Abstract

Recently, rigorous numerical techniques for treating light scattering problems with one-dimensional rough surfaces have been developed. In their usual formulation, these techniques are based on the solution of two coupled integral equations and are applicable only to surfaces whose profiles can be described by single-valued functions of a coordinate in the mean plane of the surface. In this paper we extend the applicability of the integral equation method to surfaces with multivalued profiles. A procedure for finding a parametric description of a given profile is described, and the scattering equations are established within the framework of this formalism. We then present some results of light scattering from a sequence of one-dimensional flat surfaces with defects in the form of triadic Koch curves. Beyond a certain order of the prefractal, the scattering patterns become stationary (within the numerical accuracy of the method). It can then be argued that the results obtained correspond to a surface with a fractal structure. These constitute, to our knowledge, the first rigorous calculations of light scattering from a reentrant fractal surface.

© 1997 Optical Society of America

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References

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  1. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  2. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  3. B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
    [CrossRef]
  4. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surface,” Opt. Lett. 12, 979–981 (1987).
    [CrossRef] [PubMed]
  5. 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]
  6. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [CrossRef]
  7. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  8. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 63–100.
    [CrossRef]
  9. K. K. Mei, J. G. Van Bladel, “Scattering by perfectly-conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
    [CrossRef]
  10. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 101–121.
  11. F. Moreno, F. González, J. M. Saiz, P. J. Valle, D. L. Jordan, “Experimental study of copolarized light scattering by spherical metallic particles in conducting flat substrates,” J. Opt. Soc. Am. A 10, 141–149 (1993).
    [CrossRef]
  12. P. J. Valle, F. González, F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat surfaces: study by means of the extinction theorem,” Appl. Opt. 33, 512–523 (1994).
    [CrossRef] [PubMed]
  13. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  14. T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, Mass., 1964), pp. 169–176.
  15. Ref. 14, p. 315.
  16. A. Mendoza-Suárez, “Métodos regurosos para el esparcimiento de luz por superficies rugosas y medios estratificados con perfiles arbitrarios,” Ph.D. dissertation (Centro de Investigacion Cientifica y de Educación Superior de Ensenada (1996). Procedures to deal with singular points have also been discussed by A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).
  17. Yon-Lin Kok, “General solution to the multiple-metallic-grooves scattering problem: the fast-polarization case,” Appl. Opt. 32, 2573–2581 (1993).
    [CrossRef] [PubMed]
  18. E. Jakeman, J. G. McWhirter, “Correlation function dependence of scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
    [CrossRef]
  19. C. L. Rino, “A power law phase screen model for ionospheric scintillations 1. Weak scatter,” Radio Sci. 14, 1135–1146 (1979).
    [CrossRef]
  20. C. L. Rino, “A power law phase screen model for ionospheric scintillations 2. Strong scatter,” Radio Sci. 14, 1147–1155 (1979).
    [CrossRef]
  21. M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
    [CrossRef]
  22. M. V. Berry, Z. V. Lewis, “On the Weierstrauss-Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
    [CrossRef]
  23. M. V. Berry, T. M. Blackwell, “Diffractal echoes,” J. Phys. A 14, 3101–3110 (1981).
    [CrossRef]
  24. E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
    [CrossRef]
  25. E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
    [CrossRef]
  26. E. Jakeman, “Fraunhofer scattering by a subfractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
    [CrossRef]
  27. D. L. Jaggard, Y. Kim, “Diffraction by band-limited fractal screens,” J. Opt. Soc. Am. A 4, 1055–1062 (1987).
    [CrossRef]
  28. B. J. West, “Sensing scaled scintillations,” J. Opt. Soc. Am. A 7, 1074–1100 (1990).
    [CrossRef]
  29. D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
    [CrossRef]
  30. C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
    [CrossRef]
  31. D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68, 5456–5462 (1990).
    [CrossRef]
  32. A. A. Maradudin, T. Michel, “Role of the surface height correlation function in the enhanced backscattering of light from random metallic surfaces,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 233–250 (1991).
    [CrossRef]
  33. 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]
  34. J. Feder, Fractals (Plenum, New York, 1988), p. 15.
  35. D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed., (McGraw-Hill, New York, 1972), p. 6–138.

