Abstract

We consider the scattering of optical or electromagnetic waves from perfectly conducting rough surfaces modeled by a multiscale fractal function through the use of the extended boundary condition method. This exact method is developed here through the expansion of the surface field in terms of generalized Floquet modes, which results in a closed-form solution that can be evaluated numerically. The method is validated by comparison with the Kirchhoff (approximate) method in its regime of validity and by calculation of the energy balance parameter.

© 1995 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  2. J. A. DeSanto, J. Acoust. Soc. Am. 57, 1195 (1975).
    [CrossRef]
  3. A. K. Jordan, R. H. Lang, Radio Sci. 14, 1077 (1979).
    [CrossRef]
  4. S. L. Chuang, J. A. Kong, IEEE Proc. 69, 1132 (1981).
    [CrossRef]
  5. E. Rodriguez, Y. Kim, Radio Sci. 27, 79 (1992).
    [CrossRef]
  6. J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
    [CrossRef]
  7. D. L. Jaggard, X. Sun, J. Opt. Soc. Am. A 7, 1131 (1990).
    [CrossRef]
  8. D. L. Jaggard, X. Sun, J. Appl. Phys. 68, 5456 (1990).
    [CrossRef]
  9. D. L. Jaggard, in Electromagnatic Symmetry, C. D. Baum, H. Kritikos, eds. (Taylor and Francis, Washington, D.C., 1995), Chap. 5, pp. 231–280.
  10. P. C. Waterman, Proc. IEEE 53, 805 (1965).
    [CrossRef]
  11. P. C. Waterman, Phys. Rev. D 3, 825 (1971).
    [CrossRef]
  12. P. C. Waterman, J. Acoust. Soc. Am. 57, 791 (1975).
    [CrossRef]
  13. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chaps. 12.4–12.6.
  14. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Chap. 6.3.
  15. D. L. Jaggard, A. R. Mickelson, Appl. Phys. 19, 405 (1979).
    [CrossRef]
  16. The notation ∑p′ [ ], instead of ∑pi′[ ], is used for simplicity. Similarly for index q (qi) in Eq. (7).
  17. Note that the scattering plots given in Fig. 1 were obtained from the scattering amplitudes Bp′ [Eq. (9)] through the method described in Ref. 3.

1992 (1)

E. Rodriguez, Y. Kim, Radio Sci. 27, 79 (1992).
[CrossRef]

1990 (2)

D. L. Jaggard, X. Sun, J. Opt. Soc. Am. A 7, 1131 (1990).
[CrossRef]

D. L. Jaggard, X. Sun, J. Appl. Phys. 68, 5456 (1990).
[CrossRef]

1981 (1)

S. L. Chuang, J. A. Kong, IEEE Proc. 69, 1132 (1981).
[CrossRef]

1979 (2)

A. K. Jordan, R. H. Lang, Radio Sci. 14, 1077 (1979).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, Appl. Phys. 19, 405 (1979).
[CrossRef]

1975 (2)

P. C. Waterman, J. Acoust. Soc. Am. 57, 791 (1975).
[CrossRef]

J. A. DeSanto, J. Acoust. Soc. Am. 57, 1195 (1975).
[CrossRef]

1971 (1)

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

1966 (1)

J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
[CrossRef]

1965 (1)

P. C. Waterman, Proc. IEEE 53, 805 (1965).
[CrossRef]

Beckmann, P.

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

Chuang, S. L.

S. L. Chuang, J. A. Kong, IEEE Proc. 69, 1132 (1981).
[CrossRef]

DeSanto, J. A.

J. A. DeSanto, J. Acoust. Soc. Am. 57, 1195 (1975).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chaps. 12.4–12.6.

Jaggard, D. L.

D. L. Jaggard, X. Sun, J. Opt. Soc. Am. A 7, 1131 (1990).
[CrossRef]

D. L. Jaggard, X. Sun, J. Appl. Phys. 68, 5456 (1990).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, Appl. Phys. 19, 405 (1979).
[CrossRef]

D. L. Jaggard, in Electromagnatic Symmetry, C. D. Baum, H. Kritikos, eds. (Taylor and Francis, Washington, D.C., 1995), Chap. 5, pp. 231–280.

Jordan, A. K.

A. K. Jordan, R. H. Lang, Radio Sci. 14, 1077 (1979).
[CrossRef]

Kim, Y.

E. Rodriguez, Y. Kim, Radio Sci. 27, 79 (1992).
[CrossRef]

Kong, J. A.

S. L. Chuang, J. A. Kong, IEEE Proc. 69, 1132 (1981).
[CrossRef]

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

Lang, R. H.

A. K. Jordan, R. H. Lang, Radio Sci. 14, 1077 (1979).
[CrossRef]

Mickelson, A. R.

D. L. Jaggard, A. R. Mickelson, Appl. Phys. 19, 405 (1979).
[CrossRef]

Rodriguez, E.

