Abstract

Both the surface and bulk nonlinear responses of a single centrosymmetric nanosphere excited by tightly focused cylindrical vector beams with different polarization rotation angles are investigated theoretically. The numerical results show that a distinctive feature of the calculated surface second harmonic (SH) radiation angular patterns is the possibility of strong scattering in the backward direction. In addition, when the polarization rotation angle takes a larger value, there is a distinct difference between the bulk SH responses stemming from different bulk nonlinear parameters. Those properties are in contrast to what was found in previous theories of SH light scattering from centrosymmetric nanospheres excited by tightly focused linearly polarized beams.

© 2012 Optical Society of America

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2012

Y. You, A. Bloomfield, J. Liu, L. Fu, S. B. Herzon, and E. C. Y. Yan, “Real-time kinetics of surfactant molecule transfer between emulsion particles probed by in situ second harmonic generation spectroscopy,” J. Am. Chem. Soc. 134, 4264–4268 (2012).
[CrossRef]

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, 121402(2011).
[CrossRef]

B. Huo, X. Wang, S. Chang, M. Zeng, and G. Zhao, “Second harmonic generation of individual centrosymmetric sphere excited by a tightly focused beam,” J. Opt. Soc. Am. B 28, 2702–2711 (2011).
[CrossRef]

2010

H. Chen, Z. Zheng, B.-F. Zhang, J. Ding, and H.-T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett. 35, 2825–2827 (2010).
[CrossRef]

J. Butet, G. Bachelier, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Three-dimensional mapping of single gold nanoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express 18, 22314–22323 (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. C 114, 4302–4308 (2010).
[CrossRef]

F. X. Wang, F. J. Rodríguez, W. M. Albers, and M. Kauranen, “Enhancement of bulk-type multipolar second-harmonic generation arising from surface morphology of metals,” New J. Phys. 12, 063009 (2010).
[CrossRef]

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

2009

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).
[CrossRef]

S. Roke, “Nonlinear optical spectroscopy of soft matter interfaces,” Chem. Phys. Chem. 10, 1380–1388 (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, 095502 (2009).
[CrossRef]

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

M. Hashimoto, K. Ashida, K. Yoshiki, and T. Araki, “Enhancement of second-harmonic generation from self-assembled monolayers on gold by excitation with a radially polarized beam,” Opt. Lett. 34, 1423–1425 (2009).
[CrossRef]

2008

2007

A. G. F. de Beer and S. Roke, “Sum frequency generation scattering from the interface of an isotropic particle: geometrical and chiral effects,” Phys. Rev. B 75, 245438 (2007).
[CrossRef]

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

2006

S. H. Jen, and H. L. Dai, “Probing molecules adsorbed at the surface of nanometer colloidal particles by optical second-harmonic generation,” J. Phys. Chem. B 110, 23000–23003 (2006).
[CrossRef]

A. Yamaguchi, M. Nakano, K. Nochi, T. Yamashita, K. Morita, and N. Teramae, “Longitudinal diffusion behavior of hemicyanine dyes across phospholipid vesicle membranes as studied by second-harmonic generation and fluorescence spectroscopies,” Anal. Bioanal. Chem. 386, 627–632 (2006).
[CrossRef]

J. S. Salafsky, “Detection of protein conformational change by optical second-harmonic generation,” J. Chem. Phys. 125, 074701 (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, 023819 (2006).
[CrossRef]

2005

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán, and B. S. Mendoza, “Single-beam and enhanced two-beam second-harmonic generation from silicon nanocrystals by use of spatially inhomogeneous femtosecond pulses,” Phys. Rev. Lett. 94, 047401 (2005).
[CrossRef]

2004

R. Bernal and J. A. Maytorena, “Second harmonic generation from centrosymmetric thin films by a focused beam with arbitrary transverse structure,” Phys. Rev. B 70, 125420 (2004).
[CrossRef]

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: the concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
[CrossRef]

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (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, 1328–1347(2004).
[CrossRef]

2003

A. Podlipensky, J. Lange, G. Seifert, H. Graener, and I. Cravetchi, “Second-harmonic generation from ellipsoidal silver nanoparticles embedded in silica glass,” Opt. Lett. 28, 716–718 (2003).
[CrossRef]

D. P. Biss and T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
[CrossRef]

W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 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, 258302 (2003).
[CrossRef]

2002

2001

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett. 78, 766–768 (2001).
[CrossRef]

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
[CrossRef]

J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. 196, 325–330 (2001).
[CrossRef]

2000

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef]

V. L. Brudny, B. S. Mendoza, and W. Luis Mochán, “Second-harmonic generation from spherical particles,” Phys. Rev. B 62, 11152–11162 (2000).
[CrossRef]

1999

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

1998

E. Hendrickx, K. Clays, and A. Persoons, “Hyper-Rayleigh scattering in isotropic solution,” Acc. Chem. Res. 31, 675–683 (1998).
[CrossRef]

1997

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

1996

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, 15–20 (1996).
[CrossRef]

1987

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

1986

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

1985

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

1968

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

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, 233402 (2009).
[CrossRef]

Albers, W. M.

