Abstract

A numerical technique for studying the generation of second-harmonic radiation in the interaction of light with two-dimensional particles of arbitrary shape is described. The medium of which the particles are composed is assumed to be homogeneous and isotropic. For the special case of cylindrical particles the numerical results are compared with results obtained with a Mie-type theory. The numerical technique is then illustrated through calculations for particles of various shapes by use of a free-electron model for nonlinear polarization. Among other things, we have found that, when a symmetrical particle is illuminated along its axis of symmetry, there is no second-harmonic radiation along that axis and that s-polarized second-harmonic light can be generated only by a mixture of s- and p-polarized illumination. The effects that the departure from cylindrical shape has on the resonances were also studied.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
    [CrossRef]
  2. Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
    [CrossRef]
  3. G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
    [CrossRef]
  4. M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
    [CrossRef]
  5. F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
    [CrossRef]
  6. F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
    [CrossRef]
  7. N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
    [CrossRef]
  8. 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]
  9. J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
    [CrossRef]
  10. P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988).
    [CrossRef]
  11. B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999–5006 (1996).
    [CrossRef]
  12. B. S. Mendoza and W. L. Mochán, “Erratum: Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999 (1996).
    [CrossRef]
  13. G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
    [CrossRef]
  14. R. T. Deck and R. K. Grygier, “Surface-plasmon enhanced harmonic generation at a rough metal surface,” Appl. Opt. 23, 3202–3213 (1984).
    [CrossRef] [PubMed]
  15. J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
    [CrossRef]
  16. A. R. McGurn, V. M. Agranovich, and T. A. Leskova, “Weak-localization effects in the generation of second harmonics of light at a randomly rough vacuum-metal grating,” Phys. Rev. B 44, 11441–11456 (1991).
    [CrossRef]
  17. K. A. O’Donnell, R. Torre, and C. S. West, “Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface,” Opt. Lett. 21, 1738–1740 (1996).
    [CrossRef] [PubMed]
  18. K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surface,” Phys. Rev. B 55, 7985–7992 (1997).
    [CrossRef]
  19. K. A. O’Donnell and R. Torre, “Second-harmonic generation from strongly rough metal surfaces,” Opt. Commun. 138, 341–344 (1997).
    [CrossRef]
  20. M. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, “Multiple scattering effects in the second harmonic generation of light in reflection from a randomly rough metal surface,” Opt. Lett. 21, 1809–1811 (1996).
    [CrossRef] [PubMed]
  21. M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
    [CrossRef]
  22. X. M. Hua and J. I. Gersten, “Theory of second harmonic generation by small metal spheres,” Phys. Rev. B 33, 3756–3764 (1986).
    [CrossRef]
  23. D. Rogovin and T. P. Shen, “Microparticle surface-enhanced second-harmonic generation,” J. Opt. Soc. Am. B 5, 1886–1889 (1988).
    [CrossRef]
  24. K. Hayata and M. Koshiba, “Theory of surface-emitting second-harmonic generation from optically trapped microspheres,” Phys. Rev. A 46, 6104–1889 (1992).
    [CrossRef] [PubMed]
  25. J. Martorell, R. Vilaseca, and R. Crobalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
    [CrossRef]
  26. 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]
  27. V. L. Brudny, B. S. Mendoza, and W. L. Mochán, “Second-harmonic generation from spherical particles,” Phys. Rev. B 62, 11152–11162 (2000).
    [CrossRef]
  28. R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).
  29. J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 661–701.
  30. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), p. 10.
  31. D. Maystre, M. Neviere, and R. Reinisch, “Nonlinear polarisation inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
    [CrossRef]
  32. C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” submitted to J. Opt. Soc. Am. B.
  33. E. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), p. 638.
  34. A. Mendoza-Suárez and E. R. Méndez, “Light scattering by reentrant fractal surfaces,” Appl. Opt. 36, 3521–3531 (1997).
    [CrossRef]
  35. C. I. Valencia and R. A. Depine, “Resonant scattering of light by an open cylindrical cavity ruled on a highly conducting flat surface,” Opt. Commun. 159, 254–265 (1999).
    [CrossRef]
  36. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  37. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 364.
  38. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  39. S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.
  40. J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Non-regularly shaped plasmon resonant nanoparticle as localized light source for near field microscopy,” J. Microsc. (Oxford) 202, 60–65 (2000).
    [CrossRef]
  41. J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. 341, 1–6 (2001).
    [CrossRef]