1996

C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

1995

1994

1993

1990

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

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

B. J. West, “Sensing scaled scintillations,” J. Opt. Soc. Am. A 7, 1074–1100 (1990).
[CrossRef]

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

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

1989

1988

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]

1987

1983

E. Jakeman, “Fraunhofer scattering by a subfractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
[CrossRef]

1982

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
[CrossRef]

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[CrossRef]

1981

M. V. Berry, T. M. Blackwell, “Diffractal echoes,” J. Phys. A 14, 3101–3110 (1981).
[CrossRef]

1980

M. V. Berry, Z. V. Lewis, “On the Weierstrauss-Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[CrossRef]

1979

C. L. Rino, “A power law phase screen model for ionospheric scintillations 1. Weak scatter,” Radio Sci. 14, 1135–1146 (1979).
[CrossRef]

C. L. Rino, “A power law phase screen model for ionospheric scintillations 2. Strong scatter,” Radio Sci. 14, 1147–1155 (1979).
[CrossRef]

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

1977

E. Jakeman, J. G. McWhirter, “Correlation function dependence of scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

1963

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly-conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

Apostol, T. M.

T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, Mass., 1964), pp. 169–176.

Berry, M. V.

M. V. Berry, T. M. Blackwell, “Diffractal echoes,” J. Phys. A 14, 3101–3110 (1981).
[CrossRef]

M. V. Berry, Z. V. Lewis, “On the Weierstrauss-Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[CrossRef]

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

Blackwell, T. M.

M. V. Berry, T. M. Blackwell, “Diffractal echoes,” J. Phys. A 14, 3101–3110 (1981).
[CrossRef]

Feder, J.

J. Feder, Fractals (Plenum, New York, 1988), p. 15.

Frangos, P.

González, F.

Hizanidis, K.

Jaggard, D. L.

Jakeman, E.

E. Jakeman, “Fraunhofer scattering by a subfractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
[CrossRef]

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
[CrossRef]

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Correlation function dependence of scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

Jordan, D. L.

Kachoyan, B. J.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

Kim, Y.

Kok, Yon-Lin

Lewis, Z. V.

M. V. Berry, Z. V. Lewis, “On the Weierstrauss-Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[CrossRef]

Macaskill, C.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

Maradudin, A. A.

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

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef]

A. A. Maradudin, T. Michel, “Role of the surface height correlation function in the enhanced backscattering of light from random metallic surfaces,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 233–250 (1991).
[CrossRef]

Maystre, D.

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

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 63–100.
[CrossRef]

McGurn, A. R.

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

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, “Correlation function dependence of scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

Mei, K. K.

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly-conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

Méndez, E. R.

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

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef]

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

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

Mendoza-Suárez, A.

A. Mendoza-Suárez, “Métodos regurosos para el esparcimiento de luz por superficies rugosas y medios estratificados con perfiles arbitrarios,” Ph.D. dissertation (Centro de Investigacion Cientifica y de Educación Superior de Ensenada (1996). Procedures to deal with singular points have also been discussed by A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).

Michel, T.

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

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef]

A. A. Maradudin, T. Michel, “Role of the surface height correlation function in the enhanced backscattering of light from random metallic surfaces,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 233–250 (1991).
[CrossRef]

Moreno, F.

Nieto-Vesperinas, M.

O’Donnell, K. A.

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

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

Rino, C. L.

C. L. Rino, “A power law phase screen model for ionospheric scintillations 1. Weak scatter,” Radio Sci. 14, 1135–1146 (1979).
[CrossRef]

C. L. Rino, “A power law phase screen model for ionospheric scintillations 2. Strong scatter,” Radio Sci. 14, 1147–1155 (1979).
[CrossRef]

Saillard, M.

Saiz, J. M.

Savaidis, S.

Sheppard, C. J. R.

C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

Soto-Crespo, J. M.

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]

Valle, P. J.

Van Bladel, J. G.

