Abstract

The theory of second harmonic generation (SHG) in three-dimensional structures consisting of arbitrary distributions of metallic spheres made of centrosymmetric materials is developed by means of multiple scattering of electromagnetic multipole fields. The electromagnetic field at both the fundamental frequency and second harmonic, as well as the scattering cross section, are calculated in a series of particular cases such as a single metallic sphere, two metallic spheres, chains of metallic spheres, and other distributions of the metallic spheres. It is shown that the linear and nonlinear optical response of all ensembles of metallic spheres is strongly influenced by the excitation of localized surface plasmon-polariton resonances. The physical origin for such a phenomenon has also been analyzed.

© 2012 OSA

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2011

G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
[CrossRef]

J. Xu and X. Zhang, “Negative electron energy loss and second-harmonic emission of nonlinear nanoparticles,” Opt. Express 19(23), 22999–23007 (2011).
[CrossRef] [PubMed]

2010

C. G. Biris and N. C. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18(16), 17165–17179 (2010).
[CrossRef] [PubMed]

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (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. E. 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]

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]

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

2009

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79(15), 155420 (2009).
[CrossRef]

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (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(23), 235109 (2009).
[CrossRef]

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (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]

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

2008

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
[CrossRef] [PubMed]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

2007

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

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

E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
[CrossRef] [PubMed]

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green’s tensor and local density of states calculations for collections of sphererical scatters,” Phys. Rev. B 75(11), 115124 (2007).
[CrossRef]

2006

K. B. Eisenthal, “Second harmonic spectroscopy of aqueous nano- and microparticle interfaces,” Chem. Rev. 106(4), 1462–1477 (2006).
[CrossRef] [PubMed]

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (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]

2005

C. C. Neacsu, 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(20), 201402 (2005).
[CrossRef]

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[CrossRef]

2004

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (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(1), 36–44 (2004).
[CrossRef]

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004).
[CrossRef]

2003

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[CrossRef] [PubMed]

2001

1999

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
[CrossRef]

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

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

1996

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
[CrossRef]

K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996).
[CrossRef]

1995

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573–4588 (1995).
[CrossRef] [PubMed]

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B Condens. Matter 51(4), 2068–2081 (1995).
[CrossRef] [PubMed]

1993

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
[CrossRef] [PubMed]

1985

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]

Aktsipetrov, O. A.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. 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]

Alexander, R. W.

Andersen, U. L.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

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]

Bai, B.

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

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

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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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(1), 013903 (2003).
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H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
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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(1), 013903 (2003).
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J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
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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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M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
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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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H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
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S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
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S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
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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 non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
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L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
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M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
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J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
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J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
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G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
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J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
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H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
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J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
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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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M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
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E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
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Fürst, J. U.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
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J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (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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Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
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R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (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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J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
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S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
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S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
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H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
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S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
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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 non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
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R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
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M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
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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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B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
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S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
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S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
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M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
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J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
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H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
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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 non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
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R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
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J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
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J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
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J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
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M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
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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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J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
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M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
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J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
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M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
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Osgood, R. M.

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
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L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
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Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004).
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D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
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Pertsch, T.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
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J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
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J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
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C. C. Neacsu, 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(20), 201402 (2005).
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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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D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
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J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
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Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
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Xu, Y. L.

Yan, E.

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N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
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X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
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E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
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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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J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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Zhang, X.

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J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
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Appl. Opt.

Appl. Phys. Lett.

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

Chem. Phys. Lett.

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
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Chem. Rev.

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Comput. Phys. Commun.

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J. Appl. Phys.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Soc. Jpn.

K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996).
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Nano Lett.

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

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

Nature

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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Opt. Express

Phys. Rev. A

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
[CrossRef]

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

Phys. Rev. B

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]

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
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G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
[CrossRef]

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Geometry of the scattering problem for an ensemble of N spheres consisting of centrosymmetric materials. Here θ n , ϕ n and r n are the coordinate of the nth sphere in the spherical coordinate system, Ω=(θ,ϕ) represents the solid angle of an arbitrary point P. The x, y and z represent three axis directions in corresponding Cartesian coordinate system.

Fig. 2
Fig. 2

The spectra of the scattering cross sections for metal spheres of radii a = 10nm (dashed line), 30 (solid line), and 50 nm (dotted line). (I): the FF; (II): the SH.

Fig. 3
Fig. 3

(Color online) The spatial profile of the amplitude of the electric field, calculated at ω=5.0 eV (panels A and B) and ω=2.7 eV (panels C and D). The radius of the sphere is a = 10 nm. A and C correspond to the FF; B and D to the SH.

