Abstract

Based on vector diffraction theory, an analytical integral representation to calculate the field gradient of focused electromagnetic fields is derived by using the differential recursion formula between the Bessel functions with different order. Within the phenomenological model for second harmonic generation (SHG) of centrosymmetric material, the second harmonic (SH) response of a single centrosymmetric spherical particle excited by a focused beam under different values of the NA is investigated theoretically. The results show that, with increasing NA, the SH radiation pattern of the surface response hardly changes. For the larger value of the NA, because the elements of the field gradient are almost the same order of magnitude and the relative magnitudes of Ey and Ez increase, the bulk response related to the parameters δ and ζ become remarkable.

© 2011 Optical Society of America

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  1. 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]
  2. F. Brown and M. Matsuoka, “Effect of adsorbed surface layers on second-harmonic light from silver,” Phys. Rev. 185, 985–987(1969).
    [CrossRef]
  3. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
    [CrossRef]
  4. M. Jacobsohn and U. Banin, “Size dependence of second harmonic generation in CdSe nanocrystal quantum dots,” J. Phys. Chem. B 104, 1–5 (2000).
    [CrossRef]
  5. 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, 534–540(2002).
    [CrossRef]
  6. 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, 201402 (2005).
    [CrossRef]
  7. B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
    [CrossRef] [PubMed]
  8. 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]
  9. 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] [PubMed]
  10. X. Vidal, A. Fedyanin, A. Molinos-Gómez, S. Rao, J. Martorell, and D. Petrov, “Nonlinear optical response from single spheres coated by a nonlinear monolayer,” Opt. Lett. 33, 699–701(2008).
    [CrossRef] [PubMed]
  11. 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]
  12. J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
    [CrossRef]
  13. V. L. Brudny, B. S. Mendoza, and W. L. Mochán, “Second-harmonic generation from spherical particles,” Phys. Rev. B 62, 11152–11162 (2000).
    [CrossRef]
  14. W. L. Mochán, J. A. Maytorena, and B. S. Mendoza, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
    [CrossRef]
  15. 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]
  16. Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
    [CrossRef] [PubMed]
  17. 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, 1098–1102 (2007).
    [CrossRef] [PubMed]
  18. X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. 39, 747–753 (2007).
    [CrossRef] [PubMed]
  19. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  20. E. Y. S. Yew and C. J. R. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express 14, 1167–1174 (2006).
    [CrossRef] [PubMed]
  21. P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]
  22. 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]
  23. 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]
  24. S. Roke, “Nonlinear optical spectroscopy of soft matter interfaces,” Chem. Phys. Chem. 10, 1380–1388 (2009).
    [CrossRef] [PubMed]
  25. T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.Ponath and G.Stegeman, eds. (Elsevier, 1991) pp. 353–416.
  26. G. B. Arfken and H. J. Weber, “Bessel functions,” in Mathematical Methods for Physicists (Academic, 2005) pp. 678–739.
  27. F. J. Rodríguez, F. X. Wang, and M. Kauranen, “Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass,” Opt. Express 16, 8704–8710 (2008).
    [CrossRef] [PubMed]
  28. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87(2000).
    [CrossRef] [PubMed]

2010

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]

2009

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

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]

Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
[CrossRef] [PubMed]

2008

2007

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, 1098–1102 (2007).
[CrossRef] [PubMed]

X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. 39, 747–753 (2007).
[CrossRef] [PubMed]

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

2006

2005

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

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, 201402 (2005).
[CrossRef]

2004

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]

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]

2003

W. L. Mochán, J. A. Maytorena, and B. S. Mendoza, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[CrossRef]

2002

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, 534–540(2002).
[CrossRef]

2001

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] [PubMed]

2000

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

M. Jacobsohn and U. Banin, “Size dependence of second harmonic generation in CdSe nanocrystal quantum dots,” J. Phys. Chem. B 104, 1–5 (2000).
[CrossRef]

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

1989

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[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]

1969

F. Brown and M. Matsuoka, “Effect of adsorbed surface layers on second-harmonic light from silver,” Phys. Rev. 185, 985–987(1969).
[CrossRef]

1959

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[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] [PubMed]

Arfken, G. B.

G. B. Arfken and H. J. Weber, “Bessel functions,” in Mathematical Methods for Physicists (Academic, 2005) pp. 678–739.

Bai, B.

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

Banin, U.

M. Jacobsohn and U. Banin, “Size dependence of second harmonic generation in CdSe nanocrystal quantum dots,” J. Phys. Chem. B 104, 1–5 (2000).
[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]

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]

Brown, F.