2001 (2)

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. 341, 1–6 (2001).
[CrossRef]

2000 (2)

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

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Non-regularly shaped plasmon resonant nanoparticle as localized light source for near field microscopy,” J. Microsc. (Oxford) 202, 60–65 (2000).
[CrossRef]

1999 (4)

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]

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

C. I. Valencia and R. A. Depine, “Resonant scattering of light by an open cylindrical cavity ruled on a highly conducting flat surface,” Opt. Commun. 159, 254–265 (1999).
[CrossRef]

G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
[CrossRef]

1997 (5)

Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surface,” Phys. Rev. B 55, 7985–7992 (1997).
[CrossRef]

K. A. O’Donnell and R. Torre, “Second-harmonic generation from strongly rough metal surfaces,” Opt. Commun. 138, 341–344 (1997).
[CrossRef]

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

A. Mendoza-Suárez and E. R. Méndez, “Light scattering by reentrant fractal surfaces,” Appl. Opt. 36, 3521–3531 (1997).
[CrossRef]

1996 (4)

1995 (1)

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

1992 (1)

K. Hayata and M. Koshiba, “Theory of surface-emitting second-harmonic generation from optically trapped microspheres,” Phys. Rev. A 46, 6104–1889 (1992).
[CrossRef] [PubMed]

1991 (1)

A. R. McGurn, V. M. Agranovich, and T. A. Leskova, “Weak-localization effects in the generation of second harmonics of light at a randomly rough vacuum-metal grating,” Phys. Rev. B 44, 11441–11456 (1991).
[CrossRef]

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

1988 (2)

D. Rogovin and T. P. Shen, “Microparticle surface-enhanced second-harmonic generation,” J. Opt. Soc. Am. B 5, 1886–1889 (1988).
[CrossRef]

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988).
[CrossRef]

1986 (2)

X. M. Hua and J. I. Gersten, “Theory of second harmonic generation by small metal spheres,” Phys. Rev. B 33, 3756–3764 (1986).
[CrossRef]

D. Maystre, M. Neviere, and R. Reinisch, “Nonlinear polarisation inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

1985 (1)

J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

1984 (2)

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

R. T. Deck and R. K. Grygier, “Surface-plasmon enhanced harmonic generation at a rough metal surface,” Appl. Opt. 23, 3202–3213 (1984).
[CrossRef] [PubMed]

1972 (1)

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

1971 (1)

J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

1968 (1)

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]

1966 (2)

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[CrossRef]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

1965 (1)

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

Agranovich, V. M.

A. R. McGurn, V. M. Agranovich, and T. A. Leskova, “Weak-localization effects in the generation of second harmonics of light at a randomly rough vacuum-metal grating,” Phys. Rev. B 44, 11441–11456 (1991).
[CrossRef]

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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

Brown, F.

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[CrossRef]

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

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]

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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

Christy, R. W.

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

Coutaz, J. L.

J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

Crobalán, R.

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

Dadap, J. I.

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]

Deck, R. T.

Depine, R. A.

C. I. Valencia and R. A. Depine, “Resonant scattering of light by an open cylindrical cavity ruled on a highly conducting flat surface,” Opt. Commun. 159, 254–265 (1999).
[CrossRef]

Downer, M. C.

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

Eisenthal, K. B.

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]

Farias, G. A.

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

Gavrilenko, V. I.

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

Gersten, J. I.

X. M. Hua and J. I. Gersten, “Theory of second harmonic generation by small metal spheres,” Phys. Rev. B 33, 3756–3764 (1986).
[CrossRef]

Grygier, R. K.

Guyot-Sionnest, P.

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988).
[CrossRef]

Hayata, K.

K. Hayata and M. Koshiba, “Theory of surface-emitting second-harmonic generation from optically trapped microspheres,” Phys. Rev. A 46, 6104–1889 (1992).
[CrossRef] [PubMed]

Heinz, T. F.

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]

Hua, X. M.

