Abstract

We have obtained local impedance boundary conditions for a metal film characterized by an isotropic, frequency-dependent, complex dielectric function ∊(ω) that occupies the region ζ(x 1) < x 3 < D. The surface-profile function ζ(x 1) is assumed to be a single-valued function of x 1 that is differentiable as many times as is necessary. The electromagnetic field in the system is assumed to be p polarized with the plane of incidence, the x 1 x 3 plane. The results are used to study the scattering of p-polarized light from and its transmission through the metal film when the surface-profile function ζ(x 1) in these calculations is assumed to be a stationary, zero-mean, Gaussian random process. These calculations are approximately four times faster than rigorous computer simulations, and their results are qualitatively and quantitatively in good agreement with those of the latter simulations.

© 1999 Optical Society of America

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References

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  1. R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
    [CrossRef]
  2. A. A. Maradudin, “The impedance boundary condition for a one-dimensional, curved metal surface,” Opt. Commun. 103, 227–234 (1993).
    [CrossRef]
  3. A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
    [CrossRef]
  4. A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
    [CrossRef]
  5. T. T. Ong, V. Celli, A. A. Maradudin, “The impedance of a curved surface,” Opt. Commun. 95, 1–4 (1993).
    [CrossRef]
  6. A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough surface,” Opt. Commun. 116, 452–467 (1995).
    [CrossRef]
  7. A. A. Maradudin, E. R. Méndez, “The utility of an impedance boundary condition in the scattering of light from one-dimensional randomly rough dielectric surfaces,” Opt. Spektrosk. 80, 459–470 (1996) [Opt. Spectrosc. 80, 409–420 (1996)].
  8. X. Wang, H. J. Simon, “Directionally scattered optical second-harmonic generation with surface plasmons,” Opt. Lett. 16, 1475–1477 (1991).
    [CrossRef] [PubMed]
  9. H. J. Simon, Y. Wang, L. B. Zhou, Z. Chen, “Coherent backscattering of optical second-harmonic generation with long-range surface plasmons,” Opt. Lett. 17, 1268–1270 (1992).
    [CrossRef] [PubMed]
  10. O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
    [CrossRef]
  11. Y. Wang, H. J. Simon, “Coherent backscattering of optical second-harmonic generation in silver films,” Phys. Rev. B 47, 13695–13699 (1993).
    [CrossRef]
  12. L. Kuang, H. J. Simon, “Diffusely scattered second-harmonic generation from a silver film due to surface plasmons,” Phys. Lett. A 197, 257–261 (1995).
    [CrossRef]
  13. Further information will be available in a paper entitled “Multiple-scattering effects in the second-harmonic generation of light in scattering from a random metal surface in the Kretschmann ATR geometry,” which is currently in preparation by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R. Méndez, I. V. Novikov.
  14. A. E. Danese, Advanced Calculus (Allyn and Bacon, Boston, 1965), Vol. I, p. 123.
  15. A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
    [CrossRef]
  16. P. W. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  17. V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
    [CrossRef]
  18. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  19. A. A. Maradudin, T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
    [CrossRef]
  20. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  21. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, p. 56, formula 45.

1997 (3)

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

1996 (1)

A. A. Maradudin, E. R. Méndez, “The utility of an impedance boundary condition in the scattering of light from one-dimensional randomly rough dielectric surfaces,” Opt. Spektrosk. 80, 459–470 (1996) [Opt. Spectrosc. 80, 409–420 (1996)].

