Abstract

We study numerically the effect of periodicity on the plasmon-assisted scattering and absorption of visible light by infinite and finite gratings of circular silver nanowires. The infinite grating is a convenient object of analysis because of the possibility to reduce the scattering problem to one period. We use the well-established method of partial separation of variables however make an important improvement by casting the resulting matrix equation to the Fredholm second-kind type, which guarantees convergence. If the silver wires have sub-wavelength radii, then two types of resonances co-exist and may lead to enhanced reflection and absorption: the plasmon-type and the grating-type. Each type is caused by different complex poles of the field function. The low-Q plasmon poles cluster near the wavelength where dielectric function equals −1. The grating-type poles make multiplets located in close proximity of Rayleigh wavelengths, tending to them if the wires get thinner. They have high Q-factors and, if excited, display intensive near-field patterns. A similar interplay between the two types of resonances takes place for finite gratings of silver wires, the sharpness of the grating-type peak getting greater for longer gratings. By tuning carefully the grating period, one can bring together two resonances and enhance the resonant scattering of light per wire by several times.

© 2011 OSA

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  1. For example, examination of the single-cylinder field Fourier expansion coefficients (177) of [11] for ka=2?a/??0 shows that they have poles at ?=?nP, for which ?(?nP)??1?cn(ka)2(4n)?1, where the azimuthal index is n?1 and cn>0 are known coefficients.
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  8. O. Kavaklioglu, “On diffraction of waves by the infinite grating of circular dielectric cylinders at oblique incidence: Floquet representation,” J. Mod. Phys. 48, 125–142 (2001).
  9. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of 2-D electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
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  13. V. O. Byelobrov, J. Ctyroky, T. M. Benson, R. Sauleau, A. Altintas, and A. I. Nosich, “Low-threshold lasing eigenmodes of an infinite periodic chain of quantum wires,” Opt. Lett. 35(21), 3634–3636 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  22. V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101(8), 087403 (2008).
    [CrossRef] [PubMed]
  23. B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. 101(14), 143902 (2008).
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  27. H. A. Ragheb and M. Hamid, “Scattering by N parallel conducting circular cylinders,” Int. J. Electron. 59(4), 407–421 (1985).
    [CrossRef]
  28. A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antenn. Propag. 40(1), 96–99 (1992).
    [CrossRef]
  29. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11(9), 2526–2538 (1994).
    [CrossRef]
  30. X. Antoine, C. Chniti, and K. Ramdani, “On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders,” J. Comput. Phys. 227(3), 1754–1771 (2008).
    [CrossRef]
  31. Here, we imply convergence in mathematical sense, as a possibility to minimise the error in the solution by inverting progressively greater matrices. Non-convergent numerical solutions are often able to provide a couple of correct digits however fail to achieve better accuracy.
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    [CrossRef]
  34. A. I. Nosich, “Radiation conditions, limiting absorption principle, and general relations in open waveguide scattering,” J. Electromagn. Waves Appl. 8(3), 329–353 (1994).
    [CrossRef]
  35. C. M. Linton, “The Green’s function for the two-dimensional Helmholtz equation in periodic domains,” J. Eng. Math. 33(4), 377–401 (1998).
    [CrossRef]
  36. V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. 105(26), 266801 (2010).
    [CrossRef] [PubMed]
  37. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
    [CrossRef] [PubMed]
  38. W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat. Nanotechnol. 6(7), 423–427 (2011).
    [CrossRef] [PubMed]

2011 (1)

W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat. Nanotechnol. 6(7), 423–427 (2011).
[CrossRef] [PubMed]

2010 (3)

2009 (1)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

2008 (4)

X. Antoine, C. Chniti, and K. Ramdani, “On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders,” J. Comput. Phys. 227(3), 1754–1771 (2008).
[CrossRef]

V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101(8), 087403 (2008).
[CrossRef] [PubMed]

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

T. Søndergaard and S. J. Bozhevolnyi, “Strip and gap plasmon polariton optical resonators,” Phys. Status Solidi B 245(1), 9–19 (2008).
[CrossRef]