X. M. Hua and J. I. Gersten, “Theory of second harmonic generation by small metal spheres,” Phys. Rev. B 33, 3756–3764 (1986).
[CrossRef]

Jha, S. S.

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]

Johnson, P. B.

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

Koshiba, M.

K. Hayata and M. Koshiba, “Theory of surface-emitting second-harmonic generation from optically trapped microspheres,” Phys. Rev. A 46, 6104–1889 (1992).
[CrossRef] [PubMed]

Kottmann, J. P.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. 341, 1–6 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Non-regularly shaped plasmon resonant nanoparticle as localized light source for near field microscopy,” J. Microsc. (Oxford) 202, 60–65 (2000).
[CrossRef]

Lee, C. H.

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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

Leskova, T. A.

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

M. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, “Multiple scattering effects in the second harmonic generation of light in reflection from a randomly rough metal surface,” Opt. Lett. 21, 1809–1811 (1996).
[CrossRef] [PubMed]

A. R. McGurn, V. M. Agranovich, and T. A. Leskova, “Weak-localization effects in the generation of second harmonics of light at a randomly rough vacuum-metal grating,” Phys. Rev. B 44, 11441–11456 (1991).
[CrossRef]

Leyva-Lucero, M.

Leyva-Lucero, M. A.

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

Lu, J. Q.

Lüpke, G.

G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
[CrossRef]

Maradudin, A. A.

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

M. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, “Multiple scattering effects in the second harmonic generation of light in reflection from a randomly rough metal surface,” Opt. Lett. 21, 1809–1811 (1996).
[CrossRef] [PubMed]

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

Martin, O. J. F.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. 341, 1–6 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Non-regularly shaped plasmon resonant nanoparticle as localized light source for near field microscopy,” J. Microsc. (Oxford) 202, 60–65 (2000).
[CrossRef]

Martorell, J.

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

Maystre, D.

D. Maystre, M. Neviere, and R. Reinisch, “Nonlinear polarisation inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

McGilp, J. F.

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

McGurn, A. R.

A. R. McGurn, V. M. Agranovich, and T. A. Leskova, “Weak-localization effects in the generation of second harmonics of light at a randomly rough vacuum-metal grating,” Phys. Rev. B 44, 11441–11456 (1991).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Méndez, E. R.

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

A. Mendoza-Suárez and E. R. Méndez, “Light scattering by reentrant fractal surfaces,” Appl. Opt. 36, 3521–3531 (1997).
[CrossRef]

M. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, “Multiple scattering effects in the second harmonic generation of light in reflection from a randomly rough metal surface,” Opt. Lett. 21, 1809–1811 (1996).
[CrossRef] [PubMed]

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Mendoza, B. S.

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[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]

B. S. Mendoza and W. L. Mochán, “Erratum: Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999 (1996).
[CrossRef]

B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999–5006 (1996).
[CrossRef]

Mendoza-Suárez, A.

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Mochán, W. 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]

B. S. Mendoza and W. L. Mochán, “Erratum: Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999 (1996).
[CrossRef]

B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999–5006 (1996).
[CrossRef]

Neviere, M.

D. Maystre, M. Neviere, and R. Reinisch, “Nonlinear polarisation inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

O’Donnell, K. A.

K. A. O’Donnell and R. Torre, “Second-harmonic generation from strongly rough metal surfaces,” Opt. Commun. 138, 341–344 (1997).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surface,” Phys. Rev. B 55, 7985–7992 (1997).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface,” Opt. Lett. 21, 1738–1740 (1996).
[CrossRef] [PubMed]

Parks, R. E.

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[CrossRef]

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

Pic, E.

J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

Reinisch, R.

D. Maystre, M. Neviere, and R. Reinisch, “Nonlinear polarisation inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

Rogovin, D.

Rudnick, J.

J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

Schultz, S.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. 341, 1–6 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Non-regularly shaped plasmon resonant nanoparticle as localized light source for near field microscopy,” J. Microsc. (Oxford) 202, 60–65 (2000).
[CrossRef]

Shan, J.

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, T. P.

Shen, Y. R.

Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
[CrossRef]

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988).
[CrossRef]

Sleeper, A. M.

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

Smith, D. R.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. 341, 1–6 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Non-regularly shaped plasmon resonant nanoparticle as localized light source for near field microscopy,” J. Microsc. (Oxford) 202, 60–65 (2000).
[CrossRef]

Stern, E. A.