F. Brown and M. Matsuoka, “Effect of adsorbed surface layers on second-harmonic light from silver,” Phys. Rev. 185, 985–987(1969).
[CrossRef]

Brown, T. G.

Brudny, V. L.

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

Canfield, B. K.

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

Cheng, J.

Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
[CrossRef] [PubMed]

Dadap, J. I.

Downer, M. C.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Eisenthal, K. B.

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]

El-Sayed, I. H.

X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. 39, 747–753 (2007).
[CrossRef] [PubMed]

El-Sayed, M. A.

X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. 39, 747–753 (2007).
[CrossRef] [PubMed]

Fedyanin, A.

Figliozzi, P.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Heinz, T. F.

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]

T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.Ponath and G.Stegeman, eds. (Elsevier, 1991) pp. 353–416.

Huang, X.

X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. 39, 747–753 (2007).
[CrossRef] [PubMed]

Hupp, J. T.

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, 534–540(2002).
[CrossRef]

Husu, H.

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

Jacobsohn, M.

M. Jacobsohn and U. Banin, “Size dependence of second harmonic generation in CdSe nanocrystal quantum dots,” J. Phys. Chem. B 104, 1–5 (2000).
[CrossRef]

Jiang, Y.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Johnson, R. C.

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, 534–540(2002).
[CrossRef]

Jung, Y.

Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
[CrossRef] [PubMed]

Kauranen, 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]

F. J. Rodríguez, F. X. Wang, and M. Kauranen, “Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass,” Opt. Express 16, 8704–8710 (2008).
[CrossRef] [PubMed]

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

Kuittinen, M.

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

Laukkanen, J.

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

Li, J.

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, 534–540(2002).
[CrossRef]

Liphardt, J.

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Litwin, J. A.

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]

Martorell, J.

Matlis, N.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Matsuoka, M.

F. Brown and M. Matsuoka, “Effect of adsorbed surface layers on second-harmonic light from silver,” Phys. Rev. 185, 985–987(1969).
[CrossRef]

Mattern, B.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Maytorena, J. A.

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]

W. L. Mochán, J. A. Maytorena, and B. S. Mendoza, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[CrossRef]

Mendoza, B. S.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

W. L. Mochán, J. A. Maytorena, and B. S. Mendoza, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[CrossRef]

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

Mochan, W. L.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Mochán, W. L.

W. L. Mochán, J. A. Maytorena, and B. S. Mendoza, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[CrossRef]

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

Molinos-Gómez, A.

Nakayama, Y.

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Neacsu, C. C.

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, 201402 (2005).
[CrossRef]

Onorato, R. M.

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Pauzauskie, P. J.

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Petrov, D.

Qian, W.

X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. 39, 747–753 (2007).
[CrossRef] [PubMed]

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Rao, S.

Raschke, M. B.

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, 201402 (2005).
[CrossRef]

Reider, G. A.

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, 201402 (2005).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

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).
[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]

F. J. Rodríguez, F. X. Wang, and M. Kauranen, “Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass,” Opt. Express 16, 8704–8710 (2008).
[CrossRef] [PubMed]

Roke, S.

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

Saykally, R. J.

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Schatz, G. C.

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, 534–540(2002).
[CrossRef]

Shan, J.

Shen, Y. R.

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

Sheppard, C. J. R.

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. 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]

Sun, L.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Tanaudommongkon, A.

Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
[CrossRef] [PubMed]

Tong, L.

Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
[CrossRef] [PubMed]

Turunen, J.

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

van Driel, H. M.

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.

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]

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]

F. J. Rodríguez, F. X. Wang, and M. Kauranen, “Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass,” Opt. Express 16, 8704–8710 (2008).
[CrossRef] [PubMed]

Wang, H.

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]

Weber, H. J.

G. B. Arfken and H. J. Weber, “Bessel functions,” in Mathematical Methods for Physicists (Academic, 2005) pp. 678–739.

White, C. W.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Withrow, S. P.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Yan, E. C. Y.

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]

Yang, C.

Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
[CrossRef] [PubMed]

Yang, N.

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] [PubMed]

Yang, P.

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Yew, E. Y. S.

Yodh, A. G.

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] [PubMed]

Youngworth, K. S.

Chem. Phys. Chem.

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

Chem. Phys. Lett.

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, 534–540(2002).
[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, 15–20(1996).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem. B

M. Jacobsohn and U. Banin, “Size dependence of second harmonic generation in CdSe nanocrystal quantum dots,” J. Phys. Chem. B 104, 1–5 (2000).
[CrossRef]

Lasers Surg. Med.