1995 (2)

A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough surface,” Opt. Commun. 116, 452–467 (1995).
[CrossRef]

L. Kuang, H. J. Simon, “Diffusely scattered second-harmonic generation from a silver film due to surface plasmons,” Phys. Lett. A 197, 257–261 (1995).
[CrossRef]

1994 (1)

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

1993 (3)

T. T. Ong, V. Celli, A. A. Maradudin, “The impedance of a curved surface,” Opt. Commun. 95, 1–4 (1993).
[CrossRef]

A. A. Maradudin, “The impedance boundary condition for a one-dimensional, curved metal surface,” Opt. Commun. 103, 227–234 (1993).
[CrossRef]

Y. Wang, H. J. Simon, “Coherent backscattering of optical second-harmonic generation in silver films,” Phys. Rev. B 47, 13695–13699 (1993).
[CrossRef]

1992 (2)

H. J. Simon, Y. Wang, L. B. Zhou, Z. Chen, “Coherent backscattering of optical second-harmonic generation with long-range surface plasmons,” Opt. Lett. 17, 1268–1270 (1992).
[CrossRef] [PubMed]

O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
[CrossRef]

1991 (1)

1990 (4)

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

A. A. Maradudin, T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

1972 (1)

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

Aktsipetrov, O. A.

O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
[CrossRef]

Celli, V.

T. T. Ong, V. Celli, A. A. Maradudin, “The impedance of a curved surface,” Opt. Commun. 95, 1–4 (1993).
[CrossRef]

Chen, Z.

Christy, R. W.

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

Danese, A. E.

A. E. Danese, Advanced Calculus (Allyn and Bacon, Boston, 1965), Vol. I, p. 123.

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, p. 56, formula 45.

Freilikher, V.

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

Garcia-Molina, R.

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

Golovkina, V. N.

O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
[CrossRef]

Johnson, P. W.

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

Kanzieper, E.

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

Kapusta, O. I.

O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
[CrossRef]

Kuang, L.

L. Kuang, H. J. Simon, “Diffusely scattered second-harmonic generation from a silver film due to surface plasmons,” Phys. Lett. A 197, 257–261 (1995).
[CrossRef]

Leskova, T. A.

O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
[CrossRef]

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

Further information will be available in a paper entitled “Multiple-scattering effects in the second-harmonic generation of light in scattering from a random metal surface in the Kretschmann ATR geometry,” which is currently in preparation by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R. Méndez, I. V. Novikov.

Leyva-Lucero, M.

Further information will be available in a paper entitled “Multiple-scattering effects in the second-harmonic generation of light in scattering from a random metal surface in the Kretschmann ATR geometry,” which is currently in preparation by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R. Méndez, I. V. Novikov.

Madrazo, A.

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, p. 56, formula 45.

Maradudin, A. A.

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

A. A. Maradudin, E. R. Méndez, “The utility of an impedance boundary condition in the scattering of light from one-dimensional randomly rough dielectric surfaces,” Opt. Spektrosk. 80, 459–470 (1996) [Opt. Spectrosc. 80, 409–420 (1996)].

A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough surface,” Opt. Commun. 116, 452–467 (1995).
[CrossRef]

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

A. A. Maradudin, “The impedance boundary condition for a one-dimensional, curved metal surface,” Opt. Commun. 103, 227–234 (1993).
[CrossRef]

T. T. Ong, V. Celli, A. A. Maradudin, “The impedance of a curved surface,” Opt. Commun. 95, 1–4 (1993).
[CrossRef]

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

A. A. Maradudin, T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

Further information will be available in a paper entitled “Multiple-scattering effects in the second-harmonic generation of light in scattering from a random metal surface in the Kretschmann ATR geometry,” which is currently in preparation by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R. Méndez, I. V. Novikov.

Maystre, D.

McGurn, A. R.

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

Méndez, E. R.

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

A. A. Maradudin, E. R. Méndez, “The utility of an impedance boundary condition in the scattering of light from one-dimensional randomly rough dielectric surfaces,” Opt. Spektrosk. 80, 459–470 (1996) [Opt. Spectrosc. 80, 409–420 (1996)].

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

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

Further information will be available in a paper entitled “Multiple-scattering effects in the second-harmonic generation of light in scattering from a random metal surface in the Kretschmann ATR geometry,” which is currently in preparation by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R. Méndez, I. V. Novikov.

