Abstract

We numerically investigated second harmonic generation from systems composed of two coupled gold nanowires. The developed method allows one to arbitrarily change the shape of the wire cross section in order to explore the generated and scattered field patterns. Our results suggest that the overall second harmonic generation is related to the electromagnetic energy per unit length stored in the gap area between the wires. These geometrical considerations make further optimization possible. As an example we discuss the possibility to select dipolar emission and/or quadrupolar emission patterns. The selection mechanism responsible for this kind of emission can be traced back to the interaction between nonlinear sources with the surface plasmon resonance of the metallic wires.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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  42. C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150-2161 (2003).
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  43. 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, 4045-4048 (1999).
    [CrossRef]
  44. J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
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  45. G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25, 955-960 (2008).
    [CrossRef]
  46. L. Cao and N. C. Panoiu, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79, 235416 (2009).
    [CrossRef]
  47. W. L. Schaich, “Second harmonic genaration by periodically-structured metal surfaces,” Phys. Rev. B 78, 195416 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  51. Y. Zeng and J. V. Moloney, “Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns,” Opt. Lett. 34, 2844-2846 (2009).
    [CrossRef] [PubMed]
  52. M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, “Enhanced second-harmonic generation by metal surfaces with nanoscale roughness: nanoscale dephasing, depolarization, and correlations,” Phys. Rev. Lett. 92, 057402 (2004).
    [CrossRef] [PubMed]
  53. A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271-5283 (1998).
    [CrossRef]
  54. M.-L. Thèye, “Investigation of the optical properties of Au by means of thin semitransparent films,” Phys. Rev. B 2, 3060-3078 (1970).
    [CrossRef]
  55. M. Centini, A. Benedetti, M. Scalora, C. Sibilia, and M. Bertolotti, “Second harmonic generation from metallic 2D scatterers,” Proc. SPIE 7354, 73540F (2009).
    [CrossRef]
  56. D. Maystre and M. Neviere, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A A39, 115-121 (1986).
    [CrossRef]
  57. A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248-263 (1980).
    [CrossRef]
  58. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green's tensor approach,” J. Opt. Soc. Am. A 18, 854-861 (2001).
    [CrossRef]
  59. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62, 5797-5807 (2000).
    [CrossRef]
  60. P. G. Etchegoin and E. C. Le Ru, “Multipolar emission in the vicinity of metallic nanostructures,” J. Phys. Condens. Matter 18, 1175-1188 (2006).
    [CrossRef]

2009 (9)

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

M. Chandra and P. K. Das, “Small-particle limit in the second harmonic generation from noble metal nanoparticles,” Chem. Phys. 358, 203-208 (2009).
[CrossRef]

S. Palomba, M. Danckwerts, and L. Novotny, “Nonlinear plasmonics with gold nanoparticle antennas,” J. Opt. A, Pure Appl. Opt. 11, 114030 (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, 257404 (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, 3603-3609 (2009).
[CrossRef] [PubMed]

L. Cao and N. C. Panoiu, “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. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (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, 2844-2846 (2009).
[CrossRef] [PubMed]

M. Centini, A. Benedetti, M. Scalora, C. Sibilia, and M. Bertolotti, “Second harmonic generation from metallic 2D scatterers,” Proc. SPIE 7354, 73540F (2009).
[CrossRef]

2008 (7)

N. Feth, S. Linden, M. W. Klein, M. Decker, F. B. P. Niesler, Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, J. V. Moloney, and M. Wegener, “Second-harmonic generation from complementary split-ring resonators,” Opt. Lett. 33, 1975-1977 (2008).
[CrossRef] [PubMed]

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

M. A. Vincenti, V. Petruzzelli, A. D'Orazio, F. Prudenzano, M. J. Bloemer, N. Akozbek, and M. Scalora, “Second harmonic generation from nanoslits in metal substrates: applications to palladium-based H sensor,” J. Nanophotonics 2, 021851 (2008).
[CrossRef]

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
[CrossRef]

G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25, 955-960 (2008).
[CrossRef]

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, 093119 (2008).
[CrossRef]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials: erratum,” Opt. Express 16, 8055 (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, 1251-1255 (2007).
[CrossRef] [PubMed]