J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

Torre, R.

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surface,” Phys. Rev. B 55, 7985–7992 (1997).
[CrossRef]

K. A. O’Donnell and R. Torre, “Second-harmonic generation from strongly rough metal surfaces,” Opt. Commun. 138, 341–344 (1997).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface,” Opt. Lett. 21, 1738–1740 (1996).
[CrossRef] [PubMed]

Valencia, C. I.

C. I. Valencia and R. A. Depine, “Resonant scattering of light by an open cylindrical cavity ruled on a highly conducting flat surface,” Opt. Commun. 159, 254–265 (1999).
[CrossRef]

Vilaseca, R.

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

West, C. S.

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surface,” Phys. Rev. B 55, 7985–7992 (1997).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface,” Opt. Lett. 21, 1738–1740 (1996).
[CrossRef] [PubMed]

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. A (1)

D. Maystre, M. Neviere, and R. Reinisch, “Nonlinear polarisation inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[CrossRef]

Chem. Phys. Lett. (1)

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. 341, 1–6 (2001).
[CrossRef]

J. Microsc. (Oxford) (1)

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Non-regularly shaped plasmon resonant nanoparticle as localized light source for near field microscopy,” J. Microsc. (Oxford) 202, 60–65 (2000).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

K. A. O’Donnell and R. Torre, “Second-harmonic generation from strongly rough metal surfaces,” Opt. Commun. 138, 341–344 (1997).
[CrossRef]

C. I. Valencia and R. A. Depine, “Resonant scattering of light by an open cylindrical cavity ruled on a highly conducting flat surface,” Opt. Commun. 159, 254–265 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

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]

Phys. Rev. A (2)

K. Hayata and M. Koshiba, “Theory of surface-emitting second-harmonic generation from optically trapped microspheres,” Phys. Rev. A 46, 6104–1889 (1992).
[CrossRef] [PubMed]

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

Phys. Rev. B (11)

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

J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988).
[CrossRef]

B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999–5006 (1996).
[CrossRef]

B. S. Mendoza and W. L. Mochán, “Erratum: Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999 (1996).
[CrossRef]

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

X. M. Hua and J. I. Gersten, “Theory of second harmonic generation by small metal spheres,” Phys. Rev. B 33, 3756–3764 (1986).
[CrossRef]

J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

A. R. McGurn, V. M. Agranovich, and T. A. Leskova, “Weak-localization effects in the generation of second harmonics of light at a randomly rough vacuum-metal grating,” Phys. Rev. B 44, 11441–11456 (1991).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surface,” Phys. Rev. B 55, 7985–7992 (1997).
[CrossRef]

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

Phys. Rev. Lett. (4)

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[CrossRef]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[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]

Prog. Surf. Sci. (1)

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

Solid State Commun. (1)

Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
[CrossRef]

Surf. Interface Anal. (1)

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

Surf. Sci. Rep. (1)

G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
[CrossRef]

Other (7)

C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” submitted to J. Opt. Soc. Am. B.

E. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), p. 638.

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).

J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 661–701.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), p. 10.

S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 364.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Illustrative diagram of the local surface geometry.

Fig. 2
Fig. 2

Schematic diagram of the scattering geometry.

Fig. 3
Fig. 3

Illustration of the three kinds of particle considered in the calculations: particles with (a) circular geometrical cross section, (b) elliptical geometrical cross section, and (c) sinusoidally perturbed geometrical cross section.

Fig. 4
Fig. 4

Scattering cross section for a cylinder illuminated with p-polarized light. (a) Fundamental frequency, (b) second-harmonic frequency. R is the radius of the cylinder.

Fig. 5
Fig. 5

Scattering cross section for a cylinder illuminated with s-polarized light. (a) Fundamental frequency, (b) second-harmonic frequency. R is the radius of the cylinder.

Fig. 6
Fig. 6

Differential scattering cross sections for a cylinder illuminated with p-polarized light of wavelength λ=0.25 μm. (a) Fundamental frequency, (b) second-harmonic frequency. The analytical results are shown with solid curves and the numerical results are shown with filled circles. The radius of the cylinder is 0.5 μm.

