Abstract

Electromagnetic scattering from a stack of two one-dimensional rough surfaces separating homogeneous media is modeled with a rigorous integral formulation solved by the method of moments. We present an efficient numerical method for computing the field scattered by such rough layers, in reflection as well as in transmission. We call this method propagation-inside-layer expansion (PILE) due to its straightforward physical interpretation. To our knowledge, it is the first method able to handle problems for this configuration with a huge number of unknowns. We study the convergence of this method versus a coupling condition and validate it by comparison with results from the literature.

© 2006 Optical Society of America

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  1. C. Amra, G. Albrand, and P. Roche, "Theory and application of antiscattering single layers: antiscattering antireflection coatings," Appl. Opt. 16, 2695-2702 (1986).
    [CrossRef]
  2. G. V. Rozhnov, "Electromagnetic wave diffraction by multilayer media with rough interface," Sov. Phys. JETP 69, 646-651 (1989).
  3. C. Amra, "Light scattering from multilayer optics. I. Tools of investigation," J. Opt. Soc. Am. A 11, 197-210 (1994).
    [CrossRef]
  4. C. Amra, "Light scattering from multilayer optics. II. Application to experiment," J. Opt. Soc. Am. A 11, 211-226 (1994).
    [CrossRef]
  5. I. Ohlidal and K. Navratil, "Scattering of light from multilayer systems with rough boundaries," Prog. Opt. 34, 251-334 (1995).
  6. H. Kaplan, "Black coatings are critical in optical design," Photonics Spectra 31, 48-50 (1997).
  7. R. Garcia-Llamas, L. E. Regalado, and C. Amra, "Scattering of light from a two-layer system with a rough surface," J. Opt. Soc. Am. A 16, 2713-2719 (1999).
    [CrossRef]
  8. I. M. Fuks and A. G. Voronovich, "Wave diffraction by rough interfaces in an arbitrary plane-layered medium," Waves Random Media 10, 253-272 (2000).
    [CrossRef]
  9. I. M. Fuks, "Wave diffraction by a rough boundary of an arbitrary plane-layered media," IEEE Trans. Antennas Propag. 49, 630-639 (2001).
    [CrossRef]
  10. Z. Otremba and J. Piskozub, "Modelling of the optical contrast of an oil film on a sea surface," Opt. Express 9, 411-416 (2001).
    [CrossRef] [PubMed]
  11. D. Goodman and Conyers, Ground Penetrating Radar, An Introduction for Archaeologists (Altamira, 1997).
  12. T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
    [CrossRef]
  13. M. Saillard and A. Sentenac, "Rigorous solution for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
    [CrossRef]
  14. K. F. Warnick and W. C. Chew, "Numerical simulation methods for rough surface scattering: topical review," Waves Random Media 11, R1-R30 (2001).
    [CrossRef]
  15. F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
    [CrossRef]
  16. L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001).
    [CrossRef]
  17. J. Q. Lu, A. A. Maradudin, and T. Michel, "Enhanced backscattering from a rough dielectric film on a reflecting substrate," J. Opt. Soc. Am. B 8, 311-318 (1991).
    [CrossRef]
  18. V. Freilikher, E. Kanzieper, and A. A. Maradudin, "Coherent scattering enhancement in systems bounded by rough surfaces," Phys. Rep. 288, 127-204 (1997).
    [CrossRef]
  19. N. Déchamps, "Numerical methods for electromagnetic wave scattering from one-dimensional rough surfaces," Ph.D. thesis (IREENA, Polytech'Nantes, Université de Nantes, 2004).
  20. 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]
  21. E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
    [CrossRef]
  22. M. Saillard and G. Toso, "Electromagnetic scattering from bounded or infinite subsurface bodies," Radio Sci. 32, 1347-1359 (1997).
    [CrossRef]
  23. L. Tsang, C. H. Chan, K. Pak, and H. Sangani, "Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method," IEEE Trans. Antennas Propag. 43, 851-859 (1995).
    [CrossRef]
  24. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).
  25. D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward: a new method for computing low-grazing angle scattering," IEEE Trans. Antennas Propag. 44, 722-729 (1996).
    [CrossRef]
  26. D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward method for scattering from imperfect conductors," IEEE Trans. Antennas Propag. 46, 101-107 (1998).
    [CrossRef]
  27. H. T. Chou and J. T. Johnson, "A novel acceleration of scattering from rough surfaces with the forward-backward method," Radio Sci. 33, 1277-1287 (1998).
    [CrossRef]
  28. A. Iodice, "Forward-backward method for scattering from dielectric rough surfaces," IEEE Trans. Antennas Propag. 50, 901-911 (2002).
    [CrossRef]
  29. D. A. Kapp and G. S. Brown, "A new numerical method for rough-surface scattering calculations," IEEE Trans. Antennas Propag. 44, 711-722 (1996).
    [CrossRef]
  30. L. Tsang, C. H. Chang, and H. Sangani, "A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large-scale random rough surface problems: TM case," Electron. Lett. 29, 166-167 (1993).
    [CrossRef]
  31. Q. Li, C. H. Chan, and L. Tsang, "Monte-Carlo simulations of wave scattering from lossy dielectric random rough surfaces using the physics-based two-grid method and the canonical-grid method," IEEE Trans. Antennas Propag. 47, 752-763 (1999).
    [CrossRef]
  32. V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 36, 414-439 (1990).
    [CrossRef]
  33. N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 7-12 (1992).
    [CrossRef]
  34. R. Coifman, V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: a pedestrian description," IEEE Trans. Antennas Propag. 35, 634-641 (1993).
  35. D. J. Donohue, H.-C. Ku, and D. R. Thompson, "Application of iterative moment-method solutions to ocean surfaces radar scattering," IEEE Trans. Antennas Propag. 46, 121-132 (1998).
    [CrossRef]
  36. J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
    [CrossRef]
  37. A. Madrazo and 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]
  38. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, 1996), Chap. 10.
  39. I. Simonsen and A. A. Maradudin, "Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates," Opt. Commun. 162, 99-111 (1999).
    [CrossRef]
  40. M. Saillard and D. Maystre, "Scattering from metallic and dielectric rough surfaces," J. Opt. Soc. Am. A 7, 982-990 (1990).
    [CrossRef]
  41. M. Saillard and D. Maystre, "Scattering from random rough surfaces: a beam simulation method," J. Opt. (Paris) 19, 173-176 (1988).
    [CrossRef]
  42. X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, "Electromagnetic scattering from a circular target above or below rough surface," Electromagn. Waves 40, 207-227 (2003).
    [CrossRef]

