Abstract

Theoretical study of sensing properties of lattice resonances supported by arrays of gold nanoparticles expressed in terms of the figure of merit (FOM) is reported. Analytical expressions for the FOM for surface and bulk refractive index changes are derived to establish the relationship between the sensing performance and design parameters and to allow for the design of nanoparticle arrays with optimal sensing performance. It is demonstrated that lattice resonances exhibit about two orders of magnitude higher bulk FOM than localized surface plasmon (LSP) resonance and that the surface FOM provided by lattice resonances and LSP resonances are comparable.

© 2013 Optical Society of America

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  1. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
    [CrossRef]
  2. K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
    [CrossRef]
  3. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt.40(11), 2281–2291 (1993).
    [CrossRef]
  4. S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120(23), 10871–10875 (2004).
    [CrossRef] [PubMed]
  5. S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys.121(24), 12606–12612 (2004).
    [CrossRef] [PubMed]
  6. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
    [CrossRef]
  7. B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010).
    [CrossRef]
  8. E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
    [CrossRef] [PubMed]
  9. Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
    [CrossRef]
  10. B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett.101(14), 143902 (2008).
    [CrossRef] [PubMed]
  11. P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
    [CrossRef] [PubMed]
  12. J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev.108(2), 462–493 (2008).
    [CrossRef] [PubMed]
  13. C. F. H. Bohren, D. R, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).
  14. P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6(12), 4370–4379 (1972).
    [CrossRef]
  15. A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation damping in surface-enhanced Raman-scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
    [CrossRef]
  16. F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys.79(4), 1267–1290 (2007).
    [CrossRef]
  17. P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases3(3), FD4–FD11 (2008).
    [CrossRef] [PubMed]

2011

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
[CrossRef]

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

2010

B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010).
[CrossRef]

2008

Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett.101(14), 143902 (2008).
[CrossRef] [PubMed]

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev.108(2), 462–493 (2008).
[CrossRef] [PubMed]

P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases3(3), FD4–FD11 (2008).
[CrossRef] [PubMed]

2007

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys.79(4), 1267–1290 (2007).
[CrossRef]

2005

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

2004

S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120(23), 10871–10875 (2004).
[CrossRef] [PubMed]

S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys.121(24), 12606–12612 (2004).
[CrossRef] [PubMed]

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

1993

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt.40(11), 2281–2291 (1993).
[CrossRef]

1986

K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
[CrossRef]

1982

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation damping in surface-enhanced Raman-scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Auguie, B.

B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010).
[CrossRef]

Auguié, B.

B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett.101(14), 143902 (2008).
[CrossRef] [PubMed]

Barnes, W. L.

B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010).
[CrossRef]

B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett.101(14), 143902 (2008).
[CrossRef] [PubMed]

Bendana, X. M.

B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010).
[CrossRef]

Brongersma, S. H.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Carron, K. T.

K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Chu, Y. Z.

Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Crego-Calama, M.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Crozier, K. B.

Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

de Abajo, F. J. G.

B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Fluhr, W.

K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
[CrossRef]

García de Abajo, F. J.

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys.79(4), 1267–1290 (2007).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Gómez Rivas, J.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Gordon, J. P.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation damping in surface-enhanced Raman-scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

Gunnarsson, L.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

Hicks, E. M.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

Homola, J.

P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases3(3), FD4–FD11 (2008).
[CrossRef] [PubMed]

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev.108(2), 462–493 (2008).
[CrossRef] [PubMed]

Janel, N.

S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120(23), 10871–10875 (2004).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Käll, M.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

Kasemo, B.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

Kvasnicka, P.

P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases3(3), FD4–FD11 (2008).
[CrossRef] [PubMed]

Lehmann, H. W.

K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Liao, P. F.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation damping in surface-enhanced Raman-scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

Markel, V. A.

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt.40(11), 2281–2291 (1993).
[CrossRef]

Meier, M.

K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
[CrossRef]

Novotny, L.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
[CrossRef]

Offermans, P.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Rindzevicius, T.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

Rodriguez, S. R. K.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Schaafsma, M. C.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Schatz, G. C.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys.121(24), 12606–12612 (2004).
[CrossRef] [PubMed]

S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120(23), 10871–10875 (2004).
[CrossRef] [PubMed]

Schonbrun, E.

Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Spears, K. G.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Van Duyne, R. P.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
[CrossRef]

Wokaun, A.

