Abstract

Recently a modification of the Kirchhoff approximation was presented to permit the calculation of multiple-scattered light on rough surfaces with infinite slopes. A brief description of the method, with examples of the scattering of light from rough surfaces with rectangular grooves, is presented. It is shown that, for a surface with random groove depths with a constant probability distribution of the groove depths, the backscatter intensity can increase or decrease, depending on the width of this probability distribution.

© 2005 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  2. C. J. R. Sheppard, “Scattering by fractals with an outer scale,” Opt. Commun. 122, 178–188 (1996).
    [CrossRef]
  3. D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
    [CrossRef]
  4. N. C. Bruce, V. Ruiz-Cortés, J. C. Dainty, “Calculations of the grazing incidence scattering from random rough surfaces using the Kirchhoff approximation,” Opt. Commun. 106, 123–126 (1994).
    [CrossRef]
  5. N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
    [CrossRef]
  6. A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21–34 (1991).
    [CrossRef]
  7. N. C. Bruce, “Scattering from infinitely sloped surfaces by use of the Kirchhoff approximation,” Appl. Opt. 42, 2398–2406 (2003).
    [CrossRef] [PubMed]

2003 (1)

1996 (1)

C. J. R. Sheppard, “Scattering by fractals with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

1994 (1)

N. C. Bruce, V. Ruiz-Cortés, J. C. Dainty, “Calculations of the grazing incidence scattering from random rough surfaces using the Kirchhoff approximation,” Opt. Commun. 106, 123–126 (1994).
[CrossRef]

1991 (2)

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

1990 (1)

Beckmann, P.

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

Bruce, N. C.

N. C. Bruce, “Scattering from infinitely sloped surfaces by use of the Kirchhoff approximation,” Appl. Opt. 42, 2398–2406 (2003).
[CrossRef] [PubMed]

N. C. Bruce, V. Ruiz-Cortés, J. C. Dainty, “Calculations of the grazing incidence scattering from random rough surfaces using the Kirchhoff approximation,” Opt. Commun. 106, 123–126 (1994).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

Chen, J. S.

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

Dainty, J. C.

N. C. Bruce, V. Ruiz-Cortés, J. C. Dainty, “Calculations of the grazing incidence scattering from random rough surfaces using the Kirchhoff approximation,” Opt. Commun. 106, 123–126 (1994).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

Ishimaru, A.

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

Jaggard, D. L.

Ruiz-Cortés, V.

N. C. Bruce, V. Ruiz-Cortés, J. C. Dainty, “Calculations of the grazing incidence scattering from random rough surfaces using the Kirchhoff approximation,” Opt. Commun. 106, 123–126 (1994).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, “Scattering by fractals with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

Spizzichino, A.

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

Sun, X.

Appl. Opt. (1)

J. Mod. Opt. (1)

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

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

Opt. Commun. (2)

N. C. Bruce, V. Ruiz-Cortés, J. C. Dainty, “Calculations of the grazing incidence scattering from random rough surfaces using the Kirchhoff approximation,” Opt. Commun. 106, 123–126 (1994).
[CrossRef]

C. J. R. Sheppard, “Scattering by fractals with an outer scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

Waves Random Media (1)

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

Other (1)

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

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Figures (10)

Fig. 1
Fig. 1

Geometry of the problem studied here.

Fig. 2
Fig. 2

Geometry used for calculation of the phase change for double-scattered light from two grooves of different depths.

Fig. 3
Fig. 3

Constant probability distribution of depths between a maximum and a minimum value. The total distribution can be divided into pairs of grooves with depths separated by Δh.

Fig. 4
Fig. 4

Effect of a change in the groove depth on the energy that is double scattered.

Fig. 5
Fig. 5

Results of the calculations for a random distribution of depths of width Δh = 1.443λ and a period a = 8λ. The incidence angle is 30°: (a) single scatter; (b) double scatter; (c) total scatter. Note the different scales for the double-scatter graph.

Fig. 6
Fig. 6

Results for the setup in Fig. 5 but with pairs of consecutive grooves that have a difference in groove depth equal to Δh.

Fig. 7
Fig. 7

Results for pairs of grooves of various separations with the depth difference equal to Δh. The surface period is a = 8λ, and the incidence angle is 30°. (a) Single scatter for all cases (there are no discernible changes for the cases studied on this scale). (b) Total scatter for all cases (there are no discernible changes for the cases studied on this scale). Double scatter for (c), Δh = 1.443λ, (d) Δh = 1.517λ, (e) Δh = 1.590λ, (f) Δh = 1.662λ, and (g) Δh = 1.734λ.

Fig. 8
Fig. 8

Double-scattered intensity for a surface with a = 8λ and Δh = 1.443λ: incidence angles (a) 30°, (b) 32°, (c) 35°. Note that the backscatter minimum at −30° moves to smaller scatter angles as the incidence angle is increased.