X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. 39, 747–753 (2007).
[CrossRef] [PubMed]

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 noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[CrossRef] [PubMed]

Y. Jung, L. Tong, A. Tanaudommongkon, J. Cheng, and C. Yang, “In vitro and In vivo nonlinear optical imaging of silicon nanowires,” Nano Lett. 9, 2440–2444 (2009).
[CrossRef] [PubMed]

Nature

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, 1098–1102 (2007).
[CrossRef] [PubMed]

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

New J. Phys.

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]

Opt. Express

Opt. Lett.

Phys. Rev.

F. Brown and M. Matsuoka, “Effect of adsorbed surface layers on second-harmonic light from silver,” Phys. Rev. 185, 985–987(1969).
[CrossRef]

Phys. Rev. B

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]

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, 201402 (2005).
[CrossRef]

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

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

W. L. Mochán, J. A. Maytorena, and B. S. Mendoza, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[CrossRef]

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]

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]

Phys. Rev. Lett.

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochan, 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] [PubMed]

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] [PubMed]

Proc. R. Soc. A

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Other

T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.Ponath and G.Stegeman, eds. (Elsevier, 1991) pp. 353–416.

G. B. Arfken and H. J. Weber, “Bessel functions,” in Mathematical Methods for Physicists (Academic, 2005) pp. 678–739.

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of SH response from a spherical particle excited by a focused beam. An x-polarization plane wave is focused by to an aplanatic lens L, and the emergent light illuminates a nanoparticle. The origin O of the Cartesian coordinate system and the center of the particle are taken at the beam focus. k and K are the wave vectors of the fundamental incident beam and the scattered SH radiation, respectively. r f and r p are the position vectors in the detection point and the particle. α is the maximal angle determined by the NA of the illumination objective.

Fig. 2
Fig. 2

Surface SH radiation patterns of a spherical particle excited by a focused beam with different NA. The radius of the particle is 150 nm and the center of this particle is located at the beam focus. The fundamental wavelength is 800 nm and the incident light is linearly polarized along the x axis. The inset axes shows the laboratory frame.

Fig. 3
Fig. 3

(a)–(c) Normalized intensities of the different field components in the focal plane when NA = 0.9 and (d)–(f) the distributions of the surface nonlinear polarization in the focal plane for different values of the NA. (a)  | E x | , (b)  | E y | , (c)  | E z | , (d)  NA = 0.01 , (e)  NA = 0.5 , (f)  NA = 0.9 . The incident light is linearly polarized along the x axis. The circles in (a)–(c) denote the location of the particle in the focal plane.

Fig. 4
Fig. 4

Surface SH radiation patterns corresponding to each of the surface nonlinear susceptibility elements. (a)  χ , (b)  χ , (c)  χ .

Fig. 5
Fig. 5

Bulk SH radiation patterns corresponding to the parameters γ and δ of a single isotropic centrosymmetric nanoparticle excited by a focused beam with different NA. The scale bars denote the order of magnitude in arbitrary units.

Fig. 6
Fig. 6

Intensity distributions of the elements of E in the focal plane when (a)  NA = 0.01 and (b)  NA = 0.9 , respectively. The incident light is linearly polarized along the x axis. The scale bars denote the order of magnitude in arbitrary units. The circle denotes the location of the nanoparticle in the focal plane.

Fig. 7
Fig. 7

SH radiation patterns induced by each of the elements of P γ when NA = 0.01 and 0.9, respectively. (a)  E x E x x in P γ , x , (b)  E y E y x in P γ , x , (c)  E z E z x in P γ , x , (d)  E x E x y in P γ , y , (e)  E y E y y in P γ , y , (f)  E z E z y in P γ , y , (g)  E x E x z in P γ , z , (h)  E y E y z in P γ , z , (i)  E z E z z in P γ , z . The scale bars denote the order of magnitude in arbitrary units.

Fig. 8
Fig. 8

SH radiation patterns induced by each of the elements of P δ when NA = 0.01 and 0.9, respectively. (a)  E x E x x in P δ , x , (b)  E y E x y in P δ , x , (c)  E z E x z in P δ , x , (d)  E x E y x in P δ , y , (e)  E y E y y in P δ , y , (f)  E z E y z in P δ , y , (g)  E x E z x in P δ , z , (h)  E y E z y in P δ , z , (i)  E z E z z in P δ , z . The scale bars denote the order of magnitude in arbitrary units.

