## Scattering from fractally corrugated surfaces

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

http://dx.doi.org/10.1364/JOSAA.7.001131

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### Abstract

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

**History**

Original Manuscript: April 10, 1989

Manuscript Accepted: February 6, 1990

Published: June 1, 1990

**Citation**

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

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-7-6-1131

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### References

- B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983), 1–34.
- D. L. Jaggard, X. Sun, “Scattering from bandlimited fractal fibers,” IEEE Trans. Antennas Propag. 37, 1591–1597 (1989). [CrossRef]
- P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
- F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech, Dedham, Mass., 1982), Vol. 2, Chap. 12.
- 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]
- 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]
- 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. 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]
- D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B 31, 179–186 (1983). [CrossRef]
- 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 function that is used here becomes a simple smooth sine function in the limit where D→ 1.
- 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 sinc(υyLy) for angles outside the x–z plane, where 2Ly is the patch size in the y direction. For this more general case, υx= k(sin θi− sin θs cos ϕs), υy= −k sin θs, and υz remains the same.
- 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 (0.05λ) surfaces are considered, the solution is clearly valid except at large scattering angles.
- See, e.g., A. R. Mickelson, D. L. Jaggard, “Electromagnetic wave propagation in almost periodic media,” IEEE Trans. Antenna Propag. AP-27, 34–40 (1979);D. L. Jaggard, A. R. Mickelson, “The reflection of electromagnetic waves from almost periodic structures,” Appl. Phys. 19, 405–418 (1979). [CrossRef]
- J. Teixeira, “Experimental methods for studying fractal aggregates,” in On Growth and Form, H. E. Stanley, N. Ostrowsky, eds. (Nijhoff, Boston, Mass., 1986).
- 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.
- D. L. Jaggard, Y. Kim, “Diffraction by bandlimited fractal screens,” J. Opt. Soc. Am. A 4, 1055–1062 (1987). [CrossRef]
- D. L. Jaggard, “On fractal electrodynamics,” in Recent Advances in Electromagnetic Research, H. N. Kritikos, D. L. Jaggard, eds. (Springer-Verlag, New York, 1990). [CrossRef]
- 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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