2007 (3)

2006 (2)

R. Gómez-Medina, M. Laroche, and J. J. Sáenz, “Extraordinary optical reflection from sub-wavelength cylinder arrays,” Opt. Express 14(9), 3730–3737 (2006).
[CrossRef] [PubMed]

M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, “Tuning the optical response of nanocylinder arrays: an analytical study,” Phys. Rev. B 74(24), 245422 (2006).
[CrossRef]

2005 (3)

S. Zou and G. C. Schatz, “Silver nanoparticle array structures that produce giant enhancements in electromagnetic fields,” Chem. Phys. Lett. 403(1-3), 62–67 (2005).
[CrossRef]

E. M. Hicks, S. 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]

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

2004 (3)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of 2-D electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[CrossRef]

S. 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. 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]

2001 (3)

O. Kavaklioglu, “On diffraction of waves by the infinite grating of circular dielectric cylinders at oblique incidence: Floquet representation,” J. Mod. Phys. 48, 125–142 (2001).

J. Kottmann, O. Martin, D. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross-section,” Phys. Rev. B 64(23), 235402 (2001).
[CrossRef]

J. Kottmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express 8(12), 655–663 (2001).
[CrossRef] [PubMed]

1998 (1)

C. M. Linton, “The Green’s function for the two-dimensional Helmholtz equation in periodic domains,” J. Eng. Math. 33(4), 377–401 (1998).
[CrossRef]

1994 (2)

A. I. Nosich, “Radiation conditions, limiting absorption principle, and general relations in open waveguide scattering,” J. Electromagn. Waves Appl. 8(3), 329–353 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11(9), 2526–2538 (1994).
[CrossRef]

1992 (1)

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antenn. Propag. 40(1), 96–99 (1992).
[CrossRef]

1985 (1)

H. A. Ragheb and M. Hamid, “Scattering by N parallel conducting circular cylinders,” Int. J. Electron. 59(4), 407–421 (1985).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6(12), 4370–4379 (1972).
[CrossRef]

1971 (1)

1970 (1)

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5(11), 1351–1360 (1970).
[CrossRef]

1965 (1)

1962 (1)

V. Twersky, “On scattering of waves by the infinite grating of circular cylinders,” IRE Trans. Antennas Propag. 10(6), 737–765 (1962).
[CrossRef]

1952 (1)

V. Twersky, “On a multiple scattering theory of the finite grating and the Wood anomalies,” J. Appl. Phys. 23(10), 1099–1118 (1952).
[CrossRef]

1907 (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 79(532), 399–416 (1907).
[CrossRef]

Albaladejo, S.

M. Laroche, S. Albaladejo, R. Carminati, and J. J. Sáenz, “Optical resonances in one-dimensional dielectric nanorod arrays: field-induced fluorescence enhancement,” Opt. Lett. 32(18), 2762–2764 (2007).
[CrossRef] [PubMed]

M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, “Tuning the optical response of nanocylinder arrays: an analytical study,” Phys. Rev. B 74(24), 245422 (2006).
[CrossRef]

Altintas, A.

Antoine, X.

X. Antoine, C. Chniti, and K. Ramdani, “On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders,” J. Comput. Phys. 227(3), 1754–1771 (2008).
[CrossRef]

Aubard, J.

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

Auguié, B.

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

Aussenegg, F. R.

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

Barnes, W. L.

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

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Benson, T. M.

Boriskina, S. V.

Bozhevolnyi, S. J.

T. Søndergaard and S. J. Bozhevolnyi, “Strip and gap plasmon polariton optical resonators,” Phys. Status Solidi B 245(1), 9–19 (2008).
[CrossRef]

Byelobrov, V. O.

Carminati, R.

Chniti, C.

X. Antoine, C. Chniti, and K. Ramdani, “On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders,” J. Comput. Phys. 227(3), 1754–1771 (2008).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6(12), 4370–4379 (1972).
[CrossRef]

Ctyroky, J.

