Abstract

Second Harmonic Generation (SHG) is a widely used tool to study surfaces. Here we investigate SHG from spherical nanoparticles consisting of a dielectric core (radius 100 nm) and a metallic shell of variable thickness. Plasmonic resonances occur that depend on the thickness of the nanoshells and boost the intensity of the Second Harmonic (SH) signal. The origin of the resonances is studied for the fundamental harmonic and the second harmonic frequencies. Mie resonances at the fundamental harmonic frequency dominate resonant effects of the SH-signal at low shell thickness. Resonances excited by a dipole emitting at SH frequency close to the surface explain the enhancement of the SHG-process at a larger shell thickness. All resonances are caused by surface plasmon polaritons, which run on the surface of the spherical particle and are in resonance with the circumference of the sphere. Because their wavelength critically depends on the properties of the metallic layer SHG resonances of core-shell nanoparticles can be easily tuned by varying the thickness of the shell.

© 2013 OSA

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2013 (1)

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[CrossRef]

2012 (3)

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. B29(8), 2213–2221 (2012).
[CrossRef]

G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett.3(19), 2877–2881 (2012).
[CrossRef]

S. Roke and G. Gonella, “Nonlinear light scattering and spectroscopy of particles and droplets in liquids,” Annu. Rev. Phys. Chem.63(1), 353–378 (2012).
[CrossRef] [PubMed]

2011 (7)

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. B84(12), 121402 (2011).
[CrossRef]

S. Wunderlich, B. Schürer, C. Sauerbeck, W. Peukert, and U. Peschel, “Molecular Mie model for second harmonic generation and sum frequency generation,” Phys. Rev. B84(23), 235403 (2011).
[CrossRef]

B. Schürer, M. Hoffmann, S. Wunderlich, L. Harnau, U. Peschel, M. M. Ballauff, and W. Peukert, “Second harmonic light scattering from spherical polyelectrolyte brushes,” J. Phys. Chem. C115(37), 18302–18309 (2011).
[CrossRef]

W. Gan, B. Xu, and H.-L. Dai, “Activation of Thiols at a Silver Nanoparticle Surface,” Angew. Chem. Int. Ed. Engl.50(29), 6622–6625 (2011).
[CrossRef] [PubMed]

R. Vácha, S. W. Rick, P. Jungwirth, A. G. F. de Beer, H. B. de Aguiar, J.-S. Samson, and S. Roke, “The orientation and charge of water at the hydrophobic oil droplet-water interface,” J. Am. Chem. Soc.133(26), 10204–10210 (2011).
[CrossRef] [PubMed]

L. H. Haber, S. J. J. Kwok, M. Semeraro, and K. B. Eisenthal, “Probing the colloidal gold nanoparticle/aqueous interface with second harmonic generation,” Chem. Phys. Lett.507(1-3), 11–14 (2011).
[CrossRef]

H. B. de Aguiar, M. L. Strader, A. G. F. de Beer, and S. Roke, “Surface structure of sodium dodecyl sulfate surfactant and oil at the oil-in-water droplet liquid/liquid interface: a manifestation of a nonequilibrium surface state,” J. Phys. Chem. B115(12), 2970–2978 (2011).
[CrossRef] [PubMed]

2010 (10)

A. G. F. de Beer and S. Roke, “Obtaining molecular orientation from second harmonic and sum frequency scattering experiments in water: Angular distribution and polarization dependence,” J. Chem. Phys.132(23), 234702 (2010).
[CrossRef] [PubMed]

C.-L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Bioconjugation of barium titanate nanocrystals with immunoglobulin G antibody for second harmonic radiation imaging probes,” Biomaterials31(8), 2272–2277 (2010).
[CrossRef] [PubMed]

H. B. Aguiar, A. G. F. de Beer, M. L. Strader, and S. Roke, “The interfacial tension of nanoscopic oil droplets in water is hardly affected by SDS surfactant,” J. Am. Chem. Soc.132(7), 2122–2123 (2010).
[CrossRef] [PubMed]

B. Schürer, S. Wunderlich, C. Sauerbeck, U. Peschel, and W. Peukert, “Probing colloidal interfaces by angle-resolved second harmonic light scattering,” Phys. Rev. B82(24), 241404 (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. B82(23), 235403 (2010).
[CrossRef]

