Abstract

We study theoretically the nonlinear generation and scattering of sum-frequency electromagnetic radiation at randomly rough metallic surfaces. The approach permits the calculation of the scattered sum-frequency field for surfaces whose profiles are invariant in one direction, but are otherwise quite arbitrary, and we assume a fairly general form for the nonlinear polarization. The surfaces studied have a relatively large roughness scale and high slopes, which leads to substantial amounts of multiple scattering. We find that the mean angular distribution of the scattered sum-frequency light displays a well-defined minimum in a direction that depends on the angles of incidence of the excitation fields and their frequencies. The observed features are due to destructive interference between waves that have been multiply scattered in the valleys of the surface.

© 2011 Optical Society of America

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  1. A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  2. A. R. McGurn and A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
    [CrossRef]
  3. E. R. Méndez and K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  4. K. A. O’Donnell and E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  5. A. A. Maradudin and E. R. Méndez, “Enhanced backscattering of light from a weakly rough, random metal surface,” Appl. Opt. 32, 3335–3343 (1993).
    [CrossRef] [PubMed]
  6. C. S. West and K. A. O’Donnell, “Observations of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
    [CrossRef]
  7. A. A. Maradudin, A. R. McGurn, and E. R. Méndez, “Surface plasmon polariton mechanism for enhanced backscattering of light from one-dimensional randomly rough metal surfaces,” J. Opt. Soc. Am. A 12, 2500–2506 (1995).
    [CrossRef]
  8. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  9. V. M. Agranovich and V. E. Kravtsov, “Effects of weak localization of photons in nonlinear optics: difference frequency parametric mixing,” Phys. Lett. A 131, 378–392 (1988).
    [CrossRef]
  10. 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]
  11. A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392–400(1995).
    [CrossRef]
  12. 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]
  13. 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]
  14. 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]
  15. K. A. O’Donnell and R. Torre, “Second-harmonic generation from strongly rough metal surfaces,” Opt. Commun. 138, 341–344 (1997).
    [CrossRef]
  16. C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003).
    [CrossRef]
  17. C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21, 36–44 (2004).
    [CrossRef]
  18. C. I. Valencia and E. R. Méndez, “Weak localization effects in the second-harmonic light scattered by random systems of particles,” Opt. Commun. 282, 1706–1709 (2009).
    [CrossRef]
  19. J. A. Maytorena, B. S. Mendoza, and W. L. Mochán, “Theory of surface sum frequency generation spectroscopy,” Phys. Rev. B 57, 2569–2579 (1998).
    [CrossRef]
  20. R. P. Feynman, The Feynman Lectures on Physics (Addison-Wesley, 1964), Vol  II, Chap. 10.
  21. 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]
  22. 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]
  23. E. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), p. 638.
  24. R. García-Molina, A. A. Maradudin, and T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
    [CrossRef]
  25. R. A. Depine and J. M. Simon, “Surface impedance boundary condition for metallic diffraction gratings in the optical and infrared range,” Opt. Acta 30, 313–322 (1983).
    [CrossRef]
  26. 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, 1982), pp. 661–701.
  27. B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999–5006 (1996).
    [CrossRef]
  28. “Erratum,” Phys. Rev. B 61, 16243 (2000).
    [CrossRef]

2009

C. I. Valencia and E. R. Méndez, “Weak localization effects in the second-harmonic light scattered by random systems of particles,” Opt. Commun. 282, 1706–1709 (2009).
[CrossRef]

2004

2003

2000

“Erratum,” Phys. Rev. B 61, 16243 (2000).
[CrossRef]

1999

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]

1998

J. A. Maytorena, B. S. Mendoza, and W. L. Mochán, “Theory of surface sum frequency generation spectroscopy,” Phys. Rev. B 57, 2569–2579 (1998).
[CrossRef]

1997

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

1996

1995

1993

1991

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

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

R. García-Molina, A. A. Maradudin, and T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

1988

V. M. Agranovich and V. E. Kravtsov, “Effects of weak localization of photons in nonlinear optics: difference frequency parametric mixing,” Phys. Lett. A 131, 378–392 (1988).
[CrossRef]

1987

1986

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

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

1983

R. A. Depine and J. M. Simon, “Surface impedance boundary condition for metallic diffraction gratings in the optical and infrared range,” Opt. Acta 30, 313–322 (1983).
[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]

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]

V. M. Agranovich and V. E. Kravtsov, “Effects of weak localization of photons in nonlinear optics: difference frequency parametric mixing,” Phys. Lett. A 131, 378–392 (1988).
[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]

Born, E.

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

Celli, V.

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[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]

Depine, R. A.

