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

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Corrections

Dwight L. Jaggard and Xiaoguang Sun, "Scattering from fractally corrugated surfaces: erratum," J. Opt. Soc. Am. A 8, 699-699 (1991)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-8-4-699

References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983), 1–34.
  2. D. L. Jaggard, X. Sun, “Scattering from bandlimited fractal fibers,” IEEE Trans. Antennas Propag. 37, 1591–1597 (1989).
    [CrossRef]
  3. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  4. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech, Dedham, Mass., 1982), Vol. 2, Chap. 12.
  5. 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]
  6. 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]
  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, 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, 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 function that 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 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.
  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 (0.05λ) surfaces are considered, the solution is clearly valid except at large scattering angles.
  13. 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]
  14. J. Teixeira, “Experimental methods for studying fractal aggregates,” in On Growth and Form, H. E. Stanley, N. Ostrowsky, 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, 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, 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]

1989 (2)

1988 (2)

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]

1987 (1)

1985 (1)

1984 (1)

D. H. Berman, “Scintillation behind non-Gaussian fractal phase screens,” J. Acoust. Soc. Am. 76, Suppl. 1, S94 (A) (1984).
[CrossRef]

1983 (1)

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]

1979 (1)

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]

Beckmann, P.

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

Berman, D. H.

D. H. Berman, “Scintillation behind non-Gaussian fractal phase screens,” J. Acoust. Soc. Am. 76, Suppl. 1, S94 (A) (1984).
[CrossRef]

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]

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]

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech, Dedham, Mass., 1982), Vol. 2, Chap. 12.

Hollins, R. C.

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]

Ishimaru, A.

Jaggard, D. L.

D. L. Jaggard, X. Sun, “Scattering from bandlimited fractal fibers,” IEEE Trans. Antennas Propag. 37, 1591–1597 (1989).
[CrossRef]

D. L. Jaggard, Y. Kim, “Diffraction by bandlimited fractal screens,” J. Opt. Soc. Am. A 4, 1055–1062 (1987).
[CrossRef]

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]

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]

Jakeman, E.

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]

Jordan, D. L.

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]

Kim, Y.

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983), 1–34.

Mickelson, A. R.

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]

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech, Dedham, Mass., 1982), Vol. 2, Chap. 12.

Nieto-Vesperinas, M.

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, “Scattering from bandlimited fractal fibers,” IEEE Trans. Antennas Propag. 37, 1591–1597 (1989).
[CrossRef]

Teixeira, J.

J. Teixeira, “Experimental methods for studying fractal aggregates,” in On Growth and Form, H. E. Stanley, N. Ostrowsky, eds. (Nijhoff, Boston, Mass., 1986).

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]

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech, Dedham, Mass., 1982), Vol. 2, Chap. 12.

Winebrenner, D. P.

Appl. Phys. B (1)

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]

IEEE Trans. Antenna Propag. (1)

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]

IEEE Trans. Antennas Propag. (1)

D. L. Jaggard, X. Sun, “Scattering from bandlimited fractal fibers,” IEEE Trans. Antennas Propag. 37, 1591–1597 (1989).
[CrossRef]

J. Acoust. Soc. Am. (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]

D. H. Berman, “Scintillation behind non-Gaussian fractal phase screens,” J. Acoust. Soc. Am. 76, Suppl. 1, S94 (A) (1984).
[CrossRef]

J. Opt. Soc. Am. A (3)

Radio Sci. (1)

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 (9)

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.

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.

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, “On fractal electrodynamics,” in Recent Advances in Electromagnetic Research, H. N. Kritikos, D. L. Jaggard, eds. (Springer-Verlag, New York, 1990).
[CrossRef]

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983), 1–34.

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Figures (7)

Fig. 1
Fig. 1

Samples of fractally corrugated surfaces: a gently undulating fractal surface (top) and a rough fractal surface (bottom).

Fig. 2
Fig. 2

Sample plots of the band-limited roughness fractal function fr(x) with specified fractal dimensions D increasing from top to bottom. The roughness increases with increasing D. On the left-hand side are curves with a large spatial bandwidth, while on the right-hand side curves with a more restricted bandwidth are shown.

Fig. 3
Fig. 3

Geometry of electromagnetic wave scattering in a plane from a rough surface, where the subscripts i and s indicate parameters associated with incident and scattered waves, respectively. Here k represents the wave vector and θi = 30°.

Fig. 4
Fig. 4

Patterns of the scattering coefficient from fractally corrugated surfaces with fractal dimensions (a) D = 1.05, (b) D = 1.30, (c) D = 1.50, (d) D = 1.70. Note that these are polar plots, and the labels provide a decibel scale for reference.

Fig. 5
Fig. 5

Concept of the conservation of momenta used in scattering mechanism, where β is the spatial frequency of the rough surface.

Fig. 6
Fig. 6

Patterns of the scattering from rough surfaces of fractal dimension D = 1.50 with patch sizes (a) 2.5λ, (b) 10λ, (c) 40λ, (d) 160λ. Note that these are polar plots, and the labels provide a decibel scale for reference.