Mendoza-Suárez, A.

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

Michel, T.

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

A. A. Maradudin, T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

Novikov, I. V.

Further information will be available in a paper entitled “Multiple-scattering effects in the second-harmonic generation of light in scattering from a random metal surface in the Kretschmann ATR geometry,” which is currently in preparation by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R. Méndez, I. V. Novikov.

Novikova, N. N.

O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
[CrossRef]

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, p. 56, formula 45.

Ong, T. T.

T. T. Ong, V. Celli, A. A. Maradudin, “The impedance of a curved surface,” Opt. Commun. 95, 1–4 (1993).
[CrossRef]

Saillard, M.

Simon, H. J.

L. Kuang, H. J. Simon, “Diffusely scattered second-harmonic generation from a silver film due to surface plasmons,” Phys. Lett. A 197, 257–261 (1995).
[CrossRef]

Y. Wang, H. J. Simon, “Coherent backscattering of optical second-harmonic generation in silver films,” Phys. Rev. B 47, 13695–13699 (1993).
[CrossRef]

H. J. Simon, Y. Wang, L. B. Zhou, Z. Chen, “Coherent backscattering of optical second-harmonic generation with long-range surface plasmons,” Opt. Lett. 17, 1268–1270 (1992).
[CrossRef] [PubMed]

X. Wang, H. J. Simon, “Directionally scattered optical second-harmonic generation with surface plasmons,” Opt. Lett. 16, 1475–1477 (1991).
[CrossRef] [PubMed]

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, p. 56, formula 45.

Wang, X.

Wang, Y.

Y. Wang, H. J. Simon, “Coherent backscattering of optical second-harmonic generation in silver films,” Phys. Rev. B 47, 13695–13699 (1993).
[CrossRef]

H. J. Simon, Y. Wang, L. B. Zhou, Z. Chen, “Coherent backscattering of optical second-harmonic generation with long-range surface plasmons,” Opt. Lett. 17, 1268–1270 (1992).
[CrossRef] [PubMed]

Zhou, L. B.

Ann. Phys. (1)

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

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

J. Stat. Phys. (1)

A. A. Maradudin, T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

Opt. Commun. (5)

A. Mendoza-Suárez, E. R. Méndez, “Derivation of an impedance boundary condition for one-dimensional, curved, reentrant surfaces,” Opt. Commun. 134, 241–250 (1997).
[CrossRef]

T. T. Ong, V. Celli, A. A. Maradudin, “The impedance of a curved surface,” Opt. Commun. 95, 1–4 (1993).
[CrossRef]

A. A. Maradudin, “The impedance boundary condition at a two-dimensional rough surface,” Opt. Commun. 116, 452–467 (1995).
[CrossRef]

A. A. Maradudin, “The impedance boundary condition for a one-dimensional, curved metal surface,” Opt. Commun. 103, 227–234 (1993).
[CrossRef]

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

Opt. Lett. (2)

Opt. Spektrosk. (1)

A. A. Maradudin, E. R. Méndez, “The utility of an impedance boundary condition in the scattering of light from one-dimensional randomly rough dielectric surfaces,” Opt. Spektrosk. 80, 459–470 (1996) [Opt. Spectrosc. 80, 409–420 (1996)].

Phys. Lett. A (2)

O. A. Aktsipetrov, V. N. Golovkina, O. I. Kapusta, T. A. Leskova, N. N. Novikova, “Anderson localization effects in the second harmonic generation at a weakly rough metal surface,” Phys. Lett. A 170, 231–234 (1992).
[CrossRef]

L. Kuang, H. J. Simon, “Diffusely scattered second-harmonic generation from a silver film due to surface plasmons,” Phys. Lett. A 197, 257–261 (1995).
[CrossRef]

Phys. Rep. (2)

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

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

Phys. Rev. B (2)

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

Y. Wang, H. J. Simon, “Coherent backscattering of optical second-harmonic generation in silver films,” Phys. Rev. B 47, 13695–13699 (1993).
[CrossRef]

Physica A (1)

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

Other (3)

Further information will be available in a paper entitled “Multiple-scattering effects in the second-harmonic generation of light in scattering from a random metal surface in the Kretschmann ATR geometry,” which is currently in preparation by T. A. Leskova, M. Leyva-Lucero, A. A. Maradudin, E. R. Méndez, I. V. Novikov.

