Abstract

We numerically study second harmonic generation from dipole gold nanoantennas by analyzing the different contributions of bulk and surface nonlinear terms. We focus our attention to the properties of the emitted field related to the different functional expressions of the two terms. The second harmonic field exhibits different far and near field patterns if both nonlinear contributions are taken into account or if only one of them is considered. This effect persists despite of the model used to estimate the parameters of the nonlinear sources and it is strictly related to the resonant behavior of the plasmonic nanostructure at the fundamental frequency field and to its linear properties at the second harmonic frequency. We show that the excitation of localized surface plasmon polaritons in these structures can remarkably modify the nonlinear response of the system by enhancing surface and/or bulk contributions, creating regimes where bulk nonlinear terms dominate over surface linear terms and vice versa. Finally, the results of our calculations suggest a method that could be implemented to experimentally extract information on the relevance of bulk and surface contributions by measuring and analyzing the generated far field second harmonic patterns in metal nanoantennas and, more in general, in plasmonic nanostructures.

© 2011 OSA

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2011 (3)

2010 (7)

A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “Engineering the Second Harmonic Generation Pattern from Coupled Gold Nanowires,” J. Opt. Soc. Am. B 27(3), 408 (2010).
[CrossRef]

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[CrossRef] [PubMed]

2009 (10)

J. I. Dadap, H. B. de Aguiar, and S. Roke, “Nonlinear light scattering from clusters and single particles,” J. Chem. Phys. 130(21), 214710 (2009).
[CrossRef] [PubMed]

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
[CrossRef]

J. Li, A. Salandrino, and N. Engheta, “Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas,” Phys. Rev. B 79(19), 195104 (2009).
[CrossRef]

W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas,” J. Phys. Chem. C 113(33), 14672–14675 (2009).
[CrossRef]

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103(25), 257404 (2009).
[CrossRef] [PubMed]

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

A. Belardini, M. C. Larciprete, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, A. Toma, D. Chiappe, and F. Buatier de Mongeot, “Tailored second harmonic generation from self-organized metal nano-wires arrays,” Opt. Express 17(5), 3603–3609 (2009).
[CrossRef] [PubMed]

Y. H. Joo, S. H. Song, R. Magnusson, and R. Magnusson, “Long-range surface plasmon-polariton waveguide sensors with a Bragg grating in the asymmetric double-electrode structure,” Opt. Express 17(13), 10606–10611 (2009).
[CrossRef]

Y. Zeng and J. V. Moloney, “Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns,” Opt. Lett. 34(18), 2844–2846 (2009).
[CrossRef] [PubMed]

2008 (4)

W. L. Schaich, “Second harmonic genaration by periodically-structured metal surfaces,” Phys. Rev. B 78(19), 195416 (2008).
[CrossRef]

H. Guo, T. P. Meyrath, T. Zentgraf, N. Liu, L. Fu, H. Schweizer, and H. Giessen, “Optical resonances of bowtie slot antennas and their geometry and material dependence,” Opt. Express 16(11), 7756-7766 (2008).

H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16(12), 9144–9154 (2008).
[CrossRef] [PubMed]

M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008).
[CrossRef]

2007 (1)

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

2006 (5)

A. Rasmussen and V. Deckert, “Surface– and tip–enhanced Raman scattering of DNA components,” J. Raman Spectrosc. 37(1-3), 311–317 (2006).
[CrossRef]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Lett. 6(4), 683–688 (2006).
[CrossRef] [PubMed]

L. J. Sherry, R. Jin, C. A. Mirkin, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy of single silver triangular nanoprisms,” Nano Lett. 6(9), 2060–2065 (2006).
[CrossRef] [PubMed]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

2005 (3)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[CrossRef]

K. Li, M. I. Stockman, and D. J. Bergman, “Enhanced second harmonic generation in a self-similar chain of metal nanospheres,” Phys. Rev. B 72(15), 153401 (2005).
[CrossRef]

A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B 109(43), 20522–20528 (2005).
[CrossRef] [PubMed]

2003 (1)