R. A. Depine and J. M. Simon, “Surface impedance boundary condition for metallic diffraction gratings in the optical and infrared range,” Opt. Acta 30, 313–322 (1983).
[CrossRef]

Feynman, R. P.

R. P. Feynman, The Feynman Lectures on Physics (Addison-Wesley, 1964), Vol  II, Chap. 10.

García-Molina, R.

R. García-Molina, A. A. Maradudin, and T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Heiderich, A.

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392–400(1995).
[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]

Kravtsov, V. E.

V. M. Agranovich and V. E. Kravtsov, “Effects of weak localization of photons in nonlinear optics: difference frequency parametric mixing,” Phys. Lett. A 131, 378–392 (1988).
[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]

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]

R. García-Molina, A. A. Maradudin, and T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[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.

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, A. R. McGurn, and E. R. Méndez, “Surface plasmon polariton mechanism for enhanced backscattering of light from one-dimensional randomly rough metal surfaces,” J. Opt. Soc. Am. A 12, 2500–2506 (1995).
[CrossRef]

A. A. Maradudin and E. R. Méndez, “Enhanced backscattering of light from a weakly rough, random metal surface,” Appl. Opt. 32, 3335–3343 (1993).
[CrossRef] [PubMed]

R. García-Molina, A. A. Maradudin, and T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

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

A. R. McGurn and A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Maynard, R.

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392–400(1995).
[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]

Maytorena, J. A.

J. A. Maytorena, B. S. Mendoza, and W. L. Mochán, “Theory of surface sum frequency generation spectroscopy,” Phys. Rev. B 57, 2569–2579 (1998).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, A. R. McGurn, and E. R. Méndez, “Surface plasmon polariton mechanism for enhanced backscattering of light from one-dimensional randomly rough metal surfaces,” J. Opt. Soc. Am. A 12, 2500–2506 (1995).
[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]

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

A. R. McGurn and A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Méndez, E. R.

C. I. Valencia and E. R. Méndez, “Weak localization effects in the second-harmonic light scattered by random systems of particles,” Opt. Commun. 282, 1706–1709 (2009).
[CrossRef]

C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21, 36–44 (2004).
[CrossRef]

C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003).
[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]

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, A. R. McGurn, and E. R. Méndez, “Surface plasmon polariton mechanism for enhanced backscattering of light from one-dimensional randomly rough metal surfaces,” J. Opt. Soc. Am. A 12, 2500–2506 (1995).
[CrossRef]

A. A. Maradudin and E. R. Méndez, “Enhanced backscattering of light from a weakly rough, random metal surface,” Appl. Opt. 32, 3335–3343 (1993).
[CrossRef] [PubMed]

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

E. R. Méndez and K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

K. A. O’Donnell and E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

Mendoza, B. S.

C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21, 36–44 (2004).
[CrossRef]

C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003).
[CrossRef]

J. A. Maytorena, B. S. Mendoza, and W. L. Mochán, “Theory of surface sum frequency generation spectroscopy,” Phys. Rev. B 57, 2569–2579 (1998).
[CrossRef]

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

Michel, T.

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

Mochán, W. L.

J. A. Maytorena, B. S. Mendoza, and W. L. Mochán, “Theory of surface sum frequency generation spectroscopy,” Phys. Rev. B 57, 2569–2579 (1998).
[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]

O’Donnell, K. A.

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]

Simon, J. M.

R. A. Depine and J. M. Simon, “Surface impedance boundary condition for metallic diffraction gratings in the optical and infrared range,” Opt. Acta 30, 313–322 (1983).
[CrossRef]

Sipe, J. E.

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, 1982), pp. 661–701.

Stegeman, G. I.

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, 1982), pp. 661–701.

Torre, R.

Valencia, C. I.

van Tiggelen, B. A.

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392–400(1995).
[CrossRef]

West, C. S.

Wolf, E.

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

Ann. Phys.

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

Appl. Opt.

Appl. Phys. A

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. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

R. A. Depine and J. M. Simon, “Surface impedance boundary condition for metallic diffraction gratings in the optical and infrared range,” Opt. Acta 30, 313–322 (1983).
[CrossRef]

Opt. Commun.

E. R. Méndez and K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392–400(1995).
[CrossRef]

C. I. Valencia and E. R. Méndez, “Weak localization effects in the second-harmonic light scattered by random systems of particles,” Opt. Commun. 282, 1706–1709 (2009).
[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]

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

Opt. Lett.

Phys. Lett. A

V. M. Agranovich and V. E. Kravtsov, “Effects of weak localization of photons in nonlinear optics: difference frequency parametric mixing,” Phys. Lett. A 131, 378–392 (1988).
[CrossRef]

Phys. Rep.