A. E. Danese, Advanced Calculus (Allyn and Bacon, Boston, 1965), Vol. I, p. 123.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, p. 56, formula 45.

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Figures (6)

Fig. 1
Fig. 1

Metal film for which the impedance boundary conditions on the surfaces x 3 = D and x 3 = ζ(x 1) are sought.

Fig. 2
Fig. 2

(a) [Re Kij(0)(x 1|ω)exact - Re Kij(0)(x 1|ω)imp/Re Kij(0)(x 1|ω)exact, (b) [Im Kij(0)(x 1|ω)exact - Im Kij(0)(x 1|ω)imp]/Im Kij(0)(x 1|ω)exact for a silver film as functions of the period p when the surface profile function is given by ζ(x 1) = ζ0 cos(2πx 1/p). ∊(ω) = -4.28 + i0.21, ζ0 = 14.142 nm, D = 88 nm, and x 1 = 0. The Kij(0)(x 1|ω)imp are calculated on the basis of Eq. (3.11), while the Kij(0)(x 1|ω)exact are calculated on the basis of Eqs. (5.1) and (5.5).

Fig. 3
Fig. 3

Same as Fig. 2, but for x 1 = 0.392p.

Fig. 4
Fig. 4

Same as Fig. 2, but for x 1 = 0.5p.

Fig. 5
Fig. 5

(a) 〈∂R/∂θ s incoh, (b) 〈∂T/∂θ t incoh as functions of θ s and θ t , respectively, in the case in which a p-polarized beam of light of wavelength λ = 394.7 nm is incident upon a free-standing silver film whose illuminated surface is planar and whose back surface is a one-dimensional random surface characterized by δ = 10 nm and a = 500 nm. ∊(ω) = -4.28 + i0.21, θ0 = 10°, D = 88 nm; solid curve, exact result; dashed curve, result obtained with the use of impedance matrix (3.11).

Fig. 6
Fig. 6

Same as Fig. 4, but with a = 250 nm.

Equations (125)