Fig. 7
Fig. 7

The scattering coefficient versus sin[(θs − 30°)/2], both in decibel scales, for fractal dimensions (a) D = 1.05, (b) D = 1.30, (c) D = 1.50, (d) D = 1.70. The envelope slopes of coupling sidelobes vary monotonically with the fractal dimension, while the background slope is constant for varying D.

Tables (2)

Tables Icon

Table 1 Rms Slopes σs of the Fractal Function in Eq. (1) in Units of K0σ for Different Values of Fractal Dimensions D and Frequency Scaling Parameters b with a Fixed Value of N = 6

Tables Icon

Table 2 Correlation Phase K0Γ of the Fractal Function in Eq. (1) for Different Values of Fractal Dimensions D and Frequency Scaling Parameters b with a Fixed Value of N = 6

Equations (31)

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f r ( x ) = C σ n = 0 N 1 ( D 1 ) n sin ( K 0 b n x + ϕ n ) ,
C = { 2 [ 1 ( D 1 ) 2 ] [ 1 ( D 1 ) 2 N ] } 1 / 2 = { 2 D ( 2 D ) [ 1 ( D 1 ) 2 N ] } 1 / 2
f r ( x ) 1 D 1 f r ( b x ) ,
σ s = K 0 σ { [ 1 ( D 1 ) 2 ] [ 1 ( D 1 ) 2 N ] [ 1 b 2 N ( D 1 ) 2 N ] [ 1 b 2 ( D 1 ) 2 ] } 1 / 2 .
ρ ( τ ) = f r ( x ) , f r ( x + τ ) f r ( x ) , f r ( x ) = [ 1 ( D 1 ) 2 ] [ 1 ( D 1 ) 2 N ] n = 0 N 1 ( D 1 ) 2 n cos ( K 0 b n τ ) ,
E sc = ikL exp ( i k R 0 ) 2 π R 0 L L ( p f r q ) exp [ i υ x x + i υ z f r ( x ) ] d x ,
p = ( 1 R ) sin θ i + ( 1 + R ) sin θ s ,
q = ( 1 + R ) cos θ s ( 1 R ) cos θ i ,
υ x = k ( sin θ i sin θ s ) ,
υ z = k ( cos θ i + cos θ s ) ,
R + = 1 R = 1 ,
E sc 0 = i 2 k L 2 exp ( i k R 0 ) cos θ i π R 0 .
γ = E sc E sc 0 = 1 4 L cos θ i ( ( q + p υ x υ z ) × L L exp [ i υ x x + i υ z f r ( x ) ] d x { i p υ z exp [ i υ x x + i υ z f r ( x ) ] } L L ) .
γ ± = ± sec θ i 1 + cos ( θ i + θ s ) cos θ i + cos θ s × m 1 , m 2 , , m N 1 = + exp ( i n = 0 N 1 m n ϕ n ) × n = 0 N 1 J m n [ C ( D 1 ) n υ z σ ] sinc [ ( υ x + K 0 n = 0 N 1 m n b n ) L ] ,
γ ± ± n = 0 N 1 J 0 [ 2 C ( D 1 ) n k σ cos θ i ] ,
J 0 ( x ) = 1 1 4 x 2 + 1 64 x 4
γ ± ± [ 1 2 ( k σ ) 2 cos 2 θ i ] ,
k sc = k sp + i m i β i ,
| β i | = K 0 b i .
γ = p υ x + q υ z 4 υ z L cos θ i L L exp [ i υ x x + i υ z f r ( x ) ] d x .
p + = 2 sin θ s p = 2 sin θ i ,
q + = 2 cos θ s q = 2 cos θ i .
p ± υ x + q ± υ z 2 υ z = ± 1 + cos ( θ i + θ s ) cos θ i + cos θ s .
I = 1 2 L L L exp [ i υ x x + i υ z f r ( x ) ] d x
= 1 2 L L L exp ( i υ x x ) × exp [ i υ z σ C n = 0 N 1 ( D 1 ) n sin ( K 0 b n x + ϕ n ) ] d x
= 1 2 L L L exp ( i υ x x ) × n = 0 N 1 exp [ i C ( D 1 ) n υ z σ sin ( K 0 b n x + ϕ n ) ] d x .
exp ( i u sin ϕ ) = m = + J m ( u ) exp ( i m ϕ ) ,
I = 1 2 L L L exp ( i υ x x ) n = 0 N 1 m = + exp ( i m ϕ n ) × J m [ C ( D 1 ) n υ z σ ] exp ( i m b n K 0 x ) d x
= 1 2 L L L exp ( i υ x x ) m 1 , m 2 , , m N 1 = + exp ( i n = 0 N 1 m n ϕ n ) × n = 0 N 1 J m n [ C ( D 1 ) n υ z σ ] exp ( i n = 0 N 1 m n b n K 0 x ) d x
= m 1 , m 2 , , m N 1 = + exp ( i n = 0 N 1 m n ϕ n ) n = 0 N 1 J m n [ C ( D 1 ) n υ z σ ] × 1 2 L L L exp [ i ( υ x + n = 0 N 1 m n b n K 0 ) x ] d x
= m 1 , m 2 , , m N 1 = + exp ( i n = 0 N 1 m n ϕ n ) n = 0 N 1 × J m n [ C ( D 1 ) n υ z σ ] sinc [ ( υ x + K 0 n = 0 N 1 m n b n ) L ] .

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