2006 (5)

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027-1030 (2006).
[CrossRef]

A. Datta and S. K. Pati, “Dipolar interactions and hydrogen bonding in supramolecular aggregates: understanding cooperative phenomena for 1st hyperpolarizability,” Chem. Soc. Rev. 35, 1305-1323 (2006).
[CrossRef]

P. G. Etchegoin and E. C. Le Ru, “Multipolar emission in the vicinity of metallic nanostructures,” J. Phys. Condens. Matter 18, 1175-1188 (2006).
[CrossRef]

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

2005 (1)

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, 153401 (2005).
[CrossRef]

2004 (5)

B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12, 5418-5423 (2004).
[CrossRef] [PubMed]

A. Datta and S. K. Pati, “Charge-transfer induced large nonlinear optical properties of small Al clusters: Al4M4 (M=Li, Na, and K),” J. Phys. Chem. A 108, 9527-9530 (2004).
[CrossRef]

C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21, 36-44 (2004).
[CrossRef]

T. Kitahara, A. Sugawara, H. Sano, and G. Mizutani, “Optical second-harmonic spectroscopy of Au nanowires,” J. Appl. Phys. 95, 5002-5005 (2004).
[CrossRef]

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, “Enhanced second-harmonic generation by metal surfaces with nanoscale roughness: nanoscale dephasing, depolarization, and correlations,” Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

2003 (3)

C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150-2161 (2003).
[CrossRef]

A. Datta and S. K. Pati, “Dipole orientation effects on nonlinear optical properties of organic molecular aggregates,” J. Chem. Phys. 118, 8420-8427 (2003).
[CrossRef]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

2001 (1)

2000 (2)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

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, 4045-4048 (1999).
[CrossRef]

1998 (2)

F. W. Vance, B. I. Lemon, and J. T. Hupp, “Enormous hyper-Rayleigh scattering from nanocrystalline gold particle suspensions,” J. Phys. Chem. B 102, 10091-10093 (1998).
[CrossRef]

A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271-5283 (1998).
[CrossRef]

1992 (1)

S. Di Bella, M. A. Ratner, and T. J. Marks, “Design of chromophoric molecular assemblies with large second-order optical nonlinearities: a theoretical analysis of the role of intermolecular interactions,” J. Am. Chem. Soc. 114, 5842-5849 (1992).
[CrossRef]

1991 (1)

H. E. Katz, G. Scheller, T. M. Putvinski, M. L. Schilling, W. L. Wilson, and C. E. D. Chidsey, “Polar orientation of dyes in robust multilayers by zirconium phosphate-phosphonate interlayers,” Science 254, 1485-1487 (1991).
[CrossRef] [PubMed]

1990 (1)

S. Ramasesha and I. D. L. Albert “Model exact study of dc-electric-field-induced second harmonic generation coefficients in polyene systems,” Phys. Rev. B 42, 8587-8594 (1990).
[CrossRef]

1988 (1)

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408-7414 (1988).
[CrossRef]

1986 (2)

M. Corvi and L. W. Schaich, “Hydrodynamic-model calculation of second-harmonic generation at a metal surface,” Phys. Rev. B 33, 3688-3695 (1986).
[CrossRef]

D. Maystre and M. Neviere, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A A39, 115-121 (1986).
[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, 4900-4905 (1985).
[CrossRef]

1984 (2)

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

J. Zyss, J. F. Nicoud, and M. Coquillay, “Chirality and hydrogen bonding in molecular crystals for phase-matched second-harmonic generation: N-(4-nitrophenyl)-(L)-prolinol (NPP),” J. Chem. Phys. 81, 4160-4163 (1984).
[CrossRef]

1983 (1)

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870-1885 (1983).
[CrossRef]

1982 (1)

G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Appl. Phys. Lett. 41, 906-908 (1982).
[CrossRef]

1980 (1)

A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248-263 (1980).
[CrossRef]

1979 (1)

1970 (1)

M.-L. Thèye, “Investigation of the optical properties of Au by means of thin semitransparent films,” Phys. Rev. B 2, 3060-3078 (1970).
[CrossRef]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
[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, 986-989 (1966).
[CrossRef]

1965 (2)

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

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029-1031 (1965).
[CrossRef]

1964 (1)

E. Adler, “Nonlinear optical frequency polarization in a dielectric,” Phys. Rev. 134, A728-A733 (1964).
[CrossRef]

Abdenour, A.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027-1030 (2006).
[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, 093119 (2008).
[CrossRef]

Adler, E.