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

N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: The role of separation and relative orientation,” Opt. Express18(7), 6545–6554 (2010).
[CrossRef] [PubMed]

H. Lu, X. Liu, R. Zhou, Y. Gong, and D. Mao, “Second-harmonic generation from metal-film nanohole arrays,” Appl. Opt.49(12), 2347–2351 (2010).
[CrossRef] [PubMed]

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A81(5), 053850 (2010).
[CrossRef]

S.-H. Jen, H.-L. Dai, and G. Gonella, “The effect of particle size in second harmonic generation from the surface of spherical colloidal particles. II: The nonlinear Rayleigh−Gans−Debye model,” J. Phys. Chem. C114(10), 4302–4308 (2010).
[CrossRef]

2009 (6)

F. B. P. Niesler, N. Feth, S. Linden, J. Niegemann, J. Gieseler, K. Busch, and M. Wegener, “Second-harmonic generation from split-ring resonators on a GaAs substrate,” Opt. Lett.34(13), 1997–1999 (2009).
[CrossRef] [PubMed]

M. Chandra and P. K. Das, ““Small-particle limit” in the second harmonic generation from noble metal nanoparticles,” Chem. Phys.358(3), 203–208 (2009).
[CrossRef]

M. Chandra and P. K. Das, “Size dependence and dispersion behavior of the first hyperpolarizability of copper nanoparticles,” Chem. Phys. Lett.476(1-3), 62–64 (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. B80(23), 233402 (2009).
[CrossRef]

A. G. F. de Beer, H. B. de Aguiar, J. F. W. Nijsen, and S. Roke, “Detection of buried microstructures by nonlinear light scattering spectroscopy,” Phys. Rev. Lett.102(9), 095502 (2009).
[CrossRef] [PubMed]

J. Griffin, A. K. Singh, D. Senapati, E. Lee, K. Gaylor, J. Jones-Boone, and P. C. Ray, “Sequence-Specific HCV RNA Quantification Using the Size-Dependent Nonlinear Optical Properties of Gold Nanoparticles,” Small5(7), 839–845 (2009).
[CrossRef] [PubMed]

2008 (3)

M. Subir, J. Liu, and K. B. Eisenthal, “Protonation at the Aqueous Interface of Polymer Nanoparticles with Second Harmonic Generation,” J. Phys. Chem. C112(40), 15809–15812 (2008).
[CrossRef]

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(17), 1975–1977 (2008).
[CrossRef] [PubMed]

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B78(20), 205322 (2008).
[CrossRef]

2007 (1)

J. Jiang, K. B. Eisenthal, and R. Yuste, “Second harmonic generation in neurons: electro-optic mechanism of membrane potential sensitivity,” Biophys. J.93(5), L26–L28 (2007).
[CrossRef] [PubMed]

2006 (4)

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

J. Nappa, I. Russier-Antoine, E. Benichou, Ch. Jonin, and P.-F. Brevet, “Second harmonic generation from small gold metallic particles: From the dipolar to the quadrupolar response,” J. Chem. Phys.125(18), 184712 (2006).
[CrossRef] [PubMed]

B. S. Mendoza and W. L. Mochán, “Second harmonic surface response of a composite,” Opt. Mater.29(1), 1–5 (2006).
[CrossRef]

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett.97(1), 017402 (2006).
[CrossRef] [PubMed]

2005 (1)

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. B71(16), 165407 (2005).
[CrossRef]

2004 (3)

J.-P. Abid, J. Nappa, H. H. Girault, and P.-F. Brevet, “Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles,” J. Chem. Phys.121(24), 12577–12582 (2004).
[CrossRef] [PubMed]

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B70(24), 245434 (2004).
[CrossRef]

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B70(11), 115106 (2004).
[CrossRef]

2003 (2)

W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68(8), 085318 (2003).
[CrossRef]

S. Roke, W. G. Roeterdink, J. E. G. J. Wijnhoven, A. V. Petukhov, A. W. Kleyn, and M. Bonn, “Vibrational sum frequency scattering from a submicron suspension,” Phys. Rev. Lett.91(25), 258302 (2003).
[CrossRef] [PubMed]

2002 (2)