F. X. Wang, F. J. Rodríguez, W. M. Albers, and M. Kauranen, “Enhancement of bulk-type multipolar second-harmonic generation arising from surface morphology of metals,” New J. Phys. 12, 063009 (2010).
[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, 233402 (2009).
[CrossRef]

Angerer, W. E.

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
[CrossRef]

Araki, T.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier Academic, 2005).

Ashida, K.

Bachelier, G.

J. Butet, G. Bachelier, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Three-dimensional mapping of single gold nanoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express 18, 22314–22323 (2010).
[CrossRef]

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

Benichou, E.

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. 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, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Three-dimensional mapping of single gold nanoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express 18, 22314–22323 (2010).
[CrossRef]

Bernal, R.

R. Bernal and J. A. Maytorena, “Second harmonic generation from centrosymmetric thin films by a focused beam with arbitrary transverse structure,” Phys. Rev. B 70, 125420 (2004).
[CrossRef]

Bertorelle, F.

Biss, D. P.

Bloembergen, N.

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

Bloomfield, A.

Y. You, A. Bloomfield, J. Liu, L. Fu, S. B. Herzon, and E. C. Y. Yan, “Real-time kinetics of surfactant molecule transfer between emulsion particles probed by in situ second harmonic generation spectroscopy,” J. Am. Chem. Soc. 134, 4264–4268 (2012).
[CrossRef]

Bonn, M.

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: the concept of effective susceptibility,” Phys. Rev. B 70, 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, 258302 (2003).
[CrossRef]

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, 15–20 (1996).
[CrossRef]

Brevet, P.

J. Butet, G. Bachelier, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Three-dimensional mapping of single gold nanoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express 18, 22314–22323 (2010).
[CrossRef]

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

Brown, T.

Brown, T. G.

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. B 68, 085318 (2003).
[CrossRef]

V. L. Brudny, B. S. Mendoza, and W. Luis Mochán, “Second-harmonic generation from spherical particles,” Phys. Rev. B 62, 11152–11162 (2000).
[CrossRef]

Butet, J.

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. 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, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Three-dimensional mapping of single gold nanoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express 18, 22314–22323 (2010).
[CrossRef]

Chang, R. K.

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

Chang, S.

Chen, H.

Chen, W.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

Clays, K.

E. Hendrickx, K. Clays, and A. Persoons, “Hyper-Rayleigh scattering in isotropic solution,” Acc. Chem. Res. 31, 675–683 (1998).
[CrossRef]

Corbalán, R.

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J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
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P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán, and B. S. Mendoza, “Single-beam and enhanced two-beam second-harmonic generation from silicon nanocrystals by use of spatially inhomogeneous femtosecond pulses,” Phys. Rev. Lett. 94, 047401 (2005).
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P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán, and B. S. Mendoza, “Single-beam and enhanced two-beam second-harmonic generation from silicon nanocrystals by use of spatially inhomogeneous femtosecond pulses,” Phys. Rev. Lett. 94, 047401 (2005).
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Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (2004).
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E. Hendrickx, K. Clays, and A. Persoons, “Hyper-Rayleigh scattering in isotropic solution,” Acc. Chem. Res. 31, 675–683 (1998).
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Radenovic, A.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
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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, 023819 (2006).
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Rodríguez, F. J.

F. X. Wang, F. J. Rodríguez, W. M. Albers, and M. Kauranen, “Enhancement of bulk-type multipolar second-harmonic generation arising from surface morphology of metals,” New J. Phys. 12, 063009 (2010).
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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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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, 258302 (2003).
[CrossRef]

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A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (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, 095502 (2009).
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[CrossRef]

A. G. F. de Beer and S. Roke, “Sum frequency generation scattering from the interface of an isotropic particle: geometrical and chiral effects,” Phys. Rev. B 75, 245438 (2007).
[CrossRef]

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: the concept of effective susceptibility,” Phys. Rev. B 70, 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, 258302 (2003).
[CrossRef]

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J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. 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, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Three-dimensional mapping of single gold nanoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express 18, 22314–22323 (2010).
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J. S. Salafsky, “Detection of protein conformational change by optical second-harmonic generation,” J. Chem. Phys. 125, 074701 (2006).
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Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Seifert, G.

Shan, J.

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, 023819 (2006).
[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, 1328–1347(2004).
[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, 4045–4048 (1999).
[CrossRef]

Shen, Y. R.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

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. B 80, 233402 (2009).
[CrossRef]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

Stegeman, G. I.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

Stiopkin, I.