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

Hx; t=(0, H2x1, x3|ω, 0)exp-iωt,
Ex; t=(E1x1, x3|ω, 0, E3x1, x3|ω)exp-iωt,
E1x1, x3|ω=-icωx3 H2x1, x3|ω,
E3x1, x3|ω=icωx1 H2x1, x3|ω,
2x12+2x32+ωω2c2H2x1, x3|ω=0.
2x12+2x32+ωω2c2Gx1, x3|x1, x3=-4πδx1-x1δx3-x3,
Gx1, x3|x1, x3=2K01dωx1-x12+x3-x321/2
=-dk expikx1-x1-βk, ω|x3-x3|βk, ω,
dω=cω1-ω1/2  Re dω>0, Im dω>0,
βk, ω=k2-ωω2c21/2  Re βk, ω>0, Imβk, ω<0.
Rdx1Rdx3fx1, x32x12+2x32gx1, x3-gx1, x3×2x12+2x32fx1, x3=Cdsfx1, x3n gx1, x3-g(x1,x3) n fx1, x3,
n=nˆ·xˆ1x1+xˆ3x3.
-4πH2x1, x3|ωθD-x3θx3-ζx1=-dx1H2x1, x3|ωx3 Gx1, x3|x1, x3-Gx1, x3|x1, x3x3 H2x1, x3|ωx3=D-ΓdsH2x1, x3|ω1ϕx1-ζ1x1x1+x3×Gx1, x3|x1, x3-Gx1, x3|x1, x31ϕx1-ζ1x1x1+x3×H2x1, x3|ω,
ϕx1=1+ζ1x121/2.
ζnx1dnζx1/dx1n.
ds=ϕx1dx1.
θD-x3θx3-ζx1H2x1, x3|ω=-14π-dx1x3 Gx1, x3|x1, x3x3=D×H1x1|ω-Gx1, x3|x1, x3x3=DL1x1|ω+14π-dx1N Gx1, x3|x1, x3x3=ζx1×H2x1|ω-Gx1, x3|x1, x3x3=ζx1L2x1|ω,
N=-ζ1x1x1+x3
H1x1|ω=H2x1, x3|ω|x3=D,
L1x1|ω=x3 H2x1, x3|ω|x3=D,
H2x1|ω=H2x1, x3|ω|x3=ζx1,
L2x1|ω=N H2x1, x3|ω|x3=ζx1.
H1x1|ω=-12π P -dx1N11x1|x1H1x1|ω+12π-dx1M11x1|x1L1x1|ω+12π-dx1N12x1|x1H2x1|ω-12π-dx1M12x1|x1L2x1|ω,
H2x1|ω=-12π-dx1N21x1|x1H1x1|ω+12π-dx1M21x1|x1L1x1|ω+12π P -dx1N22x1|x1H2x1|ω-12π-dx1M22x1|x1L2x1|ω.
M11x1|x1=Gx1, x3|x1, x3x3=Dx3=D=2K01d|x1-x1|,
M12x1|x1=Gx1, x3|x1, x3x3=Dx3=ζx1=2K01dx1-x12+D-ζx121/2,
M21x1|x1=Gx1, x3|x1, x3x3=ζx1x3=D=2K01dx1-x12+ζx1-D21/2,
M22x1|x1=Gx1, x3|x1, x3x3=ζx1x3=ζx1=2K01dx1-x1)2+ζx1-ζx121/2,
N11x1|x1=x3 Gx1, x3|x1, x3x3=Dx3=D=0,
N12(x1|x1)=N Gx1, x3|x1, x3x3=Dx3=ζx1=2dK11dx1-x12+D-ζx121/2x1-x12+D-ζx121/2×x1-x1ζ1x1+D-ζx1,
N21x1|x1=x3 Gx1, x3|x1, x3x3=ζx1x3=D=2dK11dx1-x12+ζx1-D21/2x1-x12+ζx1-D21/2×ζx1-D,
