Abstract

A surface integral formulation for the second-harmonic generation (SHG) from periodic metallic–dielectric nanostructures is described. This method requires the discretization of the scatterers’ surface in the unit cell only. All the physical quantities involved in this problem are derived in the unit cell by applying specific periodic boundary conditions both at the fundamental and the second-harmonic (SH) frequencies. Both the fundamental and the SH electric fields are computed using the method of moments and periodic Green’s function evaluated with the Ewald’s method. The accuracy of the method is carefully assessed using two specific cases, namely the surface plasmon enhancement of SHG from a gold film and the SHG from L-shaped nanoparticle arrays. These two examples emphasize the accuracy and versatility of the proposed method, which can be applied to a broad range of periodic metallic structures, including plasmonic arrays on arbitrary substrates and metamaterials.

© 2013 Optical Society of America

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2013 (7)

K. Thyagarajan, J. Butet, and O. J. F. Martin, “Augmenting second harmonic generation using Fano resonances in plasmonic systems,” Nano Lett. 13, 1847–1851 (2013).

J. Butet, K. Thyagarajan, and O. J. F. Martin, “Ultrasensitive optical shape characterization of gold nanoantennas using second harmonic generation,” Nano Lett. 13, 1787–1792 (2013).

R. Czaplicki, H. Husu, R. Siikanen, J. Mäkitalo, M. Kauranen, J. Laukkanen, J. Lehtolahti, and M. Kuittinen, “Enhancement of second-harmonic generation from metal nanoparticles by passive elements,” Phys. Rev. Lett. 110, 093902 (2013).
[CrossRef]

A. Rose, D. Huang, and D. R. Smith, “Nonlinear interference and unidirectional wave mixing in metamaterials,” Phys. Rev. Lett. 110, 063901 (2013).
[CrossRef]

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
[CrossRef]

J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles: erratum,” Opt. Express 21, 10205–10206 (2013).
[CrossRef]

C. Forestiere, A. Capretti, and G. Miano, “Surface integral method for second harmonic generation in metal nanoparticles including both local-surface and nonlocal-bulk sources,” J. Opt. Soc. Am. B 30, 2355–2364 (2013).
[CrossRef]

2012 (16)

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. 60, 4766–4776 (2012).
[CrossRef]

A. Slablab, L. Le Xuan, M. Zielinski, Y. de Wilde, V. Jacques, D. Chauvat, and J.-F. Roch, “Second-harmonic generation from coupled plasmon modes in a single dimer of gold nanospheres,” Opt. Express 20, 220–227 (2012).
[CrossRef]

J. Berthelot, G. Bachelier, M. Song, P. Rai, G. Colas des Francs, A. Dereux, and A. Bouhelier, “Silencing and enhancement of second-harmonic generation in optical gap antennas,” Opt. Express 20, 10498–10508 (2012).
[CrossRef]

K. Thyagarajan, S. Rivier, A. Lovera, and O. J. F. Martin, “Enhanced second-harmonic generation from double resonant plasmonic antennae,” Opt. Express 20, 12860–12865 (2012).
[CrossRef]

A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles,” Opt. Express 20, 15797–15806 (2012).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
[CrossRef]

Y. P. Chen, W. E. I. Sha, W. C. H. Choy, L. Jiang, and W. C. Chew, “Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function,” Opt. Express 20, 20210–20221 (2012).
[CrossRef]

C. Forestiere, G. Iadarola, G. Rubinacci, A. Tamburrino, L. Dal Negro, and G. Miano, “Surface integral formulations for the design of plasmonic nanostructures,” J. Opt. Soc. Am. A 29, 2314–2327 (2012).
[CrossRef]

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12, 673–677 (2012).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with multipolar second harmonic generation from spherical metallic nanoparticles,” Nano Lett. 12, 1697–1701 (2012).
[CrossRef]

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[CrossRef]

S. Roke and G. Gonella, “Nonlinear light scattering and spectroscopy of particles and droplets in liquids,” Annu. Rev. Phys. Chem. 63, 353–378 (2012).
[CrossRef]

V. K. Valev, “Characterization of nanostructured plasmonic surfaces with second harmonic generation,” Langmuir 28, 15454–15471 (2012).
[CrossRef]