Fig. 9
Fig. 9

Conditions for an interference minimum. The solid line is the left-hand side of Eq. (14) for a = 8λ and Δh = 1.443λ. Symbols represent the right-hand side of Eq. (14): open triangles, θinc = 30°; open squares, θinc = 28°; crosses, θinc = 32°. Where these curves cross gives the scatter angle for the interference minimum. For an incidence angle of 32°, the phase difference between the interference minimum and the phase at the backscatter direction is approximately 0.5, which means that at −32° (the backscatter angle) there is an interference maximum.

Fig. 10
Fig. 10

Variation of the double-scattered intensity with wavelength for an incident angle of 30°: (a) a = 8λ and Δh = 1.443λ; (b) 2% reduction in wavelength; (c) 4% reduction in wavelength; (d) 5.5% reduction in wavelength.

Equations (23)

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Φ S ( x s , y s ) = ( 1 + R ) Φ inc ( x s , y s ) ,
Φ S ( x s , y s ) n = i ( 1 - R ) ( k inc · n ) Φ inc ( x s , y s ) .
Φ sc = Φ S ( x s , y s ) n = 1 4 π Φ inc ( x s , y s ) [ ( 1 + R ) ( k sc · n ) H 1 ( 1 ) ( k r ) - i ( 1 - R ) ( k inc · n ) H 0 ( 1 ) ( k r ) ] d S ,
k inc = k sin θ inc x - k     cos θ inc y ,
k sc = k sin θ sc x + k cos θ sc y ,
n = - sin β x + cos β y = - d y d S x + d y d S y .
Φ sc = - 1 4 π Φ inc ( x s , y s ) [ ( 1 + R ) sin θ sc H 1 ( 1 ) ( k r ) - i × ( 1 - R ) sin θ inc H 0 ( 1 ) ( k r ) ] d y - 1 4 π Φ inc ( x s , y s ) × [ ( 1 + R ) cos θ sc H 1 ( 1 ) ( k r ) + i ( 1 - R ) cos θ inc H 0 ( 1 ) ( k r ) ] d x
Φ sc ( 1 ) ( x 2 , y 2 ) = - 1 4 π Φ inc ( x s , y s ) [ ( 1 + R 1 ) × sin θ 12 H 1 ( 1 ) ( k r 12 ) - i ( 1 - R 1 ) × sin θ inc H 1 ( 1 ) ( k r 12 ) ] d y 1 - 1 4 π Φ inc ( x s , y s ) [ ( 1 + R 1 ) × cos θ 12 H 1 ( 1 ) ( k r 12 ) - i ( 1 - R 1 ) × cos θ inc H 1 ( 1 ) ( k r 12 ) ] d x 1 ,
Φ sc = - 1 4 π Φ sc ( 1 ) ( x 2 , y 2 ) [ ( 1 + R 2 ) sin θ sc H 1 ( 1 ) ( k r ) - i × ( 1 - R 2 ) sin θ 12 H 1 ( 1 ) ( k r ) ] d y 2 - 1 4 π Φ sc ( 1 ) ( x 2 , y 2 ) × [ ( 1 + R 2 ) cos θ sc H 1 ( 1 ) ( k r ) - i × ( 1 - R 2 ) cos θ 12 H 1 ( 1 ) ( k r ) ] d x 2 ,
Δ ϕ = k ( AB + CD + DE + EG ) = k ( AB + CD + DE + CH - CF ) ,
AB = CH = a sin ( θ inc ) , CD = DE = Δ h cos ( θ inc ) , CF = 2 Δ h tan ( θ inc ) sin ( θ inc ) .
Δ ϕ = 2 k [ a sin ( θ inc ) + Δ h ( 1 cos θ inc - sin 2 θ inc cos θ inc ) ] = 2 k [ a sin ( θ inc ) + Δ h cos θ inc ] .
Δ ϕ a = 2 k a sin θ inc .
Δ ϕ a = 2 k a sin θ inc = 2 m π .
a λ = m 2 sin θ inc .
Δ h c λ = m 2 cos θ inc ;
Δ h d λ = m + ( 1 / 2 ) 2 cos θ inc .
Δ h c λ = 0.577 , 1.155 , 1.732 , , Δ h d λ = 0.289 , 0.866 , 1.443 , .
Δ ϕ = k a sin ( θ inc ) + k a sin ( θ ) + 2 k Δ h cos θ = ( m + 1 2 ) 2 π ,
a λ sin ( θ ) + 2 Δ h λ cos θ = ( m + 1 2 ) - a λ sin ( θ inc ) .
2 [ a λ sin ( θ ) + Δ h λ cos ( θ ) ] = m , 2 [ a λ sin ( θ ) + Δ h λ cos ( θ ) ] = m ( 1 + [ P 100 ] ) .
( m + 1 2 ) = m ( 1 + [ P 100 ] ) ,
[ P 100 ] = 1 / 2 m .

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