Fig. 7
Fig. 7

Differential scattering cross sections for a cylinder illuminated with s-polarized light of wavelength λ=0.57 μm. (a) Fundamental frequency, (b) second-harmonic frequency. The analytical results are shown with solid curves, and the numerical results are shown with filled circles. The radius of the cylinder is 0.5 μm.

Fig. 8
Fig. 8

Scattering cross sections for particles of different shapes illuminated with p-polarized light. (a) Fundamental frequency, (b) second-harmonic frequency. The three kinds of profile depicted in Fig. 3 are considered. The parameters are R=0.5 μm, T=0.5 μm, θR=0°, h=0.05 μm, and f=5.

Fig. 9
Fig. 9

Scattering cross sections for particles with elliptical geometrical cross section and different orientations illuminated with p-polarized light. (a) Fundamental frequency, (b) second-harmonic frequency. The orientations considered are θR=0°, θR=45°, and θR=90°. The parameters of the ellipse are R and T.

Fig. 10
Fig. 10

Second-harmonic scattering cross sections for elliptical particles with different aspect ratios, illuminated with p-polarized light. Values of R and semiaxis T apply to both figures.

Fig. 11
Fig. 11

Second-harmonic differential scattering cross sections for elliptical particles with different orientations illuminated with p-polarized light of wavelength λ=1.0 μm. The parameters of the ellipse are R=0.5 μm and T=0.25 μm.

Fig. 12
Fig. 12

Scattering cross sections for nanoparticles with circular and elliptical geometrical cross sections illuminated with p-polarized light. (a) Fundamental frequency, (b) second-harmonic frequency. A cylinder of radius R and ellipses with values of R and T are shown. Orientations considered are θR=0°, θR=45°, and θR=90°.

Fig. 13
Fig. 13

Scattering cross sections for nanoparticles with circular and sinusoidally perturbed geometrical cross sections illuminated with p-polarized light. (a) Fundamental frequency, (b) second-harmonic frequency. A cylinder of radius R (solid curves) and a sinusoidally perturbed particle with h=1 nm (filled squares) are considered.

Equations (78)

Equations on this page are rendered with MathJax. Learn more.

