Abstract

A new method for scattering from fractally corrugated conducting surfaces is formulated with use of the extended boundary condition method. Here we expand the fields through generalized Floquet modes and obtain analytical closed-form expressions for the scattering amplitudes for both horizontal and vertical polarization of the incident optical and electromagnetic waves. The accuracy of the proposed method is checked in several ways, such as comparison with approximate methods previously presented in the literature (Rayleigh and Kirchhoff methods) and calculation of the energy-balance parameter. Finally, numerical scattering results from fractal surfaces are provided.

© 1997 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983).
  2. E. Rodriguez, Y. Kim, “A unified perturbation expansion for surface scattering,” Radio Sci. 27, 79–93 (1992).
    [CrossRef]
  3. J. A. Sanchez-Gil, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces,” J. Opt. Soc. Am. A 12, 1547–1558 (1995).
    [CrossRef]
  4. D. P. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
    [CrossRef]
  5. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  6. 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]
  7. M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
    [CrossRef]
  8. M. Nieto-Vesperinas, J. C. Dainty, Scattering in Volumes and Surfaces (Elsevier, Amsterdam, 1990).
  9. D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, London, 1991).
  10. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  11. K. Pak, L. Tsang, C. H. Chan, J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
    [CrossRef]
  12. E. Bahar, M. El-Shenawee, “Enhanced backscatter from one-dimensional random rough surfaces: stationary-phase approximations to full-wave solutions,” J. Opt. Soc. Am. A 12, 151–161 (1995).
    [CrossRef]
  13. T. T. Ong, V. Celli, “Cross-polarized light scattering from a wavy surface with a rough overlayer,” J. Opt. Soc. Am. A 11, 2545–2549 (1994).
    [CrossRef]
  14. M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
    [CrossRef]
  15. H. Faure-Geors, D. Maystre, “Improvement of the Kirschhoff approximation for metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 1675–1685 (1990).
    [CrossRef]
  16. M. Nieto-Vesperinas, J. A. Sanchez-Gil, A. J. Sant, J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15, 1261–1263 (1990).
    [CrossRef] [PubMed]
  17. J. A. DeSanto, “Exact spectral formalism for rough-surface scattering,” J. Opt. Soc. Am. A 2, 2202–2207 (1985).
    [CrossRef]
  18. D. Maystre, Selected Papers on Diffraction Gratings, Vol. MS83 of SPIE Milestones Series (SPIE, Bellingham, Wash., 1993).
  19. J. A. DeSanto, “Scattering from a sinusoid: derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
    [CrossRef]
  20. A. K. Jordan, R. H. Lang, “Electromagnetic scattering patterns from sinusoidal surfaces,” Radio Sci. 14, 1077–1088 (1979).
    [CrossRef]
  21. D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
    [CrossRef]
  22. D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68, 5456–5462 (1990).
    [CrossRef]
  23. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
    [CrossRef]
  24. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  25. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  26. S. L. Chuang, J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
    [CrossRef]
  27. J. A. Sanchez, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  28. D. L. Jaggard, “Fractal Electrodynamics: wave interactions with discretely self-similar structures,” in Symmetry in Electrodynamics, C. Baum, H. Kritikos, eds. (Taylor & Francis, London, 1995).
  29. A. R. Mickelson, D. L. Jaggard, “Electromagnetic wave propagation in almost periodic media,” IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
    [CrossRef]
  30. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Sec. 6.3.
  31. Notation Σp′[], instead of Σpi′[], is used for simplicity here, as well as in all cases where pi′ is a subscript. Similarly for index q(qi), defined in Eq. (9). Furthermore, note in Eq. (10) that the values pi′ (i=1,2,…,M) determine the particular scattering (Floquet) mode p′ under consideration, with corresponding amplitude Bp′ (see Ref. 22, Sec. III, for more details).
  32. In other words, Fourier harmonics corresponding to all tones of fractal function (1).
  33. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Sec. 12.4.
  34. In those applications the matrices associated with the EBCM become ill-conditioned if the axial ratio (ratio of the major to the minor axis) of the scatterer is much greater than ∼5 or if the scatterer has corners.
  35. S. Savaidis, P. Frangos, D. L. Jaggard, K. Hizanidis, “Scattering from fractally corrugated surfaces: an exact approach,” Opt. Lett. 20, 2357–2359 (1995).
    [CrossRef] [PubMed]
  36. Note that the scattering plots given in Figs. 2 and 3 were obtained from the scattering amplitudes Bp′ through the method described in Ref. 20.
  37. Note that for fractal surfaces of small corrugations, the Kirchhoff (approximate) method21 provides the same results (for the scattering coefficients) for both kinds of polarization. This is due to the approximate approach of this method.
  38. K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TM polarization,” IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
    [CrossRef]
  39. J. W. Wright, “Backscattering from capillary waves with application to sea clutter,” IEEE Trans. Antennas Propag. AP-14, 749–754 (1966).
    [CrossRef]