E. Rodriguez, Y. Kim, Radio Sci. 27, 79 (1992).
[CrossRef]

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, J. Appl. Phys. 68, 5456 (1990).
[CrossRef]

D. L. Jaggard, X. Sun, J. Opt. Soc. Am. A 7, 1131 (1990).
[CrossRef]

Waterman, P. C.

P. C. Waterman, J. Acoust. Soc. Am. 57, 791 (1975).
[CrossRef]

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

P. C. Waterman, Proc. IEEE 53, 805 (1965).
[CrossRef]

Wright, J. W.

J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
[CrossRef]

Appl. Phys. (1)

D. L. Jaggard, A. R. Mickelson, Appl. Phys. 19, 405 (1979).
[CrossRef]

IEEE Proc. (1)

S. L. Chuang, J. A. Kong, IEEE Proc. 69, 1132 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
[CrossRef]

J. Acoust. Soc. Am. (2)

J. A. DeSanto, J. Acoust. Soc. Am. 57, 1195 (1975).
[CrossRef]

P. C. Waterman, J. Acoust. Soc. Am. 57, 791 (1975).
[CrossRef]

J. Appl. Phys. (1)

D. L. Jaggard, X. Sun, J. Appl. Phys. 68, 5456 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

Phys. Rev. D (1)

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Proc. IEEE (1)

P. C. Waterman, Proc. IEEE 53, 805 (1965).
[CrossRef]

Radio Sci. (2)

A. K. Jordan, R. H. Lang, Radio Sci. 14, 1077 (1979).
[CrossRef]

E. Rodriguez, Y. Kim, Radio Sci. 27, 79 (1992).
[CrossRef]

Other (6)

D. L. Jaggard, in Electromagnatic Symmetry, C. D. Baum, H. Kritikos, eds. (Taylor and Francis, Washington, D.C., 1995), Chap. 5, pp. 231–280.

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

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chaps. 12.4–12.6.

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

The notation ∑p′ [ ], instead of ∑pi′[ ], is used for simplicity. Similarly for index q (qi) in Eq. (7).

Note that the scattering plots given in Fig. 1 were obtained from the scattering amplitudes Bp′ [Eq. (9)] through the method described in Ref. 3.

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Figures (1)

Fig. 1
Fig. 1

(a) Scattering diagram (polar) for a fractally corrugated surface with parameters Dr = 1 + α =1.3, = 0.5, L = 40λ (L is the surface patch size), χ = λ/Λ = 0.2, b = 1.8122, M = 2. The angle of the incident wave is θi = 30°. Note that, because the values of χ and were chosen to be rather small, it appears that Kirchhoff assumptions are satisfied, so the two methods yield similar scattering results, especially for the main scattering lobe at θs = 30° (indistinguishable graphs). (b) Same as for (a), except that the fractal surface parameters are now Dr = 1 + α = 1.3, = 1.0, L = 40λ, χ = λ/Λ = 2.0, b = 1.8122, M = 2. Because of the above values of χ and , Kirchhoff’s assumptions are violated, and the corresponding method, as opposed to the EBCM proposed here, becomes inaccurate.

Equations (15)

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z = f ( x ) = σ C n = 0 M - 1 α n sin ( K b n x + φ n ) ,
D r = 1 + α .
k sin  θ s = k sin  θ i + nK ,
i = 1 M n i = κ .
E i y - - d S g ( r , r ) n ^ S E y ( r ) = E y ( r )             z > f ( x ) ,
= 0             z < f ( x ) ,
g ( r , r ) = j 4 H 0 ( 1 ) ( k r - r ) = j 4 π - d k x 1 k z exp [ j k x ( x - x ) + j k z z - z ] ,
d S n ^ S E y ( r ) = k d x exp ( j k i x x ) × q 1 q 2 q M a q s exp ( j qK x ) ,
E y ( r ) = E i y ( r ) + E y sc ( r ) = E i y ( r ) + p = - B p exp ( j k p + r ) ,
B p = - j k 4 π k z p ( - 1 ) κ ( p ) exp ( j p Φ ) q a q s × exp ( - j q Φ ) ( - 1 ) κ ( q ) n = 0 M - 1 J p n - q n ( k z p σ C α n ) ,
E i y ( r ) - p = - A p exp ( j k p - r ) = 0 ,
A p ' = j k 4 π k z p exp ( j p Φ ) q a q s exp ( - j q Φ ) × n = 0 M - 1 J p n - q n ( k z p σ C α n ) .
a q s = j 4 π a q s exp ( - j q Φ ) ,
[ A p ] = [ Q D - ( k ) ] [ a q s ] ,
Q D , p q - ( k ) = k k z p exp ( j p Φ ) n = 0 M - 1 J p n - q n ( k z p σ C α n ) .

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