K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
[CrossRef]

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation damping in surface-enhanced Raman-scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Yang, T.

Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Zhang, Y. C.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Zou, S. L.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys.121(24), 12606–12612 (2004).
[CrossRef] [PubMed]

S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120(23), 10871–10875 (2004).
[CrossRef] [PubMed]

ACS Nano

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011).
[CrossRef] [PubMed]

Appl. Phys. Lett.

Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Biointerphases

P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases3(3), FD4–FD11 (2008).
[CrossRef] [PubMed]

Chem. Rev.

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev.108(2), 462–493 (2008).
[CrossRef] [PubMed]

J. Chem. Phys.

S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120(23), 10871–10875 (2004).
[CrossRef] [PubMed]

S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys.121(24), 12606–12612 (2004).
[CrossRef] [PubMed]

J. Mod. Opt.

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt.40(11), 2281–2291 (1993).
[CrossRef]

J. Opt. Soc. Am. B.

K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986).
[CrossRef]

Nano Lett.

E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005).
[CrossRef] [PubMed]

Nat. Photonics

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
[CrossRef]

Nature

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Phys. Rev. B

B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation damping in surface-enhanced Raman-scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett.101(14), 143902 (2008).
[CrossRef] [PubMed]

Rev. Mod. Phys.

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys.79(4), 1267–1290 (2007).
[CrossRef]

Other

C. F. H. Bohren, D. R, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).

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Figures (9)

Fig. 1
Fig. 1

Dependence of the (a) real and (b) imaginary part of the inverse polarizability on wavelength for NPs of three different sizes. The solid lines represent the electrostatic approximation and the dotted line represents the Mie solution. (c) Dependence of the real and imaginary part of the inverse polarizability on wavelength for three different shape factors L of a spheroid with r = 40 nm. (d) Dependence of the shape factor L on the major/minor axis ratio a/b for prolate and oblate NPs excited by a light wave with different polarizations.

Fig. 2
Fig. 2

The real and imaginary parts of the lattice sum as a function of the normalized wavelength calculated for two arrays with a different number of NPs.

Fig. 3
Fig. 3

Transmission spectrum having characteristic resonances for three different gold NP arrays. The vertical lines indicate the lattice resonance λr ( Re{ 1/α }=Re{ G } ), the localized surface plasmon resonance λLSP ( Re{ 1/α }0 ) and the Rayleigh anomaly ( λ RA =Λ n m ). Parameters of the arrays: N × N = 2000 × 2000, L = 1/6, (a) Λ = 500 nm, r = 40 nm, (b) Λ = 400 nm, r = 40 nm (since λ RA < λ LSP , lattice resonance is not excited), (c) Λ = 500 nm, r = 15 nm (since Re{ 1/α }>max( Re{ G } ) , lattice resonance is not excited).

Fig. 4
Fig. 4

Transmission as a function of wavelength calculated for (a) three different radii of a spherical NP, for Λ = 450 nm, N × N = 2.104 × 2.104, (b) three different shape factors L of a spheroid, for Λ = 450 nm, N × N = 2.104 × 2.104, r = 40 nm, (c) three different periods of an array, for N × N = 2.104 × 2.104, r = 40 nm, and (d) three different numbers of spherical NPs, for Λ = 500 nm, r = 40 nm. The solid lines represent the analytical solutions obtained using the electrostatic approximation and the dotted lines represent the Mie solution.

Fig. 5
Fig. 5

(top) Real part of the lattice sum and inverse polarizability and (bottom) transmission as a function of wavelength for two refractive indices (a) of a bulk medium (solid line nm = 1.33, dashed line nm = 1.335) and (b) within a layer of thickness Δ = 10nm (solid line ns = 1.33, dashed line nm = 1.35). Parameters of the array: Λ = 450 nm, N × N = 2.104 × 2.104, r = 40 nm.

Fig. 6
Fig. 6

(a) FOMS and (b) FOMB as a function of δ calculated for different resonant wavelengths. Parameters of the array: N × N = 2000 × 2000, L = 1/6 (λLSP = 590 nm).

Fig. 7
Fig. 7

FOMB as a function of parameter δ for an array of gold NPs. The solid lines represent the theoretical values calculated using Eq. (16) for different radii of NP. The dotted lines represent the empirical dependence determined experimentally [11]. Parameters of the arrays used in simulations: height 50 nm, period 300 – 600 nm, nm = 1.45.