N22x1|x1=N Gx1, x3|x1, x3x3=ζx1x3=ζx1=2dK11dx1-x12+ζx1-ζx121/2x1-x12+ζx1-ζx121/2×x1-x1ζ1x1+ζx1-ζx1.
-dx1M11x1|x1-M12x1|x1-M21x1|x1M22x1|x1L1x1|ωL2x1|ω=-dx12πδx1-x1-N12x1|x1-N21x1|x1-2πδx1-x1+PN22x1|x1H1x1|ωH2x1|ω,
L1x1|ωL2x1|ω=-dx1K11x1|x1K12x1|x1K21x1|x1K22x1|x1×H1x1|ωH2x1|ω.
-dx1M11x1|x1-M12x1|x1-M21x1|x1M22x1|x1×K11x1|x1K12x1|x1K21x1|x1K21x1|x1=2πδx1-x1-N12x1|x1-N21x1|x1-2πδx1-x1+PN22x1|x1.
Lix1|ω=j=12-dx1Kijx1|x1Hjx1|ω=j=12-dx1Kijx1|x1Hjx1|ω+x1-x1ddx1 Hjx1|ω+12x1-x12×d2dx12 Hjx1|ω+=j=12n=01n! Kijnx1|ωdndx1n Hjx1|ω,
Kijnx1|ω=-dx1Kijx1|x1x1-x1n.
L1x1|ωL2x1|ω=K110x1|ωK120x1|ωK210x1|ωK220x1|ωH1x1|ωH2x1|ω,
Kij0x1|ω=-dx1Kijx1|x1.
-dx1M11x1|x1-M12x1|x1-M21x1|x1M22x1{x1×K110x1|ωK120x1|ωK210x1|ωK220x1|ω=2π 00-2π+-dx1×  0-N12x1|x1-N21x1|x1PN22x1|x1.
n=01n!-dx1M11x1|x1-M12x1|x1-M21x1|x1M22x1|x1x1-x1n×dndx1nK110x1|ωK120x1|ωK210x1|ωK220x1|ω=2π 00-2π+-dx1  0-N12x1|x1-N21x1|x1PN22x1|x1.
A110x1|ω-A120x1|ω-A210x1|ωA220x1|ωK110x1|ωK120x1|ωK210x1|ωK220x1|ω=B11x1|ωB12x1|ωB21x1|ωB22x1|ω-n=11n!A11nx1|ω-A12nx1|ω-A21nx1|ωA22nx1|ωdndx1n×K110x1|ωK120x1|ωK210x1|ωK220x1|ω,
Aijnx1|ω=-dx1Mijx1|x1x1-x1n,
B11x1|ω=2π,
B12x1|ω=--dx1N12x1|x1,
B21x1|ω=--dx1N21x1|x1,
B22x1|ω=-2π+P -dx1N22x1|x1.
A110x1|ω=2πd,
A120x1|ω=2π da exp-βˆ×1-dζ22βζ12a43+1βˆβζ1a2+Od2,
A210x1|ω=2πd exp-β,
A220x1|ω=2πdϕ+2πd3ϕ432ζ12ζ22ϕ3-38ζ22ϕ3-12ζ1ζ3ϕ+Od5,
A111x1|ω=0,
A121x1|ω=2π d2βζ1a3 exp-βˆ-2π d3ζ1ζ22a5 exp-βˆ×31+βˆ+6β2ζ12a2β4ζ14βˆa4+Od4,
A211x1|ω=0,
A221x1|ω=-2π 32d3ζ1ζ2ϕ5+Od5,
A112x1|ω=2πd3,
A122x1|ω=2π d3a3 exp-βˆ1+βˆ+ β2ζ12a2+Od4,
A212x1|ω=2πd3 exp-β1+β,
A222x1|ω=2π d3ϕ3+Od5;
B11x1|ω=2π,
B12x1|ω=-2π βaβˆ exp-βˆ1+dζ22a2βˆβ1-βζ1a2×2βˆ-3βˆ2-βζ1a41βˆ+1βˆ3+Od2,
B21x1|ω=2π exp-β,
B22x1|ω=-2π+πd ζ2ϕ3+3π d3ϕ5154ζ12ζ23ϕ4-58ζ23ϕ4-52ζ1ζ2ζ3ϕ2+ζ44+Od5,
ax1=1+ζ12x1-D-ζx1ζ2x11/2,