R. García-Molina, A. A. Maradudin, and T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Phys. Rev.

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. B

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

“Erratum,” Phys. Rev. B 61, 16243 (2000).
[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]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

J. A. Maytorena, B. S. Mendoza, and W. L. Mochán, “Theory of surface sum frequency generation spectroscopy,” Phys. Rev. B 57, 2569–2579 (1998).
[CrossRef]

Other

R. P. Feynman, The Feynman Lectures on Physics (Addison-Wesley, 1964), Vol  II, Chap. 10.

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, 1982), pp. 661–701.

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

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the scattering geometry. Fields of optical frequencies ω 1 and ω 2 are incident on a randomly rough surface. The nonlinear interaction between the fields and the surface produces (among other fields) a scattered field at the SF ω 3 = ω 1 + ω 2 .

Fig. 2
Fig. 2

Surface profile function and the local system of coordinates ( x , y , z ) . The curve Γ represents the interface between media I and II, and the vector r s = ( x 1 , ζ ( x 1 ) ) represents points on the surface.

Fig. 3
Fig. 3

Far-field angle-resolved mean scattering distribution for a surface characterized by the roughness parameters a = 3.4 μm and δ = 1.85 μm . (a) Linear scattering for λ 1 = 1064 nm and p-polarized illumination with θ 01 = 5 ° . (b) Linear scattering for λ 2 = 850 nm and p-polarized illumination with θ 02 = 10 ° . (c) SFG ( λ 3 = 472.5 nm ). (d) Linear scattering for λ 3 = 472.5 nm and p-polarized illumination with θ 0 e = 3.32 ° . The sampling on the surface, of length L = 63.84 μm , was Δ x = 42.56 nm and the curves show the results of averaging over N p = 5000 realizations of the random surface.

Fig. 4
Fig. 4

Effect of changing the sign of the angles of incidence.

Fig. 5
Fig. 5

Illustrative diagram of the double scattering nonlinear processes in the valleys of the surface.

Fig. 6
Fig. 6

Illustrative diagram of the coherent and incoherent scattering processes. For simplicity, we consider that the angles of incidence are equal.

Fig. 7
Fig. 7

Normalized far-field angle-resolved mean SF scattering distribution for a randomly rough surface. The parameters are the same as for Fig. 3, but for angles of incidence (a)  θ 01 = 5 ° and θ 02 = 15 ° and (b)  θ 01 = 15 ° and θ 02 = 10 ° .

Fig. 8
Fig. 8

Far-field angle-resolved mean scattering distribution for a surface characterized by the roughness parameters a = 4.0 μm and δ = 0.5 μm . (a) Linear scattering for λ 1 = 1064 nm and p-polarized illumination with θ 01 = 5 ° . (b) Linear scattering for λ 2 = 850 nm and p-polarized illumination with θ 02 = 10 ° . (c) SFG ( λ 3 = 472.5 nm ). (d) Linear scattering for λ 3 = 472.5 nm and p-polarized illumination with θ 0 e = 3.32 ° . The sampling on the surface, of length L = 63.84 μm , was Δ x = 42.56 nm and the curves show the results of averaging over N p = 5000 realizations of the random surface.

Fig. 9
Fig. 9

Normalized far-field angle-resolved mean SF scattering distribution for a randomly rough surface. The parameters are the same as for Fig. 8, but for angles of incidence (a)  θ 01 = 5 ° and θ 02 = 15 ° and (b)  θ 01 = 15 ° and θ 02 = 10 ° .

Equations (97)