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Dal Negro, L.

Elsherbeni, A. Z.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antenn. Propag. 40(1), 96–99 (1992).
[CrossRef]

Felbacq, D.

Félidj, N.

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

García de Abajo, F. J. G.

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

Giannini, V.

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Gómez Rivas, J.

V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. 105(26), 266801 (2010).
[CrossRef] [PubMed]

Gomez-Medina, R.

M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, “Tuning the optical response of nanocylinder arrays: an analytical study,” Phys. Rev. B 74(24), 245422 (2006).
[CrossRef]

Gómez-Medina, R.

Grigorenko, A. N.

V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101(8), 087403 (2008).
[CrossRef] [PubMed]

Gunnarsson, L.

E. M. Hicks, S. 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]

Hamid, M.

H. A. Ragheb and M. Hamid, “Scattering by N parallel conducting circular cylinders,” Int. J. Electron. 59(4), 407–421 (1985).
[CrossRef]

Hessel, A.

Hicks, E. M.

E. M. Hicks, S. 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]

Hohenau, A.

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

Janel, N.

S. 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 the noble metals,” Phys. Rev. 6(12), 4370–4379 (1972).
[CrossRef]

Käll, M.

E. M. Hicks, S. 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. 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]

Kavaklioglu, O.

O. Kavaklioglu, “On diffraction of waves by the infinite grating of circular dielectric cylinders at oblique incidence: Floquet representation,” J. Mod. Phys. 48, 125–142 (2001).

Kerr, D. W.

Kishk, A. A.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antenn. Propag. 40(1), 96–99 (1992).
[CrossRef]

Kottmann, J.

J. Kottmann, O. Martin, D. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross-section,” Phys. Rev. B 64(23), 235402 (2001).
[CrossRef]

J. Kottmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express 8(12), 655–663 (2001).
[CrossRef] [PubMed]

Kravets, V. G.

V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101(8), 087403 (2008).
[CrossRef] [PubMed]

Krenn, J. R.

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of 2-D electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[CrossRef]

Laroche, M.

Laurent, G.

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

Lévi, G.

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (2005).
[CrossRef] [PubMed]

Linton, C. M.

C. M. Linton, “The Green’s function for the two-dimensional Helmholtz equation in periodic domains,” J. Eng. Math. 33(4), 377–401 (1998).
[CrossRef]

Ma, R.-M.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Martin, O.

J. Kottmann, O. Martin, D. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross-section,” Phys. Rev. B 64(23), 235402 (2001).
[CrossRef]

Martin, O. J. F.

Maystre, D.

Nosich, A. I.

V. O. Byelobrov, J. Ctyroky, T. M. Benson, R. Sauleau, A. Altintas, and A. I. Nosich, “Low-threshold lasing eigenmodes of an infinite periodic chain of quantum wires,” Opt. Lett. 35(21), 3634–3636 (2010).
[CrossRef] [PubMed]

A. I. Nosich, “Radiation conditions, limiting absorption principle, and general relations in open waveguide scattering,” J. Electromagn. Waves Appl. 8(3), 329–353 (1994).
[CrossRef]

Odom, T. W.

W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat. Nanotechnol. 6(7), 423–427 (2011).
[CrossRef] [PubMed]

Olaofe, G. O.

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5(11), 1351–1360 (1970).
[CrossRef]

Oliner, A. A.

Oulton, R. F.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Palmer, C. H.

Ragheb, H. A.

H. A. Ragheb and M. Hamid, “Scattering by N parallel conducting circular cylinders,” Int. J. Electron. 59(4), 407–421 (1985).
[CrossRef]

Ramdani, K.

X. Antoine, C. Chniti, and K. Ramdani, “On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders,” J. Comput. Phys. 227(3), 1754–1771 (2008).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 79(532), 399–416 (1907).
[CrossRef]

Rindzevicius, T.

E. M. Hicks, S. 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]

Saenz, J. J.

M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, “Tuning the optical response of nanocylinder arrays: an analytical study,” Phys. Rev. B 74(24), 245422 (2006).
[CrossRef]

Sáenz, J. J.