R. C. Johnson, J. Li, J. T. Hupp, and G. C. Schatz, “Hyper-Rayleigh scattering studies of silver, copper, and platinum nanoparticle suspensions,” Chem. Phys. Lett.356(5-6), 534–540 (2002).
[CrossRef]

E. C. Hao, G. C. Schatz, R. C. Johnson, and J. T. Hupp, “Hyper-Rayleigh scattering from silver nanoparticles,” J. Chem. Phys.117(13), 5963 (2002).
[CrossRef]

2001 (1)

Y. Liu, E. C. Y. Yan, and K. B. Eisenthal, “Effects of bilayer surface charge density on molecular adsorption and transport across liposome bilayers,” Biophys. J.80(2), 1004–1012 (2001).
[CrossRef] [PubMed]

2000 (1)

H. Wang, T. Troxler, A.-G. Yeh, and H.-L. Dai, “In situ, nonlinear optical probe of surfactant adsorption on the durface of microparticles in colloids,” Langmuir16(6), 2475–2481 (2000).
[CrossRef]

1999 (1)

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

1998 (3)

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

E. C. Y. Yan, Y. Liu, and K. B. Eisenthal, “New method for determination of surface potential of microscopic particles by second harmonic generation,” J. Chem. Phys. B102(33), 6331–6336 (1998).
[CrossRef]

F. W. Vance, B. I. Lemon, and J. T. Hupp, “Enormous hyper-Rayleigh scattering from nanocrystalline gold particle suspensions,” J. Chem. Phys. B102(50), 10091–10093 (1998).
[CrossRef]

1997 (1)

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A55(6), 4520–4525 (1997).
[CrossRef]

1996 (1)

H. Wang, E. C. Y. 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]

1993 (1)

D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D28, 169–175 (1993).
[CrossRef]

1987 (1)

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir3(1), 85–93 (1987).
[CrossRef]

1986 (1)

Y. Shen, “Surface 2nd harmonic-generation - a new technique for surface studies,” Annu. Rev. Mater. Sci.16(1), 69–86 (1986).
[CrossRef]

1982 (1)

G. S. Agarwal and S. S. Jha, “Theory of second harmonic generation at a metal surface with surface plasmon excitation,” Solid State Commun.41(6), 499–501 (1982).
[CrossRef]

1980 (2)

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt.19(9), 1505–1509 (1980).
[CrossRef] [PubMed]

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B21(10), 4389–4402 (1980).
[CrossRef]

1977 (1)

D. L. Mills, “Theory of second harmonic generation by surface polaritons on metals,” Solid State Commun.24(9), 669–671 (1977).
[CrossRef]

1976 (1)

H. J. Simon and J. K. Guha, “Directional surface plasmon scattering from silver films,” Opt. Commun.18(3), 391–394 (1976).
[CrossRef]

1975 (1)

H. J. Simon, D. E. Mitchell, and J. G. Watson, “Second harmonic generation with surface plasmons in alkali metals,” Opt. Commun.13(3), 294–298 (1975).
[CrossRef]

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(26), 1531–1534 (1974).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1971 (2)

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I: multipole expansion and ray-optical solutions,” IEEE Trans. Antenn. Propag.19(3), 378–390 (1971).
[CrossRef]

J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B4(12), 4274–4290 (1971).
[CrossRef]

1969 (1)

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop.13(3), 302–313 (1969).
[CrossRef]

Abid, J.-P.

J.-P. Abid, J. Nappa, H. H. Girault, and P.-F. Brevet, “Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles,” J. Chem. Phys.121(24), 12577–12582 (2004).
[CrossRef] [PubMed]

Agarwal, G. S.

G. S. Agarwal and S. S. Jha, “Theory of second harmonic generation at a metal surface with surface plasmon excitation,” Solid State Commun.41(6), 499–501 (1982).
[CrossRef]

Aguiar, H. B.

H. B. Aguiar, A. G. F. de Beer, M. L. Strader, and S. Roke, “The interfacial tension of nanoscopic oil droplets in water is hardly affected by SDS surfactant,” J. Am. Chem. Soc.132(7), 2122–2123 (2010).
[CrossRef] [PubMed]

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. B80(23), 233402 (2009).
[CrossRef]

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. B80(23), 233402 (2009).
[CrossRef]

Antoine, R.

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Bachelier, G.

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. B82(23), 235403 (2010).
[CrossRef]

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

Ballauff, M. M.