2004 (1)

T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

2003 (1)

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, "Electromagnetic scattering from a circular target above or below rough surface," Electromagn. Waves 40, 207-227 (2003).
[CrossRef]

2002 (1)

A. Iodice, "Forward-backward method for scattering from dielectric rough surfaces," IEEE Trans. Antennas Propag. 50, 901-911 (2002).
[CrossRef]

2001 (4)

M. Saillard and A. Sentenac, "Rigorous solution for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

K. F. Warnick and W. C. Chew, "Numerical simulation methods for rough surface scattering: topical review," Waves Random Media 11, R1-R30 (2001).
[CrossRef]

I. M. Fuks, "Wave diffraction by a rough boundary of an arbitrary plane-layered media," IEEE Trans. Antennas Propag. 49, 630-639 (2001).
[CrossRef]

Z. Otremba and J. Piskozub, "Modelling of the optical contrast of an oil film on a sea surface," Opt. Express 9, 411-416 (2001).
[CrossRef] [PubMed]

2000 (1)

I. M. Fuks and A. G. Voronovich, "Wave diffraction by rough interfaces in an arbitrary plane-layered medium," Waves Random Media 10, 253-272 (2000).
[CrossRef]

1999 (3)

R. Garcia-Llamas, L. E. Regalado, and C. Amra, "Scattering of light from a two-layer system with a rough surface," J. Opt. Soc. Am. A 16, 2713-2719 (1999).
[CrossRef]

Q. Li, C. H. Chan, and L. Tsang, "Monte-Carlo simulations of wave scattering from lossy dielectric random rough surfaces using the physics-based two-grid method and the canonical-grid method," IEEE Trans. Antennas Propag. 47, 752-763 (1999).
[CrossRef]

I. Simonsen and A. A. Maradudin, "Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates," Opt. Commun. 162, 99-111 (1999).
[CrossRef]

1998 (3)

D. J. Donohue, H.-C. Ku, and D. R. Thompson, "Application of iterative moment-method solutions to ocean surfaces radar scattering," IEEE Trans. Antennas Propag. 46, 121-132 (1998).
[CrossRef]

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward method for scattering from imperfect conductors," IEEE Trans. Antennas Propag. 46, 101-107 (1998).
[CrossRef]

H. T. Chou and J. T. Johnson, "A novel acceleration of scattering from rough surfaces with the forward-backward method," Radio Sci. 33, 1277-1287 (1998).
[CrossRef]

1997 (4)

M. Saillard and G. Toso, "Electromagnetic scattering from bounded or infinite subsurface bodies," Radio Sci. 32, 1347-1359 (1997).
[CrossRef]