×E(r|2ω)=2iωcH(r|2ω),
×H(r|2ω)=-2iωcD(r|2ω),
  D(r|2ω)=0,
  H(r|2ω)=0.
D(r|2ω)=II(2ω)E(r|2ω)+4πPNL(r|2ω),
PNL(r|2ω)=α[E(r|ω)  ]E(r|ω)+βE(r|ω)[  E(r|ω)]+γ[E(r|ω)  E(r|ω)],
Et(r|2ω)z=tEz(r|2ω)-2iωc[zˆ×Ht(r|2ω)],
Et(I)(rs|2ω)-Et(II)(rs|2ω)=-4πtPzs(rs|2ω),
Pzs(rs|2ω)=limτ0-ττPzNL(rs, z|2ω)II(z|2ω)dz.
Ht(I)(rs|2ω)-Ht(II)(rs|2ω)=4π2iωczˆ×Pts(rs|2ω),
Pts(rs|2ω)=limτ0-ττPtNL(rs, z|2ω)dz.
Pzs(rs|2ω)=χzzzs[Dz(rs|ω)]2+χztts[Et(rs|ω)  Et(rs|ω)],
Pts(rs|2ω)=χttzs[Et(rs|ω)Dz(rs|ω)],
E2(r|2ω)x3=-i 2ωc H1(r|2ω),
E2(r|2ω)x1=i 2ωc H3(r|2ω),
H3(r|2ω)x1-H1(r|2ω)x3=i 2ωc [II(2ω)E2(r|2ω)+4πP2NL(r|2ω)],
H2(r|2ω)x3=i 2ωc [II(2ω)E1(r|2ω)+4πP1NL(r|2ω)],
H2(r|2ω)x1=-i 2ωc [II(2ω)E3(r|2ω)+4πP3NL(r|2ω)],
E3(r|2ω)x1-E1(r|2ω)x3
=-i 2ωc H2(r|2ω)
rs(t)=[ξ(t), η(t)],
Z=[-η(t), ξ(t)],X=[ξ(t), η(t)],
Z=-η(t) x1+ξ(t) x3,
X=ξ(t) x1+η(t) x3.
F(r)Xr=rs=dF(t)dt,
ψs,p(R)(t|Ω)=ψs,p(R)(r|Ω)|r=rs,
Υs,p(R)(t|Ω)=ψs,p(R)(r|Ω)Zr=rs.
2x12+2x32+2ωc2II(2ω)ψs(II)(r|2ω)
=-4π2ωc2P2NL(r|2ω).
ψs(I)(t|2ω)-ψs(II)(t|2ω)=0,
Υs(I)(t|2ω)-Υs(II)(t|2ω)=-4π2ωc2ϕ(t)Pys(t|2ω).
2x12+2x32+2ωc2II(2ω)ψp(II)(r|2ω)=0.
ψp(I)(t|2ω)-ψp(II)(t|2ω)
=4π 2iωc Pxs(t|2ω),
1I(2ω) Υp(I)(t|2ω)-1II(2ω) Υp(II)(t|2ω)
=-4π 2iωcdPzs(t|2ω)dt+ϕ(t)PxNL(t|2ω)II(2ω).
P2NL(r|2ω)=α icωII(ω)ψp(II)(r|ω)x1ψs(II)(r|ω)x3-ψp(II)(r|ω)x3ψs(II)(r|ω)x1,
ϕ(t)Pys(t|2ω)=-χttzsciωψs(I)(t|ω) dψp(I)(t|ω)dt.
ϕ(t)PxNL(t|2ω)=-(α/2+γ) c2ω21I2(ω)ddtΥp(I)(t|ω)ϕ(t)2+1II2(ω)ddt1ϕ(t)dψp(I)(t|ω)dt2-α2II(ω)ddt [ψp(I)(t|ω)]2+γ ddt [ψs(I)(t|ω)]2,
Pxs(t|2ω)=χttzsI(ω)cω21ϕ2(t) Υp(I)(t|ω) dψp(I)(t|ω)dt,
Pzs(t|2ω)=-χzzzscω21ϕ(t)dψp(I)(t|ω)dt2-χztts1I2(ω)cω2Υp(I)(t|ω)ϕ(t)2-[ψs(I)(t|ω)]2.
ψs,p(I)(r|Ω)=ψs,p(I)(r|Ω)inc+14πΓGΩI(r|r)Zr=rs(t)×ψs,p(I)(t|Ω)-GΩI[r|rs(t)]Υs,p(I)(t|Ω)dt,
GΩI(r|r)=iπH0(1)nI(Ω) Ωc |r-r|,
ψs,p(I)(t|Ω)=ψs,p(I)(t|Ω)inc+14πlimτ0ΓGΩI(rs+|r)Zr=rs(t)ψs,p(I)(t|Ω)-GΩI[rs+|rs(t)]Υs,p(I)(t|Ω)dt,
0=-14πlimτ0ΓGΩII(rs+|r)Zr=rs(t)ψs,p(II)(t|Ω)-GΩII[rs+|rs(t)]Υs,p(II)(t|Ω)dt-14πSBGΩII(rs+|rB)Fs,pNL(rB|Ω)dSB,