βx1|ω=D-ζx1dω,
βˆx1|ω=βx1|ω1-ζ12x1a2x11/2=βx1|ω1-D-ζx1ζ2x11/2ax1.
K0x1|ω=1d100-ϕ+0-1aexp-βˆdββˆ+ϕ2ϕ exp-βdζ22ϕ2+.
K0x1|ω=1daϕa-ϕ exp-β+βˆ1aϕa+ϕ exp-β+βˆ-1a1+βϕβˆ-dζ22ϕ3exp-βˆ2exp-β-1+dζ22ϕ3-βaβˆ exp-β+βˆ.
K0x1|ω=1dωcothDdω-cschDdωcschDdω-cothDdω,
H1,2x1|ω=-dq2π Hˆ1,2qexpiqx1,
L1,2x1|ω=-dq2π Lˆ1,2qexpiqx1;
expγζx1=-dQ2π Iγ|QexpiQx1,
ζx1expγζx1=-dQ2πiQγ Iγ|QexpiQx1.
0=-dk2πexpikx1-βk, ωx3βk, ω×-expβk, ωDβk, ωHˆ1k-Lˆ1k+-dq2π I(βk, ω|k-q) kq-εωω2/c2βk, ω×Hˆ2q-Lˆ2q.
0=-expβk, ωD-dq2π 2πδk-qβq, ωHˆ1q-Lˆ1q+-dq2π I(βk, ω|k-q)×kq-ωω2/c2βk, ω Hˆ2q-Lˆ2q.
0=-dk2πexpikx1+βk, ωx3βk, ωexp-βk, ωD×βk, ωHˆ1k+Lˆ1k--dq2π I(-βk, ω|k-q)×kq-ωω2/c2βk, ω Hˆ2q+Lˆ2q.
0=exp-βk, ωD-dq2π×2πδk-qβq, ωHˆ1q+Lˆ1q--dq2π I(-βk, ω|k-q)kq-ωω2/c2βk, ω×Hˆ2q+Lˆ2q.
-dq2π -2πδk-qexpβk, ωDI(βk, ω|k-q)-2πδk-qexp-βk, ωDI(-βk, ω|k-q) Lˆ1qLˆ2q=-dq2π-2πδk-qβk, ωexpβk, ωDI(βk, ω|k-q)kq-ωω2/c2βk, ω2πδk-qβk, ωexp-βk, ωD-I(-βk, ω|k-q)kq-ωω2/c2βk, ωHˆ1qHˆ2q.
Lˆ1qLˆ2q=-dp2π Kˆ11q|pKˆ12q|pKˆ21q|pKˆ22q|pHˆ1pHˆ2p.
Hˆ1,2q=-dx1 exp-iqx1H1,2x1|ω,
Lˆ1,2q=-dx1 exp-iqx1L1,2x1|ω,
Lix1|ω=j=12-dx1×-dq2π-dp2π expiqx1Kˆijq|p×exp-ipx1Hjx|ω,  i=1, 2.
Kijx1|x1=-dq2π-dp2π expiqx1Kˆijq|p×exp-ipx1.
-dq2π-2πδk-qexpβk, ωDI(βk, ω|k-q)-2πδk-qexp-βk, ωDI(-βk, ω|k-q)Kˆ11q|pKˆ12q|pKˆ21q|pKˆ22q|p=-2πδk-pβk, ωexpβk, ωDI(βk, ω|k-p)kp-ωω2/c2βk, ω2πδk-pβk, ωexp-βk, ωD-I(-βk, ω|k-p)kp-ωω2/c2βk, ω.
Kij0x1|ω=-dx1Kijx1|x1=-dq2π expiqx1Kijq|0.
-dq2π-2πδk-qexpβk, ωDI(βk, ω|k-q)-2πδk-qexp-βk, ωDI(-βk, ω|k-q)Kˆ11q|0Kˆ12q|0Kˆ21q|0Kˆ22q|0=-2πδkexpD/dωdω-I(βk, ω|k)ωω2/c2βk, ω2πδkexp-D/dωdωI(-βk, ω|k)ωω2/c2βk, ω,
Kij0x1|ω=n=- Kˆijnωexpi 2πnp x1.
Kˆijq|0=-dx1 exp-iqx1Kij0x1|ω=n=- 2πδq-2πnpKˆijnω.
Iγ|Q=-dx1 exp-iQx1+γζx1=n=-n-1/2pn+1/2pdx1 exp-iQx1+γζx1=n=- exp-iQnp-p/2p/2dx1 exp-iQx1+γζx1=n=- 2πδQ-2πnpnγ,