A. Madrazo and 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]

H. Kaplan, "Black coatings are critical in optical design," Photonics Spectra 31, 48-50 (1997).

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

1996 (2)

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward: a new method for computing low-grazing angle scattering," IEEE Trans. Antennas Propag. 44, 722-729 (1996).
[CrossRef]

D. A. Kapp and G. S. Brown, "A new numerical method for rough-surface scattering calculations," IEEE Trans. Antennas Propag. 44, 711-722 (1996).
[CrossRef]

1995 (3)

L. Tsang, C. H. Chan, K. Pak, and H. Sangani, "Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method," IEEE Trans. Antennas Propag. 43, 851-859 (1995).
[CrossRef]

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]

I. Ohlidal and K. Navratil, "Scattering of light from multilayer systems with rough boundaries," Prog. Opt. 34, 251-334 (1995).

1994 (3)

C. Amra, "Light scattering from multilayer optics. I. Tools of investigation," J. Opt. Soc. Am. A 11, 197-210 (1994).
[CrossRef]

C. Amra, "Light scattering from multilayer optics. II. Application to experiment," J. Opt. Soc. Am. A 11, 211-226 (1994).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

1993 (2)

R. Coifman, V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: a pedestrian description," IEEE Trans. Antennas Propag. 35, 634-641 (1993).

L. Tsang, C. H. Chang, and H. Sangani, "A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large-scale random rough surface problems: TM case," Electron. Lett. 29, 166-167 (1993).
[CrossRef]

1992 (1)

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 7-12 (1992).
[CrossRef]

1991 (1)

1990 (2)

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

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 36, 414-439 (1990).
[CrossRef]

1989 (1)

G. V. Rozhnov, "Electromagnetic wave diffraction by multilayer media with rough interface," Sov. Phys. JETP 69, 646-651 (1989).

1988 (2)

E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

M. Saillard and D. Maystre, "Scattering from random rough surfaces: a beam simulation method," J. Opt. (Paris) 19, 173-176 (1988).
[CrossRef]

1986 (1)

C. Amra, G. Albrand, and P. Roche, "Theory and application of antiscattering single layers: antiscattering antireflection coatings," Appl. Opt. 16, 2695-2702 (1986).
[CrossRef]

Albrand, G.

C. Amra, G. Albrand, and P. Roche, "Theory and application of antiscattering single layers: antiscattering antireflection coatings," Appl. Opt. 16, 2695-2702 (1986).
[CrossRef]

Amra, C.

Ao, C. O.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Brown, G. S.

D. A. Kapp and G. S. Brown, "A new numerical method for rough-surface scattering calculations," IEEE Trans. Antennas Propag. 44, 711-722 (1996).
[CrossRef]

Chan, C. H.

Q. Li, C. H. Chan, and L. Tsang, "Monte-Carlo simulations of wave scattering from lossy dielectric random rough surfaces using the physics-based two-grid method and the canonical-grid method," IEEE Trans. Antennas Propag. 47, 752-763 (1999).
[CrossRef]

L. Tsang, C. H. Chan, K. Pak, and H. Sangani, "Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method," IEEE Trans. Antennas Propag. 43, 851-859 (1995).
[CrossRef]

Chang, C. H.

L. Tsang, C. H. Chang, and H. Sangani, "A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large-scale random rough surface problems: TM case," Electron. Lett. 29, 166-167 (1993).
[CrossRef]

Chew, W. C.

K. F. Warnick and W. C. Chew, "Numerical simulation methods for rough surface scattering: topical review," Waves Random Media 11, R1-R30 (2001).
[CrossRef]

Chou, H. T.

H. T. Chou and J. T. Johnson, "A novel acceleration of scattering from rough surfaces with the forward-backward method," Radio Sci. 33, 1277-1287 (1998).
[CrossRef]

Coifman, R.

R. Coifman, V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: a pedestrian description," IEEE Trans. Antennas Propag. 35, 634-641 (1993).

Conyers,

D. Goodman and Conyers, Ground Penetrating Radar, An Introduction for Archaeologists (Altamira, 1997).

Déchamps, N.

N. Déchamps, "Numerical methods for electromagnetic wave scattering from one-dimensional rough surfaces," Ph.D. thesis (IREENA, Polytech'Nantes, Université de Nantes, 2004).

DeRaad, L. L.

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward method for scattering from imperfect conductors," IEEE Trans. Antennas Propag. 46, 101-107 (1998).
[CrossRef]

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward: a new method for computing low-grazing angle scattering," IEEE Trans. Antennas Propag. 44, 722-729 (1996).
[CrossRef]

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Donohue, D. J.