Sánchez-Gil, J. A.

Sauleau, R.

Schatz, G. C.

S. Zou and G. C. Schatz, “Silver nanoparticle array structures that produce giant enhancements in electromagnetic fields,” Chem. Phys. Lett. 403(1-3), 62–67 (2005).
[CrossRef]

E. M. Hicks, S. 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. 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. 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]

Schedin, F.

V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101(8), 087403 (2008).
[CrossRef] [PubMed]

Schultz, S.

J. Kottmann, O. Martin, D. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross-section,” Phys. Rev. B 64(23), 235402 (2001).
[CrossRef]

Smith, D.

J. Kottmann, O. Martin, D. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross-section,” Phys. Rev. B 64(23), 235402 (2001).
[CrossRef]

Søndergaard, T.

T. Søndergaard and S. J. Bozhevolnyi, “Strip and gap plasmon polariton optical resonators,” Phys. Status Solidi B 245(1), 9–19 (2008).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Spears, K. G.

E. M. Hicks, S. 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]

Tayeb, G.

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of 2-D electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[CrossRef]

Twersky, V.

V. Twersky, “On scattering of waves by the infinite grating of circular cylinders,” IRE Trans. Antennas Propag. 10(6), 737–765 (1962).
[CrossRef]

V. Twersky, “On a multiple scattering theory of the finite grating and the Wood anomalies,” J. Appl. Phys. 23(10), 1099–1118 (1952).
[CrossRef]

Van Duyne, R. P.

E. M. Hicks, S. 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]

Vecchi, G.

V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. 105(26), 266801 (2010).
[CrossRef] [PubMed]

Yasumoto, K.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of 2-D electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[CrossRef]

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Zhang, X.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Zhou, W.

W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat. Nanotechnol. 6(7), 423–427 (2011).
[CrossRef] [PubMed]

Zou, S.

E. M. Hicks, S. 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. Zou and G. C. Schatz, “Silver nanoparticle array structures that produce giant enhancements in electromagnetic fields,” Chem. Phys. Lett. 403(1-3), 62–67 (2005).
[CrossRef]

S. 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. 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]

Appl. Opt. (1)

Chem. Phys. Lett. (1)

S. Zou and G. C. Schatz, “Silver nanoparticle array structures that produce giant enhancements in electromagnetic fields,” Chem. Phys. Lett. 403(1-3), 62–67 (2005).
[CrossRef]

IEEE Trans. Antenn. Propag. (2)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of 2-D electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[CrossRef]

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antenn. Propag. 40(1), 96–99 (1992).
[CrossRef]

Int. J. Electron. (1)

H. A. Ragheb and M. Hamid, “Scattering by N parallel conducting circular cylinders,” Int. J. Electron. 59(4), 407–421 (1985).
[CrossRef]

IRE Trans. Antennas Propag. (1)

V. Twersky, “On scattering of waves by the infinite grating of circular cylinders,” IRE Trans. Antennas Propag. 10(6), 737–765 (1962).
[CrossRef]

J. Appl. Phys. (1)

V. Twersky, “On a multiple scattering theory of the finite grating and the Wood anomalies,” J. Appl. Phys. 23(10), 1099–1118 (1952).
[CrossRef]

J. Chem. Phys. (3)

N. Félidj, G. Laurent, J. Aubard, G. Lévi, A. Hohenau, J. R. Krenn, and F. R. Aussenegg, “Grating-induced plasmon mode in gold nanoparticle arrays,” J. Chem. Phys. 123(22), 221103 (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(23), 10871–10875 (2004).
[CrossRef] [PubMed]

S. 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. Comput. Phys. (1)

X. Antoine, C. Chniti, and K. Ramdani, “On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders,” J. Comput. Phys. 227(3), 1754–1771 (2008).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. I. Nosich, “Radiation conditions, limiting absorption principle, and general relations in open waveguide scattering,” J. Electromagn. Waves Appl. 8(3), 329–353 (1994).
[CrossRef]