1995 (4)

1994 (1)

1993 (1)

1992 (1)

E. Rodriguez, Y. Kim, “A unified perturbation expansion for surface scattering,” Radio Sci. 27, 79–93 (1992).
[CrossRef]

1991 (1)

1990 (4)

1989 (1)

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. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[CrossRef]

1985 (2)

1981 (1)

S. L. Chuang, J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

1979 (2)

A. K. Jordan, R. H. Lang, “Electromagnetic scattering patterns from sinusoidal surfaces,” Radio Sci. 14, 1077–1088 (1979).
[CrossRef]

A. R. Mickelson, D. L. Jaggard, “Electromagnetic wave propagation in almost periodic media,” IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[CrossRef]

1975 (2)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

J. A. DeSanto, “Scattering from a sinusoid: derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[CrossRef]

1971 (2)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TM polarization,” IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[CrossRef]

1966 (1)

J. W. Wright, “Backscattering from capillary waves with application to sea clutter,” IEEE Trans. Antennas Propag. AP-14, 749–754 (1966).
[CrossRef]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
[CrossRef]

Bahar, E.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Celli, V.

Chan, C. H.

Chen, M. F.

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[CrossRef]

Chuang, S. L.

S. L. Chuang, J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

Dainty, J. C.

M. Nieto-Vesperinas, J. A. Sanchez-Gil, A. J. Sant, J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15, 1261–1263 (1990).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, J. C. Dainty, Scattering in Volumes and Surfaces (Elsevier, Amsterdam, 1990).

D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, London, 1991).

DeSanto, J. A.

J. A. DeSanto, “Exact spectral formalism for rough-surface scattering,” J. Opt. Soc. Am. A 2, 2202–2207 (1985).
[CrossRef]

J. A. DeSanto, “Scattering from a sinusoid: derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[CrossRef]

El-Shenawee, M.

Faure-Geors, H.

Frangos, P.

Fung, A. K.

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[CrossRef]

Hizanidis, K.

Ishimaru, A.

D. P. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
[CrossRef]

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Sec. 12.4.

Jaggard, D. L.

S. Savaidis, P. Frangos, D. L. Jaggard, K. Hizanidis, “Scattering from fractally corrugated surfaces: an exact approach,” Opt. Lett. 20, 2357–2359 (1995).
[CrossRef] [PubMed]

D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
[CrossRef]

D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68, 5456–5462 (1990).
[CrossRef]

A. R. Mickelson, D. L. Jaggard, “Electromagnetic wave propagation in almost periodic media,” IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[CrossRef]

D. L. Jaggard, “Fractal Electrodynamics: wave interactions with discretely self-similar structures,” in Symmetry in Electrodynamics, C. Baum, H. Kritikos, eds. (Taylor & Francis, London, 1995).

Johnson, J.

Jordan, A. K.

A. K. Jordan, R. H. Lang, “Electromagnetic scattering patterns from sinusoidal surfaces,” Radio Sci. 14, 1077–1088 (1979).
[CrossRef]

Kim, Y.

E. Rodriguez, Y. Kim, “A unified perturbation expansion for surface scattering,” Radio Sci. 27, 79–93 (1992).
[CrossRef]

Knotts, M. E.

Kong, J. A.

S. L. Chuang, J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Sec. 6.3.

Lang, R. H.