E. Adler, “Nonlinear optical frequency polarization in a dielectric,” Phys. Rev. 134, A728-A733 (1964).
[CrossRef]

Akozbek, N.

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T. Kitahara, A. Sugawara, H. Sano, and G. Mizutani, “Optical second-harmonic spectroscopy of Au nanowires,” J. Appl. Phys. 95, 5002-5005 (2004).
[CrossRef]

Moloney, J. V.

Neviere, M.

D. Maystre and M. Neviere, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A A39, 115-121 (1986).
[CrossRef]

Nevière, M.

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870-1885 (1983).
[CrossRef]

Nicoud, J. F.

J. Zyss, J. F. Nicoud, and M. Coquillay, “Chirality and hydrogen bonding in molecular crystals for phase-matched second-harmonic generation: N-(4-nitrophenyl)-(L)-prolinol (NPP),” J. Chem. Phys. 81, 4160-4163 (1984).
[CrossRef]

Niesler, F. B. P.

Novotny, L.

S. Palomba, M. Danckwerts, and L. Novotny, “Nonlinear plasmonics with gold nanoparticle antennas,” J. Opt. A, Pure Appl. Opt. 11, 114030 (2009).
[CrossRef]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

Osgood, R. M.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027-1030 (2006).
[CrossRef]

Palomba, S.

S. Palomba, M. Danckwerts, and L. Novotny, “Nonlinear plasmonics with gold nanoparticle antennas,” J. Opt. A, Pure Appl. Opt. 11, 114030 (2009).
[CrossRef]

Panoiu, N. C.

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

Panoiu, N. -C.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027-1030 (2006).
[CrossRef]

Parks, R. E.

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029-1031 (1965).
[CrossRef]

Pati, S. K.

A. Datta and S. K. Pati, “Dipolar interactions and hydrogen bonding in supramolecular aggregates: understanding cooperative phenomena for 1st hyperpolarizability,” Chem. Soc. Rev. 35, 1305-1323 (2006).
[CrossRef]

A. Datta and S. K. Pati, “Charge-transfer induced large nonlinear optical properties of small Al clusters: Al4M4 (M=Li, Na, and K),” J. Phys. Chem. A 108, 9527-9530 (2004).
[CrossRef]

A. Datta and S. K. Pati, “Dipole orientation effects on nonlinear optical properties of organic molecular aggregates,” J. Chem. Phys. 118, 8420-8427 (2003).
[CrossRef]

Paulus, M.

M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green's tensor approach,” J. Opt. Soc. Am. A 18, 854-861 (2001).
[CrossRef]

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

Petruzzelli, V.

M. A. Vincenti, V. Petruzzelli, A. D'Orazio, F. Prudenzano, M. J. Bloemer, N. Akozbek, and M. Scalora, “Second harmonic generation from nanoslits in metal substrates: applications to palladium-based H sensor,” J. Nanophotonics 2, 021851 (2008).
[CrossRef]

Podolskiy, V. A.

S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

Polli, D.

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, 093119 (2008).
[CrossRef]

Prangsma, J. C.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

Prudenzano, F.

M. A. Vincenti, V. Petruzzelli, A. D'Orazio, F. Prudenzano, M. J. Bloemer, N. Akozbek, and M. Scalora, “Second harmonic generation from nanoslits in metal substrates: applications to palladium-based H sensor,” J. Nanophotonics 2, 021851 (2008).
[CrossRef]

Putvinski, T. M.

H. E. Katz, G. Scheller, T. M. Putvinski, M. L. Schilling, W. L. Wilson, and C. E. D. Chidsey, “Polar orientation of dyes in robust multilayers by zirconium phosphate-phosphonate interlayers,” Science 254, 1485-1487 (1991).
[CrossRef] [PubMed]

Quail, J. C.