B. Schürer, M. Hoffmann, S. Wunderlich, L. Harnau, U. Peschel, M. M. Ballauff, and W. Peukert, “Second harmonic light scattering from spherical polyelectrolyte brushes,” J. Phys. Chem. C115(37), 18302–18309 (2011).
[CrossRef]

Benichou, E.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[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. B29(8), 2213–2221 (2012).
[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. B82(23), 235403 (2010).
[CrossRef]

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

J. Nappa, I. Russier-Antoine, E. Benichou, Ch. Jonin, and P.-F. Brevet, “Second harmonic generation from small gold metallic particles: From the dipolar to the quadrupolar response,” J. Chem. Phys.125(18), 184712 (2006).
[CrossRef] [PubMed]

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. B71(16), 165407 (2005).
[CrossRef]

Bennemann, K. H.

D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D28, 169–175 (1993).
[CrossRef]

Berg, O.

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

Bonn, M.

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B70(11), 115106 (2004).
[CrossRef]

S. Roke, W. G. Roeterdink, J. E. G. J. Wijnhoven, A. V. Petukhov, A. W. Kleyn, and M. Bonn, “Vibrational sum frequency scattering from a submicron suspension,” Phys. Rev. Lett.91(25), 258302 (2003).
[CrossRef] [PubMed]

Borguet, E.

H. Wang, E. C. Y. 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]

Brevet, P. F.

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Brevet, P.-F.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[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. B29(8), 2213–2221 (2012).
[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. B82(23), 235403 (2010).
[CrossRef]

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

J. Nappa, I. Russier-Antoine, E. Benichou, Ch. Jonin, and P.-F. Brevet, “Second harmonic generation from small gold metallic particles: From the dipolar to the quadrupolar response,” J. Chem. Phys.125(18), 184712 (2006).
[CrossRef] [PubMed]

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. B71(16), 165407 (2005).
[CrossRef]

J.-P. Abid, J. Nappa, H. H. Girault, and P.-F. Brevet, “Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles,” J. Chem. Phys.121(24), 12577–12582 (2004).
[CrossRef] [PubMed]

Broyer, M.

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Brudny, V. L.

W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68(8), 085318 (2003).
[CrossRef]

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I: multipole expansion and ray-optical solutions,” IEEE Trans. Antenn. Propag.19(3), 378–390 (1971).
[CrossRef]

Buitenhuis, J.

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

Busch, K.

Butet, J.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[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. B29(8), 2213–2221 (2012).
[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. B82(23), 235403 (2010).
[CrossRef]

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

Chandra, M.

M. Chandra and P. K. Das, ““Small-particle limit” in the second harmonic generation from noble metal nanoparticles,” Chem. Phys.358(3), 203–208 (2009).
[CrossRef]

M. Chandra and P. K. Das, “Size dependence and dispersion behavior of the first hyperpolarizability of copper nanoparticles,” Chem. Phys. Lett.476(1-3), 62–64 (2009).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Corbalán, R.

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A55(6), 4520–4525 (1997).
[CrossRef]

Dadap, J. I.

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B78(20), 205322 (2008).
[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]

Dai, H.-L.

G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett.3(19), 2877–2881 (2012).
[CrossRef]

W. Gan, B. Xu, and H.-L. Dai, “Activation of Thiols at a Silver Nanoparticle Surface,” Angew. Chem. Int. Ed. Engl.50(29), 6622–6625 (2011).
[CrossRef] [PubMed]

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. B84(12), 121402 (2011).
[CrossRef]

S.-H. Jen, H.-L. Dai, and G. Gonella, “The effect of particle size in second harmonic generation from the surface of spherical colloidal particles. II: The nonlinear Rayleigh−Gans−Debye model,” J. Phys. Chem. C114(10), 4302–4308 (2010).
[CrossRef]

H. Wang, T. Troxler, A.-G. Yeh, and H.-L. Dai, “In situ, nonlinear optical probe of surfactant adsorption on the durface of microparticles in colloids,” Langmuir16(6), 2475–2481 (2000).
[CrossRef]

Das, P. K.

M. Chandra and P. K. Das, “Size dependence and dispersion behavior of the first hyperpolarizability of copper nanoparticles,” Chem. Phys. Lett.476(1-3), 62–64 (2009).
[CrossRef]

M. Chandra and P. K. Das, ““Small-particle limit” in the second harmonic generation from noble metal nanoparticles,” Chem. Phys.358(3), 203–208 (2009).
[CrossRef]

Dave, J. V.