D. J. Donohue, H.-C. Ku, and D. R. Thompson, "Application of iterative moment-method solutions to ocean surfaces radar scattering," IEEE Trans. Antennas Propag. 46, 121-132 (1998).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Engheta, N.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 7-12 (1992).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Freilikher, V.

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

Freilikher, V. D.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

Fuks, I. M.

I. M. Fuks, "Wave diffraction by a rough boundary of an arbitrary plane-layered media," IEEE Trans. Antennas Propag. 49, 630-639 (2001).
[CrossRef]

I. M. Fuks and A. G. Voronovich, "Wave diffraction by rough interfaces in an arbitrary plane-layered medium," Waves Random Media 10, 253-272 (2000).
[CrossRef]

Gan, Y.-B.

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, "Electromagnetic scattering from a circular target above or below rough surface," Electromagn. Waves 40, 207-227 (2003).
[CrossRef]

Garcia-Llamas, R.

Goodman, D.

D. Goodman and Conyers, Ground Penetrating Radar, An Introduction for Archaeologists (Altamira, 1997).

Guérin, C.-A.

T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Harrington, F.

F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
[CrossRef]

Holliday, D.

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward method for scattering from imperfect conductors," IEEE Trans. Antennas Propag. 46, 101-107 (1998).
[CrossRef]

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward: a new method for computing low-grazing angle scattering," IEEE Trans. Antennas Propag. 44, 722-729 (1996).
[CrossRef]

Iodice, A.

A. Iodice, "Forward-backward method for scattering from dielectric rough surfaces," IEEE Trans. Antennas Propag. 50, 901-911 (2002).
[CrossRef]

Johnson, J. T.

H. T. Chou and J. T. Johnson, "A novel acceleration of scattering from rough surfaces with the forward-backward method," Radio Sci. 33, 1277-1287 (1998).
[CrossRef]

Kanzieper, E.

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

Kaplan, H.

H. Kaplan, "Black coatings are critical in optical design," Photonics Spectra 31, 48-50 (1997).

Kapp, D. A.

D. A. Kapp and G. S. Brown, "A new numerical method for rough-surface scattering calculations," IEEE Trans. Antennas Propag. 44, 711-722 (1996).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Ku, H.-C.

D. J. Donohue, H.-C. Ku, and D. R. Thompson, "Application of iterative moment-method solutions to ocean surfaces radar scattering," IEEE Trans. Antennas Propag. 46, 121-132 (1998).
[CrossRef]

Li, L.-W.

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, "Electromagnetic scattering from a circular target above or below rough surface," Electromagn. Waves 40, 207-227 (2003).
[CrossRef]

Li, Q.

Q. Li, C. H. Chan, and L. Tsang, "Monte-Carlo simulations of wave scattering from lossy dielectric random rough surfaces using the physics-based two-grid method and the canonical-grid method," IEEE Trans. Antennas Propag. 47, 752-763 (1999).
[CrossRef]

Lu, J. Q.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

J. Q. Lu, A. A. Maradudin, and T. Michel, "Enhanced backscattering from a rough dielectric film on a reflecting substrate," J. Opt. Soc. Am. B 8, 311-318 (1991).
[CrossRef]

Madrazo, A.

A. Madrazo and 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]

Maradudin, A. A.

I. Simonsen and A. A. Maradudin, "Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates," Opt. Commun. 162, 99-111 (1999).
[CrossRef]

A. Madrazo and 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, and A. A. Maradudin, "Coherent scattering enhancement in systems bounded by rough surfaces," Phys. Rep. 288, 127-204 (1997).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

J. Q. Lu, A. A. Maradudin, and T. Michel, "Enhanced backscattering from a rough dielectric film on a reflecting substrate," J. Opt. Soc. Am. B 8, 311-318 (1991).
[CrossRef]

Maystre, D.

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

M. Saillard and D. Maystre, "Scattering from random rough surfaces: a beam simulation method," J. Opt. (Paris) 19, 173-176 (1988).
[CrossRef]

Michel, T.

Murphy, W. D.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 7-12 (1992).
[CrossRef]

Navratil, K.

I. Ohlidal and K. Navratil, "Scattering of light from multilayer systems with rough boundaries," Prog. Opt. 34, 251-334 (1995).

O'Donnell, K. A.

Ohlidal, I.

I. Ohlidal and K. Navratil, "Scattering of light from multilayer systems with rough boundaries," Prog. Opt. 34, 251-334 (1995).

Otremba, Z.

Pak, K.

L. Tsang, C. H. Chan, K. Pak, and H. Sangani, "Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method," IEEE Trans. Antennas Propag. 43, 851-859 (1995).
[CrossRef]

Piskozub, J.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Pustilnik, M.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

Regalado, L. E.

Roche, P.

