Abstract

We present the frequency-domain boundary element formulation for solving surface second-harmonic generation from nanoparticles of virtually arbitrary shape and material. We use the Rao-Wilton-Glisson basis functions and Galerkin’s testing, which leads to very accurate solutions for both near and far fields. This is verified by a comparison to a solution obtained via multipole expansion for the case of a spherical particle. The frequency-domain formulation allows the use of experimentally measured linear and nonlinear material parameters or the use of parameters obtained using ab-initio principles. As an example, the method is applied to a non-centrosymmetric L-shaped gold nanoparticle to illustrate the formation of surface nonlinear polarization and the second-harmonic radiation properties of the particle. This method provides a theoretically well-founded approach for modelling nonlinear optical phenomena in nanoparticles.

© 2011 OSA

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    [CrossRef]
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    [CrossRef]
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  4. F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
    [CrossRef]
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  6. H. Husu, J. Mäkitalo, R. Siikanen, G. Genty, H. Pietarinen, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Spectral control in anisotropic resonance-domain metamaterials,” Opt. Lett. 36, 2375–2377 (2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  43. S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
    [CrossRef] [PubMed]
  44. H. Husu, J. Mäkitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonances in L-shaped gold nanoparticles,” Opt. Express 18, 16601–16606 (2010).
    [CrossRef] [PubMed]
  45. A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996).
    [CrossRef]
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2011 (4)

2010 (7)

2009 (5)

A. Kern and O. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[CrossRef]

L. Cao, N. Panoiu, R. Bhat, and R. Osgood Jr, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79, 235416 (2009).
[CrossRef]

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

A. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

2008 (2)

2007 (3)

S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

B. Bai and J. Turunen, “Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials,” J. Opt. Soc. Am. B 24, 1105–1112 (2007).
[CrossRef]

L. Cao, N. Panoiu, and R. Osgood, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007).
[CrossRef]

2006 (2)

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. D. Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14, 9988–9999 (2006).
[CrossRef] [PubMed]

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res. 63, 243–278 (2006).
[CrossRef]

2004 (3)

J. Dadap, J. Shan, and T. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21, 1328–1347 (2004).
[CrossRef]

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (2004).
[CrossRef]

S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
[CrossRef]

2003 (2)

K. Kelly, E. Coronado, L. Zhao, and G. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
[CrossRef]

2002 (1)

1996 (2)

A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996).
[CrossRef]

J. Dewitz, W. Hübner, and K. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Zeitschrift für Physik D Atoms, Molecules and Clusters 37, 75–84 (1996).
[CrossRef] [PubMed]

1989 (1)

Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

1987 (2)

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

J. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

1986 (2)

P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986).
[CrossRef]

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

1982 (1)

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

1980 (1)

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389 (1980).
[CrossRef]

1972 (1)

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

Abajo, F. J. G. D.

Ahorinta, R.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Aizpurua, J.

G. Bryant, F. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef] [PubMed]

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. D. Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14, 9988–9999 (2006).
[CrossRef] [PubMed]

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
[CrossRef]

Albers, W.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Bachelier, G.

Bai, B.

Benedetti, A.

Benichou, E.

Bennemann, K.

J. Dewitz, W. Hübner, and K. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Zeitschrift für Physik D Atoms, Molecules and Clusters 37, 75–84 (1996).
[CrossRef] [PubMed]

Bertolotti, M.

Bhat, R.

L. Cao, N. Panoiu, R. Bhat, and R. Osgood Jr, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79, 235416 (2009).
[CrossRef]

Biris, C.

C. Biris and N. Panoiu, “Excitation of linear and nonlinear cavity modes upon interaction of femtosecond pulses with arrays of metallic nanowires,” Appl. Phys. A pp. 1–5 (2011).

C. Biris and N. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
[CrossRef]

C. Biris and N. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18, 17165–17179 (2010).
[CrossRef] [PubMed]

Bonn, M.

S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
[CrossRef]

Brevet, P.

Bryant, G.

G. Bryant, F. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef] [PubMed]

Bryant, G. W.