Fig. 4
Fig. 4

The spectra of the scattering cross sections for the metallic two-sphere system. The radii of two spheres are taken as a = 10 nm. (I) and (II) correspond to the separation distance d = 22nm, (III) and (IV) to d = 35nm. Solid line and dashed line correspond to the case at the incident direction of the wave perpendicular or parallel to the longitudinal axis of the system, respectively. (I) and (III): the FF; (II)and (IV): the SH.

Fig. 5
Fig. 5

(Color online) The spatial profile of the amplitude of the electric field for the two-sphere system, calculated at ω=4.1 eV for the normal incidence with d = 22nm (panels A and B) and d = 35nm (panels C and D); calculated at ω=6.54 eV for the parallel incidence with d = 22nm (panels E and F) and d = 35nm (panels G and H). The radii of the spheres are a = 10 nm. A, C, E and G correspond to the FF; B, D, F and H to the SH. Here A, B, C, D, E, F, G and H correspond to the points marked in Fig. 3.

Fig. 6
Fig. 6

(Color online) The top two panels show the spectra of the scattering cross section corresponding to a chain of N = 12 metallic spheres under the parallel incidence. The radius of the sphere is a = 10 nm and the separation distance is d = 22nm. (I): the FF; (II): the SH. The spatial profile of the amplitude of the electric field, calculated at ω=5.3 eV ((A) and (B)) and ω=2.82 eV ((C) and (D)), is presented in the bottom panels. The panels A and C correspond to the FF, whereas the panels B and D correspond to the SH. Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

Fig. 7
Fig. 7

(Color online) The same as in Fig. 6, but for the normal incidence. The spatial profile of the amplitude of the electric field, calculated at ω=5.4 eV ((A) and (B)) and ω=2.59 eV ((C) and (D)). Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

Fig. 8
Fig. 8

(I) The 3D sample and cross sections. (II) and (III) The spectra of the scattering cross sections corresponding to the cluster as shown in (I) under the incidence of the plane wave from the left side of the sample with a = 10 nm and d = 22nm. (II): the FF; (III): the SH.

Fig. 9
Fig. 9

The spatial profiles of the amplitude of the electric field, calculated at ω=2.21 eV (corresponding to the points A and B in Fig. 7 (II) and (III)). (a) and (b) correspond to the FF and the SH fields at the cross-section as shown by shadow in Fig. 7 (I), respectively, whereas (c) and (d) correspond to the FF and the SH fields at the cross-section as shown by dot-dashed lines in Fig. 7 (I). The other parameters are identical with those in Fig. 7.

Equations (48)