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop.13(3), 302–313 (1969).
[CrossRef]

de Aguiar, H. B.

H. B. de Aguiar, M. L. Strader, A. G. F. de Beer, and S. Roke, “Surface structure of sodium dodecyl sulfate surfactant and oil at the oil-in-water droplet liquid/liquid interface: a manifestation of a nonequilibrium surface state,” J. Phys. Chem. B115(12), 2970–2978 (2011).
[CrossRef] [PubMed]

R. Vácha, S. W. Rick, P. Jungwirth, A. G. F. de Beer, H. B. de Aguiar, J.-S. Samson, and S. Roke, “The orientation and charge of water at the hydrophobic oil droplet-water interface,” J. Am. Chem. Soc.133(26), 10204–10210 (2011).
[CrossRef] [PubMed]

A. G. F. de Beer, H. B. de Aguiar, J. F. W. Nijsen, and S. Roke, “Detection of buried microstructures by nonlinear light scattering spectroscopy,” Phys. Rev. Lett.102(9), 095502 (2009).
[CrossRef] [PubMed]

de Beer, A. G. F.

H. B. de Aguiar, M. L. Strader, A. G. F. de Beer, and S. Roke, “Surface structure of sodium dodecyl sulfate surfactant and oil at the oil-in-water droplet liquid/liquid interface: a manifestation of a nonequilibrium surface state,” J. Phys. Chem. B115(12), 2970–2978 (2011).
[CrossRef] [PubMed]

R. Vácha, S. W. Rick, P. Jungwirth, A. G. F. de Beer, H. B. de Aguiar, J.-S. Samson, and S. Roke, “The orientation and charge of water at the hydrophobic oil droplet-water interface,” J. Am. Chem. Soc.133(26), 10204–10210 (2011).
[CrossRef] [PubMed]

A. G. F. de Beer and S. Roke, “Obtaining molecular orientation from second harmonic and sum frequency scattering experiments in water: Angular distribution and polarization dependence,” J. Chem. Phys.132(23), 234702 (2010).
[CrossRef] [PubMed]

H. B. Aguiar, A. G. F. de Beer, M. L. Strader, and S. Roke, “The interfacial tension of nanoscopic oil droplets in water is hardly affected by SDS surfactant,” J. Am. Chem. Soc.132(7), 2122–2123 (2010).
[CrossRef] [PubMed]

A. G. F. de Beer, H. B. de Aguiar, J. F. W. Nijsen, and S. Roke, “Detection of buried microstructures by nonlinear light scattering spectroscopy,” Phys. Rev. Lett.102(9), 095502 (2009).
[CrossRef] [PubMed]

Decker, M.

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(5), 1717–1721 (2010).
[CrossRef] [PubMed]

Eisenthal, K. B.

L. H. Haber, S. J. J. Kwok, M. Semeraro, and K. B. Eisenthal, “Probing the colloidal gold nanoparticle/aqueous interface with second harmonic generation,” Chem. Phys. Lett.507(1-3), 11–14 (2011).
[CrossRef]

M. Subir, J. Liu, and K. B. Eisenthal, “Protonation at the Aqueous Interface of Polymer Nanoparticles with Second Harmonic Generation,” J. Phys. Chem. C112(40), 15809–15812 (2008).
[CrossRef]

J. Jiang, K. B. Eisenthal, and R. Yuste, “Second harmonic generation in neurons: electro-optic mechanism of membrane potential sensitivity,” Biophys. J.93(5), L26–L28 (2007).
[CrossRef] [PubMed]

Y. Liu, E. C. Y. Yan, and K. B. Eisenthal, “Effects of bilayer surface charge density on molecular adsorption and transport across liposome bilayers,” Biophys. J.80(2), 1004–1012 (2001).
[CrossRef] [PubMed]

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]

E. C. Y. Yan, Y. Liu, and K. B. Eisenthal, “New method for determination of surface potential of microscopic particles by second harmonic generation,” J. Chem. Phys. B102(33), 6331–6336 (1998).
[CrossRef]

H. Wang, E. C. Y. 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]

Feth, N.