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003).
[CrossRef] [PubMed]

2002 (1)

H. Tamaru, H. Kuwata, H. T. Miyazaki, and K. Miyano, “Resonant light scattering from individual Ag nanoparticles and particle pairs,” Appl. Phys. Lett. 80(10), 1826–1828 (2002).
[CrossRef]

1999 (1)

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[CrossRef]

1998 (1)

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998).
[CrossRef]

1985 (1)

J. C. Quail and H. J. Simon, “Second harmonic generation from silver and aluminium films in total internal reflection,” Phys. Rev. B 31(8), 4900–4905 (1985).
[CrossRef]

1984 (1)

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30(6), 3002–3015 (1984).
[CrossRef]

1966 (1)

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16(22), 986–989 (1966).
[CrossRef]

1965 (1)

S. Jha, “Theory of optical harmonic generation at a metal surface,” Phys. Rev. 140(6A), A2020–A2030 (1965).
[CrossRef]

Adam, P.-M.

M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008).
[CrossRef]

Ahorinta, R.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Aizpurua, J.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003).
[CrossRef] [PubMed]

Akozbek, N.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[CrossRef]

Aktsipetrov, O. A.

V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
[CrossRef] [PubMed]

V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
[CrossRef]

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Albers, W. M.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Allegrini, M.

M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008).
[CrossRef]

Ameloot, M.

V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
[CrossRef]

V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
[CrossRef] [PubMed]

Averitt, R. D.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998).
[CrossRef]

Avlasevich, Y.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
[CrossRef]

Bachelier, G.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

Bai, B.

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J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10(5), 1717–1721 (2010).
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A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
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Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
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T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103(25), 257404 (2009).
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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
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B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
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M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008).
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Laukkanen, J.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
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H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
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N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16(22), 986–989 (1966).
[CrossRef]

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T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103(25), 257404 (2009).
[CrossRef] [PubMed]

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J. Li, A. Salandrino, and N. Engheta, “Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas,” Phys. Rev. B 79(19), 195104 (2009).
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K. Li, M. I. Stockman, and D. J. Bergman, “Enhanced second harmonic generation in a self-similar chain of metal nanospheres,” Phys. Rev. B 72(15), 153401 (2005).
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C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Lett. 6(4), 683–688 (2006).
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M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
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Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
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W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas,” J. Phys. Chem. C 113(33), 14672–14675 (2009).
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H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16(12), 9144–9154 (2008).
[CrossRef] [PubMed]

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Mirkin, C. A.

L. J. Sherry, R. Jin, C. A. Mirkin, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy of single silver triangular nanoprisms,” Nano Lett. 6(9), 2060–2065 (2006).
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H. Tamaru, H. Kuwata, H. T. Miyazaki, and K. Miyano, “Resonant light scattering from individual Ag nanoparticles and particle pairs,” Appl. Phys. Lett. 80(10), 1826–1828 (2002).
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Miyazaki, H. T.

H. Tamaru, H. Kuwata, H. T. Miyazaki, and K. Miyano, “Resonant light scattering from individual Ag nanoparticles and particle pairs,” Appl. Phys. Lett. 80(10), 1826–1828 (2002).
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A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
[CrossRef]

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Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
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[CrossRef] [PubMed]

V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
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A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
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C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Lett. 6(4), 683–688 (2006).
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H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

Oldenburg, S. J.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998).
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Panoiu, N. C.

Petkov, V.

V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
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M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008).
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J. C. Quail and H. J. Simon, “Second harmonic generation from silver and aluminium films in total internal reflection,” Phys. Rev. B 31(8), 4900–4905 (1985).
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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
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Rasmussen, A.

A. Rasmussen and V. Deckert, “Surface– and tip–enhanced Raman scattering of DNA components,” J. Raman Spectrosc. 37(1-3), 311–317 (2006).
[CrossRef]

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F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
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J. I. Dadap, H. B. de Aguiar, and S. Roke, “Nonlinear light scattering from clusters and single particles,” J. Chem. Phys. 130(21), 214710 (2009).
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Roppo, V.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
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J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10(5), 1717–1721 (2010).
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J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
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A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B 109(43), 20522–20528 (2005).
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W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas,” J. Phys. Chem. C 113(33), 14672–14675 (2009).
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L. J. Sherry, R. Jin, C. A. Mirkin, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy of single silver triangular nanoprisms,” Nano Lett. 6(9), 2060–2065 (2006).
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Silhanek, A. V.