Fig. 9
Fig. 9

(a)–(c) Total radiation patterns of the bulk SH response of a cubic centrosymmetric nanoparticle excited by a focused beam with different NA, and (d) the SH radiation pattern corresponding to ζ when NA = 0.9 .

Equations (16)

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P ( 2 ω ) ( r p ) = P e ( 2 ω ) + P s ( 2 ω ) + P b ( 2 ω ) = χ ( 2 ) : E ( r p ) E ( r p ) + χ s ( 2 ) : E ( r p ) E ( r p ) δ ( r p h ( r p ) ) + χ b ( 2 ) E ( r p ) E ( r p ) .
P s , i ( 2 ω ) ( r p ) = δ ( r p a ) j m E j E m ( δ j i χ s m + δ j , m χ s i + χ s i s j s m ) ,
P b , i ( 2 ω ) ( r p ) = P γ , i + P δ , i + P ζ , i = γ i ( E ( r p ) · E ( r p ) ) + δ ( E ( r p ) · ) E i + ζ E i i E i ,
E x ( ω ) = i A ( I 0 + I 2 cos 2 ϕ p ) , E y ( ω ) = i A I 2 sin 2 ϕ p , E z ( ω ) = 2 A I 1 cos ϕ p ,
I 0 = 0 α cos θ sin θ ( 1 + cos θ ) J 0 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ , I 1 = 0 α cos θ sin 2 θ J 1 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ I 2 = 0 α cos θ sin θ ( 1 cos θ ) J 2 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ ,
d d x [ x n J n ( x ) ] = x n J n 1 ( x ) .
p I = ( p I 0 p I 1 p I 2 ) = ( I 0 r p I 1 r p I 2 r p 1 r p I 0 θ p 1 r p I 1 θ p 1 r p I 2 θ p 1 r p sin θ p I 0 ϕ p 1 r p sin θ p I 1 ϕ p 1 r p sin θ p I 2 ϕ p ) = k ( ( sin θ p I 1 + i cos θ p I 0 ' ) sin θ p I 0 ' ' + i cos θ p I 1 ' I 1 k r p sin θ p I 1 ' ' + i cos θ p I 2 ' 2 I 2 k r p ( cos θ p I 1 i sin θ p I 0 ' ) cos θ p I 0 ' ' i sin θ p I 1 ' I 1 cos θ p k r p sin θ p cos θ p I 1 ' ' i sin θ p I 2 ' 2 cos θ p I 2 k r p sin θ p 0 0 0 ) .
p E = ( E x x p E y x p E z x p E x y p E y y p E z y p E x z p E y z p E z z p ) = M ( E x r p E y r p E z r p 1 r p E x θ p 1 r p E y θ p 1 r p E z θ p 1 r p sin θ p E x ϕ p 1 r p sin θ p E y ϕ p 1 r p sin θ p E z ϕ p ) ,
p E = A k ( i [ cos ϕ p ( I 1 + cos 2 ϕ p I 1 ' ' ) 2 I 2 cos 3 ϕ p k r p sin θ p ] i ( cos ϕ p sin 2 ϕ p I 1 ' ' 2 sin 3 ϕ p I 2 k r p sin θ p ) 2 ( cos 2 ϕ p I 0 ' ' cos 2 ϕ p I 1 k r p sin θ p ) i [ sin ϕ p ( I 1 + cos 2 ϕ p I 1 ' ' ) 2 I 2 sin 3 ϕ p k r p sin θ p ] i ( sin ϕ p sin 2 ϕ p I 1 ' ' + 2 cos 3 ϕ p I 2 k r p sin θ p ) ( sin 2 ϕ p I 0 ' ' + 2 cos 2 ϕ p I 1 k r p sin θ p ) ( I 0 ' + I 2 ' cos 2 ϕ p ) sin 2 ϕ p I 2 ' 2 i I 1 ' cos ϕ p )