A. K. Jordan, R. H. Lang, “Electromagnetic scattering patterns from sinusoidal surfaces,” Radio Sci. 14, 1077–1088 (1979).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983).

Maradudin, A. A.

Maystre, D.

H. Faure-Geors, D. Maystre, “Improvement of the Kirschhoff approximation for metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 1675–1685 (1990).
[CrossRef]

D. Maystre, Selected Papers on Diffraction Gratings, Vol. MS83 of SPIE Milestones Series (SPIE, Bellingham, Wash., 1993).

D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, London, 1991).

Mendez, E. R.

Michel, T. R.

Mickelson, A. R.

A. R. Mickelson, D. L. Jaggard, “Electromagnetic wave propagation in almost periodic media,” IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[CrossRef]

Neureuther, A. R.

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TM polarization,” IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[CrossRef]

Nieto-Vesperinas, M.

O’Donnell, K. A.

Ong, T. T.

Pak, K.

Rodriguez, E.

E. Rodriguez, Y. Kim, “A unified perturbation expansion for surface scattering,” Radio Sci. 27, 79–93 (1992).
[CrossRef]

Sanchez, J. A.

Sanchez-Gil, J. A.

Sant, A. J.

Savaidis, S.

Soto-Crespo, J. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Sun, X.

D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68, 5456–5462 (1990).
[CrossRef]

D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
[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]

Tsang, L.

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
[CrossRef]

Winebrenner, D. P.

Wright, J. W.

J. W. Wright, “Backscattering from capillary waves with application to sea clutter,” IEEE Trans. Antennas Propag. AP-14, 749–754 (1966).
[CrossRef]

Zaki, K. A.

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TM polarization,” IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

A. R. Mickelson, D. L. Jaggard, “Electromagnetic wave propagation in almost periodic media,” IEEE Trans. Antennas Propag. AP-27, 34–40 (1979).
[CrossRef]

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TM polarization,” IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[CrossRef]

J. W. Wright, “Backscattering from capillary waves with application to sea clutter,” IEEE Trans. Antennas Propag. AP-14, 749–754 (1966).
[CrossRef]

J. Acoust. Soc. Am. (3)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

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. A. DeSanto, “Scattering from a sinusoid: derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[CrossRef]

J. Appl. Phys. (1)

D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68, 5456–5462 (1990).
[CrossRef]

J. Opt. Soc. Am. A (11)

D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
[CrossRef]

J. A. Sanchez, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

J. A. DeSanto, “Exact spectral formalism for rough-surface scattering,” J. Opt. Soc. Am. A 2, 2202–2207 (1985).
[CrossRef]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[CrossRef]

K. Pak, L. Tsang, C. H. Chan, J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

E. Bahar, M. El-Shenawee, “Enhanced backscatter from one-dimensional random rough surfaces: stationary-phase approximations to full-wave solutions,” J. Opt. Soc. Am. A 12, 151–161 (1995).
[CrossRef]

T. T. Ong, V. Celli, “Cross-polarized light scattering from a wavy surface with a rough overlayer,” J. Opt. Soc. Am. A 11, 2545–2549 (1994).
[CrossRef]

M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
[CrossRef]

H. Faure-Geors, D. Maystre, “Improvement of the Kirschhoff approximation for metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 1675–1685 (1990).
[CrossRef]

J. A. Sanchez-Gil, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces,” J. Opt. Soc. Am. A 12, 1547–1558 (1995).
[CrossRef]

D. P. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Proc. IEEE (2)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
[CrossRef]

S. L. Chuang, J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

Radio Sci. (3)

A. K. Jordan, R. H. Lang, “Electromagnetic scattering patterns from sinusoidal surfaces,” Radio Sci. 14, 1077–1088 (1979).
[CrossRef]

E. Rodriguez, Y. Kim, “A unified perturbation expansion for surface scattering,” Radio Sci. 27, 79–93 (1992).
[CrossRef]

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[CrossRef]

Other (13)

M. Nieto-Vesperinas, J. C. Dainty, Scattering in Volumes and Surfaces (Elsevier, Amsterdam, 1990).

D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, London, 1991).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

D. Maystre, Selected Papers on Diffraction Gratings, Vol. MS83 of SPIE Milestones Series (SPIE, Bellingham, Wash., 1993).