Fukui, M.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B21(10), 4389–4402 (1980).
[CrossRef]

Galletto, P.

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Gan, W.

G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett.3(19), 2877–2881 (2012).
[CrossRef]

W. Gan, B. Xu, and H.-L. Dai, “Activation of Thiols at a Silver Nanoparticle Surface,” Angew. Chem. Int. Ed. Engl.50(29), 6622–6625 (2011).
[CrossRef] [PubMed]

Gaylor, K.

J. Griffin, A. K. Singh, D. Senapati, E. Lee, K. Gaylor, J. Jones-Boone, and P. C. Ray, “Sequence-Specific HCV RNA Quantification Using the Size-Dependent Nonlinear Optical Properties of Gold Nanoparticles,” Small5(7), 839–845 (2009).
[CrossRef] [PubMed]

Gieseler, J.

Girault, H. H.

J.-P. Abid, J. Nappa, H. H. Girault, and P.-F. Brevet, “Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles,” J. Chem. Phys.121(24), 12577–12582 (2004).
[CrossRef] [PubMed]

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Gonella, G.

S. Roke and G. Gonella, “Nonlinear light scattering and spectroscopy of particles and droplets in liquids,” Annu. Rev. Phys. Chem.63(1), 353–378 (2012).
[CrossRef] [PubMed]

G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett.3(19), 2877–2881 (2012).
[CrossRef]

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. B84(12), 121402 (2011).
[CrossRef]

S.-H. Jen, H.-L. Dai, and G. Gonella, “The effect of particle size in second harmonic generation from the surface of spherical colloidal particles. II: The nonlinear Rayleigh−Gans−Debye model,” J. Phys. Chem. C114(10), 4302–4308 (2010).
[CrossRef]

Gong, Y.

Grange, R.

C.-L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Bioconjugation of barium titanate nanocrystals with immunoglobulin G antibody for second harmonic radiation imaging probes,” Biomaterials31(8), 2272–2277 (2010).
[CrossRef] [PubMed]

Griffin, J.

J. Griffin, A. K. Singh, D. Senapati, E. Lee, K. Gaylor, J. Jones-Boone, and P. C. Ray, “Sequence-Specific HCV RNA Quantification Using the Size-Dependent Nonlinear Optical Properties of Gold Nanoparticles,” Small5(7), 839–845 (2009).
[CrossRef] [PubMed]

Guha, J. K.

H. J. Simon and J. K. Guha, “Directional surface plasmon scattering from silver films,” Opt. Commun.18(3), 391–394 (1976).
[CrossRef]

Haber, L. H.

L. H. Haber, S. J. J. Kwok, M. Semeraro, and K. B. Eisenthal, “Probing the colloidal gold nanoparticle/aqueous interface with second harmonic generation,” Chem. Phys. Lett.507(1-3), 11–14 (2011).
[CrossRef]

Håkanson, U.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett.97(1), 017402 (2006).
[CrossRef] [PubMed]

Hansson, T.

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A81(5), 053850 (2010).
[CrossRef]

Hao, E. C.

E. C. Hao, G. C. Schatz, R. C. Johnson, and J. T. Hupp, “Hyper-Rayleigh scattering from silver nanoparticles,” J. Chem. Phys.117(13), 5963 (2002).
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Harnau, L.

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J. Griffin, A. K. Singh, D. Senapati, E. Lee, K. Gaylor, J. Jones-Boone, and P. C. Ray, “Sequence-Specific HCV RNA Quantification Using the Size-Dependent Nonlinear Optical Properties of Gold Nanoparticles,” Small5(7), 839–845 (2009).
[CrossRef] [PubMed]

Sipe, J. E.

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. B80(23), 233402 (2009).
[CrossRef]

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B21(10), 4389–4402 (1980).
[CrossRef]

Sitarski, M.

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir3(1), 85–93 (1987).
[CrossRef]

So, V. C. Y.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B21(10), 4389–4402 (1980).
[CrossRef]

Stampfli, P.

D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D28, 169–175 (1993).
[CrossRef]

Stannigel, K.