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. D. Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14, 9988–9999 (2006).
[CrossRef] [PubMed]

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
[CrossRef]

Canfield, B.

S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Cao, L.

L. Cao, N. Panoiu, R. Bhat, and R. Osgood Jr, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79, 235416 (2009).
[CrossRef]

L. Cao, N. Panoiu, and R. Osgood, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007).
[CrossRef]

Centini, M.

Chen, W.

P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986).
[CrossRef]

Christy, R.

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

Colton, D.

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 (Springer Verlag, 1998).

Coronado, E.

K. Kelly, E. Coronado, L. Zhao, and G. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Dadap, J.

de Abajo, F.

G. Bryant, F. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef] [PubMed]

de Beer, A.

A. de Beer, S. Roke, and J. Dadap, “Theory of optical second-harmonic and sum-frequency scattering from arbitrarily shaped particles,” J. Opt. Soc. Am. B 28, 1374–1384 (2011).
[CrossRef]

A. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

Dewitz, J.

J. Dewitz, W. Hübner, and K. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Zeitschrift für Physik D Atoms, Molecules and Clusters 37, 75–84 (1996).
[CrossRef] [PubMed]

Fainman, Y.

Fukui, M.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389 (1980).
[CrossRef]

Gallinet, B.

B. Gallinet, A. Kern, and O. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010).
[CrossRef]

B. Gallinet and O. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation,” Phot. Nano. Fund. Appl. 8, 278–284 (2010).
[CrossRef]

Garcia de Abajo, F. J.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
[CrossRef]

Genty, G.

Glisson, A.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Guyot-Sionnest, P.

P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986).
[CrossRef]

Hanarp, P.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
[CrossRef]

Hänninen, I.

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res. 63, 243–278 (2006).
[CrossRef]

Harrington, R.

R. Harrington, Field computation by moment methods (Wiley-IEEE Press, 1993).

Heinz, T.

Heinz, T. F.

T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman (Elsevier, Amsterdam, 1991) p. 353.

Hoyer, W.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Hübner, W.

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (2004).
[CrossRef]

J. Dewitz, W. Hübner, and K. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Zeitschrift für Physik D Atoms, Molecules and Clusters 37, 75–84 (1996).
[CrossRef] [PubMed]

Husu, H.

Jackson, J.

J. Jackson, Classical electrodynamics (John Wiley & Sons inc., 1999).

Johnson, P.

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

Jonin, C.

Jorgenson, R. E.

A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996).
[CrossRef]

Käll, M.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
[CrossRef]

Kauranen, M.

H. Husu, J. Mäkitalo, R. Siikanen, G. Genty, H. Pietarinen, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Spectral control in anisotropic resonance-domain metamaterials,” Opt. Lett. 36, 2375–2377 (2011).
[CrossRef] [PubMed]

H. Husu, J. Mäkitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonances in L-shaped gold nanoparticles,” Opt. Express 18, 16601–16606 (2010).
[CrossRef] [PubMed]

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Kelly, K.

K. Kelly, E. Coronado, L. Zhao, and G. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Kern, A.

Koch, S.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 (Springer Verlag, 1998).

Kuittinen, M.

Kujala, S.

S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau and E. M. Lifshits, The electrodynamics of continuous media (Pergamon, Oxford, 1960).

Laukkanen, J.

Lederer, F.

Lehtolahti, J.

Lifshits, E. M.

L. D. Landau and E. M. Lifshits, The electrodynamics of continuous media (Pergamon, Oxford, 1960).

Liu, J.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Mäkitalo, J.

Martin, O.

Mizrahi, V.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

Moloney, J.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
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Nakagawa, W.

Osgood, R.

L. Cao, N. Panoiu, and R. Osgood, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007).
[CrossRef]

Osgood Jr, R.

L. Cao, N. Panoiu, R. Bhat, and R. Osgood Jr, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79, 235416 (2009).
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Panoiu, N.

C. Biris and N. Panoiu, “Excitation of linear and nonlinear cavity modes upon interaction of femtosecond pulses with arrays of metallic nanowires,” Appl. Phys. A pp. 1–5 (2011).