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N = [ ζ ( x 1 ) x ^ 1 + x ^ 3 ] = ϕ ( x 1 ) n ^ ,
T = x ^ 1 + ζ ( x 1 ) x ^ 3 = ϕ ( x 1 ) t ^ ,
ϕ ( x 1 ) = [ 1 + ( ζ ( x 1 ) ) 2 ] 1 / 2
N = [ ζ ( x 1 ) x 1 + x 3 ] .
× E ( r | ω 3 ) = i ω 3 c H ( r | ω 3 ) ,
× H ( r | ω 3 ) = i ω 3 c D ( r | ω 3 ) ,
· D ( r | ω 3 ) = 0 ,
· H ( r | ω 3 ) = 0 ,
D ( r | ω 3 ) = ϵ ( ω 3 ) E ( r | ω 3 ) + 4 π P ( r | ω 3 )
E t ( r | ω 3 ) z = t E z ( r | ω 3 ) ( i ω 3 c ) [ z ^ × H t ( r | ω 3 ) ] ,
E t ( I ) ( r s | ω 3 ) E t ( II ) ( r s | ω 3 ) = 4 π t P z s ( r s | ω 3 ) ,
H t ( I ) ( r s | ω 3 ) H t ( II ) ( r s | ω 3 ) = 4 π ( i ω 3 c ) [ z ^ × P t s ( r s | ω 3 ) ] ,
P ( r | ω 3 ) = n ( r ) e χ ( ω 3 ) χ ( ω 1 ) [ ( E ( r | ω 1 ) · ) E ( r | ω 2 ) ω 1 ω 2 E ( r | ω 1 ) × × E ( r | ω 2 ) ] + 1 2 e χ ( ω 1 ) χ ( ω 2 ) · ( n ( r ) E ( r | ω 1 ) E ( r | ω 2 ) ) + ω 1 ω 2 1 2 e χ ( ω 1 ) χ ( ω 2 ) ( ω 2 ω 1 ω 3 ) × ( n ( r ) E ( r | ω 1 ) × E ( r | ω 2 ) ) ,
E ( r | ω 1 ) × × E ( r | ω 2 ) = E ( r | ω 2 ) ( E ( r | ω 1 ) · E ( r | ω 2 ) ) ( E ( r | ω 1 ) · ) E ( r | ω 2 ) ,
· ( E ( r | ω 1 ) E ( r | ω 2 ) ) = ( · E ( r | ω 1 ) ) E ( r | ω 2 ) + ( E ( r | ω 1 ) · ) E ( r | ω 2 ) ,
× ( E ( r | ω 1 ) × E ( r | ω 2 ) ) = ( · E ( r | ω 2 ) ) E ( r | ω 1 ) ( · E ( r | ω 1 ) ) E ( r | ω 2 ) + ( E ( r | ω 2 ) · ) E ( r | ω 1 ) ( E ( r | ω 1 ) · ) E ( r | ω 2 ) ,
× ( E ( r | ω 1 ) × E ( r | ω 2 ) ) = ( · E ( r | ω 2 ) ) E ( r | ω 1 ) ( · E ( r | ω 1 ) ) E ( r | ω 2 ) + ( E ( r | ω 2 ) · ) E ( r | ω 1 ) ( E ( r | ω 1 ) · ) E ( r | ω 2 ) ,
P ( r | ω 3 ) = n ( r ) e χ ( ω 3 ) χ ( ω 2 ) ω 2 ω 1 E ( r | ω 1 ) ( E ( r | ω 1 ) · E ( r | ω 2 ) ) + n ( r ) e χ ( ω 2 ) ( χ ( ω 1 ) 2 χ ( ω 3 ) ω 3 ω 1 ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) 1 8 π e χ ( ω 2 ) ( · E ( r | ω 1 ) ) E ( r | ω 2 ) + n ( r ) 4 e χ ( ω 1 ) χ ( ω 2 ) ( ω 2 ω 1 ω 3 ) [ ( E ( r | ω 2 ) · ) E ( r | ω 1 ) + ( 1 1 4 π n ( r ) χ ( ω 1 ) ) ( · E ( r | ω 1 ) ) E ( r | ω 2 ) ] 1 4 e χ ( ω 1 ) χ ( ω 2 ) ( ω 2 ω 1 ω 3 ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) + ω 1 ω 2 .
P ( r | ω 3 ) = γ e ( r | ω 1 ) E ( r | ω 1 ) ( E ( r | ω 1 ) · E ( r | ω 2 ) ) + ( β e ( r | ω 1 ) + β m ( r | ω 1 ) ) ( · E ( r | ω 1 ) ) E ( r | ω 2 ) + ( α e ( r | ω 1 ) + α m ( r ) ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) + α m ( r ) n ( r ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) + ω 1 ω 2 ,