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
[CrossRef]

B.-L. Wang, M.-L. Ren, J.-F. Li, and Z.-Y. Li, “Plasmonic coupling effect between two gold nanospheres for efficient second-harmonic generation,” J. Appl. Phys. 112, 083102 (2012).
[CrossRef]

2011 (9)

A. M. Kern and O. J. F. Martin, “Excitation and reemission of molecules near realistic plasmonic nanostructures,” Nano Lett. 11, 482–487 (2011).
[CrossRef]

Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. 11, 5519–5523 (2011).
[CrossRef]

C. Awada, C. Jonin, F. Kessi, P. M. Adam, S. Kostcheev, R. Bachelot, P. Royer, M. Samah, I. Russier-Antoine, E. Benichou, G. Bachelier, and P.-F. Brevet, “Polarized second harmonic response of square, hexagonal and random arrays of gold metallic nanocylinders,” Opt. Mater. 33, 1440–1444 (2011).
[CrossRef]

M. A. Vincenti, D. Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19, 2064–2078 (2011).
[CrossRef]

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, 36–45 (2011).
[CrossRef]

A. Farhang and O. J. F. Martin, “Plasmon delocalization onset in finite sized nanostructures,” Opt. Express 19, 11387–11396 (2011).
[CrossRef]

J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express 19, 23386–23399 (2011).
[CrossRef]

A. Benedetti, M. Centini, M. Bertolotti, and C. Sibilia, “Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis,” Opt. Express 19, 26752–26767 (2011).
[CrossRef]

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint, “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett. 11, 61–65 (2011).
[CrossRef]

2010 (8)

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, 043828 (2010).
[CrossRef]

E. H. Barakat, M.-P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express 18, 6530–6536(2010).
[CrossRef]

B. Gallinet, A. M. Kern, and O. J. F. 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).
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G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
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J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interferences between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (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, 1717–1721 (2010).
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A. M. Kern and O. J. F. Martin, “Pitfalls in the determination of optical cross sections from surface integral equation simulations,” IEEE Trans. Antennas Propag. 58, 2158–2161 (2010).
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B. Gallinet and O. J. F. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation,” Photon. Nanostr. Fundam. Appl. 8, 278–284 (2010).
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2009 (5)

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic generation and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
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Y. Zeng, W. Hoyer, J. J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (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, 233402 (2009).
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A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulation of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
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P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009).
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2008 (3)

2007 (2)

2006 (8)

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, “A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,” J. Opt. A 8, S278–S284 (2006).
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B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, “Chirality arising from small defects in gold nanoparticle arrays,” Opt. Express 14, 950–955 (2006).
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I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
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W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express 14, 9570–9575 (2006).
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I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” PIER 63, 243–278 (2006).

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microw. Theory Tech. 54, 3688–3697 (2006).
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M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic meta-materials,” Science 313, 502–504 (2006).
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2005 (3)

C. C. Neascu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71, 201402 (2005).
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T. Laroche, F. I. Baida, and D. Van Labeke, “Three-dimensional finite-difference time-domain study of enhanced second-harmonic generation at the end of a apertureless scanning near-field optical microscope metal tip,” J. Opt. Soc. Am. B 22, 1045–1051 (2005).
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R. Naraoka, H. Okawa, K. Hashimoto, and K. Kajikawa, “Surface plasmon resonance enhanced second-harmonic generation in Kretschmann configuration,” Opt. Commun. 248, 249–256 (2005).
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2004 (3)

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric materials: small-particle limit,” J. Opt. Soc. Am. B 21, 1328–1347 (2004).
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2003 (1)

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).
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2002 (2)

1988 (2)

V. Mizrahi and J. E. Sipe, “Phenomenological treatment of surface second-harmonic generation,” J. Opt. Soc. Am. B 5, 660–667 (1988).
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1986 (2)

M. Corvi and W. L. Schaich, “Hydrodynamic-model calculation of second-harmonic generation at a metal surface,” Phys. Rev. B 33, 3688–3695 (1986).
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J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (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, 4389–4402 (1980).
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1977 (1)

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
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1974 (1)

