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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 7, Iss. 6 — Jun. 1, 1990
  • pp: 1131–1139

Scattering from fractally corrugated surfaces

Dwight L. Jaggard and Xiaoguang Sun  »View Author Affiliations

JOSA A, Vol. 7, Issue 6, pp. 1131-1139 (1990)

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We consider the problem of scattering of optical or electromagnetic waves from a family of irregular rough surfaces characterized by band-limited fractal functions. This method provides a unified and realistic method for examining rough surfaces without the use of random or periodic functions. We relate the angular distribution and the amount of energy in the specularly scattered field to the fractal characteristics of the surfaces by finding their analytical expressions under the Kirchhoff limit and calculating the scattering patterns.

© 1990 Optical Society of America

Dwight L. Jaggard and Xiaoguang Sun, "Scattering from fractally corrugated surfaces," J. Opt. Soc. Am. A 7, 1131-1139 (1990)

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983), 1–34.
  2. D. L. Jaggard and X. Sun, "Scattering from bandlimited fractal fibers," IEEE Trans. Antennas Propag. 37,1591–1597 (1989). [CrossRef]
  3. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  4. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech, Dedham, Mass., 1982), Vol. 2, Chap. 12.
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  6. M. F. Chen and 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]
  7. 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]
  8. J. M. Soto-Crespo and M. Nieto-Vesperinas, "Electromagnetic scattering from very rough random surfaces and deep reflection gratings," J. Opt. Soc. Am. A 6,367–384 (1989). [CrossRef]
  9. D. L. Jordan, R. C. Hollins, and E. Jakeman, "Experimental measurements of non-Gaussian scattering by a fractal diffuser," Appl. Phys. B 31, 179–186 (1983). [CrossRef]
  10. Note that this roughness fractal function is related to but different from the Weierstrass function that has been used previously by the authors and others (e.g., Refs. 2 and 16). The functionthat is used here becomes a simple smooth sine function in the limit where D → 1.
  11. The formulations of this section are correct for observation angles in the x-z plane. Equation (6) [and hence Eqs. (12)-(14)] should be multiplied by smc(vyLy) for angles outside the x-z plane, where 2Ly is the patch size in the y direction. For this more general case, vx = fe(sin sin θS cos øS), vy = -k sin θS, and vz remains the same.
  12. The validity of the Kirchhoff approximation for the chosen parameters in the numerical calculation has been checked against the Kirchhoff criterion, λ2h/Λ3 « cos θ (Ref. 3, page 48). Since small-rms-height (0.05X) surfaces are considered, the solution is clearly valid except at large scattering angles.
  13. See, e.g., A. R. Mickelson and D. L. Jaggard, "Electromagnetic wave propagation in almost periodic media," IEEE Trans. Antenna Propag. AP-27, 34-40 (1979); D. L. Jaggard and A. R. Mickelson, "The reflection of electromagnetic waves from almost periodic structures," Appl. Phys. 19,405–418 (1979). [CrossRef]
  14. J. Teixeira, "Experimental methods for studying fractal aggregates," in On Growth and Form, H. E. Stanley and N. Os-trowsky, eds. (Nijhoff, Boston, Mass., 1986).
  15. Here, our model is a one-dimensional cut through a surface, while in Ref. 14 the dimension Ds is a surface fractal dimension. The relation is given by Ds = D + 1.
  16. D. L. Jaggard and Y. Kim, "Diffraction by bandlimited fractal screens," J. Opt. Soc. Am. A 4, 1055–1062 (1987). [CrossRef]
  17. D. L. Jaggard, "On fractal electrodynamics," in Recent Advances in Electromagnetic Research, H. N. Kritikos and D. L. Jaggard, eds. (Springer-Verlag, New York, 1990). [CrossRef]
  18. D. H. Berman, "Scintillation behind non-Gaussian fractal phase screens," J. Acoust. Soc. Am. 76, Suppl. 1, S94 (A) (1984). [CrossRef]

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