D. L. Jaggard, “Fractal Electrodynamics: wave interactions with discretely self-similar structures,” in Symmetry in Electrodynamics, C. Baum, H. Kritikos, eds. (Taylor & Francis, London, 1995).

Note that the scattering plots given in Figs. 2 and 3 were obtained from the scattering amplitudes Bp′ through the method described in Ref. 20.

Note that for fractal surfaces of small corrugations, the Kirchhoff (approximate) method21 provides the same results (for the scattering coefficients) for both kinds of polarization. This is due to the approximate approach of this method.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Sec. 6.3.

Notation Σp′[], instead of Σpi′[], is used for simplicity here, as well as in all cases where pi′ is a subscript. Similarly for index q(qi), defined in Eq. (9). Furthermore, note in Eq. (10) that the values pi′ (i=1,2,…,M) determine the particular scattering (Floquet) mode p′ under consideration, with corresponding amplitude Bp′ (see Ref. 22, Sec. III, for more details).

In other words, Fourier harmonics corresponding to all tones of fractal function (1).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Sec. 12.4.

In those applications the matrices associated with the EBCM become ill-conditioned if the axial ratio (ratio of the major to the minor axis) of the scatterer is much greater than ∼5 or if the scatterer has corners.

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Figures (4)

Fig. 1
Fig. 1

Geometry of the fractal rough surface scattering problem under consideration.

Fig. 2
Fig. 2

Polar scattering patterns for horizontal (TE) and vertical (TM) polarization with increasing values of the effective rms surface height . Surface parameters are Dr=1+a=1.5, M=2, and b=1.3797; χ=λ/Λ=0.2, θi=60°, and patch size L=40λ. These patterns, as well as those of Fig. 3, are plotted from the scattering amplitudes of the EBCM presented here through the method of Ref. 20.

Fig. 3
Fig. 3

Same as in Fig. 2, except that the scattering patterns are shown here with increasing surface fractal dimension Dr=1+a. Surface parameters are kσ=1.0, M=2, and b=1.3797; χ=λ/Λ=1.0, θi=30°, and patch size L=40λ.

Fig. 4
Fig. 4

Relative reflected power in the specular direction P(0,0), Eq. (31), versus angle of incidence θi, for several values of the fractal dimension Dr=1+a. (a) Horizontal (TE) polarization, (b) vertical (TM) polarization. Surface parameters are M=6, a=0.1, 0.5, and 0.9; kσ=0.3; χ=0.75; b=1.3797.

Tables (2)

Tables Icon

Table 1 Energy Balance Parameter for Horizontal (Vertical) Polarization, Based on the Proposed EBCM Method

Tables Icon

Table 2 Percent Relative Error for the Energy Balance Parameter (EBCM and Rayleigh Methods)

Equations (53)