H. J. Simon, D. E. Mitchell, and J. G. Watson, “Optical second-harmonic generation with surface plasmons in silver films,” Phys. Rev. Lett. 33, 1531–1534 (1974).
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1973 (1)

G. R. Cowper, “Gaussian quadrature formulas for triangle,” Int. J. Numer. Methods Eng. 7, 405–408 (1973).
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1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
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1968 (1)

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C. Awada, C. Jonin, F. Kessi, P. M. Adam, S. Kostcheev, R. Bachelot, P. Royer, M. Samah, I. Russier-Antoine, E. Benichou, G. Bachelier, and P.-F. Brevet, “Polarized second harmonic response of square, hexagonal and random arrays of gold metallic nanocylinders,” Opt. Mater. 33, 1440–1444 (2011).
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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, 233402 (2009).
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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, 043828 (2010).
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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, 233402 (2009).
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K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint, “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett. 11, 61–65 (2011).
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Awada, C.

C. Awada, C. Jonin, F. Kessi, P. M. Adam, S. Kostcheev, R. Bachelot, P. Royer, M. Samah, I. Russier-Antoine, E. Benichou, G. Bachelier, and P.-F. Brevet, “Polarized second harmonic response of square, hexagonal and random arrays of gold metallic nanocylinders,” Opt. Mater. 33, 1440–1444 (2011).
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Ayala-Orozco, C.

Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. 11, 5519–5523 (2011).
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Bachelier, G.

J. Berthelot, G. Bachelier, M. Song, P. Rai, G. Colas des Francs, A. Dereux, and A. Bouhelier, “Silencing and enhancement of second-harmonic generation in optical gap antennas,” Opt. Express 20, 10498–10508 (2012).
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C. Awada, C. Jonin, F. Kessi, P. M. Adam, S. Kostcheev, R. Bachelot, P. Royer, M. Samah, I. Russier-Antoine, E. Benichou, G. Bachelier, and P.-F. Brevet, “Polarized second harmonic response of square, hexagonal and random arrays of gold metallic nanocylinders,” Opt. Mater. 33, 1440–1444 (2011).
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G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interferences between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (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, 1717–1721 (2010).
[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).
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Bachelot, R.

C. Awada, C. Jonin, F. Kessi, P. M. Adam, S. Kostcheev, R. Bachelot, P. Royer, M. Samah, I. Russier-Antoine, E. Benichou, G. Bachelier, and P.-F. Brevet, “Polarized second harmonic response of square, hexagonal and random arrays of gold metallic nanocylinders,” Opt. Mater. 33, 1440–1444 (2011).
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Baida, F. I.

Barakat, E. H.

Baumberg, J. J.

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
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G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
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Benichou, E.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with multipolar second harmonic generation from spherical metallic nanoparticles,” Nano Lett. 12, 1697–1701 (2012).
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J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
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C. Awada, C. Jonin, F. Kessi, P. M. Adam, S. Kostcheev, R. Bachelot, P. Royer, M. Samah, I. Russier-Antoine, E. Benichou, G. Bachelier, and P.-F. Brevet, “Polarized second harmonic response of square, hexagonal and random arrays of gold metallic nanocylinders,” Opt. Mater. 33, 1440–1444 (2011).
[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, 1717–1721 (2010).
[CrossRef]

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

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[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).
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Bernal, M.-P.

Berthelot, J.

Bertolotti, M.

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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).
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Blagovic, K.

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microw. Theory Tech. 54, 3688–3697 (2006).
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Blair, S.

Bloemer, M. J.

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, 043828 (2010).
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I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microw. Theory Tech. 54, 3688–3697 (2006).
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J. Berthelot, G. Bachelier, M. Song, P. Rai, G. Colas des Francs, A. Dereux, and A. Bouhelier, “Silencing and enhancement of second-harmonic generation in optical gap antennas,” Opt. Express 20, 10498–10508 (2012).
[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]

Brevet, P.-F.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with multipolar second harmonic generation from spherical metallic nanoparticles,” Nano Lett. 12, 1697–1701 (2012).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
[CrossRef]

C. Awada, C. Jonin, F. Kessi, P. M. Adam, S. Kostcheev, R. Bachelot, P. Royer, M. Samah, I. Russier-Antoine, E. Benichou, G. Bachelier, and P.-F. Brevet, “Polarized second harmonic response of square, hexagonal and random arrays of gold metallic nanocylinders,” Opt. Mater. 33, 1440–1444 (2011).
[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, 1717–1721 (2010).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

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

Brueck, S. R. J.