Stegeman, G. I.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B21(10), 4389–4402 (1980).
[CrossRef]

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J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B4(12), 4274–4290 (1971).
[CrossRef]

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H. B. de Aguiar, M. L. Strader, A. G. F. de Beer, and S. Roke, “Surface structure of sodium dodecyl sulfate surfactant and oil at the oil-in-water droplet liquid/liquid interface: a manifestation of a nonequilibrium surface state,” J. Phys. Chem. B115(12), 2970–2978 (2011).
[CrossRef] [PubMed]

H. B. Aguiar, A. G. F. de Beer, M. L. Strader, and S. Roke, “The interfacial tension of nanoscopic oil droplets in water is hardly affected by SDS surfactant,” J. Am. Chem. Soc.132(7), 2122–2123 (2010).
[CrossRef] [PubMed]

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M. Subir, J. Liu, and K. B. Eisenthal, “Protonation at the Aqueous Interface of Polymer Nanoparticles with Second Harmonic Generation,” J. Phys. Chem. C112(40), 15809–15812 (2008).
[CrossRef]

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H. Wang, T. Troxler, A.-G. Yeh, and H.-L. Dai, “In situ, nonlinear optical probe of surfactant adsorption on the durface of microparticles in colloids,” Langmuir16(6), 2475–2481 (2000).
[CrossRef]

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R. Vácha, S. W. Rick, P. Jungwirth, A. G. F. de Beer, H. B. de Aguiar, J.-S. Samson, and S. Roke, “The orientation and charge of water at the hydrophobic oil droplet-water interface,” J. Am. Chem. Soc.133(26), 10204–10210 (2011).
[CrossRef] [PubMed]

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S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
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[CrossRef]

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F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B80(23), 233402 (2009).
[CrossRef]

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H. Wang, T. Troxler, A.-G. Yeh, and H.-L. Dai, “In situ, nonlinear optical probe of surfactant adsorption on the durface of microparticles in colloids,” Langmuir16(6), 2475–2481 (2000).
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H. J. Simon, D. E. Mitchell, and J. G. Watson, “Second harmonic generation with surface plasmons in alkali metals,” Opt. Commun.13(3), 294–298 (1975).
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Wijnhoven, J. E. G. J.

S. Roke, W. G. Roeterdink, J. E. G. J. Wijnhoven, A. V. Petukhov, A. W. Kleyn, and M. Bonn, “Vibrational sum frequency scattering from a submicron suspension,” Phys. Rev. Lett.91(25), 258302 (2003).
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[CrossRef]

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B. Schürer, S. Wunderlich, C. Sauerbeck, U. Peschel, and W. Peukert, “Probing colloidal interfaces by angle-resolved second harmonic light scattering,” Phys. Rev. B82(24), 241404 (2010).
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Small (1)

J. Griffin, A. K. Singh, D. Senapati, E. Lee, K. Gaylor, J. Jones-Boone, and P. C. Ray, “Sequence-Specific HCV RNA Quantification Using the Size-Dependent Nonlinear Optical Properties of Gold Nanoparticles,” Small5(7), 839–845 (2009).
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Figures (8)

Fig. 1
Fig. 1

a) typical experimental setup for angle-resolved SHG from colloidal core-shell particles. b) Angular scattering pattern I sp ( θ ) (FH s-polarized, SH p-polarized) and I pp ( θ ) (FH and SH p-polarized) for a silica particle (radius 100 nm) with a silver shell of 6.3 nm thickness (compare Fig. 2).

Fig. 2
Fig. 2

Angle-resolved SH-intensity I pp ( θ ) and I sp ( θ ) (a,b) and mean intensity, averaged over the detection angle (c,d) for a growing silver shell on a silica particle. For the metallic shell either idealized silver without plasmon losses ϵ shell 400 nm =4.4 , ϵ shell 800 nm =30.8 (a,b,c) or realistic values, including plasmon losses ϵ shell 400 nm =4.4+0.2i , ϵ shell 800 nm =30.8+0.4i , are used (d).

Fig. 3
Fig. 3

Mean electric field E r ( ϑ ) ϑ induced by a scattered plane wave (fundamental harmonic wavelength) at the inner ( r in ) and outer ( r out ) interface of a sphere with silica core of radius 100 nm and a silver shell of variable thickness with ( ϵ shell 800 nm =30.8+0.4i ) and without ( ϵ shell 800 nm =30.8 ) damping. Additionally, the extinction at λ=800 nm is given as a function of the shell thickness.