J. Eng. Math. (1)

C. M. Linton, “The Green’s function for the two-dimensional Helmholtz equation in periodic domains,” J. Eng. Math. 33(4), 377–401 (1998).
[CrossRef]

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O. Kavaklioglu, “On diffraction of waves by the infinite grating of circular dielectric cylinders at oblique incidence: Floquet representation,” J. Mod. Phys. 48, 125–142 (2001).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nano Lett. (1)

E. M. Hicks, S. 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. Nanotechnol. (1)

W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat. Nanotechnol. 6(7), 423–427 (2011).
[CrossRef] [PubMed]

Nature (1)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6(12), 4370–4379 (1972).
[CrossRef]

Phys. Rev. B (2)

M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, “Tuning the optical response of nanocylinder arrays: an analytical study,” Phys. Rev. B 74(24), 245422 (2006).
[CrossRef]

J. Kottmann, O. Martin, D. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross-section,” Phys. Rev. B 64(23), 235402 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. 105(26), 266801 (2010).
[CrossRef] [PubMed]

V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101(8), 087403 (2008).
[CrossRef] [PubMed]

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

Phys. Status Solidi B (1)

T. Søndergaard and S. J. Bozhevolnyi, “Strip and gap plasmon polariton optical resonators,” Phys. Status Solidi B 245(1), 9–19 (2008).
[CrossRef]

Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 79(532), 399–416 (1907).
[CrossRef]

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G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5(11), 1351–1360 (1970).
[CrossRef]

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

Other (3)

For example, examination of the single-cylinder field Fourier expansion coefficients (177) of [11] for ka=2?a/??0 shows that they have poles at ?=?nP, for which ?(?nP)??1?cn(ka)2(4n)?1, where the azimuthal index is n?1 and cn>0 are known coefficients.

Here, we imply convergence in mathematical sense, as a possibility to minimise the error in the solution by inverting progressively greater matrices. Non-convergent numerical solutions are often able to provide a couple of correct digits however fail to achieve better accuracy.

X. Antoine, K. Ramdani, and B. Thierry, “Etude numerique de la resolution par equations integrales de la diffraction multiple par des disques,” in Proc. Congres Français d’Acoustique, Lyon (2010).

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Figures (10)

Fig. 1
Fig. 1

Infinite periodic grating of circular cylinders illuminated by a normally incident plane wave.

Fig. 2
Fig. 2

Reflectance (left panels) and absorbance (right panels) of infinite grating of silver wires as a function of the wavelength and wire radius for the normal incidence of the H-polarized plane wave. The period is p=450 nm for the upper panels and p=350 nm for the lower ones.

Fig. 3
Fig. 3

Reflectance (left) and absorbance (right) of the infinite grating of silver wires as function of the wavelength and period for the normal incidence of the H-polarized plane wave. Wires’ radii are a = 70 nm.

Fig. 4
Fig. 4

Near-field patterns: (a) in the P-resonance ( λ=340.8 nm ) and (b) in the G-resonance ( λ=451.35 nm ) for the grating of silver wires with radii a = 90 nm and period p = 450 nm, and (c) in the combined P-G resonance for the grating of a = 90 nm and p = 350 nm ( λ=362.45 nm ). See Figs. 2 and 3 for the corresponding values of reflectance and absorbance of these gratings. A plane wave is incident from the upper half-space.

Fig. 5
Fig. 5

Finite periodic grating of circular cylinders illuminated by a plane wave.

Fig. 6
Fig. 6

TSCS (left panels) and ACS (right panels) per silver wire versus the wavelength for the normal incidence ( φ 0 =π/2 ) at the gratings of M = 100 wires with radii a = 30 nm (upper panels) and 70 nm (lower panels).

Fig. 9
Fig. 9

Near-field amplitude (a, c) and phase (b, d) patterns in two combined P-G resonances for the H-wave normally incident ( φ 0 =π/2 ) at the grating of M = 25 silver nanowires with a = 30 nm, p = 349 nm, at the wavelength 348 nm (a, b) and with a = 70 nm, p = 391 nm, at the wavelength λ = 394 nm (c, d).

