Abstract

The spectral amplitudes of the scattered and transmitted fields for plane-wave incidence on an arbitrary rough interface in one dimension are derived exactly and simply by using Green’s theorem. Results are stated in terms of integrals on values of the field and its normal derivative on the interface. The energy constraint is derived, and individual energy contributions in each region are also related to the surface-field values. The latter contributions can be calculated from coupled linear equations that are also derived using Green’s theorem. The interface separates media of different but constant densities and sound speeds (acoustics) or different dielectrics (electromagnetics).

© 1985 Optical Society of America

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References

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  1. J. A. DeSanto, “Scattering from a sinusoid: Derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
    [CrossRef]
  2. J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
    [CrossRef]
  3. C. H. Wilcox, Scattering Theory for Diffraction Gratings (Springer, New York, 1984).
    [CrossRef]
  4. V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
    [CrossRef]
  5. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [CrossRef]
  6. J. Nakayama, H. Ogura, B. Matsumoto, “A probabilistic theory of scattering from a random rough surface, Radio Sci. 15, 1049–1057 (1980).
    [CrossRef]
  7. J. Nakayama, “Scattering from a randomly rough surface: a new formulation,” AP-S (1983), preprint.
  8. N. R. Hill, P. C. Wuenschel, “Numerical modeling of refraction arrivals in complex areas,” Geophysics. 50, 90–98 (1985).
    [CrossRef]
  9. J. A. DeSanto, “Scattering of scalar waves from a rough interface using a single integral equation,” Wave Motion 5, 125–135 (1983).
    [CrossRef]
  10. P. M. van den Berg, A. T. deHoop, “Reflection and transmission of electromagnetic waves at a rough interface between two different media,” IEEE Trans. GE-22, 42–52 (1984).
  11. J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, to be published), Vol. 23.

1985 (1)

N. R. Hill, P. C. Wuenschel, “Numerical modeling of refraction arrivals in complex areas,” Geophysics. 50, 90–98 (1985).
[CrossRef]

1984 (1)

P. M. van den Berg, A. T. deHoop, “Reflection and transmission of electromagnetic waves at a rough interface between two different media,” IEEE Trans. GE-22, 42–52 (1984).

1983 (1)

J. A. DeSanto, “Scattering of scalar waves from a rough interface using a single integral equation,” Wave Motion 5, 125–135 (1983).
[CrossRef]

1981 (1)

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

1980 (1)

J. Nakayama, H. Ogura, B. Matsumoto, “A probabilistic theory of scattering from a random rough surface, Radio Sci. 15, 1049–1057 (1980).
[CrossRef]

1977 (1)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

1975 (2)

J. A. DeSanto, “Scattering from a sinusoid: Derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[CrossRef]

Brown, G. S.

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, to be published), Vol. 23.

Celli, V.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[CrossRef]

deHoop, A. T.

P. M. van den Berg, A. T. deHoop, “Reflection and transmission of electromagnetic waves at a rough interface between two different media,” IEEE Trans. GE-22, 42–52 (1984).

DeSanto, J. A.

J. A. DeSanto, “Scattering of scalar waves from a rough interface using a single integral equation,” Wave Motion 5, 125–135 (1983).
[CrossRef]

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

J. A. DeSanto, “Scattering from a sinusoid: Derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[CrossRef]

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, to be published), Vol. 23.

Hill, N. R.

N. R. Hill, P. C. Wuenschel, “Numerical modeling of refraction arrivals in complex areas,” Geophysics. 50, 90–98 (1985).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[CrossRef]

Matsumoto, B.

J. Nakayama, H. Ogura, B. Matsumoto, “A probabilistic theory of scattering from a random rough surface, Radio Sci. 15, 1049–1057 (1980).
[CrossRef]

Nakayama, J.

J. Nakayama, H. Ogura, B. Matsumoto, “A probabilistic theory of scattering from a random rough surface, Radio Sci. 15, 1049–1057 (1980).
[CrossRef]

J. Nakayama, “Scattering from a randomly rough surface: a new formulation,” AP-S (1983), preprint.

Ogura, H.

J. Nakayama, H. Ogura, B. Matsumoto, “A probabilistic theory of scattering from a random rough surface, Radio Sci. 15, 1049–1057 (1980).
[CrossRef]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[CrossRef]

van den Berg, P. M.

P. M. van den Berg, A. T. deHoop, “Reflection and transmission of electromagnetic waves at a rough interface between two different media,” IEEE Trans. GE-22, 42–52 (1984).

Wilcox, C. H.

C. H. Wilcox, Scattering Theory for Diffraction Gratings (Springer, New York, 1984).
[CrossRef]

Wuenschel, P. C.