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
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K. Thyagarajan, J. Butet, and O. J. F. Martin, “Augmenting second harmonic generation using Fano resonances in plasmonic systems,” Nano Lett. 13, 1847–1851 (2013).

J. Butet, K. Thyagarajan, and O. J. F. Martin, “Ultrasensitive optical shape characterization of gold nanoantennas using second harmonic generation,” Nano Lett. 13, 1787–1792 (2013).

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with multipolar second harmonic generation from spherical metallic nanoparticles,” Nano Lett. 12, 1697–1701 (2012).
[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, 1717–1721 (2010).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

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

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A. Benedetti, M. Centini, M. Bertolotti, and C. Sibilia, “Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis,” Opt. Express 19, 26752–26767 (2011).
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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, 043828 (2010).
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Chen, Y. P.

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Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
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C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
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M. Corvi and W. L. Schaich, “Hydrodynamic-model calculation of second-harmonic generation at a metal surface,” Phys. Rev. B 33, 3688–3695 (1986).
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G. R. Cowper, “Gaussian quadrature formulas for triangle,” Int. J. Numer. Methods Eng. 7, 405–408 (1973).
[CrossRef]

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I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microw. Theory Tech. 54, 3688–3697 (2006).
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R. Czaplicki, H. Husu, R. Siikanen, J. Mäkitalo, M. Kauranen, J. Laukkanen, J. Lehtolahti, and M. Kuittinen, “Enhancement of second-harmonic generation from metal nanoparticles by passive elements,” Phys. Rev. Lett. 110, 093902 (2013).
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Dal Negro, L.

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A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic generation and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
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de Ceglia, D.

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, 043828 (2010).
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de Wilde, Y.

Dereux, A.

Duboisset, J.

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, 1717–1721 (2010).
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M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic meta-materials,” Science 313, 502–504 (2006).
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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).
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Fainman, Y.

Fan, W.

Fang, N. X.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint, “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett. 11, 61–65 (2011).
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Forestiere, C.

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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, 4389–4402 (1980).
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K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint, “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett. 11, 61–65 (2011).
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B. Gallinet and O. J. F. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation,” Photon. Nanostr. Fundam. Appl. 8, 278–284 (2010).
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B. Gallinet, A. M. Kern, and O. J. F. 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).
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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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Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. 11, 5519–5523 (2011).
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Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. 11, 5519–5523 (2011).
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Hanninen, I.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Sketch of the geometry used for the computation of the SHG from a gold slab of thickness L. The upper medium is made of glass and the lower medium is air. The incident wave is a TM plane wave at 45° incidence propagating in the glass medium. The mesh used for the computation is shown in the inset. (b) The tangential component of the fundamental electric field evaluated using Eq. (14) 1 nm below the glass/gold interface as a function of the incident wavelength for different slab thicknesses L. The amplitude of the electric field is normalized to the amplitude of the incident electric field. (c) Reflected SH intensity as a function of the fundamental wavelength for the same gold slab, evaluated in the far-field 50 μm above the slab, in the glass medium.

Fig. 2.
Fig. 2.

(a) and (b) Sketch of the nanoparticle array sample A and sample B discussed in the text. The nanoparticle arm length and width are 250 and 100 nm, respectively. The nanoparticle thickness is 20 nm. The unit cell dimension is 1μm×1μm. Computed SH intensities as a function of the incident polarization for (c) sample A and (d) sample B considering the SH signal polarized along the vector v (squares) and along the vector u (circles).

Fig. 3.
Fig. 3.

Near-field distributions of fundamental intensities evaluated for (a) and (b) the u-component, and (c) and (d) the v-component of the electric field computed for sample A considering (a) and (c) a u-polarized, and (b) and (d) a v-polarized incident wave. The same color scale is used for all the plots but some have been multiplied by 2 for clarity. Near-field distributions of the SH intensity computed for sample A considering (e) a u-polarized and (f) a v-polarized incident wave shown in a logarithmic scale. The SH intensity in (e) has been multiplied by 10 for clarity.