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408-7414 (1988).
[CrossRef]

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

Quelin, X.

S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

Rakic, A. D.

Ramasesha, S.

S. Ramasesha and I. D. L. Albert “Model exact study of dc-electric-field-induced second harmonic generation coefficients in polyene systems,” Phys. Rev. B 42, 8587-8594 (1990).
[CrossRef]

Ratner, M. A.

S. Di Bella, M. A. Ratner, and T. J. Marks, “Design of chromophoric molecular assemblies with large second-order optical nonlinearities: a theoretical analysis of the role of intermolecular interactions,” J. Am. Chem. Soc. 114, 5842-5849 (1992).
[CrossRef]

Reinisch, R.

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870-1885 (1983).
[CrossRef]

Rivoal, J. C.

S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

Roke, S.

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

Russier-Antoine, I.

Safonov, V. P.

S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

Sandtke, M.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

Sano, H.

T. Kitahara, A. Sugawara, H. Sano, and G. Mizutani, “Optical second-harmonic spectroscopy of Au nanowires,” J. Appl. Phys. 95, 5002-5005 (2004).
[CrossRef]

Scalora, M.

M. Centini, A. Benedetti, M. Scalora, C. Sibilia, and M. Bertolotti, “Second harmonic generation from metallic 2D scatterers,” Proc. SPIE 7354, 73540F (2009).
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M. A. Vincenti, V. Petruzzelli, A. D'Orazio, F. Prudenzano, M. J. Bloemer, N. Akozbek, and M. Scalora, “Second harmonic generation from nanoslits in metal substrates: applications to palladium-based H sensor,” J. Nanophotonics 2, 021851 (2008).
[CrossRef]

Schaich, L. W.

M. Corvi and L. W. Schaich, “Hydrodynamic-model calculation of second-harmonic generation at a metal surface,” Phys. Rev. B 33, 3688-3695 (1986).
[CrossRef]

Schaich, W. L.

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

Scheller, G.

H. E. Katz, G. Scheller, T. M. Putvinski, M. L. Schilling, W. L. Wilson, and C. E. D. Chidsey, “Polar orientation of dyes in robust multilayers by zirconium phosphate-phosphonate interlayers,” Science 254, 1485-1487 (1991).
[CrossRef] [PubMed]

Schilling, M. L.

H. E. Katz, G. Scheller, T. M. Putvinski, M. L. Schilling, W. L. Wilson, and C. E. D. Chidsey, “Polar orientation of dyes in robust multilayers by zirconium phosphate-phosphonate interlayers,” Science 254, 1485-1487 (1991).
[CrossRef] [PubMed]

Segerink, F. B.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

Shan, J.

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, 4045-4048 (1999).
[CrossRef]

Shen, Y. R.

Sibilia, C.

Simon, H. J.

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408-7414 (1988).
[CrossRef]

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

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J. E. Sipe and G. I. Stegeman, in Nonlinear Optical Response of Metal Surfaces, Surface Polaritons, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), pp. 661-701.

Sleeper, A. M.

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029-1031 (1965).
[CrossRef]

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G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Appl. Phys. Lett. 41, 906-908 (1982).
[CrossRef]

J. E. Sipe and G. I. Stegeman, in Nonlinear Optical Response of Metal Surfaces, Surface Polaritons, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), pp. 661-701.

Stockman, M. I.

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, 153401 (2005).
[CrossRef]

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, “Enhanced second-harmonic generation by metal surfaces with nanoscale roughness: nanoscale dephasing, depolarization, and correlations,” Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

Sugawara, A.

T. Kitahara, A. Sugawara, H. Sano, and G. Mizutani, “Optical second-harmonic spectroscopy of Au nanowires,” J. Appl. Phys. 95, 5002-5005 (2004).
[CrossRef]

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M.-L. Thèye, “Investigation of the optical properties of Au by means of thin semitransparent films,” Phys. Rev. B 2, 3060-3078 (1970).
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Toma, A.