V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
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V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
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J. C. Quail and H. J. Simon, “Second harmonic generation from silver and aluminium films in total internal reflection,” Phys. Rev. B 31(8), 4900–4905 (1985).
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F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
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Träutlein, D.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103(25), 257404 (2009).
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B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
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V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
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V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

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L. J. Sherry, R. Jin, C. A. Mirkin, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy of single silver triangular nanoprisms,” Nano Lett. 6(9), 2060–2065 (2006).
[CrossRef] [PubMed]

A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B 109(43), 20522–20528 (2005).
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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
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V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
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V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
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V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

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M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[CrossRef]

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A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[CrossRef] [PubMed]

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V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
[CrossRef]

V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
[CrossRef] [PubMed]

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F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Wang, H.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
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M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
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S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998).
[CrossRef]

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A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B 109(43), 20522–20528 (2005).
[CrossRef] [PubMed]

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T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103(25), 257404 (2009).
[CrossRef] [PubMed]

Yu, Z.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
[CrossRef]

Zavelani-Rossi, M.

M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008).
[CrossRef]

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
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Y. Zeng and J. V. Moloney, “Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns,” Opt. Lett. 34(18), 2844–2846 (2009).
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W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas,” J. Phys. Chem. C 113(33), 14672–14675 (2009).
[CrossRef]

Zentgraf, T.

Zhang, W.

W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas,” J. Phys. Chem. C 113(33), 14672–14675 (2009).
[CrossRef]

Zheng, X.

V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
[CrossRef] [PubMed]

V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials,” Opt. Mater. Express 1(1), 36–45 (2011).
[CrossRef]

Zinovev, A. V.

A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B 109(43), 20522–20528 (2005).
[CrossRef] [PubMed]

Zou, S. L.

A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B 109(43), 20522–20528 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008).
[CrossRef]

H. Tamaru, H. Kuwata, H. T. Miyazaki, and K. Miyano, “Resonant light scattering from individual Ag nanoparticles and particle pairs,” Appl. Phys. Lett. 80(10), 1826–1828 (2002).
[CrossRef]

Chem. Phys. Lett. (1)

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998).
[CrossRef]

J. Chem. Phys. (1)

J. I. Dadap, H. B. de Aguiar, and S. Roke, “Nonlinear light scattering from clusters and single particles,” J. Chem. Phys. 130(21), 214710 (2009).
[CrossRef] [PubMed]

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

J. Phys. Chem. B (1)

A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B 109(43), 20522–20528 (2005).
[CrossRef] [PubMed]

J. Phys. Chem. C (1)

W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas,” J. Phys. Chem. C 113(33), 14672–14675 (2009).
[CrossRef]

J. Raman Spectrosc. (1)

A. Rasmussen and V. Deckert, “Surface– and tip–enhanced Raman scattering of DNA components,” J. Raman Spectrosc. 37(1-3), 311–317 (2006).
[CrossRef]

Nano Lett. (5)

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Lett. 6(4), 683–688 (2006).
[CrossRef] [PubMed]

L. J. Sherry, R. Jin, C. A. Mirkin, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy of single silver triangular nanoprisms,” Nano Lett. 6(9), 2060–2065 (2006).
[CrossRef] [PubMed]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

Nat. Photonics (1)

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Rep. (1)

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Phys. Rev. (1)

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Phys. Rev. A (1)

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
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Phys. Rev. B (8)

J. Li, A. Salandrino, and N. Engheta, “Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas,” Phys. Rev. B 79(19), 195104 (2009).
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F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
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C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010).
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[CrossRef]

Phys. Rev. Lett. (7)

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003).
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V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
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V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques,” Phys. Rev. Lett. 106(22), 226803 (2011).
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J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
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Science (2)

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic of the system under examination. (b) Simplified representation: when the k vector lays on the x-z plane, a mirror-like perfect electric conductor (perfect magnetic conductor) can be placed in the middle of the antenna for TE (TM) plane wave illumination mode. k denotes the wave vector and indicates the direction of propagation of the pump field. A and B vectors represent the directions of oscillation of the electric field for the TE (s) and TM (p) mode respectively.