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

f(x)=σCn=0M-1an sin(Kbnx+ϕn),
C=[2(1-a2)/(1-a2M)]1/2.
Dr=1+a.
k sin θs=k sin θi+nK,
i=1M|ni|=κ.
ψi(r)+-dS[ψ(r)nˆSg(r, r)
-g(r, r)nˆSψ(r, r)]=ψ(r),z>f(x),
=0,z<f(x),
g(r, r)=j4H0(1)(k|r-r|)=j4π-dkx 1kz×exp[jkx(x-x)+jkz|z-z|],
Eiy-dSg(r, r)nˆSEy(r)
=Ey(r),z>f(x),
=0,z<f(x).
dSnˆSEy(r)=kdxq1q2  qMαqs exp(jk sin θqx)=kdx exp(jkixx)q1q2  qMαqs×exp(jqKx),
Ey(r)=Eiy(r)+Eysc(r)=Eiy(r)+p=-Bp exp(jkp+r),
Bp=-jk4πkzp(-1)κ(p) exp(jpΦ)qαqs×exp(-jqΦ)(-1)κ(q)n=0M-1Jpn-qn(kzpσCan),
αqs=j4παqs exp(-jqΦ),
[Bp]=-[QD+(k)][αqs],
QD,pq+(k)=(-1)κ(p)(-1)κ(q) kkzp×exp(jpΦ)n=0M-1Jpn-qn(kzpσCan).
Eiy(r)-p=-Ap exp(jkp-r)=0,
Ap=jk4πkzpexp(jpΦ)q αqs×exp(-jqΦ)n=0M-1Jpn-qn(kzpσCan).
[Ap]=[QD-(k)][αqs],
QD,pq-(k)=kkzpexp(jpΦ)n=0M-1Jpn-qn(kzpσCan).
[Bp]=-[QD+(k)][QD-(k)]-1[Ap].
Hiy(r)+-dSHy(r)nˆSg(r, r)
=Hy(r),z>f(x),
=0,z<f(x).
Hy(r)=Hiy(r)+Hysc(r)=Hy(r)+p=-Bp exp(jkp+r),
[Bp]=[QN+(k)][βqs],
βqs=14πβqs exp(-jqΦ),
QN,pq+(k)=(-1)κ(p)(-1)κ(q) exp(jpΦ) k2-kxpkxqkzp2×n=0M-1Jpn-qn(kzpσCan),
Hiy(r)-p=-Ap exp(jkp-r)=0,
[Ap]=[QN-(k)][βqs],
QN,pq-(k)=exp(jpΦ) k2-kxpkxqkzp2×n=0M-1Jpn-qn(kzpσCan).
[Bp]=[QN+(k)][QN-(k)]-1[Ap],
[αqs]=[QD-(k)]-1[Ap],
[βqs]=[QN-(k)]-1[Ap],
e=p=-|Bp|2 cos θpcos θi,
W=|Bp|2 cos θpcos θi
Eiy(r)-jk4π- dkx 1kzexp(jkxx+jkzz)
×qiαqs-dx exp[-j(kx-kix-qK)x-jkzz]=Ey(r).
Iq=-dx exp[-j(kx-kix-qK)x-jkzz],
exp(-jkzz)=exp-jkzf(x)=exp-jkzσC n=0M-1 an sin(Knx+ϕn)=pi=-(-1)κ(p) exp(jpΦ+jpKx)×n=0M-1 Jpn(kzσCan),
Iq=p=-(-1)κ(p) exp(jpΦ)-dx×expjkixx+j(q+p)×Kx-jkxxn=0M-1 Jpn(kzσCan)=p=-(-1)κ(p) exp(jpΦ)δkx-kix-(q+p)K×n=0M-1 Jpn(kzσCan).
Eiy(r)-jk4πp(-1)κ(p) exp(jpΦ)
×qαqs - dkx 1kzexp(jkxx+jkzz)×δkx-kix-(q+p)n=0M-1 Jpn(kzσCan)=Ey(r),
Eiy(r)-jk4πp(-1)κ(p) exp(jpΦ)
×qαqs 1kz,qpexp(jkx,qpx+jkz,qpz)×n=0M-1 Jpn(kz,qpσCan)=Ey(r),
Hy(r)=exp(jkixx)q1q2  qMβqs exp(jqKx).
dSnˆSg(r, r)=dx-df(x)dxx+zg(r, r),
Hiy(r)-14π- dkxkzexp(jkxx+jkzz)×q βqs- dxkx df(x)dx-kz×exp-j(kx-kix-qK)x-jkzz=Hy(r).
Iq=- dxkx df(x)dx-kz×exp-j(kx-kix-qK)x-jkzz=kx-dx df(x)dxexp-j(kx-kix-qK)×x-jkzz-kz×-dx exp-j(kx-kix-qK)x-jkzz.
-dx df(x)dxexp-j(kx-kix-qK)x-jkzz=j- dx exp(jkxx)kzddx{expj(kix+qK)x-jkzz}+kix+qKkz-dx exp-j(kx-kix-qK)x-jkzz.
Iq1=p=-(-1)κ(p)×exp(jpΦ)jkz- dx exp(-jkxx) ddx×exp{j[kix+(q+p)K]x} n=0M-1Jpn(kzσCan)=p=-(-1)κ(p) exp(jpΦ)-kxkz×δ[kx-kix-(q+p)K]n=0M-1Jpn(kzσCan).

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