Fig. 8
Fig. 8

FOMS and FOMB as a function of NP radius for (a, b) different numbers of spherical NPs, and (c, d) different shape factor L of a spheroid for N × N = 2000 × 2000. The period of the array is set according to Eq. (6) to satisfy the resonance condition at λr = 800 nm. The solid lines represent the analytical solutions using Eqs. (15) and (16), dots represent the exact solution using the electrostatic approximation, and crosses represent the exact solution using the Mie theory.

Fig. 9
Fig. 9

(a) FOMS and (b) FOMB at the optimized NP size as a function of the resonant wavelength for different NP aspect ratios, N × N = 2000 × 2000. (Inset) The optimized NP radius as a function of the resonant wavelength. Dots represent FOMS and FOMB of a LSP supported by a non-ordered array of non-interacting particles.

Equations (28)

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 1/ α es = 3 r 3 ( L+ ε m ε ε m ),
G= n0 ( k 2 + ) exp( ik R n ) R n ,
G 4 π 2 2 Λ 3 2π/kΛ1 +i( 2πk Λ 2 2 k 3 3 + 2 π 2 ( Λ πN ) 3 ( 2π/kΛ1 ) 2 ) 118 Λ 3 .
T= | 1+ 2πik/ Λ 2 1/α G | 2 .
Re{ 1/α }=Re{ G }.
δ= λ r Λ n m 1 ( 4 π 2 2 Re { 1/α } λ r Λ 3 ) 2 .
T= | 1+ i4 π 2 Λ 3 λ RA / λ r dRe { 1/α G } λ r dλ ( λ λ r )+iIm { 1/α G } λ r | 2
W=2 | Im{ 1/α G } dRe{ 1/α G } / dλ | λ r  .
S B = d λ r d n m ( dRe{ G } / d n m dRe{ G1/α } / dλ ) λ r ,
( dRe{ G } d n m ) λ r = 1 n m dRe{ G } dδ ( δ+1 ),
( dRe{ G } d n m ) λ r 2 π 2 n m Λ 3 2 δ 3 .
S s = d λ r d n S ( dRe{ 1/α } / d n S dRe{ 1/α G } / dλ ) λ r .
1/ α es = 3f r 3 [ ε 2 +( ε ε s )( L ( 1 ) f L ( 2 ) ) ][ ε m +( ε s ε m ) L ( 2 ) ]+f L ( 2 ) ε s ( ε ε s ) ( ε s ε m )[ ε s +( ε ε s )( L ( 1 ) f L ( 2 ) ) ]+f ε s ( ε ε s ) ,
( dRe{ 1/α } d n S ) λ r = 2 3 n m ( 11/f )( r 3 Re { 1/α } λ r 2 + 9f r 3 ( L L 2 ) ).
FO M S = 1 3 n m B g i / g r A ,
FO M B = 1 2 n m d g r g r dδ δ+1 A g i / g r ,
A= Im{ 1/ α es } Re{ 1/ α es } ,B=( 11/f )( r 3 Re{ 1/ α es }+ 9f( L L 2 ) r 3 Re{ 1/ α es } ).
FO M B [ 4 n m δ( δ 2 + 1 ( 2πNδ ) 3 A ) ] 1 ,
FO M S [ 3 n m B ( δ 2 + 1 ( 2πNδ ) 3 A ) ] 1 .
δ S,B opt K S,B / ( πN ) 3/4 ,
r S,B opt λ r n m 3 4 π 2 ( L+ ε m Re { ε } λ r ε m ) δ S,B opt 2 3 .
G= i 2π n0 Q k 2 Q x 2 k 2 Q 2 exp( iQ R n ) d 2 Q.
G = 2πi Λ 2 g k 2 g x 2 k 2 g 2 G 0 ,
G 0 = i 2π Q k 2 Q x 2 k 2 Q 2 d 2 Q= 2i k 3 3 .
G 4 π 2 2 Λ 3 2π/kΛ1 +i( 2πk Λ 2 2 k 3 3 ) 118 Λ 3 ,
G N = i N 2 2π Λ 2 g Q k 2 Q x 2 k 2 Q 2 sinc( ( Qg ) NΛ 2π ) d 2 Q G 0 ,
Im{ G N } π 2N Λ 2 0 k k 2 Q y k 2 Q y 2 1 ( Q y g 1 ) 2 + ( π/ NΛ ) 2 2 d Q y +Im{ G },
Im{ G N } 2 π 2 ( Λ πN ) 3 ( 2π / kΛ 1 ) 2 +Im{ G }.

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