C. Amra, G. Albrand, and P. Roche, "Theory and application of antiscattering single layers: antiscattering antireflection coatings," Appl. Opt. 16, 2695-2702 (1986).
[CrossRef]

Rokhlin, V.

R. Coifman, V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: a pedestrian description," IEEE Trans. Antennas Propag. 35, 634-641 (1993).

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 7-12 (1992).
[CrossRef]

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 36, 414-439 (1990).
[CrossRef]

Rozhnov, G. V.

G. V. Rozhnov, "Electromagnetic wave diffraction by multilayer media with rough interface," Sov. Phys. JETP 69, 646-651 (1989).

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, 1996), Chap. 10.

Saillard, M.

M. Saillard and A. Sentenac, "Rigorous solution for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

M. Saillard and G. Toso, "Electromagnetic scattering from bounded or infinite subsurface bodies," Radio Sci. 32, 1347-1359 (1997).
[CrossRef]

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

M. Saillard and D. Maystre, "Scattering from random rough surfaces: a beam simulation method," J. Opt. (Paris) 19, 173-176 (1988).
[CrossRef]

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

Sangani, H.

L. Tsang, C. H. Chan, K. Pak, and H. Sangani, "Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method," IEEE Trans. Antennas Propag. 43, 851-859 (1995).
[CrossRef]

L. Tsang, C. H. Chang, and H. Sangani, "A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large-scale random rough surface problems: TM case," Electron. Lett. 29, 166-167 (1993).
[CrossRef]

Sentenac, A.

M. Saillard and A. Sentenac, "Rigorous solution for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

Simonsen, I.

I. Simonsen and A. A. Maradudin, "Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates," Opt. Commun. 162, 99-111 (1999).
[CrossRef]

St-Cyr, G. J.

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward method for scattering from imperfect conductors," IEEE Trans. Antennas Propag. 46, 101-107 (1998).
[CrossRef]

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward: a new method for computing low-grazing angle scattering," IEEE Trans. Antennas Propag. 44, 722-729 (1996).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Thompson, D. R.

D. J. Donohue, H.-C. Ku, and D. R. Thompson, "Application of iterative moment-method solutions to ocean surfaces radar scattering," IEEE Trans. Antennas Propag. 46, 121-132 (1998).
[CrossRef]

Thorsos, E. I.

E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

Toso, G.

M. Saillard and G. Toso, "Electromagnetic scattering from bounded or infinite subsurface bodies," Radio Sci. 32, 1347-1359 (1997).
[CrossRef]

Tsang, L.

Q. Li, C. H. Chan, and L. Tsang, "Monte-Carlo simulations of wave scattering from lossy dielectric random rough surfaces using the physics-based two-grid method and the canonical-grid method," IEEE Trans. Antennas Propag. 47, 752-763 (1999).
[CrossRef]

L. Tsang, C. H. Chan, K. Pak, and H. Sangani, "Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method," IEEE Trans. Antennas Propag. 43, 851-859 (1995).
[CrossRef]

L. Tsang, C. H. Chang, and H. Sangani, "A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large-scale random rough surface problems: TM case," Electron. Lett. 29, 166-167 (1993).
[CrossRef]

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Vassiliou, M. S.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 7-12 (1992).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Voronovich, A. G.

I. M. Fuks and A. G. Voronovich, "Wave diffraction by rough interfaces in an arbitrary plane-layered medium," Waves Random Media 10, 253-272 (2000).
[CrossRef]

Wandzura, S.

R. Coifman, V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: a pedestrian description," IEEE Trans. Antennas Propag. 35, 634-641 (1993).

Wang, C.-F.

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, "Electromagnetic scattering from a circular target above or below rough surface," Electromagn. Waves 40, 207-227 (2003).
[CrossRef]

Wang, X.

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, "Electromagnetic scattering from a circular target above or below rough surface," Electromagn. Waves 40, 207-227 (2003).
[CrossRef]

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K. F. Warnick and W. C. Chew, "Numerical simulation methods for rough surface scattering: topical review," Waves Random Media 11, R1-R30 (2001).
[CrossRef]

West, C. S.

Yurkevich, I.