Fig. 4.
Fig. 4.

Near-field distributions of fundamental intensities evaluated for (a) and (b) the u-component, and (c) and (d) the v-component of the electric field computed for sample B considering (a) and (c) a u-polarized, and (b) and (d) a v-polarized incident wave. For comparison, the color scale is identical to the one in Figs. 3(a)3(d). Near-field distributions of the SH intensity computed for sample B considering (e) a u-polarized and (f) a v-polarized incident wave shown in a logarithmic scale.

Equations (37)

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Uk(rt)=eik·tUk(r).
(iωμnVnΩdSG¯n,k(r,r)·Jn,k(r)+VnΩdS[×G¯n,k(r,r)]·Mn,k(r))=(En,kinc(r)),rVnΩ
(iωεnVnΩdSG¯n,k(r,r)·Mn,k(r)VnΩdS[×G¯n,k(r,r)]·Jn,k(r))=(Hn,kinc(r)),rVnΩ.
En,kinc(r)=iωμnVnΩdVG¯n,k(r,r)·j(r)
Hn,kinc(r)=VnΩdV[×j(r)]·G¯n,k(r,r)
G¯n,k(r,r)=teik·tG¯n(rt,r),
Jn=iαifin,
Mn=iβifin,
[niωμnDnnKnnKnniωεnDn]·[{α}{β}]=n[q(E),nq(H),n],
Di,jn=VnΩdSfin(r)·VnΩdSG¯n(r,r)·fjn(r),
Ki,jn=VnΩdSfin(r)·VnΩdS[×G¯n(r,r)]·fjn(r),
qi(E),n=VnΩdSfin(r)·Eninc(r),
qi(H),n=VnΩdSfin(r)·Hninc(r).
Enscat(r)=iiωμnVnΩdSG¯n(r,r)·αifin(r)VnΩdS[×G¯n(r,r)]·βifin(r),
Hnscat(r)=iiωεnVnΩdSG¯n(r,r)·βifin(r)+VnΩdS[×G¯n(r,r)]·αifin(r).
P(r+)=χ(2):E(r)E(r).
Mm=Em×n^m,·Jm=iωεmn^m·Em,
PK(r+t)=eiK.tPK(r+),
(ΔESH)=(EdSH(r+)EmSH(r))=1εP,
(ΔHSH)=(HdSH(r+)HmSH(r))=2ωP×n^m,
JmSH=iαim,SHfim,
MmSH=iβim,SHfim,
JdSH=iγid,SHfid,
Md,SH=iδid,SHfid,
[m,di2ωμnDSH,nmKSH,ndKSH,nm,dKSH,nmi2ωεnDSH,ndi2ωεnDSH,n0FF]·[{αSH}{βm,SH}{δd,SH}]=d[b(1),n0b(2),n],
Fi,j=VnΩdSfid(r)·f(r)jd,
bi(1),d=12εSiΩdS·fid(r)P(r),
bi(2),d=1εlplSiΩSlΩdSfid(r)·(n^d×fld(r)),
EmSH(r)=ii2ωμmVmΩdSG¯mSH(r,r)·αiSHfim(r)VmΩdS[×G¯mSH(r,r)]·βim,SHfim(r),
HmSH(r)=ii2ωεmVmΩdSG¯mSH(r,r)·βim,SHfim(r)+VmΩdS[×G¯mSH(r,r)]·αiSHfim(r)
EdSH(r)=ii2ωμdVdΩdSG¯dSH(r,r)·αiSHfid(r)VdΩdS[×G¯dSH(r,r)]·δid,SHfid(r),
HdSH(r)=ii2ωεdVdΩdSG¯dSH(r,r)·δid,SHfid(r)+VdΩdS[×G¯dSH(r,r)]·αiSHfid(r)
Jn(rt)=eik·tJn(r),
Mn(rt)=eik·tMn(r),
Jm,dSH(rt)=eiK·tJm,dSH(r),
Mm,dSH(rt)=eiK·tMm,dSH(r).
Ej(2ω)=k,lAjklEk(ω)El(ω).

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