nγ=1p-p/2p/2dx1 exp-i 2πnp x1+γζx1.
n=--δm,n expβmωDm-nβmω-δm,n exp-βmωDm-n-βmω×Kˆ11nωKˆ12nωKˆ21nωKˆ22nω=-δm,01dω expD/dω1d2ωmβmωβmωδm,01dω exp-D/dω-1d2ωm-βmωβmω,  m=0, ±1, ±2, ,
βmω=β2πmp, ω=2πmp2+1d2ω1/2 Re βmω>0, Im βmω<0.
ζx1=ζ0 cos2πx1p,
ζx1ζx1=δ2W|x1-x1|,
g|k|=-dx1 exp-ikx1W|x1|.
g|k|=πa exp-a2k2/4,
|dζ2|  |ϕ/d|
A120x1|ω=2 -dx1K01dx1-x12+D-ζx121/2.
1dx1-x12+D-ζx121/2=a2u2-2βζ1u+β21/2+d2ζ1ζ2u3a2u2-2βζ1u+β21/2+d21213 ζ1ζ3+14 ζ22u4-13 βζ3u3a2u2-2βζ1u+β21/2-18ζ12ζ22u6a2u2-2βζ1u+β23/2+Od3,
ax1=1+ζ12x1-D-ζx1ζ2x11/2,
βx1|ω=D-ζx1dω,
ζnx1=dnζx1dx1n.
A120x1|ω=2dI1x1|ω-d2ζ1ζ2I2x1|ω+Od3,
I1x1|ω=-duK0a2u2-2βζ1u+β21/2,
I2x1|ω=-duu3K1a2u2-2βζ1u+β21/2a2u2-2βζ1u+β21/2.
I1x1|ω=2a0dzK0z2+βˆ21/2,
I2x1|ω=2a40dz3βζ1a z2+βζ1a3K1z2+βˆ21/2z2+βˆ21/2,
βˆ=β1-ζ12a21/2=β 1-βdζ21/2a.
0dx Kνx2+β21/2x2+β21/2ν cos xy=π2 β1/2-ν1+y21/2ν-1/4Kν-1/2β1+y21/2
I1x1|ω=πa exp-βˆ,
I2x1|ω=π βζ1a5 exp-βˆ3+1βˆβζ1a2.
A120x1|ω=2π da exp-βˆ1-dζ22βζ12a4×3+1βˆβζ1a2+Od2.
-B12x1|ω=2d2-dx1×K11dx1-x12+D-ζx121/21dx1-x12+D-ζx121/2×x1-x1ζx1+D-ζx1.
x1-x1ζ1x1+D-ζx1=dβ+12dζ2u2+13d2ζ3u3+Od3
-B12x1|ω=2βJ1x1|ω+dζ2βζ1J2x1|ω+J3x1|ω+Od2,
J1x1|ω=-du K1a2u2-2βζ1u+β21/2a2u2-2βζ1u+β21/2,
J2x1|ω=-duu3K1a2u2-2βζ1u+β21/2a2u2-2βζ1u+β2-K1a2u2-2βζ1u+β21/2a2u2-2βζ1u+β23/2=--duu3K1a2u2-2βζ1u+β21/2a2u2-2βζ1u+β2,
J3x1|ω=-duu2K1a2u2-2βζ1u+β21/2a2u2-2βζ1u+β21/2.
J1x1|ω=2a0dz K1z2+βˆ21/2z2+βˆ21/2,
J2x1|ω=-2βζ1a50dzβζ1a2+3z2×K2z2+βˆ21/2z2+βˆ2,
J3x1|ω=2a30dzβζ1a2+z2K1z2+βˆ21/2z2+βˆ21/2.
J1x1|ω=πaβˆexp-βˆ.
J2x1|ω=-π βζ1a5βˆ exp-βˆ31-1βˆ+βζ1a21βˆ+1βˆ2,
J3x1|ω=πa3 exp-βˆ1+βζ1a21βˆ.
B12x1|ω=-2π βaβˆ exp-βˆ-π dζ2a3 exp-βˆ×1-βζ1a22βˆ-3βˆ2-βζ1a41βˆ2+1βˆ3+Od2.

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