γ e ( r | ω i ) = 4 κ 12 ϵ ( r | ω 3 ) 1 ϵ ( r | ω i ) 1 ω 1 ω 2 ω i 2 ,
β e ( r | ω i ) = 2 κ 12 1 ϵ ( r | ω i ) ,
β m ( r | ω i ) = κ 12 ( ω 2 ω 1 ω 3 ) ( 1 1 ϵ ( r | ω i ) 1 ) ,
α e ( r | ω i ) = 2 κ 12 ( 1 2 ω 3 ω i ϵ ( r | ω 3 ) 1 ϵ ( r | ω i ) 1 ) ,
α m ( r ) = κ 12 ( ω 1 ω 2 ω 3 ) ,
κ 12 = 1 4 n ( r ) e ( ϵ ( r | ω 1 ) 1 4 π ) ( ϵ ( r | ω 2 ) 1 4 π ) .
( α e ( r | ω 1 ) + α m ( r ) ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) + α m ( r ) n ( r ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) = α e m ( ω 1 ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) ,
α e m ( ω i ) = 4 κ 12 ( ω i ω 3 ϵ ( ω 3 ) 1 ϵ ( ω i ) 1 ω 3 ω i ) .
P ( r | ω 3 ) = γ e ( ω 1 ) E ( r | ω 1 ) ( E ( r | ω 1 ) · E ( r | ω 2 ) ) + ( β e ( ω 1 ) + β m ( ω 1 ) ) ( · E ( r | ω 1 ) ) E ( r | ω 2 ) + α e m ( ω 1 ) ( E ( r | ω 2 ) · ) E ( r | ω 1 ) + ω 1 ω 2 ,
κ 12 = e 3 n b 4 m o 2 ω 1 2 ω 2 2 ,
α e m ( ω i ) = 0 ,
β e ( ω i ) = 2 κ 12 ω i 2 ω p 2 ,
β m ( ω i ) = κ 12 ( ω 2 ω 1 ω 3 ) ( 1 + ω i 2 ω p 2 ) ,
γ e = 4 κ 12 ω 1 ω 2 ω 3 2 .
P ( r | ω 3 ) = n ( z ) e χ ( ω 3 ) ( χ ( ω 1 ) E ( r | ω 1 ) · E ( r | ω 2 ) + χ ( ω 2 ) E ( r | ω 2 ) · E ( r | ω 1 ) ) + 1 2 e χ ( ω 1 ) χ ( ω 2 ) · n ( z ) ( E ( r | ω 1 ) E ( r | ω 2 ) + E ( r | ω 2 ) E ( r | ω 1 ) ) 1 2 e χ ( ω 1 ) χ ( ω 2 ) ( ω 2 ω 1 ω 3 ) × n ( z ) ( E ( r | ω 1 ) × E ( r | ω 2 ) ) .
P s ( r s | ω 3 ) d z P ( r | ω 3 ) ,
P z s ( r s | ω 3 ) = χ z z z s ( ω 1 , ω 2 ) D z ( r s | ω 1 ) D z ( r s | ω 2 ) + ω 1 ω 2 ,
χ z z z s ( ω 1 , ω 2 ) = κ 12 ϵ ( ω 1 ) ϵ ( ω 2 ) a ( ω 1 , ω 2 ) ,
a ( ω 1 , ω 2 ) = 2 [ 1 + ( 1 ϵ ( ω 3 ) ) ϵ ( ω 1 ) ϵ ( ω 2 ) [ ϵ ( ω 2 ) log ( ϵ ( ω 3 ) / ϵ ( ω 1 ) ) + C.P. ] ( ϵ ( ω 1 ) ϵ ( ω 2 ) ) ( ϵ ( ω 2 ) ϵ ( ω 3 ) ) ( ϵ ( ω 3 ) ϵ ( ω 1 ) ) ] .
P t s ( r s | ω 3 ) = χ t t z s ( ω 1 , ω 2 ) E t ( r s | ω 1 ) D z ( r s | ω 2 ) + χ t z t s ( ω 1 , ω 2 ) D z ( r s | ω 1 ) E t ( r s | ω 2 ) + ω 1 ω 2 ,
χ t t z s ( ω 1 , ω 2 ) = χ t z t s ( ω 2 , ω 1 ) = κ 12 ϵ ( ω 2 ) 2 ω 1 ω 3 b ( ω 1 , ω 2 ) ,
χ t t z s ( ω 2 , ω 1 ) = χ t z t s ( ω 1 , ω 2 ) = κ 12 ϵ ( ω 1 ) 2 ω 2 ω 3 b ( ω 2 , ω 1 ) ,
b ( ω 1 , ω 2 ) = b ( ω 2 , ω 1 ) = 1 .
H 2 ( r | ω 3 ) x 3 = i ω 3 c [ ϵ ( ω 3 ) E 1 ( r | ω 3 ) + 4 π P 1 ( r | ω 3 ) ] ,