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, 257404 (2009).
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Turunen, 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, 1251-1255 (2007).
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B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12, 5418-5423 (2004).
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Valencia, C. I.

van Nieuwstadt, J. A. H.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

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F. W. Vance, B. I. Lemon, and J. T. Hupp, “Enormous hyper-Rayleigh scattering from nanocrystalline gold particle suspensions,” J. Phys. Chem. B 102, 10091-10093 (1998).
[CrossRef]

Vincenti, M. A.

M. A. Vincenti, V. Petruzzelli, A. D'Orazio, F. Prudenzano, M. J. Bloemer, N. Akozbek, and M. Scalora, “Second harmonic generation from nanoslits in metal substrates: applications to palladium-based H sensor,” J. Nanophotonics 2, 021851 (2008).
[CrossRef]

Wegener, M.

Wild, B.

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, 257404 (2009).
[CrossRef]

Wilson, W. L.

H. E. Katz, G. Scheller, T. M. Putvinski, M. L. Schilling, W. L. Wilson, and C. E. D. Chidsey, “Polar orientation of dyes in robust multilayers by zirconium phosphate-phosphonate interlayers,” Science 254, 1485-1487 (1991).
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A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248-263 (1980).
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S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

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, 093119 (2008).
[CrossRef]

Zeng, Y.

Zhang, S.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027-1030 (2006).
[CrossRef]

Zyss, J.

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, “Enhanced second-harmonic generation by metal surfaces with nanoscale roughness: nanoscale dephasing, depolarization, and correlations,” Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

J. Zyss, J. F. Nicoud, and M. Coquillay, “Chirality and hydrogen bonding in molecular crystals for phase-matched second-harmonic generation: N-(4-nitrophenyl)-(L)-prolinol (NPP),” J. Chem. Phys. 81, 4160-4163 (1984).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A (1)

D. Maystre and M. Neviere, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A A39, 115-121 (1986).
[CrossRef]

Appl. Phys. Lett. (2)

G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Appl. Phys. Lett. 41, 906-908 (1982).
[CrossRef]

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, 093119 (2008).
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Chem. Phys. (1)

M. Chandra and P. K. Das, “Small-particle limit in the second harmonic generation from noble metal nanoparticles,” Chem. Phys. 358, 203-208 (2009).
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Chem. Soc. Rev. (1)

A. Datta and S. K. Pati, “Dipolar interactions and hydrogen bonding in supramolecular aggregates: understanding cooperative phenomena for 1st hyperpolarizability,” Chem. Soc. Rev. 35, 1305-1323 (2006).
[CrossRef]

J. Am. Chem. Soc. (1)

S. Di Bella, M. A. Ratner, and T. J. Marks, “Design of chromophoric molecular assemblies with large second-order optical nonlinearities: a theoretical analysis of the role of intermolecular interactions,” J. Am. Chem. Soc. 114, 5842-5849 (1992).
[CrossRef]

J. Appl. Phys. (1)

T. Kitahara, A. Sugawara, H. Sano, and G. Mizutani, “Optical second-harmonic spectroscopy of Au nanowires,” J. Appl. Phys. 95, 5002-5005 (2004).
[CrossRef]

J. Chem. Phys. (3)

J. Zyss, J. F. Nicoud, and M. Coquillay, “Chirality and hydrogen bonding in molecular crystals for phase-matched second-harmonic generation: N-(4-nitrophenyl)-(L)-prolinol (NPP),” J. Chem. Phys. 81, 4160-4163 (1984).
[CrossRef]

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

A. Datta and S. K. Pati, “Dipole orientation effects on nonlinear optical properties of organic molecular aggregates,” J. Chem. Phys. 118, 8420-8427 (2003).
[CrossRef]

J. Nanophotonics (1)

M. A. Vincenti, V. Petruzzelli, A. D'Orazio, F. Prudenzano, M. J. Bloemer, N. Akozbek, and M. Scalora, “Second harmonic generation from nanoslits in metal substrates: applications to palladium-based H sensor,” J. Nanophotonics 2, 021851 (2008).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

S. Ducourtieux, S. Grésillon, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. P. Drachev, W. D. Bragg, V. P. Safonov, V. A. Podolskiy, Z. C. Ying, R. L. Armstrong, and V. M. Shalaev, “Percolation and fractal composites: optical studies,” J. Nonlinear Opt. Phys. Mater. 9, 105-116 (2000).