Fig. 2
Fig. 2

Linear absorption (red) and scattering (blue) cross section vs. wavelength for a nanoantenna of square section (24 x 24) nm2.

Fig. 3
Fig. 3

Linear absorption cross section (ACS- red curve) and scattering cross section (SCS-blue curve) values for TE (solid) and TM (dashed) polarization.

Fig. 4
Fig. 4

Nonlinear scattering cross section (NLSCS) under TE (a) and TM (b) illumination, calculated considering surface (green-dotted), bulk (blue-dashed) and both (red-solid) nonlinear contributions.

Fig. 5
Fig. 5

Nonlinear scattering cross section (NLSCS) for the (13 x13) nm2 antenna with a TE polarized pump calculated by neglecting bulk nonlinear contributions and considering surface non linear terms only. The a parameter has been multiplied by a ξ coefficient in order to evaluate the sensitivity of the results as a function of the value of the a parameter.

Fig. 6
Fig. 6

Front and back view of 3D second harmonic differential scattering cross section [cm2/W] for TE polarization modes (E field laying parallel to the antenna's main axis) calculated by (a) taking into account and (b) neglecting the nonlinear surface contributions, with a resulting differential nonlinear scattering cross section 20 times weaker. Black arrows indicate direction of propagation of the pump field; the nanoantenna has been interposed only to provide a reference for the 3D scattering distribution and its size has been rescaled to fit the picture size. The square cross section of the antenna is (13 x 13) nm2.

Fig. 7
Fig. 7

Front and back view of 3D second harmonic differential scattering cross section [cm2/W] for TM illumination mode calculated by (a) taking into account and (b) neglecting the nonlinear surface contributions. Note that the second harmonic generation process is 2 order of magnitude less efficient than for the TE case. The square cross section of the antenna is (13 x 13) nm2.

Equations (18)