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

Appl. Opt. (1)

C. Amra, G. Albrand, and P. Roche, "Theory and application of antiscattering single layers: antiscattering antireflection coatings," Appl. Opt. 16, 2695-2702 (1986).
[CrossRef]

Electromagn. Waves (1)

X. Wang, C.-F. Wang, Y.-B. Gan, and L.-W. Li, "Electromagnetic scattering from a circular target above or below rough surface," Electromagn. Waves 40, 207-227 (2003).
[CrossRef]

Electron. Lett. (1)

L. Tsang, C. H. Chang, and H. Sangani, "A banded matrix iterative approach to Monte Carlo simulations of scattering of waves by large-scale random rough surface problems: TM case," Electron. Lett. 29, 166-167 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (10)

Q. Li, C. H. Chan, and L. Tsang, "Monte-Carlo simulations of wave scattering from lossy dielectric random rough surfaces using the physics-based two-grid method and the canonical-grid method," IEEE Trans. Antennas Propag. 47, 752-763 (1999).
[CrossRef]

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 7-12 (1992).
[CrossRef]

R. Coifman, V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: a pedestrian description," IEEE Trans. Antennas Propag. 35, 634-641 (1993).

D. J. Donohue, H.-C. Ku, and D. R. Thompson, "Application of iterative moment-method solutions to ocean surfaces radar scattering," IEEE Trans. Antennas Propag. 46, 121-132 (1998).
[CrossRef]

L. Tsang, C. H. Chan, K. Pak, and H. Sangani, "Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method," IEEE Trans. Antennas Propag. 43, 851-859 (1995).
[CrossRef]

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward: a new method for computing low-grazing angle scattering," IEEE Trans. Antennas Propag. 44, 722-729 (1996).
[CrossRef]

D. Holliday, L. L. DeRaad, Jr., and G. J. St-Cyr, "Forward-backward method for scattering from imperfect conductors," IEEE Trans. Antennas Propag. 46, 101-107 (1998).
[CrossRef]

A. Iodice, "Forward-backward method for scattering from dielectric rough surfaces," IEEE Trans. Antennas Propag. 50, 901-911 (2002).
[CrossRef]

D. A. Kapp and G. S. Brown, "A new numerical method for rough-surface scattering calculations," IEEE Trans. Antennas Propag. 44, 711-722 (1996).
[CrossRef]

I. M. Fuks, "Wave diffraction by a rough boundary of an arbitrary plane-layered media," IEEE Trans. Antennas Propag. 49, 630-639 (2001).
[CrossRef]

J. Acoust. Soc. Am. (1)

E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

J. Comput. Phys. (1)

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 36, 414-439 (1990).
[CrossRef]

J. Opt. (Paris) (1)

M. Saillard and D. Maystre, "Scattering from random rough surfaces: a beam simulation method," J. Opt. (Paris) 19, 173-176 (1988).
[CrossRef]

J. Opt. Soc. Am. A (5)

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

Opt. Commun. (2)

A. Madrazo and 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]

I. Simonsen and A. A. Maradudin, "Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates," Opt. Commun. 162, 99-111 (1999).
[CrossRef]

Opt. Express (1)

Photonics Spectra (1)

H. Kaplan, "Black coatings are critical in optical design," Photonics Spectra 31, 48-50 (1997).

Phys. Rep. (1)

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

Phys. Rev. B (1)

J. A. Sánchez-Gil, A. A. Maradudin, J. Q. Lu, V. D. Freilikher, M. Pustilnik, and I. Yurkevich, "Scattering of electromagnetic waves from a bounded medium with random surface," Phys. Rev. B 50, 15353-15368 (1994).
[CrossRef]

Prog. Opt. (1)

I. Ohlidal and K. Navratil, "Scattering of light from multilayer systems with rough boundaries," Prog. Opt. 34, 251-334 (1995).

Radio Sci. (2)

M. Saillard and G. Toso, "Electromagnetic scattering from bounded or infinite subsurface bodies," Radio Sci. 32, 1347-1359 (1997).
[CrossRef]

H. T. Chou and J. T. Johnson, "A novel acceleration of scattering from rough surfaces with the forward-backward method," Radio Sci. 33, 1277-1287 (1998).
[CrossRef]

Sov. Phys. JETP (1)

G. V. Rozhnov, "Electromagnetic wave diffraction by multilayer media with rough interface," Sov. Phys. JETP 69, 646-651 (1989).

Waves Random Media (4)

I. M. Fuks and A. G. Voronovich, "Wave diffraction by rough interfaces in an arbitrary plane-layered medium," Waves Random Media 10, 253-272 (2000).
[CrossRef]

T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

M. Saillard and A. Sentenac, "Rigorous solution for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

K. F. Warnick and W. C. Chew, "Numerical simulation methods for rough surface scattering: topical review," Waves Random Media 11, R1-R30 (2001).
[CrossRef]

Other (6)

F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
[CrossRef]

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

N. Déchamps, "Numerical methods for electromagnetic wave scattering from one-dimensional rough surfaces," Ph.D. thesis (IREENA, Polytech'Nantes, Université de Nantes, 2004).

D. Goodman and Conyers, Ground Penetrating Radar, An Introduction for Archaeologists (Altamira, 1997).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).

Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, 1996), Chap. 10.

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Figures (13)

Fig. 1
Fig. 1

Geometry of the problem and contour integration paths.

Fig. 2
Fig. 2

Physical interpretation of [ ( Z U ) 1 C U ( Z L ) 1 C L ] p ( Z U ) 1 .

Fig. 3
Fig. 3

Norm M c sr versus thickness H for the dielectric (DI) case: TE (top) and TM (bottom) polarizations, two plane surfaces, Δ x = 0.03 λ , ε r 0 = 1 , ε r 1 = 2.5 + 0.01 i , and ε r 2 = 8 .

Fig. 4
Fig. 4

Norm M c sr versus thickness H for the perfectly conducting (PC) case: TE (top) and TM (bottom) polarizations, two plane surfaces, Δ x = 0.03 λ , ε r 0 = 1 , ε r 1 = 2.5 + 0.01 i , and ε r 2 = i .

Fig. 5
Fig. 5

Norm M c sr versus height rms σ h for one rough surface and one plane surface: upper rough (solid curves) or lower rough (dashed curves), dielectric case for both polarizations, H = 1.5 λ , L = 12 λ , Δ x = 0.03 λ , ε r 0 = 1 , ε r 1 = 2.5 + 0.01 i , and ε r 2 = 8 .

Fig. 6
Fig. 6

Norm M c sr versus height rms σ h for both rough surfaces: identical surfaces (solid curves) or uncorrelated surfaces (dashed curves), dielectric case for both polarizations, H = 1.5 λ , L = 12 λ , Δ x = 0.03 λ , ε r 0 = 1 , ε r 1 = 2.5 + 0.01 i , and ε r 2 = 8 .

Fig. 7
Fig. 7

Norm M c sr versus total length L for perfectly conducting and dielectric cases: TE polarization, H = 0.3 λ and Δ x = 0.03 λ for both plane surfaces.

Fig. 8
Fig. 8

Influence of preconditioning: norm M c sr versus total length L (squares) and norm ( I M 0 ) 1 ( M c M 0 ) sr with b = 51 (triangles), for the perfectly conducting case and TE polarization for both plane surfaces.

Fig. 9
Fig. 9

rms error for different orders P of the PILE method. The same parameters are used as those of Fig. 9 in Ref. [22]: ε r 0 = 1 , ε r 1 = 2.5 + 0.01 i , and ε r 2 = 8 . Parameters of rough surfaces are L = 70 λ , Δ x = 0.03 λ , σ h + = 0.01 λ , σ p + = 0.014 , σ h = 0.35 λ , σ p + = 0.49 , and H = 1.5 λ , and TE polarization is used. Parameters of the Thorsos tapered incident wave are θ i = 30 ° and g = L 6 11.7 λ .

Fig. 10
Fig. 10

Bistatic cross section in TE polarization. The dashed curve is the result of Fig. 9 in Ref. [22], and the solid curve represents the PILE method at order 5.

Fig. 11
Fig. 11

Bistatic cross section in TE (top) and TM (bottom) polarizations. The dashed curves are the result of Fig. 11 in Ref. [22], and the solid curves represent the PILE method at order 5.

Fig. 12
Fig. 12

Incoherent bistatic cross section in TE polarization. The dashed curve is the result of Fig. 3 in Ref. [39], and the solid curve represents the PILE method at order 15.

Fig. 13
Fig. 13

Comparison of the modulus of the field inside the layer for different orders of the PILE method. The parameters are the same as those in Fig. 12.

Equations (41)

