Abstract

An inversion algorithm, based on the Kirchhoff approximation, for the reconstruction of a rough-surface profile from measurements of the scattered field is described. It is shown that the data can be related to the surface profile through a Fourier-transform relationship by a specific choice of data and so lead to a simple fast-Fourier-transform-based procedure. The algorithm is illustrated through the use of numerical results.

© 1991 Optical Society of America

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References

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  1. J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23, pp. 2–62.
    [CrossRef]
  2. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  3. M.-J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
    [CrossRef]
  4. B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrarily rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
    [CrossRef]
  5. 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]
  6. R. T. Prosser, “Formal solutions of inverse scattering, III,” J. Math. Phys. 21, 2648–2652 (1980).
    [CrossRef]
  7. E. Marx, T. V. Vorburger, “Direct and inverse problems for light scattered from rough surfaces,” Appl. Opt. 29, 3613–3626 (1990).
    [CrossRef] [PubMed]
  8. A. Roger, D. Maystre, “Inverse scattering method in electromagnetic optics: application to diffraction gratings,” J. Opt. Soc. Am. 70, 1483–1495 (1980).
    [CrossRef]
  9. A. M. Husier, A. Quattropani, H. D. Baltes, “Construction of grating profiles yielding prescribed diffraction efficiencies,” Opt. Commun. 41, 149–153 (1982).
    [CrossRef]
  10. R. J. Wombell, J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
    [CrossRef]
  11. J. A. DeSanto, R. J. Wombell, “Rough-surface scattering,” Waves Random Media 1, S41–S56 (1991).
    [CrossRef]
  12. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).
  13. J. A. DeSanto, “Exact spectral formalism for rough-surface scattering,” J. Opt. Soc. Am. A 2, 2202–2207 (1985).
    [CrossRef]
  14. A. V. Oppenheim, R. W. Shaffer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  15. A. Ishimaru, J. S. Chen, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,” J. Acoust. Soc. Am. 88, 1846–1850 (1990).
    [CrossRef]
  16. E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–190 (1991).
    [CrossRef]
  17. J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by surfaces of known statistics: new results,” Waves Random Media 1, S29–S40 (1991).
    [CrossRef]

1991 (4)

R. J. Wombell, J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

J. A. DeSanto, R. J. Wombell, “Rough-surface scattering,” Waves Random Media 1, S41–S56 (1991).
[CrossRef]

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–190 (1991).
[CrossRef]

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by surfaces of known statistics: new results,” Waves Random Media 1, S29–S40 (1991).
[CrossRef]

1990 (3)

1988 (1)

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 (2)

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrarily rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

1985 (1)

1982 (1)

A. M. Husier, A. Quattropani, H. D. Baltes, “Construction of grating profiles yielding prescribed diffraction efficiencies,” Opt. Commun. 41, 149–153 (1982).
[CrossRef]

1980 (2)

Baltes, H. D.

A. M. Husier, A. Quattropani, H. D. Baltes, “Construction of grating profiles yielding prescribed diffraction efficiencies,” Opt. Commun. 41, 149–153 (1982).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).

Brown, G. S.

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23, pp. 2–62.
[CrossRef]

Bruce, N. C.

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by surfaces of known statistics: new results,” Waves Random Media 1, S29–S40 (1991).
[CrossRef]

Chen, J. S.

A. Ishimaru, J. S. Chen, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,” J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

Dainty, J. C.

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by surfaces of known statistics: new results,” Waves Random Media 1, S29–S40 (1991).
[CrossRef]

M.-J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

DeSanto, J. A.

R. J. Wombell, J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

J. A. DeSanto, R. J. Wombell, “Rough-surface scattering,” Waves Random Media 1, S41–S56 (1991).
[CrossRef]

J. A. DeSanto, “Exact spectral formalism for rough-surface scattering,” J. Opt. Soc. Am. A 2, 2202–2207 (1985).
[CrossRef]

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23, pp. 2–62.
[CrossRef]

Friberg, A. T.

Husier, A. M.

A. M. Husier, A. Quattropani, H. D. Baltes, “Construction of grating profiles yielding prescribed diffraction efficiencies,” Opt. Commun. 41, 149–153 (1982).
[CrossRef]

Ishimaru, A.

A. Ishimaru, J. S. Chen, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,” J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

Jackson, D. R.

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–190 (1991).
[CrossRef]

Kachoyan, B. J.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrarily rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

Kim, M.-J.

Macaskill, C.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrarily rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

Marx, E.

Maystre, D.

Mendez, E. R.

O’Donnell, K. A.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Shaffer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Prosser, R. T.

R. T. Prosser, “Formal solutions of inverse scattering, III,” J. Math. Phys. 21, 2648–2652 (1980).
[CrossRef]

Quattropani, A.

A. M. Husier, A. Quattropani, H. D. Baltes, “Construction of grating profiles yielding prescribed diffraction efficiencies,” Opt. Commun. 41, 149–153 (1982).
[CrossRef]

Roger, A.

Sant, A. J.

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by surfaces of known statistics: new results,” Waves Random Media 1, S29–S40 (1991).
[CrossRef]

M.-J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

Shaffer, R. W.

A. V. Oppenheim, R. W. Shaffer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).

Thorsos, E. I.

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–190 (1991).
[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]

Vorburger, T. V.

Wombell, R. J.