N. R. Hill, P. C. Wuenschel, “Numerical modeling of refraction arrivals in complex areas,” Geophysics. 50, 90–98 (1985).
[CrossRef]

Geophysics. (1)

N. R. Hill, P. C. Wuenschel, “Numerical modeling of refraction arrivals in complex areas,” Geophysics. 50, 90–98 (1985).
[CrossRef]

IEEE Trans. (1)

P. M. van den Berg, A. T. deHoop, “Reflection and transmission of electromagnetic waves at a rough interface between two different media,” IEEE Trans. GE-22, 42–52 (1984).

J. Acoust. Soc. Am. (1)

J. A. DeSanto, “Scattering from a sinusoid: Derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[CrossRef]

Phys. Rev. B (2)

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Radio Sci. (2)

J. Nakayama, H. Ogura, B. Matsumoto, “A probabilistic theory of scattering from a random rough surface, Radio Sci. 15, 1049–1057 (1980).
[CrossRef]

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

Wave Motion (1)

J. A. DeSanto, “Scattering of scalar waves from a rough interface using a single integral equation,” Wave Motion 5, 125–135 (1983).
[CrossRef]

Other (3)

C. H. Wilcox, Scattering Theory for Diffraction Gratings (Springer, New York, 1984).
[CrossRef]

J. Nakayama, “Scattering from a randomly rough surface: a new formulation,” AP-S (1983), preprint.

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, to be published), Vol. 23.

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Figures (2)

Fig. 1
Fig. 1

Plane-wave scattering from an arbitrary rough interface z = s(x) separating two semi-infinite media Vj with different densities ρj and wave numbers kj (sound speeds cj). H and − h illustrate the highest and lowest surface excursion, respectively, and define the lower (upper) boundary of region A (B).

Fig. 2
Fig. 2

Contours of integration for Green’s theorem.

Equations (56)

