D.

Requicha, A. A. G.

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[Crossref]

Rindzevicius, T.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

Schatz, G. C.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

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]

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Schider, G.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

Sievert, P. R.

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” New J. Phys. 8, 264 (2006).
[Crossref]

Simsek, E.

Spears, K. G.

J. Sung, E. M. Hicks, R. P. Van Duyne, and K. G. Spears, “Nanoparticle Spectroscopy: Dipole Coupling in Two-Dimensional Arrays of L-shaped Silver Nanoparticles,” J. Phys. Chem. C 111, 10368–10376 (2007).
[Crossref]

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

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]

Su, K.-H.

Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004).
[Crossref]

Sung, J.

J. Sung, E. M. Hicks, R. P. Van Duyne, and K. G. Spears, “Nanoparticle Spectroscopy: Dipole Coupling in Two-Dimensional Arrays of L-shaped Silver Nanoparticles,” J. Phys. Chem. C 111, 10368–10376 (2007).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Ch. 7 (Artech House Antennas and Propagation Library, 2000).

Togan, E.

Van Duyne, R. P.

J. Sung, E. M. Hicks, R. P. Van Duyne, and K. G. Spears, “Nanoparticle Spectroscopy: Dipole Coupling in Two-Dimensional Arrays of L-shaped Silver Nanoparticles,” J. Phys. Chem. C 111, 10368–10376 (2007).
[Crossref]

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

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]

Weeber, J. C.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

Wei, Q.-H.

Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004).
[Crossref]

Yablonovitch, E.

J. Lu, C. Petre, J. Conway, and E. Yablonovitch, “Numerical optimization of a grating coupler for the efficient excitation of surface plasmons at an Ag-SiO2 interface,” arXiv:physics/0703036v1 [physics.optics] (2007).

Yang, T.

Zhang, X.

Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004).
[Crossref]

Zhao, L. L.

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Zou, S.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

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]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

P. Ghenuche, I. G. Cormack, G. Badenes, P. Loza-Alvarez, and R. Quidant, “Cavity resonances in finite plasmonic chains,” Appl. Phys. Lett. 90, 041109 (2007).
[Crossref]

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[Crossref]

J. Appl. Geophys. (1)

S. J. Radzeviciusa, C.-C. Chenb, L. Peters Jr., and J. J. Daniels, “Near-field dipole radiation dynamics through FDTD modeling,” J. Appl. Geophys. 52, 75–91 (2003).
[Crossref]

J. Chem. Phys. (1)

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. Opt. Soc. Am. (3)

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

J. Phys. Chem. C (1)

J. Sung, E. M. Hicks, R. P. Van Duyne, and K. G. Spears, “Nanoparticle Spectroscopy: Dipole Coupling in Two-Dimensional Arrays of L-shaped Silver Nanoparticles,” J. Phys. Chem. C 111, 10368–10376 (2007).
[Crossref]

Nano Lett. (2)

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, 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, 1065–1070 (2005).
[Crossref] [PubMed]

Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004).
[Crossref]

Nature Mater. (1)

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 in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003).
[Crossref]

New J. Phys. (1)

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” New J. Phys. 8, 264 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Chem. B (1)

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

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 65, 193408 (2002).
[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, R16356–16359 (2000).
[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]

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]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[Crossref]

A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Phys. Rev. B 76, 201403(R) (2007).
[Crossref]

Phys. Rev. Lett. (2)

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999).
[Crossref]

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000).
[Crossref] [PubMed]

Other (3)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Ch. 7 (Artech House Antennas and Propagation Library, 2000).

J. Lu, C. Petre, J. Conway, and E. Yablonovitch, “Numerical optimization of a grating coupler for the efficient excitation of surface plasmons at an Ag-SiO2 interface,” arXiv:physics/0703036v1 [physics.optics] (2007).

L. Novotny L. and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Ch. 10.

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Figures (10)

Fig. 1.
Fig. 1.

Scanning electron micrographs of an array of gold nanoparticle chains on a glass substrate.

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Fig. 2.
Fig. 2.

Power transmission and reflection spectra for an array of gold nanoparticle chains at an air/glass interface. The incidence angle θ is 0° (red), 13° (wine), 25° (brown), and 35° (olive) for the transmission spectra, and 48° (green), 55° (blue), and 61° (magenta) for the reflection spectra. The TIR critical angle is 41.5°. (a) Experimental results; (b) FDTD simulation results. Inset: schematic diagram for illumination with s-polarization and definition of x, y and z directions.

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Fig. 3.
Fig. 3.

Dispersion relation for an array of gold nanoparticle chains at an air/glass interface. Orange: experimental results; Green: FDTD simulation results.

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Fig. 4.
Fig. 4.

Electric field E at an air/glass interface for an s-polarized plane wave incident from the glass side towards the air side (no metal nanoparticles). (a) |E|2/incident power; (b) phase of E compared to the electric field of the incident plane wave at the interface.

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Fig. 5.
Fig. 5.

Simulation of extinction in the first irreducible Brillouin Zone for an array of gold nanoparticle chains at an air/glass interface. The two white lines are, left: the light line of air, right: the light line of glass.

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Fig. 6.
Fig. 6.

Simulated dispersion relations for an array of gold nanoparticle chains. Green: 55 nm tall nanoparticles at an air/glass interface; blue: 30 nm tall nanoparticles at an air/glass interface; purple: 55 nm tall nanoparticles in a uniform glass environment.

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Fig. 7.
Fig. 7.

The y=0 cross section of the Ey field pattern from a line of point dipoles. The point dipoles in the line were spaced by 320 nm along y axis at an air/glass interface. One point dipole was sitting at origin. The point dipoles were polarized in +y direction and continuously oscillating at a frequency of 1.6 c/µm. The amplitude of Ey has been raised to the power of 0.6 in the plot in order to improve the visual contrast. Inset: definitions of nanoparticle chain and nanoparticle line in this paper.

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Fig. 8.
Fig. 8.

Amplitude (blue, in log scale) and phase (green) of Ey along x-axis. Ey was the field from a line of point dipoles. The point dipoles in the line were spaced by 320 nm along y axis. One point dipole was sitting at origin. The point dipoles were polarized in +y direction and continuously oscillating at a frequency of 1.6 c/µm. (a) The point dipole line was at an air/glass interface; (b) the point dipole line was in a uniform glass medium.

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Fig. 9.
Fig. 9.

Eindu for an array of point dipole lines oscillating at a frequency of 1.6 c/µm. The point dipole lines were polarized in +y direction and were spaced by 140 nm in x direction. kx spans the first irreducible Brillouin Zone from 0 to π/d with a fixed interval. (a) The point dipoles were on an air/glass interface; (b) the point dipoles were in a uniform glass medium.

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Fig. 10.
Fig. 10.

Eindu for a single chain of point dipoles on an air/glass interface, which are oscillating at a frequency of 1.6 c/µm. The point dipoles were polarized in +y direction and were spaced by 140 nm in x direction. kx spans the first irreducible Brillouin Zone from 0 to π/d with a fixed interval.

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Equations (2)

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

E indu = ∑ n = − ∞ , n ≠ 0 ∞ E − n exp ( i k x · n · 140 nm )
= ∑ n = 1 ∞ E n exp ( − i k x · n · 140 nm ) + E n exp ( i k x · n · 140 nm )

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