J. A. DeSanto, R. J. Wombell, “Rough-surface scattering,” Waves Random Media 1, S41–S56 (1991).
[CrossRef]

R. J. Wombell, J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

Appl. Opt. (1)

Inverse Probl. (1)

R. J. Wombell, J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

J. Acoust. Soc. Am. (3)

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrarily rough surface,” J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[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]

A. Ishimaru, J. S. Chen, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,” J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

J. Math. Phys. (1)

R. T. Prosser, “Formal solutions of inverse scattering, III,” J. Math. Phys. 21, 2648–2652 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

A. M. Husier, A. Quattropani, H. D. Baltes, “Construction of grating profiles yielding prescribed diffraction efficiencies,” Opt. Commun. 41, 149–153 (1982).
[CrossRef]

Waves Random Media (3)

J. A. DeSanto, R. J. Wombell, “Rough-surface scattering,” Waves Random Media 1, S41–S56 (1991).
[CrossRef]

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–190 (1991).
[CrossRef]

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by surfaces of known statistics: new results,” Waves Random Media 1, S29–S40 (1991).
[CrossRef]

Other (3)

A. V. Oppenheim, R. W. Shaffer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23, pp. 2–62.
[CrossRef]

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).

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Figures (6)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

(a) Scattering and illumination angles θs± and θi± corresponding to fixed q and varying p for q = 0.1 (outer locus), q = 0.5, q = 1.0, q = 1.5, and q = 1.9 (inner locus). The solid curves correspond to the positive sign and the dashed curves to the negative. The central square indicates the region over which data can be physically measured. (b) Sampling density for θi and θi for uniform sampling in p with q = 1.75.

Fig. 3
Fig. 3

Reconstruction r+ (solid curves) of a surface (dashed curves) with π/λ = 0.1 and l/λ = 1.0 as q increases: (a) q = 1.1, (b) q = 1.5, (c) q = 1.75, (d) q = 1.9.

Fig. 4
Fig. 4

(a) Reconstruction r (solid curve) of a surface (dashed curve) with σ/λ = 0.1 and l/λ = 1.0, with q = 1.75. (b) Reconstruction r ¯ = ( r + + r ) / 2.

Fig. 5
Fig. 5

Reconstruction r+ (solid curves) of a surface (dashed curves) with l/λ = 1.0, using q = 1.75 as σ/λ increases: (a) σ/λ = 0.15, unwrapped from the left, (b) σ/λ = 0.175, unwrapped from the left, (c) σ/λ = 0.2, unwrapped from the left, (d) σ/λ = 0.2, unwrapped from the left to x/λ = 0 and from the right to x/λ = 3.0.

Fig. 6
Fig. 6

Reconstruction r+ (solid curves) of a surface (dashed curves) with σ/λ = 0.4 and l/λ = 1.0 with q = 1.25 (a) from Kirchhoff data and (b) from exact data.

Equations (26)

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ψ s ( x , z ) = k 4 π 1 m F ( μ , α ) exp [ i k ( μ x + m z ) ] d μ ,
F ( μ , α ) = N ( x , α ) exp { i k [ μ x + m s ( x ) ] } d x ,
N ( x , α ) = ( i k ) 1 n ψ ( x , z ) | z = s ( x ) ,
lim k r ( 2 π r / k ) 1 / 2 exp ( i k r ) ψ s ( x , z ) = ( 1 / 2 ) exp ( i π / 4 ) F ( μ , α ) ,
N ka ( x , α ) = 2 ( i k ) 1 n ψ o ( x , z ) | z = s ( x ) .
F k a ( μ , α ) = N k a ( x , α ) exp [ i k ( μ x + m s ( x ) ] d x .
ψ o ( x , z , θ i ) = exp [ i k ( α x β z ) ] ,
F k a ( μ , α ) = 2 [ β + α s ( x ) ] exp { i k [ α x β s ( x ) ] } × exp { i k [ μ x + m s ( x ) ] } d x .
F ( μ , α ) = 2 ( 1 + β m α μ β + m ) exp [ i k ( μ α ) x ] × exp [ i k ( β + m ) s ( x ) ] d x .
p = p ( θ s , θ i ) ( μ α ) ,
q = q ( θ s , θ i ) ( m + β ) .
F ( μ , α ) = [ ( p 2 + q 2 ) / q ] D ( θ s , θ i ) ,
D ( θ s , θ i ) = exp ( i k p x ) exp [ i k q s ( x ) ] d x ,
p = sin θ s sin θ i = 2 cos ( θ s + θ i 2 ) sin ( θ s θ i 2 ) ,
q = cos θ s + cos θ i = 2 cos ( θ s + θ i 2 ) cos ( θ s θ i 2 )
θ s θ i = 2 arctan ( p / q ) + 2 π j ,
θ s + θ i = 2 arccos [ ± ( p 2 + q 2 ) 1 / 2 / 2 ] + 4 π j ,
θ s ± = arccos [ ± ( p 2 + q 2 ) 1 / 2 / 2 ] + arctan ( p / q ) + π n ,
θ i ± = arccos [ ± ( p 2 + q 2 ) 1 / 2 / 2 ] arctan ( p / q ) + π n ,
| ( p 2 + q 2 ) 1 / 2 / 2 | 1
0 p 2 , 0 q 2 ,
| p | ( 4 q 2 ) 1 / 2 .
D ± ( p , q ) = D [ θ s ± ( p , q ) , θ i ± ( p , q ) ] = { exp [ i k q s ( x ) ] } ( p ) ,
R ± ( x , q ) = 1 [ D ± ( p , q ) ] ( x , q ) = exp [ i k q r ± ( x , q ) ] .
r ± ( x , q ) = 1 k q arctan { [ R ± ( x , q ) ] [ R ± ( x , q ) ] } ,
r ¯ ( x , q ) = [ r + ( x , q ) + r ( x , q ) ] / 2 .

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