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( x 2 + z 2 + k 1 2 ) ϕ 1 ( x , z ) = 0 .
ϕ A ( x , y ) = D exp [ i k 1 ( α x ) β z ) ] + A ( μ ) exp [ i k 1 ( μ x + m z ) ] d μ ,
G ± ( x , z ) = exp [ i k 1 ( ± m z μ x ) ] ,
( x 2 + z 2 + k 2 2 ) ϕ 2 ( x , z ) = 0 ,
ϕ B ( x , z ) = B ( p ) exp [ i k 2 ( p x q z ) ] d p ,
H ± ( x , z ) = exp [ i k 2 ( ± q z p x ) ] .
C [ ϕ 1 ( x , z ) n 1 G ± ( x , z ) G ± ( x , z ) n 1 ϕ 1 ( x , z ) ] d τ = 0 ,
12 ± + 23 ± + 345 ± + 51 ± = 0 .
lim L 1 2 L ( 23 ± + ± 51 ) = 0 ,
lim L 1 2 L ( 12 ± + ± 345 ) = 0 ,
12 ± = [ ϕ A ( x , H ) z G ± ( x , H ) G ± ( x , H ) ϕ A z ( x , H ) ] d x .
12 ± = 4 π i m [ D δ ( α μ ) A ( μ ) ] ,
ϕ 1 [ x , s ( x ) ] = F 1 ( x ) ,
n 1 ϕ 1 [ x , s ( x ) d τ = i k 1 N 1 ( x ) d x ,
n 1 G ± [ x , s ( x ) ] d τ = [ z s ( x ) x ] G ± ( x , z ) z = s ( x ) d x
345 ± = i k 1 { [ ± m + μ s ( x ) ] F 1 ( x ) N 1 ( x ) } × exp { i k 1 [ ± m s ( x ) μ x ] } d x .
δ ( α μ ) = ( k 1 / 4 π β D ) { [ m + μ s ( x ) ] F 1 ( x ) N 1 ( x ) } × exp { i k 1 [ m s ( x ) + μ x ] } d x
A ( μ ) = ( k 1 / 4 π m ) { [ m + μ s ( x ) ] F 1 ( x ) + N 1 ( x ) } × exp { i k 1 [ m s ( x ) + μ x ] } d x .
C [ ϕ 2 ( x , z ) n 2 H ± ( x , z ) H ± ( x , z ) n 2 ϕ 2 ( x , z ) ] d τ = 0 .
lim L 1 2 L ( 2 3 ± + 5 1 ± ) = 0 ,
lim L 1 2 L ( 1 2 ± + 3 4 5 ± ) = 0 .
1 2 ± = [ ϕ B ( x , h ) z H ± ( x , h ) + H ± ( x , h ) z ϕ B ( x , h ) ] d x .
1 2 ± = [ 4 π i q B ( p ) 0 ]
ϕ 2 [ x , s ( x ) ] = F 2 ( x ) ,
n 2 ϕ 2 [ x , s ( x ) ] d τ = i k 2 N 2 ( x ) d x ,
n 2 H ± [ x , s ( x ) ] d τ = [ z s ( x ) x ] H ± ( x , z ) z = s ( x ) d x ,
3 4 5 ± = i k 2 { [ ± q + p s ( x ) ] F 2 ( x ) N 2 ( x ) } × exp { i k 2 [ ± q s ( x ) p x ] } d x .
B ( p ) = ( k 2 / 4 π q ) { [ q + p s ( x ) ] F 2 ( x ) N 2 ( x ) } × exp { i k 2 [ q s ( x ) p x ] } d x
0 = { [ q p s ( x ) ] F 2 ( x ) + N 2 ( x ) } × exp { i k 2 [ q s ( x ) + p x ] } d x .
F 1 ( x ) = ρ F 2 ( x ) F ( x ) ,
N 1 ( x ) = K N 2 ( x ) N ( x ) ,
δ ( α μ ) = ( k 1 / 4 π β D ) { [ m + μ s ( x ) } × exp { i k 1 [ m s ( x ) μ x ] } d x
0 = { [ q p s ( x ) ] ρ 1 F ( x ) + K 1 N ( x ) } × exp { i k 2 [ q s ( x ) + p x ] } d x ,
A ( μ ) = ( k 1 / 4 π m ) { [ m μ s ( x ) ] F ( x ) + N ( x ) } × exp { i k 1 [ m s ( x ) + μ x ] } d x
B ( ρ ) = ( k 2 / 4 π q ) { [ q + p s ( x ) ] ρ 1 F ( x ) K 1 N ( x ) } × exp { i k 2 [ q s ( x ) p x ] } d x .
J z sc ( x , z ) = ρ 1 [ ϕ s c ( x , z ) z ϕ sc * ( x , z ) ϕ sc * ( x , z ) z ϕ sc ( x , z ) ] ,
J z sc ( x , z ) = i k 1 ρ 1 d μ d u A ( μ ) A * ( μ ) ( m + m * ) × exp { i k 1 [ ( μ μ ) x + ( m m * ) z ] } .
J z in ( x , z ) = ρ 1 [ ϕ in ( x , z ) z ϕ in * ( x , z ) ϕ in * ( x , z ) z ϕ in ( x , z ) ] ,
J z in ( x , z ) = 2 i k 1 β ρ 1 D 2 .
J z tr ( x , z ) = ρ 2 [ ϕ B ( x , z ) z ϕ B * ( x , z ) ϕ B * ( x , z ) z ϕ B ( x , z ) ] .
J z tr ( x , z ) = i k 2 ρ 2 d p d p B ( p ) B * ( p ) ( q + q * ) × exp { i k 2 [ ( p p ) x + ( q * q ) z ] } .
J z sc ( x , H ) = J z in ( x , H ) + J z tr ( x , h ) ,
ρ K d p d p B ( p ) B * ( p ) ( q + q * ) × exp { i k 2 [ ( p p ) x + ( q q * ) h ] } + d μ d μ A ( μ ) A * ( μ ) ( m + m * ) × exp { i k 1 [ ( μ μ ) x + ( m m * ) H ] } = 2 β D 2 .
D = ( π / k 1 L ) 1 / 2
ρ d p | B ( p ) | 2 Re ( q ) + d μ | A ( μ ) | 2 Re ( m ) = β ,
C [ ϕ 1 ( x , z ) n 1 ϕ 1 * ( x , z ) ϕ 1 * ( x , z ) n 1 ϕ 1 ( x , z ) ] d τ = 0 .
lim L 1 2 L ( 12 + 345 ) = 0
12 = 2 i { 2 L k 1 β D 2 + D Re d μ A ( μ ) ( β m ) × exp [ i k 1 ( β + m ) H ] sin [ k 1 ( μ α ) L ] / ( μ α ) d μ d μ ¯ A ( μ ) A * ( μ ¯ ) ( m + m ¯ * ) × exp [ i k 1 ( m m ¯ * ) H ] sin [ k 1 ( μ u ¯ ) L ] / ( μ μ ¯ ) } ,
12 = 4 π i β 4 π i d μ | A ( μ ) | 2 Re ( m ) .
345 = 2 i k Re F 1 ( x ) N 1 * ( x ) d x .
2 π d μ | A ( μ ) | 2 Re ( m ) 2 π β = k 1 Re F ( x ) N * ( x ) d x ,
C [ ϕ 2 ( x , z ) n 2 ϕ 2 * ( x , z ) ϕ 2 * ( x , z ) n 2 ϕ ( x , z ) ] d τ = 0 .
lim L 1 2 L [ 1 2 + 3 4 5 ] = 0 .
1 2 = 4 π i d p | B ( p ) | 2 Re ( q ) ,
3 4 5 = 2 i k 1 ρ 1 Re F ( x ) N * ( x ) d x .
2 π ρ d p | B ( p ) | 2 Re ( q ) = k 1 Re F ( x ) N * ( x ) d x ,

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