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x n = L 2 + ( n 1 2 ) Δ x , ζ ± ( x n ) , n = 1 , , N ,
ψ 0 ( r ) r S + = ψ 1 ( r ) r S + ,
ψ 0 ( r ) n + r S + = 1 ρ 10 ψ 1 ( r ) n + r S + ,
ψ 1 ( r ) r S = ψ 2 ( r ) r S ,
ψ 1 ( r ) n r S = 1 ρ 21 ψ 2 ( r ) n r S ,
n ± = ζ ± x ̂ + z ̂ 1 + ( ζ ± ) 2 ,
g j ( r , r ) = i 4 H 0 ( 1 ) ( k j r r ) ,
X 4 N × 1 = [ X + X ] ,
X + t = [ ψ 0 ( r 1 + ) ψ 0 ( r N + ) ψ 0 ( r 1 + ) n + ψ 0 ( r N + ) n + ] ,
X t = [ ψ 1 ( r 1 ) ψ 1 ( r N ) ψ 1 ( r 1 ) n ψ 1 ( r N ) n ] ,
Z 4 N × 4 N = [ Z U C U C L Z L ] ,
Z 2 N × 2 N U = [ A + B + C + ρ 10 D + ] , Z 2 N × 2 N L = [ A B C ρ 21 D ] ,
C 2 N × 2 N U = [ 0 0 E F ] , C 2 N × 2 N L = [ G ρ 10 H 0 0 ] .
Z N × N L = B , C 2 N × N U = [ 0 F ] , C N × 2 N L = [ G ρ 10 H ] ,
Z N × N L = A , C 2 N × N U = [ 0 E ] , C N × 2 N L = [ G ρ 10 H ] .
ψ sc ( r ) = S + d s + [ ψ 0 ( r + ) g 1 ( r + , r ) n + g 1 ( r + , r ) ψ 0 ( r + ) n + ] S d s [ ψ 1 ( r ) g 1 ( r , r ) n g 1 ( r , r ) ψ 1 ( r ) n ] .
Z 1 = [ T U V W ] ,
T = [ Z U C U ( Z L ) 1 C L ] 1 ,
U = T C U ( Z L ) 1 ,
V = ( Z L ) 1 C L T ,
W = ( Z L ) 1 ( Z L ) 1 C L T C U ( Z L ) 1 .
( X + X ) = Z 1 ( b + b ) = ( T b + + U b V b + + W b ) .
X + = T b + = [ Z U C U ( Z L ) 1 C L ] 1 b + .
X + = [ I ( Z U ) 1 C U ( Z L ) 1 C L ] 1 ( Z U ) 1 b + = ( I M c ) 1 ( Z U ) 1 b + ,
M c = ( Z U ) 1 C U ( Z L ) 1 C L .
M c sr = ( Z U ) 1 C U ( Z L ) 1 C L sr < 1
X + ( P ) = ( p = 0 P M c p ) ( Z U ) 1 b + = p = 0 P Y + ( p ) ,
Y + ( 0 ) = ( Z U ) 1 b + , Y + ( p ) = M c Y + ( p 1 ) for p > 0 .
X = V b + = [ ( Z L ) 1 C L ] T b + = [ ( Z L ) 1 C L ] X + .
( I M c ) 1 = [ I M 0 ( M c M 0 ) ] 1 = { ( I M 0 ) [ I ( I M 0 ) 1 ( M c M 0 ) ] } 1 = [ I ( I M 0 ) 1 ( M c M 0 ) ] 1 ( I M 0 ) 1 .
( I M 0 ) 1 ( M c M 0 ) sr < 1 .
( I M 0 ) 1 ( M c M 0 ) sr < M c sr < 1 .
X + ( P ) = { p = 0 P [ ( I M 0 ) 1 ( M c M 0 ) ] p } ( I M 0 ) 1 ( Z U ) 1 b + .
A m n + = { i Δ x k 0 4 H 1 ( 1 ) ( k 0 r n + r m + ) r n + r m + × { ζ + ( x n ) ( x n x m ) [ ζ + ( x n ) ζ + ( x m ) ] } for m n + 1 2 Δ x 4 π ζ + ( x m ) 1 + ( ζ + ( x m ) ) 2 for m = n } ,
B m n + = { γ n + i Δ x 4 H 0 ( 1 ) ( k 0 r n + r m + ) for m n γ n + i Δ x 4 [ 1 + i 2 π log ( e γ 2 k 0 Δ x 2 e γ n + ) ] for m = n } ,
C m n + = { i Δ x k 1 4 H 1 ( 1 ) ( k 1 r n + r m + ) r n + r m + × { ζ + ( x n ) ( x n x m ) [ ζ + ( x n ) ζ + ( x m ) ] } for m n 1 2 Δ x 4 π ζ + ( x m ) 1 + ( ζ + ( x m ) ) 2 for m = n } ,
D m n + = { γ n + i Δ x 4 H 0 ( 1 ) ( k 1 r n + r m + ) for m n γ n + i Δ x 4 [ 1 + i 2 π log ( e γ 2 k 1 Δ x 2 e γ n + ) ] for m = n } ,
E m n = + i Δ x k 1 4 H 1 ( 1 ) ( k 1 r n r m + ) r n r m + { ζ ( x n ) ( x n x m ) [ ζ ( x n ) ζ + ( x m ) ] } m , n ,
F m n = γ n i Δ x 4 H 0 ( 1 ) ( k 1 r m + r n ) m , n ,
G m n = + i Δ x k 1 4 H 1 ( 1 ) ( k 1 r n + r m ) r n + r m { ζ + ( x n ) ( x n x m ) [ ζ + ( x n ) ζ ( x m ) ] } m , n ,
H m n = γ n + i Δ x 4 H 0 ( 1 ) ( k 1 r m r n + ) m , n ,

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