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, 023819 (2006).
[CrossRef]

Sun, L.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán, and B. S. Mendoza, “Single-beam and enhanced two-beam second-harmonic generation from silicon nanocrystals by use of spatially inhomogeneous femtosecond pulses,” Phys. Rev. Lett. 94, 047401 (2005).
[CrossRef]

Teramae, N.

A. Yamaguchi, M. Nakano, K. Nochi, T. Yamashita, K. Morita, and N. Teramae, “Longitudinal diffusion behavior of hemicyanine dyes across phospholipid vesicle membranes as studied by second-harmonic generation and fluorescence spectroscopies,” Anal. Bioanal. Chem. 386, 627–632 (2006).
[CrossRef]

van Driel, H. M.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

Vidal, X.

Vilaseca, R.

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

Wang, F. X.

F. X. Wang, F. J. Rodríguez, W. M. Albers, and M. Kauranen, “Enhancement of bulk-type multipolar second-harmonic generation arising from surface morphology of metals,” New J. Phys. 12, 063009 (2010).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Schematic diagram of the discussed problem. (a) A CVB is focused by an infinity-corrected objective L and a centrosymmetric nanosphere in the focal region emits SH radiation excited by the focused field. The origin O of the Cartesian coordinate system and the center of the particle are taken at the focus. α is the maximal angle determined by the NA of the objective L . k⃗ and K⃗ are the wave vectors of the fundamental incident beam and the scattered SH radiation, respectively. r⃗ and r⃗ f are the position vectors in the particle and the detection point. (b) The polarization distribution of a CVB in the transversal plane shown by a dotted line in (a).

Fig. 2.
Fig. 2.

Illustrations of the cylindrical coordinate (red), the Cartesian coordinate (green), and the spherical coordinate (blue). The crystallographic axes are assumed to coincide with the laboratory axes (the Cartesian axes).

Fig. 3.
Fig. 3.

Surface SH radiation patterns of a centrosymmetric nanosphere excited by focused CVBs with different ϕ 0 . (a)  χ s , = 1 , χ s , = χ s , = 0 , (b)  χ s , = 1 , χ s , = χ s , = 0 , (c)  χ s , = 1 , χ s , = χ s , = 0 , (d)  χ s , = 59 , χ s , = 3.8 , χ s , = 7.9 in units of 10 22 m 2 / V . All the radiation patterns are normalized relative to the peak magnitude of the SH radiation power per unit solid angle stemming from χ s , .

Fig. 4.
Fig. 4.

Field distributions of focused CVBs with different ϕ 0 in the focal plane when NA = 1.4 . The circle denotes the location of the particle in the focal plane. The scale bars denote the order of magnitude in arbitrary units.

Fig. 5.
Fig. 5.

Surface SH polarizations stemming from different surface susceptibility elements. (a)  χ s , = 1 , χ s , = χ s , = 0 , (b)  χ s , = 1 , χ s , = χ s , = 0 , (c)  χ s , = 1 , χ s , = χ s , = 0 . The black arrows denote the directions of the nonlinear polarization at the local surface.

Fig. 6.
Fig. 6.

Bulk SH radiation patterns of a single centrosymmetric nanosphere excited by focused CVBs with different ϕ 0 . (a)  γ = 1 , δ = ζ = 0 , (b)  δ = 1 , γ = ζ = 0 , (c)  ζ = 1 , γ = δ = 0 . The scale bars denote the order of magnitude in arbitrary units.

Fig. 7.
Fig. 7.

Intensity distributions of the elements of E⃗ of focused CVBs with different ϕ 0 in the focal plane when NA = 1.4 . The scale bars denote the order of magnitude in arbitrary units. The circle denotes the location of the nanosphere in the focal plane.

Fig. 8.
Fig. 8.

Distributions of the bulk SH polarization stemming from different bulk parameters in the interior of a nanosphere with diameter of 100 nm.

Equations (15)