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E n 0 ( x n )= lm { a nlm 0,E J Elm ( x n )+ a nlm 0,H J Hlm ( x n ) }
J Elm ( r )= i k b × j l ( k b r ) X lm ( r ^ ), J Hlm ( r )= j l ( k b r ) X lm ( r ^ ),
[ a nlm 0,H a nlm 0,E ]=4π e i K i r n i l [ X lm * ( Ω i ) X lm * ( Ω i )( × Κ ι ) / k b ],
E n inc ( x n )= lm { a nlm inc,E J Elm ( x n )+ a nlm inc,H J Hlm ( x n ) } ,
E n sc ( x n )= lm { a nlm sc,E H Elm ( x n )+ a nlm sc,H H Hlm ( x n ) } ,
E n in ( x n )= lm { a nlm in,E J Elm s ( x n )+ a nlm inc,H J Hlm s ( x n ) }
J Elm s ( r )= i k s × j l ( k s r ) X lm ( r ^ ), J Hlm s ( r )= j l ( k s r ) X lm ( r ^ ),
H Elm ( r )= i k b × h l ( k b r ) X lm ( r ^ ), H Hlm ( r )= h l ( k b r ) X lm ( r ^ ),
E t inc + E t sc = E t in ,
D r inc + D r sc = D r in ,
H t inc + H t sc = H t in ,
B r inc + B r sc = B r in .
T l E = a nlm sc,E / a nlm inc,E , T l H = a nlm sc,H / a nlm inc,H , C l E = a nlm in,E / a nlm inc,E , C l H = a nlm in,H / a nlm inc,H ,
a nlm inc,E = a nlm 0,E + n n l m ( Ω nlm, n l m EE a n l m sc,E + Ω nlm, n l m EH a n l m sc,H ) ,
a nlm inc,H = a nlm 0,H + n n l m ( Ω nlm, n l m HE a n l m sc,E + Ω nlm, n l m HH a n l m sc,H ) ,
n l m [ δ n n δ l l δ m m T l E ( Ω nlm, n l m EE a n l m sc,E + Ω nlm, n l m EH a n l m sc,H ) ] = T l E a nlm 0,E ,
n l m [ δ n n δ l l δ m m T l H ( Ω nlm, n l m HE a n l m sc,E + Ω nlm, n l m HH a n l m sc,H ) ] = T l H a nlm 0,H .
P surface (2ω) ( r )= P s (2ω) ( θ,φ )δ(ra)= χ s (2) : E (ω) ( r ) E (ω) ( r )δ(ra),
P s (2ω) ( θ,φ )= lm G r,nlm Y lm (θ,φ) r ^ + G M,nlm X lm (θ,φ)+ G E,nlm r ^ × X lm (θ,φ) ,
E n int,(2ω) ( x n )= lm { A nlm int,E J Elm s,(2ω) ( x n )+ A nlm int,H J Hlm s,(2ω) ( x n ) } ,
E n out,(2ω) ( x n )= lm { A nlm out,E H Elm (2ω) ( x n )+ A nlm out,H H Hlm (2ω) ( x n ) } ,
E t out,(2ω) E t in,(2ω) = 4π ε 1 (2ω) t P s,r (2ω) ,
D r out,(2ω) D r in,(2ω) =4π t P s (2ω) ,
H t out,(2ω) H t in,(2ω) =4πi 2ω c r ^ × P s (2ω) ,
B r out,(2ω) B r in,(2ω) =0.
A nlm int,E = ε b (2ω) K b h l (1) ( K b a) A nlm out,E +4πi G E,nlm ε b (2ω) j l ( K 1 a) / K 1 ,
A nlm int,H = h l (1) ( K b a) j l ( K 1 a) A nlm out,H ,
χ s (2) = χ r ^ r ^ r ^ + χ r ^ ( θ ^ θ ^ + φ ^ φ ^ )+ χ ( θ ^ r ^ θ ^ + φ ^ r ^ φ ^ + θ ^ θ ^ r ^ + φ ^ φ ^ r ^ ).
P s (2ω) = r ^ ( χ E r (ω) E r (ω) + χ E t (ω) E t (ω) )+2 χ E r (ω) E t (ω) .
E n sca,(2ω) ( x n )= lm { A nlm sca,E H Elm (2ω) ( x n )+ A nlm sca,H H Hlm (2ω) ( x n ) } ,
E n loc,(2ω) ( x n )= n n [ E n out,(2ω) ( x n )+ E n sca,(2ω) ( x n ) ] ,
E n sca,(2ω) ( x n )= lm { A n lm sca,E H Elm (2ω) ( x n )+ A n lm sca,H H Hlm (2ω) ( x n ) } ,
E n loc,(2ω) ( x n )= n n lm { ( A nlm out,E + A n lm sca,E ) H Elm (2ω) ( x n )+( A nlm out,H + A n lm sca,H ) H Hlm (2ω) ( x n ) } .
E n loc,(2ω) ( x n )= nlm { A nlm loc,E J Elm (2ω) ( x n )+ A nlm loc,H J Hlm (2ω) ( x n ) } ,
A nlm loc,E = n n l m { Ω nlm, n l m EE ( A n l m out,E + A n l m sca,E )+ Ω nlm, n l m EH ( A n l m out,H + A n l m sca,H ) } ,
A nlm loc,H = n n l m { Ω nlm, n l m HE ( A n l m out,E + A n l m sca,E )+ Ω nlm, n l m HH ( A n l m out,H + A n l m sca,H ) } .
A nlm sca,E = T l E A nlm loc,E ,
A nlm sca,H = T l H A nlm loc,H .
n l m [ δ n n δ l l δ m m T l E ( Ω nlm, n l m EE A n l m sca,E + Ω nlm, n l m EH A n l m sca,H ) ] = n l m T l E [ Ω nlm, n l m EE A n l m out,E + Ω nlm, n l m EH A n l m out,H ] ,
n l m [ δ n n δ l l δ m m T l H ( Ω nlm, n l m HE A n l m sca,E + Ω nlm, n l m HH A n l m sca,H ) ] = n l m T l H [ Ω nlm, n l m EE A n l m out,E + Ω nlm, n l m EH A n l m out,H ] .
E (2ω) ( r )= nlm ( A nlm out,E + A nlm sca,E ) H Elm (2ω) ( r n )+( A nlm out,H + A nlm sca,H ) H Hlm (2ω) ( r n ) .
C s ( ϖ )= q s ( ϖ;Ω )dΩ ,
q s ( ϖ;Ω )= lim r r 2 Re{ [ E s ( ϖ )× H s ( ϖ ) ] r ^ },
lim r E s ( r )= f ( Ω ) e ikr r ,
lim r H s ( r )= r ^ × f ( Ω ) e ikr r .
f ( Ω )= n e i k b r ^ r n i l1 lm [ X lm ( Ω ) a nlm H /k r ^ × X lm ( Ω ) a nlm E /k ] ,
q s ( ϖ;Ω )= | f ( Ω ) | 2 .
ε s (ω)=1 ω p 2 ω( ω+iν ) ,

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