rB(τ1, τ2)=[ξB(τ1, τ2), ηB(τ1, τ2)].
ψs,p(I)(t|Ω)-ψs,p(II)(t|Ω)=As,p(t|Ω),
1νI(Ω) Υs,p(I)(t|Ω)-1νII(Ω) Υs,p(II)(t|Ω)=Bs,p(t|Ω),
ψs,p(I)(tm|ω)=ψs,p(I)(tm|ω)inc+n=1N[HmnI(ω)ψs,p(I)(tn|ω)-LmnI(ω)Υs,p(I)(tn|ω)],
0=n=1NHmnII(ω)ψs,p(I)(tn|ω)-νII(ω)νI(ω) LmnII(ω)Υs,p(I)(tn|ω),
HmnI(ω)=iΔtn4 nI(ω) ωc [-ηn(ξm-ξn)+ξn(ηm-ηn)] H1(1){nI(ω)(ω/c)[(ξm-ξn)2+(ηm-ηn)2]1/2}[(ξm-ξn)2+(ηm-ηn)2]1/2mn12+Δtm4πϕ2(tm) (ξmηm-ξmηm)m=n,
LmnI(ω)=iΔtn4 H0(1)nI(ω) ωc [(ξm-ξn)2+(ηm-ηn)2]1/2mniΔtm4 H0(1)ΔtmnI(ω)(ω/c)ϕ(tm)2e,m=n,
ψs,p(I)(tm|2ω)=n=1N[HmnI(2ω)ψs,p(I)(tn|2ω)-LmnI(2ω)Υs,p(I)(tn|2ω)],
Qs,p(tm|2ω)=n=1NHmnII(2ω)ψs,p(I)(tn|2ω)-νII(2ω)νI(2ω) LmnII(2ω)Υs,p(I)(tn|2ω),
Qs,p(tm|2ω)=n=1N[HmnII(2ω)As,p(tn|2ω)-νII(2ω)LmnII(2ω)Bs,p(tn|2ω)]+l=1N1j=1N2LmljII(2ω)Fs,pNL(τ1l, τ2j|2ω)×ϕ1(τ1l, τ2j)ϕ2(τ1l, τ2j).
ϕ1(τ1l, τ2j)=ξBτ1 (τ1l, τ2j)2+ηBτ1 (τ1l, τ2j)21/2,
ϕ2(τ1l, τ2j)=ξBτ2 (τ1l, τ2j)2+ηBτ2 (τ1l, τ2j)21/2.
LmljII(2ω)=iΔτ1lΔτ2j4 H0(1)nII(2ω) 2ωc×{[ξm-ξB(τ1l, τ2j)]2+[ηm-ηB(τ1l, τ2j)]2}1/2,
ψs,p(I)(r, θ|Ω)sc=i4Γ(Z uˆ)nI(Ω) Ωc×H1(1)nI(Ω) Ωc |u|ψs,p(I)(t|Ω)-H0(1)nI(Ω) Ωc |u|Υs,p(I)(t|Ω)dt,
ψs,p(I)(r, θ|Ω)sc
=exp[i(π/4)] exp[inI(Ω)(Ω/c)r][8πnI(Ω)(Ω/c)r]1/2 Ss,p(θ|Ω),
Ss,p(θ|Ω)=ΓinI(Ω) Ωc [η(t)cos θ-ξ(t)sin θ]×ψs,p(I)(t|Ω)-Υs,p(I)(t|Ω)×exp-inI(Ω) Ωc [ξ(t)cos θ+η(t)sin θ]dt.
P(Ω)sc=r0L02πSsc(r0, θ|Ω)  e^rdθ,
Ssc(r0, θ|Ω)=(c/8π)R[Esc(r0, θ|Ω)  Hsc*(r0, θ|Ω)]
Ps,p(Ω)sc=r0Lc8πνI(Ω)(Ω/c)Ri02πψs,p(I)(r0, θ|Ω)sc×ψs,p(I)(r0, θ|Ω)scr*dθ
=Lc64π2νI(Ω)(Ω/c)02π|Ss,p(θ|Ω)|2dθ.
Ps(ω)inc=σnI(ω) c8π |ψ0s|2,
Pp(ω)inc=σ1nI(ω)c8π |ψ0p|2,
Qς(ω)=Pς(ω)scPς(ω)inc=02πqς(θ|ω)dθ,
qς(θ|ω)=18πD(ω/c)1nI(ω)|Sς(θ|ω)|2|ψ0ς|2.
Qpς(2ω)=σ Pp(2ω)sc[Pς(ω)inc]2=02πqpς(θ|2ω)dθ,
qpς(θ|2ω)=12ωDνI2(ω)nI2(ω)νI(2ω)|Sp(θ|2ω)|2|ψ0ς|4.
α=0,β=e8πm0ω2,γ=e3n0(z)8m02ω4.
PtNL(x, z|2ω)βEt(x, z|ω) Ez(x, z|ω)z,
PzNL(x, z|2ω)γ Ez2(x, z|ω)z+βEz(x, xz|ω) Ez(x, z|ω)z.
χzzzs=-23 β[II(ω)-1][II(ω)-3]2II2(ω)-23lnII(ω)II(2ω),
χztts=0,
χttzs=βII(ω)-1II(ω).

Metrics