Fig. 7
Fig. 7

Per- wire TSCS (left panels) and ACS (right panels) versus the wavelength and the period for linear gratings of M = 100 silver nanowires with period p = 450 nm (upper panels) and 350 nm (lower panels).

Fig. 8
Fig. 8

Per-one wire TSCS (left panels) and ACS (right panels) versus the wavelength and the period for the gratings of M = 100 silver nanowires with radii a = 30 nm (upper panels) and 70 nm (lower panels).

Fig. 10
Fig. 10

Normalized reflective efficiency R ˜ as function of wavelength for finite and infinite gratings of silver nanowires with the radii of 70 nm and different periods. An H-polarized plane wave is at normal incidence.

Equations (18)

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U int (x,y)= n= a n J n (kνr) e inφ ,r<a,
U ext (x,y)= U 0 + n= b n [ H n (kr) e inφ + l= S nl (kp) J l (kr) e ilφ ],r>a,
S 2q (kp)=2 s=1 H 2q (skp), S 2q+1 (kp)=0,
X+AX=B,X= { x m } m= + ,A= { A mn } m,n= + ,B= { B m } m= + ,
A mn = S nm (2πσ) V m (u,ν) J n (u) F m (u,ν) J m (u) , B m = (1) m V m (u,ν) F m (u,ν) J m (u)
F m E,H = β E,H H m (u) J m (νu) H m (u) J m (νu), V m E,H = β E,H J m (u) J m (νu) J m (u) J m (νu),
S 0 (2πσ)=1 2i π [γ+log(σ/2)]+ 1 πσ + 2 π s=1 [ 1 ( σ 2 s 2 ) 1/2 + i s ] ,
S 2q (2πσ)= 2 π s=1 e 2iq θ s ( σ 2 s 2 ) 1/2 + 1 πσ + i qπ + i π s=1 q (1) s (q+s1)! (2s)!(qs)! 2 2s σ 2s D 2s ,q=1,2,...
U ext,± (x,y)= s= f s ± e ik π s x e ik τ s |y| , f s ± = (πσ τ s ) 1 n= x n J n (ka) (i π s ± τ s ) n ,
R inf = γ s <k | f s + | 2 , T inf =|1+ f 0 | 2 + γ s <k,s0 | f s | 2 ,
σ m,0 GH+ =mi m 5 (mπ1) 1 ( ν 2 1)δ(1/2 m 1/2 ) m 8 ( ν 2 1) 2 δ 2 ,
σ m,n GH+ =m i m 4n+1 ( ν 2 1) ( ν 2 +1) 1 δ n 2 2n+1 (2n)!(2n1)!(mπ2) 8 m 8n ( ν 2 1) 2 ( ν 2 +1) 2 δ 2n 4 2n [(2n)!(2n1)!] 2 ,n=1,2,...,
X+ A ˜ X= B ˜ ,X= { X (q) } q=1 M , X (q) = { x m (q) } m= + ,
A ˜ = { A ˜ (q,j) } q,j=1 M , A ˜ (q,j) = { A ˜ m,n (q,j) } m,n= + , B ˜ = { B ˜ (q) } q=1 M , B ˜ (q) = { B ˜ m (q) } m= + ,
A ˜ m,n (q,j) = H l (|qj|2πσ) V m (u,ν) J n (u) F m (u,ν) J m (u) , B ˜ m (q) = (i) m V m (u,ν) F m (u,ν) J m (u) e i2πσ(q1)cos φ 0 +im φ 0 ,
S sc = 2 πk 0 2π |Φ(φ) | 2 dφ ,Φ(φ)= n= + (i) n J n (u) [ q=1 M x n ( q ) e i2πσ(q1)cosφ ] e inφ .
S sc + S ab =(4/k)ReΦ( φ 0 +π),
R ˜ fin =1/(πMka) 0 π |Φ(φ) | 2 dφ .

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