I 1 = 0 α cos θ sin θ ( 1 + cos θ ) sin θ J 1 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ , I 0 ' = 0 α cos θ sin θ ( 1 + cos θ ) cos θ J 0 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ I 0 ' ' = 0 α cos θ sin 3 θ J 0 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ I 1 ' = 0 α cos 3 θ sin 2 θ J 1 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ , I 1 ' ' = 0 α cos θ sin 2 θ ( 1 cos θ ) J 1 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ I 2 ' = 0 α cos 3 θ sin θ ( 1 cos θ ) J 2 ( k r p sin θ sin θ p ) e i k r p cos θ cos θ p d θ .
P s , x ( 2 ω ) ( r p ) = A 2 δ ( r p a ) cos ϕ p [ χ ( I 0 + I 2 cos 2 ϕ p ) Γ 1 + χ sin θ p Γ 2 + χ sin θ p cos 2 ϕ p Γ 1 2 ] , P s , y ( 2 ω ) ( r p ) = A 2 δ ( r p a ) sin ϕ p [ 2 χ I 2 cos 2 ϕ p Γ 1 + χ sin θ p Γ 2 + χ cos 2 ϕ p sin θ p Γ 1 2 ] , P s , z ( 2 ω ) ( r p ) = A 2 δ ( r p a ) [ 2 i χ I 1 cos 2 ϕ p Γ 1 + χ cos θ p Γ 2 + χ cos θ cos 2 ϕ p Γ 1 2 ] ,
Γ 1 = [ ( I 0 + I 2 ) sin θ p 2 i I 1 cos θ p ] Γ 2 = [ I 0 2 2 I 1 2 + I 2 2 2 ( I 1 2 I 0 I 2 ) cos 2 ϕ p ] .
P b , x ( 2 ω ) ( r p ) = 2 γ ( E x E x x + E y E y x + E z E z x ) + δ ( E x E x x + E y E x y + E z E x z ) + ζ E x E x x , P b , y ( 2 ω ) ( r p ) = 2 γ ( E x E x y + E y E y y + E z E z y ) + δ ( E x E y x + E y E y y + E z E y z ) + ζ E y E y y P b , z ( 2 ω ) ( r p ) = 2 γ ( E x E x z + E y E y z + E z E z z ) + δ ( E x E z x + E y E z y + E z E z z ) + ζ E z E z z .
P b , x ( 2 ω ) ( r p ) = A 2 k cos ϕ p { 4 γ cos 2 ϕ p I 1 2 C + 2 cos 2 ϕ p δ I 1 I 2 + [ γ + δ cos 2 ϕ p + ζ ( 1 + cos 4 ϕ p ) / 2 ] I 1 I 2 + ( δ + γ cos 2 ϕ p + ζ cos 2 ϕ p ) I 2 I 1 + I 0 ( γ + δ + ζ ) [ cos 2 ϕ p ( I 1 4 I 2 C ) + 2 I 2 C + I 1 ] 2 γ I 0 I 1 ( cos 2 ϕ p + 1 ) + 2 δ I 0 I 1 2 [ γ + δ + ζ ( cos 4 ϕ p 2 sin 2 ϕ p ) ] I 2 2 C } P b , y ( 2 ω ) ( r p ) = A 2 k { sin ϕ p { 2 γ I 0 I 1 ( 1 + cos 2 ϕ p ) ( I 0 + I 2 cos 2 ϕ p ) γ I 1 2 δ I 1 I 2 ( 1 + cos 2 ϕ p ) [ cos 2 ϕ p ( γ + δ ) + δ ] I 0 I 1 [ γ + δ ( 1 + cos 2 ϕ p ) + ζ ( 1 cos 4 ϕ p ) / 2 ] I 1 I + 2 [ I 2 + ( 2 cos 2 ϕ p + 1 ) I 0 ] ( γ + δ ) I 2 C 2 ζ I 2 2 C ( cos 2 ϕ p + cos 4 ϕ p ) } + 2 γ I 1 2 C ( cos ϕ p + cos 3 ϕ p ) } P b , z ( 2 ω ) ( r p ) = i A 2 k [ γ I 0 ( I 0 + cos 2 ϕ p I 2 ) + γ I 2 ( cos 2 ϕ p I 0 + I 2 ) δ I 0 ( 1 + cos 2 ϕ p ) ( I 0 + I 2 ) + δ I 1 I 2 C ( 1 + cos 4 ϕ p sin 4 ϕ p ) + 2 cos 2 ϕ p δ I 0 I 1 C ] ,
A ( 2 ω ) ( r f ) = 1 c exp ( i K | r f r p | ) | r f r p | J ( r p ) d r p = K i exp ( i K | r f r p | ) | r f r p | P ( 2 ω ) ( r p ) d r p .
H ( 2 ω ) ( r f ) = B ( 2 ω ) ( r f ) μ 0 = × A ( 2 ω ) ( r f ) μ 0 = i K μ 0 r ^ f × A ( 2 ω ) ( r f ) , E ( 2 ω ) ( r f ) = i ε s K × B ( 2 ω ) ( r f ) = K ε s r ^ f × ( r ^ f × A ( 2 ω ) ( r f ) ) .

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