J. Opt. A, Pure Appl. Opt. (1)

S. Palomba, M. Danckwerts, and L. Novotny, “Nonlinear plasmonics with gold nanoparticle antennas,” J. Opt. A, Pure Appl. Opt. 11, 114030 (2009).
[CrossRef]

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

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

J. Phys. Chem. A (1)

A. Datta and S. K. Pati, “Charge-transfer induced large nonlinear optical properties of small Al clusters: Al4M4 (M=Li, Na, and K),” J. Phys. Chem. A 108, 9527-9530 (2004).
[CrossRef]

J. Phys. Chem. B (1)

F. W. Vance, B. I. Lemon, and J. T. Hupp, “Enormous hyper-Rayleigh scattering from nanocrystalline gold particle suspensions,” J. Phys. Chem. B 102, 10091-10093 (1998).
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J. Phys. Condens. Matter (1)

P. G. Etchegoin and E. C. Le Ru, “Multipolar emission in the vicinity of metallic nanostructures,” J. Phys. Condens. Matter 18, 1175-1188 (2006).
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Nano Lett. (2)

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, 1251-1255 (2007).
[CrossRef] [PubMed]

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027-1030 (2006).
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Opt. Express (3)

Opt. Lett. (3)

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

Phys. Rev. B (11)

M. Corvi and L. W. Schaich, “Hydrodynamic-model calculation of second-harmonic generation at a metal surface,” Phys. Rev. B 33, 3688-3695 (1986).
[CrossRef]

H. J. Simon, C. Huang, J. C. Quail, and Z. Chen, “Second-harmonic generation with surface plasmons from a silvered quartz grating,” Phys. Rev. B 38, 7408-7414 (1988).
[CrossRef]

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

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870-1885 (1983).
[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, 153401 (2005).
[CrossRef]

S. Ramasesha and I. D. L. Albert “Model exact study of dc-electric-field-induced second harmonic generation coefficients in polyene systems,” Phys. Rev. B 42, 8587-8594 (1990).
[CrossRef]

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, 235109 (2009).
[CrossRef]

M.-L. Thèye, “Investigation of the optical properties of Au by means of thin semitransparent films,” Phys. Rev. B 2, 3060-3078 (1970).
[CrossRef]

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

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

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
[CrossRef]

Phys. Rev. B. (1)

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

Phys. Rev. E (1)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

Phys. Rev. Lett. (7)

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, 4045-4048 (1999).
[CrossRef]

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Proc. IEEE (1)

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Sketch of the scattering geometry for the FF field. Inset: (up) dimensions of the trapezoid section wires; (down) dimensions of the RS wires.

Fig. 2
Fig. 2

Relative energy per unit length inside the gap region in cases of TS (solid line) and RS (dashed line) wires. NSCS for the TS (triangles) and RS (squares) wires.

Fig. 3
Fig. 3

Modulus of the normalized electric field (a),(c) y- and (b),(d) x-components normalized with respect to the incident field amplitude for gap values of (a),(b) 18 and (c),(d) 38 nm.

Fig. 4
Fig. 4

Differential scattering cross section for the TS wires when different values of the distance between wires ( g ) are considered.

Fig. 5
Fig. 5

Differential scattering cross section for the RS wires when different values of the distance between wires ( g ) are considered.

Fig. 6
Fig. 6

z-component of the SH magnetic field (ampere/meter) (snapshots in time) for the (a) TS and (b) RS wires.

Fig. 7
Fig. 7

Differential scattering cross section (left column) and magnetic field (ampere/meter) (snapshots in time, right column) calculated for different configurations obtained by varying the value of B (see inset of Fig. 1), from (a) 45 to (b) 245 and (c) 345 nm.

Fig. 8
Fig. 8

Modulus of the Poynting vector ( W / m 2 ) outside the TS wires: air gaps of (a) 18 and (b) 38 nm.