H 2 ( r | ω 3 ) x 1 = i ω 3 c [ ϵ ( ω 3 ) E 3 ( r | ω 3 ) + 4 π P 3 ( r | ω 3 ) ] ,
[ E 3 ( r | ω 3 ) x 1 E 1 ( r | ω 3 ) x 3 ] = i ω 3 c H 2 ( r | ω 3 ) ,
[ 2 x 1 2 + 2 x 3 2 + ( ω 3 c ) 2 ϵ ( ω 3 ) ] H 2 ( II ) ( r | ω 3 ) = 4 π i ω 3 c [ × P ( r | ω 3 ) ] 2 ,
[ × P ( r | ω 3 ) ] 2 = γ e ( ω 1 ) [ × E ( r | ω 1 ) [ E ( r | ω 1 ) · E ( r | ω 2 ) ] ] 2 + ( β e ( ω 1 ) + β m ( ω 1 ) ) [ × [ · E ( r | ω 1 ) ] E ( r | ω 2 ) ] 2 + α e m ( ω 1 ) [ × [ E ( r | ω 2 ) · ] E ( r | ω 1 ) ] 2 + ω 1 ω 2 ,
[ × P ( r | ω 3 ) ] 2 = 0 ,
θ ( x 3 ζ ( x 1 ) ) H 2 ( I ) ( x 1 , x 3 | Ω ) = H 0 ( x 1 , x 3 | Ω ) + 1 4 π d x 1 { N G 0 ( Ω ) ( x 1 , x 3 | x 1 , ζ ( x 1 ) ) × H ( I ) ( x 1 | Ω ) G 0 ( Ω ) ( x 1 , x 3 | x 1 , ζ ( x 1 ) ) L ( I ) ( x 1 | Ω ) } ,
θ ( ζ ( x 1 ) x 3 ) H 2 ( II ) ( x 1 , x 3 | Ω ) = 1 4 π d x 1 { N G ϵ ( Ω ) ( x 1 , x 3 | x 1 , ζ ( x 1 ) ) H ( II ) ( x 1 | Ω ) G ϵ ( Ω ) ( x 1 , x 3 | x 1 , ζ ( x 1 ) ) L ( II ) ( x 1 | Ω ) } .
H 0 ( x 1 , x 3 | Ω ) = { H 2 ( x 1 , x 3 | ω 1 , 2 ) inc if     Ω = ω 1     or     ω 2 0 if     Ω = ω 3 ,
H ( I , II ) ( x 1 | Ω ) = H 2 ( I , II ) ( x 1 , x 3 | Ω ) | x 3 = ζ ( x 1 ) ,
L ( I , II ) ( x 1 | Ω ) = N H 2 ( I , II ) ( x 1 , x 3 | Ω ) | x 3 = ζ ( x 1 ) ;
H 2 ( I ) ( x 1 | Ω ) = H 0 ( x 1 | Ω ) + 1 4 π lim η 0 + d x 1 { N G 0 ( Ω ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) × H ( I ) ( x 1 | Ω ) G 0 ( Ω ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) L ( I ) ( x 1 | Ω ) } ,
0 = 1 4 π lim η 0 + d x 1 { N G ( Ω ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) H ( II ) ( x 1 | Ω ) G ϵ ( Ω ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) L ( II ) ( x 1 | Ω ) } ,
L ( I ) ( x 1 | ω 1 , 2 ) = K ( 0 ) ( x 1 | ω 1 , 2 ) ϵ ( ω 1 , 2 ) H ( I ) ( x 1 | ω 1 , 2 ) .
K ( 0 ) ( x 1 | ω 1 , 2 ) = ϕ ( x 1 ) d ( ω 1 , 2 ) { 1 + d ( ω 1 , 2 ) 2 ζ ( x 1 ) ϕ 3 ( x 1 ) d 2 ( ω 1 , 2 ) 8 [ ζ ( x 1 ) ] 2 ϕ 6 ( x 1 ) + O ( d 3 ( ω 1 , 2 ) ) } ,
d ( ω 1 , 2 ) = c ω 1 , 2 ϵ ( ω 1 , 2 ) .
H 2 ( I ) ( x 1 | ω 1 , 2 ) = H ( x 1 | ω 1 , 2 ) inc + 1 4 π lim η 0 + d x 1 { N G 0 ( ω 1 , 2 ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) G 0 ( ω 1 , 2 ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) K ( 0 ) ( x 1 | ω 1 , 2 ) ϵ ( ω 1 , 2 ) } H ( I ) ( x 1 | ω 1 , 2 ) .
H ( x m | ω 1 , 2 ) inc = n = 1 N [ ( 1 2 δ n m 1 2 H m n ( 0 ) ) + 1 2 L m n ( 0 ) K ( 0 ) ( x n | ω 1 , 2 ) ϵ ( ω 1 , 2 ) ] H ( I ) ( x n | ω 1 , 2 ) .