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G ¯ ¯ m,B ( r , r' )=[ I ¯ ¯ + k m,B 2 ] g m,B ( r , r' )
G ¯ ¯ m,B M,TM/TE [ r , r ' ]= G ¯ ¯ m,B [ r , r ' ]+ + ( 1 ) α G ¯ ¯ m,B [ r ,( x',2 y 0 y',z' ) ]· S ¯ ¯ , S ¯ ¯ =( 1 0 0 0 +1 0 0 0 1 ); r '=( x',y',z' );
E 1,i = E 0,i + j=1,ji N/2 G ¯ ¯ 1,B M,TE/TM ( r i , r j )·[ Δ ε 1,j k 1,B 2 E 1,j ]Δ τ j + + ( 1 ) α G ¯ ¯ 1,B [ r i ,( x i ,2 y 0 y i , z i ) ]· S ¯ ¯ ·[ Δ ε 1,i k 1,B 2 E 1,i ]Δ τ i + M ¯ ¯ 1,i · E 1,i ;
E 2,l = j=1,jl N/2 G ¯ ¯ 2,B M,TM ( r l , r j )·[ Δ ε 2,j k 2,B 2 E 2,j +i2ω μ 0 J NL,j Bulk ]Δ τ j + G ¯ ¯ 2,B [ r l ,( x l ,2 y 0 y l , z l ) ]· S ¯ ¯ ·[ Δ ε 2,l k 2,B 2 E 2,l +i2ω μ 0 J NL,l Bulk ]Δ τ l + + M ¯ ¯ 2,l ·[ E 2,l + i 2ω ε 0 Δ ε 2,l J NL,l Bulk ]+ + k=1 N B /2 G ¯ ¯ 2,B M,TM ( r l , r k )·[ i2ω μ 0 J NL,k Surface ]Δ τ k ;
M ¯ ¯ m,i = 2 3 Δ ε m,i [ ( 1i k m,B R i eff ) e i k m,B R i eff 1 2 ε B 1 ] I ¯ ¯ , Δ ε m,i = ε m,i ε B , R i eff = ( 3 4π Δ τ i ) 1 3 ;
J NL Bulk =i2ω[ γδ'( E 1 · ) E 1 +γd [ E 1 · E 1 ] ];
X ^ · J NL Surface =i2ω( 4γb ) E 1,X () E 1,Y () δ( Y ); Y ^ · J NL Surface =i2ω( 2γa ) [ E 1,Y () ] 2 δ( Y ); Z ^ · J NL Surface =i2ω( 4γb ) E 1,Z () E 1,Y () δ( Y );
d=αβ; δ'=2αβ( α1 ); b= α 2 β; a= α 2 β ( ε r,ω +3 ) 2 ;
α= ω ω+i κ 0 ; β= 2ω 2ω+i κ 0 ;
SCS= S Re( E SC,ω × H SC,ω * ) n ^ dS Re( E 0,ω × H 0,ω * )
ACS= S Re( E ω × H ω * ) n ^ dS Re( E 0,ω × H 0,ω * )
NLSCS= S σ( θ,φ )dΩ ; σ( θ,φ )= 2 S inc Re( E SC,2ω × H SC,2ω * )· n ^ R 2 P 0,ω 2 ;
{ E S,x = E P,x E S,y =+ E P,y E S,z = E P,z TE pump; { E S,x =+ E P,x E S,y = E P,y E S,z =+ E P,z TM pump;
( E S · ) E S =( E S,x , E S,y , E S,z )·( E S,x x E S,y x E S,z x E S,x y E S,y y E S,z y E S,x z E S,y z E S,z z )= ={ ( E P,x , E P,y , E P,z )·( E P,x x E P,y x E P,z x E P,x y E P,y y E P,z y E P,x z E P,y z E P,z z ) TEpumpmode ( E P,x , E P,y , E P,z )·( E 1,x x E 1,y x E 1,z x E 1,x y E 1,y y E 1,z y E 1,x z E 1,y z E 1,z z ) TMpumpmode = =( [ ( E P · ) E P ] x , [ ( E P · ) E P ] y , [ ( E P · ) E P ] z )
( | E S | 2 )=( | E P | 2 x , | E P | 2 y , | E P | 2 z )=( [ ( | E P | 2 ) ] x , [ ( | E P | 2 ) ] y , [ ( | E P | 2 ) ] z )
( X ^ S,x , X ^ S,y , X ^ S,z )=±( X ^ P,x , X ^ P,y , X ^ P,z ) ( Y ^ S,x , Y ^ S,y , Y ^ S,z )=( Y ^ P,x , Y ^ P,y , Y ^ P,z ) ( Z ^ S,x , Z ^ S,y , Z ^ S,z )=( Z ^ P,x , Z ^ P,y , Z ^ P,z )
E S, X S () E S, Y S () ={ ±[ E P, X P () ][ E P, Y P () ] TEpumpmode ±[ E P, X P () ][ E P, Y P () ] TMpumpmode = E P, X P () E P, Y P () { E S, X S () E S, Y S () X ^ S,x =+ E P, X P () E P, Y P () X ^ P,x E S, X S () E S, Y S () X ^ S,y = E P, X P () E P, Y P () X ^ P,y E S, X S () E S, Y S () X ^ S,z =+ E P, X P () E P, Y P () X ^ P,z
[ E 2,Y () ] 2 = [ E 1,Y () ] 2 { [ E 2,Y () ] 2 Y ^ 2,x =+ [ E 1,Y () ] 2 Y ^ 1,x [ E 2,Y () ] 2 Y ^ 2,y = [ E 1,Y () ] 2 Y ^ 1,y [ E 2,Y () ] 2 Y ^ 2,z =+ [ E 1,Y () ] 2 Y ^ 1,z

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