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly-conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

Vincent, P.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 101–121.

West, B. J.

Ann. Phys. (N.Y.)

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

Appl. Opt.

IEEE Trans. Antennas Propag.

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly-conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

J. Acoust. Soc. Am.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[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. Appl. Phys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Correlation function dependence of scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

M. V. Berry, T. M. Blackwell, “Diffractal echoes,” J. Phys. A 14, 3101–3110 (1981).
[CrossRef]

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
[CrossRef]

Opt. Acta

E. Jakeman, “Fraunhofer scattering by a subfractal diffuser,” Opt. Acta 30, 1207–1212 (1983).
[CrossRef]

Opt. Commun.

C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

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

Opt. Lett.

Proc. R. Soc. London Ser. A

M. V. Berry, Z. V. Lewis, “On the Weierstrauss-Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[CrossRef]

Radio Sci.

C. L. Rino, “A power law phase screen model for ionospheric scintillations 1. Weak scatter,” Radio Sci. 14, 1135–1146 (1979).
[CrossRef]

C. L. Rino, “A power law phase screen model for ionospheric scintillations 2. Strong scatter,” Radio Sci. 14, 1147–1155 (1979).
[CrossRef]

Other

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 63–100.
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 101–121.

T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, Mass., 1964), pp. 169–176.

Ref. 14, p. 315.

A. Mendoza-Suárez, “Métodos regurosos para el esparcimiento de luz por superficies rugosas y medios estratificados con perfiles arbitrarios,” Ph.D. dissertation (Centro de Investigacion Cientifica y de Educación Superior de Ensenada (1996). Procedures to deal with singular points have also been discussed by A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).

A. A. Maradudin, T. Michel, “Role of the surface height correlation function in the enhanced backscattering of light from random metallic surfaces,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 233–250 (1991).
[CrossRef]

J. Feder, Fractals (Plenum, New York, 1988), p. 15.

D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed., (McGraw-Hill, New York, 1972), p. 6–138.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the scattering problem. The surface profile R(s) defines the boundary between regions I and II. Region I is a vacuum, whereas region II is filled with a homogeneous and isotropic medium characterized by its complex dielectric constant ∊(ω). The surface is illuminated by a plane wave component with wave vector k inc. The angles of incidence and scattering are denoted by θ0 and θs, respectively.

Fig. 2
Fig. 2

Surface profile defined by a cylinder on a flat substrate.

Fig. 3
Fig. 3

Scattering patterns of a perfectly conducting cylinder on a perfectly conducting flat substrate. (a) S-polarized scattering pattern corresponding to a cylinder of diameter D = 2λ, illuminated by a Gaussian beam making an angle of incidence θ0 = 34°. (b) P-polarized scattering pattern corresponding to a cylinder of diameter D = λ/2 illuminated by a Gaussian beam at normal incidence.

Fig. 4
Fig. 4

Surface profile defined by a sequence of K rectangular grooves.

Fig. 5
Fig. 5

Scattering by a perfectly conducting surface with twelve rectangular grooves. The grooves have depth h = 0.4λ and width c = 0.8λ, and the incident wave is an s-polarized plane wave with an angle of incidence θ0 = 30°. (a) Surface profile and (b) far-field diffracted amplitude plotted as a function of the scattering angle.

Fig. 6
Fig. 6

Construction of the triadic Koch curve. The segment shown in (a) is the initiator or zeroth generation of the Koch curve. The generator is shown in (b) and represents the prefractal of order 1. The prefractal of order 2 is shown in (c).

Fig. 7
Fig. 7

Central portion of the surface profile consisting of a flat substrate with a triadic Koch prefractal defect. The total length of the surface along the x 1 direction is L x = 20λ and that of the defect is L f = λ. The number of line segments N s that define the prefractal defect are (a) N s = 4, (b) N s = 16, (c) N s = 64, (d) N s = 256, corresponding to orders ν = 1,2, 3, and 4, respectively.