C. Biris and N. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
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C. Biris and N. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18, 17165–17179 (2010).
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L. Cao, N. Panoiu, R. Bhat, and R. Osgood Jr, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79, 235416 (2009).
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L. Cao, N. Panoiu, and R. Osgood, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007).
[CrossRef]

Paul, T.

Pavlyukh, Y.

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (2004).
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Peterson, A. F.

A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996).
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Petukhov, A.

S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
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Pietarinen, H.

Rao, S.

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
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S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
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Rockstuhl, C.

Rodríguez, F.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
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A. de Beer, S. Roke, and J. Dadap, “Theory of optical second-harmonic and sum-frequency scattering from arbitrarily shaped particles,” J. Opt. Soc. Am. B 28, 1374–1384 (2011).
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A. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
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S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
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Romero, I.

Russier-Antoine, I.

Sarvas, J.

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res. 63, 243–278 (2006).
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W. Schaich, “Second harmonic generation by periodically-structured metal surfaces,” Phys. Rev. B 78, 195416 (2008).

Schatz, G.

K. Kelly, E. Coronado, L. Zhao, and G. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Shan, J.

Shen, Y.

Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
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P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986).
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Sibilia, C.

Siikanen, R.

Sipe, J.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
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J. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
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J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
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J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389 (1980).
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J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389 (1980).
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J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
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J. Stratton, Electromagnetic theory (New York and London: McGraw-Hill, 1941).

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J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
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S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
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K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

Taskinen, M.

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res. 63, 243–278 (2006).
[CrossRef]

Turunen, J.

S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
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B. Bai and J. Turunen, “Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials,” J. Opt. Soc. Am. B 24, 1105–1112 (2007).
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Tyan, R.-C.

Umashankar, K.

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

Wang, F.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Wilton, D.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Wilton, D. R.

A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996).
[CrossRef]

Zeng, Y.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Zhao, L.

K. Kelly, E. Coronado, L. Zhao, and G. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Appl. Phys. A (1)

C. Biris and N. Panoiu, “Excitation of linear and nonlinear cavity modes upon interaction of femtosecond pulses with arrays of metallic nanowires,” Appl. Phys. A pp. 1–5 (2011).

IEEE Trans. Antennas Propag. (3)

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (7)

J. Phys. Chem. B (1)

K. Kelly, E. Coronado, L. Zhao, and G. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Nano Lett. (1)

G. Bryant, F. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef] [PubMed]

Nature (1)

Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phot. Nano. Fund. Appl. (1)

B. Gallinet and O. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation,” Phot. Nano. Fund. Appl. 8, 278–284 (2010).
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Phys. Rev. B (13)

L. Cao, N. Panoiu, and R. Osgood, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007).
[CrossRef]

L. Cao, N. Panoiu, R. Bhat, and R. Osgood Jr, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79, 235416 (2009).
[CrossRef]

W. Schaich, “Second harmonic generation by periodically-structured metal surfaces,” Phys. Rev. B 78, 195416 (2008).

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986).
[CrossRef]

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

A. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (2004).
[CrossRef]

S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
[CrossRef]

C. Biris and N. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
[CrossRef]

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

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389 (1980).
[CrossRef]

Phys. Rev. Lett. (2)

S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003).
[CrossRef]

Prog. Elec. Res. (1)

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res. 63, 243–278 (2006).
[CrossRef]

Zeitschrift für Physik D Atoms, Molecules and Clusters (1)

J. Dewitz, W. Hübner, and K. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Zeitschrift für Physik D Atoms, Molecules and Clusters 37, 75–84 (1996).
[CrossRef] [PubMed]

Other (6)

T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman (Elsevier, Amsterdam, 1991) p. 353.

J. Jackson, Classical electrodynamics (John Wiley & Sons inc., 1999).

J. Stratton, Electromagnetic theory (New York and London: McGraw-Hill, 1941).

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 (Springer Verlag, 1998).

R. Harrington, Field computation by moment methods (Wiley-IEEE Press, 1993).

L. D. Landau and E. M. Lifshits, The electrodynamics of continuous media (Pergamon, Oxford, 1960).

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