R p ( θ s | ω 1 , 2 ) θ s = | r p ( θ s | ω 1 , 2 ) | 2 8 π ( ω 1 , 2 / c ) L cos θ 0 ,
r p ( θ s | ω 1 , 2 ) = d x 1 [ ı ω 1 , 2 c [ ζ ( x 1 ) sin θ s cos θ s ] K ( 0 ) ( x 1 | ω 1 , 2 ) ϵ ( ω 1 , 2 ) ] H ( I ) ( x 1 | ω 1 , 2 ) × exp { i ω 1 , 2 c ( x 1 sin θ s + ζ ( x 1 ) cos θ s ) } .
R p ( θ s | ω 1 , 2 ) θ s inc = | r p ( θ s | ω 1 , 2 ) | 2 | r p ( θ s | ω 1 , 2 ) | 2 8 π ( ω 1 , 2 / c ) L cos θ 0 .
H ( I ) ( x 1 | ω 3 ) H ( II ) ( x 1 | ω 3 ) = 4 π i ω 3 c P x s ( x 1 | ω 3 ) ,
L ( I ) ( x 1 | ω 3 ) 1 ϵ ( ω 3 ) L ( II ) ( x 1 | ω 3 ) = 4 π i ω 3 c [ d P z s ( x 1 | ω 3 ) d x 1 + ϕ ( x 1 ) P x ( x 1 | ω 3 ) ϵ ( ω 3 ) ] .
ϕ ( x 1 ) P x ( x 1 | ω 3 ) = γ e c 2 ω 1 ω 2 [ d d x 1 ( L ( I ) ( x 1 | ω 1 ) L ( I ) ( x 1 | ω 2 ) ϕ 2 ( x 1 ) ) + 1 ϵ ( ω 1 ) ϵ ( ω 2 ) d d x 1 ( 1 ϕ 2 ( x 1 ) d H ( I ) ( x 1 | ω 1 ) d x 1 d H ( I ) ( x 1 | ω 2 ) d x 1 ) ] ,
P z s ( x 1 | ω 3 ) = χ z z z s ( ω 1 , ω 2 ) ( c 2 ω 1 ω 2 ) ( 1 ϕ 2 ( x 1 ) d H ( I ) ( x 1 | ω 1 ) d x 1 d H ( I ) ( x 1 | ω 2 ) d x 1 ) + ( ω 1 ω 2 ) ,
P x s ( x 1 | ω 3 ) = χ t t z s ( ω 1 , ω 2 ) ϵ ( ω 1 ) ( c 2 ω 1 ω 2 ) 1 ϕ 2 ( x 1 ) L ( I ) ( x 1 | ω 1 ) d H ( I ) ( x 1 | ω 2 ) d x 1 + χ t z t s ( ω 1 , ω 2 ) ϵ ( ω 2 ) ( c 2 ω 1 ω 2 ) 1 ϕ 2 ( x 1 ) L ( I ) ( x 1 | ω 2 ) d H ( I ) ( x 1 | ω 1 ) d x 1 + ( ω 1 ω 2 ) .
H ( I ) ( x 1 | ω 3 ) H ( II ) ( x 1 | ω 3 ) = A ( x 1 ) ,
L ( I ) ( x 1 | ω 3 ) 1 ϵ ( II ) ( ω 3 ) L ( II ) ( x 1 | ω 3 ) = B ( x 1 ) ,
H 2 ( I ) ( x 1 | ω 3 ) = 1 4 π lim η 0 + d x 1 { N G 0 ( ω 3 ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) H ( I ) ( x 1 | ω 3 ) G 0 ( ω 3 ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) L ( I ) ( x 1 | ω 3 ) } ,
0 = 1 4 π lim η 0 + d x 1 { N G ( ω 3 ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) [ H ( I ) ( x 1 | ω 3 ) A ( x 1 ) ] ϵ ( II ) ( ω 3 ) G ϵ ( ω 3 ) ( x 1 , ζ ( x 1 ) + η | x 1 , ζ ( x 1 ) ) [ L ( I ) ( x 1 | ω 3 ) B ( x 1 ) ] } .
0 = n = 1 N [ ( 1 2 δ n m 1 2 H m n ( 0 ) ) H ( I ) ( x n | ω 3 ) + 1 2 L m n ( 0 ) L ( I ) ( x n | ω 3 ) ] ,
Q ( x m ) = n = 1 N [ ( 1 2 δ n m + 1 2 H m n ( ϵ ) ) H ( I ) ( x n | ω 3 ) ϵ ( ω 3 ) 1 2 L m n ( ϵ ) L ( I ) ( x n | ω 3 ) ] ,
Q ( x m ) = n = 1 N [ ( 1 2 δ n m + 1 2 H m n ( ϵ ) ) A ( x n ) ϵ ( ω 3 ) 1 2 L m n ( ϵ ) B ( x n ) ] ,