Fig. 8
Fig. 8

Scattering by the samples with a triadic Koch prefractal for θ0 = 0°. Curves (a)–(d) show the calculated modulus of the scattering amplitude versus the scattering angle for the corresponding profiles shown in Fig. 7. The continuous curves correspond to the results obtained with p polarization, whereas those shown by broken curves correspond to s-polarized ones. A wavelength λ = 0.65 µm was assumed, and ∊(ω) = -11.36 + i0.96. The 1/e value of the intercept of the Gaussian incident beam with the x 3 axis was g = 5λ. The thick broken curve in (d) represents the scattering pattern of a Lambertian diffuser.

Fig. 9
Fig. 9

Scattering by samples with a triadic Koch prefractal for θ0 = 20°. Curves (a)–(d) show the calculated modulus of the scattering amplitude versus the scattering angle using the same parameters and notation as in Fig. 8.

Equations (40)

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

Er; t=0, Er, 0exp-iωt,
rsr=--iωcnˆR·rˆER-FR×exp-iωcrˆ·Rds,
rˆ=sin θs, cos θs,
ER=EIR,
FR=nˆR·rEIrr=R·
r=x1,x3.
Rsθs=rsr2θ0,
θ0=22π3/2wωc1-1+2 tan2 θ02w2ω/c2.
θ0=8πωcLx cos θ0,
ER=EincR+limv0+14π×-nˆR·rGIR+vnˆR|rr=R×ER-GIR+vnˆR|RFRds,
0=limv0+-14π×-nˆR·rGIIR+vnˆR|rr=R×ER-GIIR+vnˆR|RFRds.
GIr|r=iπH01ωcr-r,
GIIr|r=iπH01ncωωcr-r.
Hr; t=0, Hr, 0exp-iωt.
rpr=-iωcnˆR·rˆHR-LR×exp-iωcrˆ·Rds,
HR=HIR,
LR=nˆR·rHIrr=R.
Rpθs=rpr2θ0.
HR=HincR+limv0+14π×-nˆR·rGIR+vnˆR|r|r=R}×HR-GIR+vnˆR|RLRds,
0=limv0+-14π×-nˆR·rGIIR+vnˆR|rr=R×HR-(ω)GIIR+vnˆR|RLRds.
Rs=ξs, ηs.
ds2=dξ2+dη2.
sξ=sξj+ξjξ1+dηξdξ21/2dξ,
sη=sηj+ηjη1+dξηdη21/2dη,
N=j=0MNj.
Esm=Esminc+n=1NHmnIEsn-LmnIFsn,
n=1NHmnIIEsn-LmnIIFsn=0,
HmnII=-Δsnc2ωω/c24i-ηsnξsm-ξsn+ξsnηsm-ηsn×H11ncωω/cξsm-ξsn2+ηsm-ηsn21/2ncωω/cξsm-ξsn2+ηsm-ηsn21/2  if mn,  =12+Δs4πξsmηsm-ξsmηsm  if m=n,
LmnII=-Δs4iH01ncωωcξsm-ξsn2+ηsm-ηsn21/2  if mn,  =-Δs4iH01ncωωcΔs2e  if m=n,
rsθs=Δsn=1Niωcηsnsinθs-ξsncosθsEsn-Fsnexp-iωcξsnsinθs+ηsncosθs.
Hsm=Hsminc+n=1NHmnIHsn-LmnILsn,
n=1NHmnIIHsn-ωLmnIILsn=0.
rpθs=Δsn=1Niωcηsnsinθs-ξsncosθsHsn-Lsn×exp-iωcξsnsinθs+ηsncosθs.
Rs=RsAs-sA, 0for SA<S<SO,RsO+-D2 sin2s-sOD, D21-cos2s-sODfor SO<S<SO,RsO+s-sO, 0for SO<S<SB,
Rs=Rsj+s-sj, 0for the horizontal segments,0, ±s-sjfor the vertical segments,
Ps,pθs=14πrs,pθssample-rs,pθsflat.
Nsν=4ν,
lν=Lf3ν.
Rs=RsA+s-sAcos α,  s-sAsin α,
Dp,sν=-π/2π/2rp,sνθs-rp,sν-1θsdθs-π/2π/2rp,sν-1θsdθs,

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