Fig. 4
Fig. 4

Field distribution log 10 | E | 2 emitted by a dipole radiating at 400 nm wavelength (a) and respective field distribution log 10 | E ϑ | 2 modified by the presence of a sphere (b,c). The sphere has a silica core of radius 100 nm, and silver shell of thickness D=25 nm . The shell thickness is tuned for the resonance K=5 with 10 lobes. For the metallic shell either idealized silver without plasmon losses ϵ shell 400 nm =4.4 (b) or realistic ϵ shell 400 nm =4.4+0.2i is used (c).

Fig. 5
Fig. 5

Tangential component of the intensity of a dipole (second harmonic wavelength) placed at the surface of a spherical particle with silica core of radius 100 nm and a silver shell (with and without damping) of variable thickness (compare Fig. 4 for the geometry). a) Intensity | E ϑ ( ϑ ) | 2 of a dipole at the outer surface (without plasmonic losses, ϵ shell 400 nm =4.4 ). The intensity is detected 20 nm above the surface of the sphere and shown as a function of the scattering angle ϑ. b, c: Intensity | E ϑ ( ϑ ) | 2 ϑ of a dipole, which is placed at the inner and outer interface of a sphere, without (b) and with (c) plasmonic losses ( ϵ shell 400 nm =4.4+0.2i ), detected in 2 cm distance.

Fig. 6
Fig. 6

Resonances with 2K lobes occurring at certain values of the permittivity ε N Sphere of a solid metallic sphere in comparison to the model based on the planar plasmon for R=100 nm and λ Dipole =400 nm .

Fig. 7
Fig. 7

SH-intensity (a) and extinction spectrum (b) as a function of the fundamental harmonic wavelength. The dashed lines show the SH-scattering cross section C ext m,n (2n+1) k 2 (| a m,n SH | 2 +| b m,n SH | 2 ) . Three systems of silica particles with silver shell (including plasmon losses) of 5 nm, 7.5 nm, and 10 nm thickness are shown.

Fig. 8
Fig. 8

Effect of the variation of χ ( 2 ) on the position of the first maximum in pp- and sp-polarization ( θ pp , θ sp ), the ratio of the intensities of the first maximum ( I pp / I sp ) and the sum of the intensities of the first maxima in sp- and pp-polarization ( I pp + I sp ) for a silica particle ( R=100 nm ) with a silver shell (including plasmon losses) of R=15 nm thickness. The triangular parameter space displays the relative amount of the three independent components of χ ( 2 ) : χ zzz ( 2 ) , χ zxx ( 2 ) , and χ xxz ( 2 ) . In the corners of the triangle, only the component written next to the corner is nonzero. Along the edges the two components are nonzero, their relative magnitude becomes larger the closer the point lies to the respective corner [19,20].

Equations (9)

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P i (2) j,k χ ijk ( 2 ) E j FH E k FH
E FH ( r )= n=1 N m=n n a m,n N m,n ( 1 ) ( r )+ b m,n N m,n ( 1 ) ( r ) + c m,n N m,n ( 3 ) ( r )+ d m,n M m,n ( 3 ) ( r ).
E θ,ϕ inc ( R )+ E θ,ϕ sc, out ( R )= E θ,ϕ sc, in ( R )
E θ,ϕ sc, l ( R l )= E θ,ϕ sc, l+1 ( R l ).
E θ,ϕ sc, l ( R L )= E θ,ϕ sc, l+1 ( R L )+ E θ,ϕ pw ( R L ).
E Dipole ( r ) ={ E (j=3),Dipole ( r ) = m,n a m,n (j=3) N m,n (j=3) ( r )+ b m,n (j=3) N m,n (j=3) ( r ) if| r |>if| r d | . E (j=1),Dipole ( r ) = m,n a m,n (j=1) N m,n (j=1) ( r )+ b m,n (j=1) N m,n (j=1) ( r ) if| r |< | r d |
E θ,ϕ sc, l1 ( R l1 )= E θ,ϕ Dipole,(1) ( R l1 )+ E θ,ϕ sc, l ( R l1 ),
E θ,ϕ sc, l ( R l )+ E θ,ϕ Dipole,(3) ( R l )= E θ,ϕ sc, l+1 ( R l ).
λ SPP = λ Dipole ε Sphere + ε Surrounding ε Sphere ε Surrounding

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