Fig. 9
Fig. 9

Modulus of the Poynting vector ( W / m 2 ) outside the RS wires: air gaps of (a) 18 and (b) 38 nm.

Equations (26)

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ε r ( ω ) = ε D ( ω ) + k = 1 5 G k ω P 2 ω 0 , k 2 ω 2 i ω γ k ,
ε D ( ω ) = 1 G 0 ω P 2 ω ( ω + i γ 0 ) = 1 ω 0 2 ω ( ω + i γ 0 ) ,
J = J D + P b .e . t ,
J D t = J D n e J D + ( J D ) J D n e γ 0 J D + e 2 m n E e μ 0 m J D H ,
J D = e n t ,
J ( r , t ) = m = 1 , 2 J m ( r ) e i m ω t + c .c . ,
E ( r , t ) = m = 1 , 2 E n ( r ) e i m ω t + c .c . ,
H ( r , t ) = m = 1 , 2 H m ( r ) e i m ω t + c .c . ,
n ( r , t ) = n 0 + m = 1 , 2 n m ( r ) e i m ω t + c .c . ,
J 1 = i ( ε r , 1 1 ) ω ε 0 E 1 ,
J 2 = i 2 ω ε 0 χ r , 2 E 2 i 2 ω ε 0 β [ a b E 1 [ χ D , 1 E 1 ] + b ( a 1 ) χ D , 1 2 ( E 1 ) E 1 + b χ D , 1 4 [ E 1 E 1 ] ] ,
a = ω ω + i γ 0 ,
b = 2 ω 2 ω + i γ 0 ,
β = e 2 m ω 2 ,
J NL Bulk = i 2 ω ε 0 β χ D , 1 2 b [ ( a 1 ) ( E 1 ) E 1 + 1 2 [ E 1 E 1 ] ] .
X ̂ J NL Surface = i 2 ω ε 0 a b E 1 , X ( ) E 1 , Y ( ) β ( 1 ε D , 1 ) δ ( Y ) ,
Y ̂ J NL Surface = i 2 ω ε 0 a b [ E 1 , Y ( ) ] 2 β 4 [ ( 1 ε D , 1 ) ( ε r , 1 + 3 ) ] δ ( Y ) .
X ̂ J NL Surface = i 2 ω ε 0 a b E 1 , X ( ) E 1 , Y ( ) β ε B [ ε r , 1 ( ε B 1 ) + ε B ( 1 ε D , 1 ) ] δ ( Y ) ,
Y ̂ J NL Surface = i 2 ω ε 0 a b [ E 1 , Y ( ) ] 2 β 4 ε B 2 [ 3 ε r , 1 2 ( ε B 1 ) + ε B [ ε r , 1 ( ε B ε D , 1 ) + 3 ε B ( 1 ε D , 1 ) ] ] δ ( Y ) .
J 2 = i ( ε r , 2 1 ) 2 ω ε 0 E 2 + J NL Bulk + J NL Surface .
( ( 2 ω ) 2 c 2 I ̿ ) E 2 ( r ) = i ( 2 ω ) μ 0 J 2 ( r ) ,
E 2 ( r ) = Σ G ̿ 2 ( r , r ) ( 2 ω c ) 2 [ ε r , 2 ( r ) 1 ] E 2 ( r ) d Σ + i 2 ω μ 0 Σ G ̿ 2 ( r , r ) J NL Bulk ( r ) d Σ + i 2 ω μ 0 Σ G ̿ 2 ( r , r ) J NL Surface ( r ) d Σ ,
G ̿ 2 ( r , r ) = [ I ̿ + k 0 , 2 2 ] g 2 ( r , r ) ,
Q ( 2 ω ) = σ P ( 2 ω ) sc [ P ( ω ) inc ] 2 = 0 2 π q ( θ 2 ω ) d θ ,
W = gap [ ε 0 | E 1 ( x , y ) | 2 + μ 0 | H 1 ( x , y ) | 2 ] d x d y 2 μ 0 | H 1 , 0 | 2 g d ,
H 1 ( inc ) ( x , y ) = ( H 1 , 0 e i ( ω / c ) x + c .c . ) z ̂ .

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