P sc ( ω 3 ) ( P inc ( ω 1 ) A s ) ( P inc ( ω 2 ) A s ) A s ,
p sc ( ω 3 ) = P sc ( ω 3 ) P inc ( ω 1 ) P inc ( ω 2 ) A s .
I p ( θ s | ω 3 ) = | r p ( θ s | ω 3 ) | 2 ω 3 L cos θ 01 cos θ 02 | H 01 | 2 | H 02 | 2 ,
r p ( θ s | ω 3 ) = d x 1 [ i ω 3 c [ ζ ( x 1 ) sin θ s cos θ s ] H ( I ) ( x 1 | ω 3 ) L ( I ) ( x 1 | ω 3 ) ] × exp { i ω 3 c ( x 1 sin θ s + ζ ( x 1 ) cos θ s ) } .
I p ( θ s | ω 3 ) inc = | r p ( θ s | ω 3 ) | 2 | r p ( θ s | ω 3 ) | 2 ω 3 L cos θ 01 cos θ 02 | H 01 | 2 | H 02 | 2 .
ω 3 c sin θ sp = ω 1 c sin θ 01 + ω 2 c sin θ 02 .
A ( θ 01 , θ 02 | ω 3 ) = A ( θ 01 , θ 02 | ω 3 ) .
E × × E = 1 2 ( E · E ) ( E · ) E ,
· ( E E ) = · ( E E ) = ( · E ) E + ( E · ) E = ( · E ) E + ( E · ) E ,
P ( r | 2 ω ) = n ( r ) e χ ( 2 ω ) χ ( ω ) [ 2 ( E ( r | ω ) · ) E ( r | ω ) 1 2 ( E ( r | ω ) · E ( r | ω ) ) ] + 1 2 e χ 2 ( ω ) [ ( · E ( r | ω ) ) E ( r | ω ) + ( E ( r | ω ) · ) E ( r | ω ) ] = α ( r | ω ) ( E ( r | ω ) · ) E ( r | ω ) + γ ( r | ω ) ( E ( r | ω ) · E ( r | ω ) ) + δ ( r | ω ) ( · E ( r | ω ) ) E ( r | ω ) ,
α ( r | ω ) = 1 2 n ( r ) e ϵ ( r | ω ) 1 4 π ( ϵ ( r | ω ) 1 4 π 4 ϵ ( r | 2 ω ) 1 4 π ) ,
δ ( r | ω ) = 1 2 n 2 ( r ) e ( ϵ ( r | ω ) 1 4 π ) 2 ,
γ ( r | ω ) = 1 2 n ( r ) e ϵ ( r | ω ) 1 4 π ϵ ( r | 2 ω ) 1 4 π .
δ ( r | ω ) · E ( r | ω ) = δ ( r | ω ) 4 π χ ( ω ) · E ( r | ω ) = β ( r | ω ) · E ( r | ω ) ,
β ( r | ω ) = ϵ ( r | ω ) 1 32 π 2 n ( r ) e .
P ( r | 2 ω ) = α ( r | ω ) ( E ( r | ω ) · ) E ( r | ω ) + γ ( r | ω ) ( E ( r | ω ) · E ( r | ω ) ) + β ( r | ω ) ( · E ( r | ω ) ) E ( r | ω ) .
β e ( ω ) = e 8 π m 0 ω 2 , γ e ( ω ) = e 3 n b 4 m 0 2 ω 4 .
γ ( E ( r | ω ) · E ( r | ω ) ) = γ ( E ( ω ) · E ( ω ) ) e i 2 q · r = 2 i γ ( E ( r | ω ) · E ( r | ω ) ) q .
γ 1 e E ( r | ω 1 ) ( E ( r | ω 1 ) · E ( r | ω 2 ) ) = γ 1 e ( E ( ω 1 ) · E ( r | ω 2 ) ) e i q 1 · r = i γ 1 e ( E ( r | ω 1 ) · E ( r | ω 2 ) ) q 1 ,
γ 1 e E ( r | ω 1 ) ( E ( r | ω 1 ) · E ( r | ω 2 ) ) | SHG i γ e ( E ( r | ω ) · E ( r | ω ) ) q ,
α = 0 , γ = n b e 3 8 m 0 2 ω 4 , β = e 8 π m 0 ω 2 ,
a ( ω 1 , ω 2 ) = 2 [ ( 1 ω 1 2 1 ω 2 2 ) ϵ ( ω 1 ) ϵ ( ω 2 ) + C.P. ] + 1 ω 3 2 ϵ ( ω 1 ) ϵ ( ω 2 ) [ ϵ ( ω 2 ) log ( ϵ ( ω 3 ) ϵ ( ω 1 ) ) + C.P. ] ω p 4 ( 1 ω 1 2 1 ω 3 2 ) ( 1 ω 3 2 1 ω 2 2 ) ( 1 ω 2 2 1 ω 1 2 ) ,

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