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P⃗ ( 2 ω ) ( r⃗ ) = χ s : E⃗ ( ω ) ( r⃗ ) δ ( r h ( r ) ) + χ b : E⃗ ( ω ) ( r⃗ ) E⃗ ( ω ) ( r⃗ ) ,
P⃗ b , i ( 2 ω ) = γ i ( E⃗ ( ω ) · E⃗ ( ω ) ) + δ ( E⃗ ( ω ) · ) E i ( ω ) ( r⃗ ) + ζ E i ( ω ) i E i ( ω ) ,
γ = χ q , i j i j ( 2 ) / 2 , δ = χ q , i j j i ( 2 ) 2 χ q , i i j j ( 2 ) 2 χ q , i j i j ( 2 ) , ζ = χ q , i i i i ( 2 ) ( χ q , i j j i ( 2 ) + χ q , i i j j ( 2 ) + χ q , i j i j ( 2 ) ) .
A⃗ ( 2 ω ) = K i exp ( i K | r⃗ f r⃗ | ) | r⃗ f r⃗ | P⃗ ( 2 ω ) ( r⃗ ) d r⃗ .
E ρ ( ρ , z ) = 2 A cos ϕ 0 I 1 0 , 1 , E ϕ ( ρ , z ) = 2 A sin ϕ 0 I 1 0 , 0 , E z ( ρ , z ) = 2 i A cos ϕ 0 I 1 1 , 0 ,
I m u , v = 0 a l ( θ ) cos θ sin θ ( sin u θ ) J m ( k ρ sin θ ) e i k z cos θ d θ ,
E ρ = 2 A cos ϕ 0 I 1 0 , 1 = 2 A cos ϕ 0 [ ρ ^ ( k I 0 1 , 1 1 ρ I 1 0 , 1 ) + i k z ^ I 1 0 , 2 ] , E ϕ = 2 A sin ϕ 0 I 1 0 , 0 = 2 A sin ϕ 0 { ρ ( k I 0 1 , 0 I 1 0 , 0 / ρ ) + z ^ i k I 1 0 , 1 } , E z = 2 i A cos ϕ 0 I 0 1 , 0 = 2 i A cos ϕ 0 ( ρ k I 1 2 , 0 + z ^ i k I 1 1 , 1 ) .
P⃗ s ( 2 ω ) = [ χ s , E r E r + χ s , ( E θ 2 + E ϕ 2 ) ] r ^ + χ s , E r ( E θ θ ^ + E ϕ ϕ ^ ) .
( E r E θ E ϕ ) = ( sin θ 0 cos θ cos θ 0 sin θ 0 1 1 ) ( E ρ E ϕ E z ) .
P⃗ s ( 2 ω ) = [ χ s , E r E r + χ s , ( E θ 2 + E ϕ 2 ) ] r ^ + χ s , E r ( E θ θ ^ + E ϕ ϕ ^ ) = { χ s , ( sin θ E ρ + cos θ E z ) 2 + χ s , [ ( cos θ E ρ + sin θ E z ) 2 + E ϕ 2 ] } r ^ + 2 χ s , ( sin θ E ρ + cos θ E z ) [ ( cos θ E ρ + sin θ E z ) θ ^ + E ϕ ϕ ^ ] .
= ρ ^ ρ + ϕ ^ 1 ρ ϕ + z ^ z .
( E ρ E ϕ E z ) = ( cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) ( E x E y E z ) .
P⃗ b , γ ( 2 ω ) = γ ( E⃗ · E⃗ ) = 2 ( E ρ + E ϕ + E z ) ( ρ ^ ρ + z ^ z ) ( E ρ + E ϕ + E z ) , P⃗ b , δ ( 2 ω ) = δ ( E⃗ ( r⃗ ) · ) E⃗ = ( E ρ ρ + E z z ) ( E ρ ρ ^ + E ϕ ϕ ^ + E z z ^ ) , P⃗ b , ζ ( 2 ω ) = ρ ^ { sin ϕ cos ϕ [ ( sin ϕ cos ϕ ) ( E ρ E ϕ ρ E ϕ E ρ ρ ) ( sin ϕ + cos ϕ ) ( E ρ E ρ ρ E ϕ E ϕ ρ ) ] + ( sin ϕ + cos ϕ ) E ρ E ρ ρ } + 1 ρ ϕ ^ { sin ϕ cos ϕ [ 2 ( sin ϕ cos ϕ ) E ρ E ϕ + ( sin ϕ + cos ϕ ) ( E ρ E ρ E ϕ E ϕ ) ] ( sin ϕ cos ϕ ) E ρ E ϕ } + z ^ E z E z z .
P⃗ b , γ ( 2 ω ) = 2 ( E ρ E z ) ( ρ ^ ρ z ^ z ) ( E ρ + E z ) , P⃗ b , δ ( 2 ω ) = ( E ρ ρ + E z z ) ( E ρ ρ ^ + E z z ^ ) , P⃗ b , ζ ( 2 ω ) = ρ ^ ( sin 3 ϕ + cos 3 ϕ ) E ρ E ρ ρ + 1 ρ ϕ ^ sin ϕ cos ϕ ( sin ϕ + cos ϕ ) E ρ E ρ + z ^ E z E z z .
P⃗ b , γ ( 2 ω ) = γ ( E⃗ · E⃗ ) = 2 γ E ϕ ( ρ ^ ρ + z ^ z ) E ϕ , P⃗ b , δ ( 2 ω ) = 0 , P⃗ b , ζ ( 2 ω ) = ζ sin ϕ cos ϕ ( sin ϕ + cos ϕ ) E ϕ ( ρ ^ E ϕ ρ ϕ ^ 1 ρ E ϕ ) .

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