Abstract

A simple reformulation of the double-scatter Kirchhoff approximation is presented to extend the use of this calculation method to infinitely sloped surfaces. Examples are presented for square grooves and lines on perfectly conducting surfaces. The results presented show the accuracy of the method and the errors produced by not including higher-order scattering in the calculations.

© 2003 Optical Society of America

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References

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  1. A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
    [CrossRef]
  2. 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]
  3. 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]
  4. J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Hilger, London, 1991).
  5. D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
    [CrossRef]
  6. 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]
  7. N. C. Bruce, A. J. Sant, “The Mueller matrix for rough surface scattering using the Kirchhoff approximation,” Opt. Commun. 88, 471–484 (1992).
    [CrossRef]
  8. C. J. R. Sheppard, “Scattering by fractal surfaces with an outer scale,” Opt. Commun. 122, 178–188 (1996).
    [CrossRef]
  9. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  10. A. Mendoza-Suárez, E. R. Méndez, “Light scattering by a reentrant fractal surface,” Appl. Opt. 36, 3521–3531 (1997).
    [CrossRef] [PubMed]

1997

1996

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

1992

N. C. Bruce, A. J. Sant, “The Mueller matrix for rough surface scattering using the Kirchhoff approximation,” Opt. Commun. 88, 471–484 (1992).
[CrossRef]

1991

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

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

1988

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]

Bruce, N. C.

N. C. Bruce, A. J. Sant, “The Mueller matrix for rough surface scattering using the Kirchhoff approximation,” Opt. Commun. 88, 471–484 (1992).
[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]

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

Dainty, J. C.

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]

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

Jaggard, D. L.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Mendez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Méndez, E. R.

Mendoza-Suárez, A.

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Hilger, London, 1991).

Sant, A. J.

N. C. Bruce, A. J. Sant, “The Mueller matrix for rough surface scattering using the Kirchhoff approximation,” Opt. Commun. 88, 471–484 (1992).
[CrossRef]

Sheppard, C. J. R.

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

Sun, X.

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]

Ann. Phys.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Appl. Opt.

J. Acoust. Soc. Am.

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[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]

J. Mod. Opt.

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

Opt. Commun.

N. C. Bruce, A. J. Sant, “The Mueller matrix for rough surface scattering using the Kirchhoff approximation,” Opt. Commun. 88, 471–484 (1992).
[CrossRef]

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

Waves Random Media

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

J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Hilger, London, 1991).

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Figures (11)

Fig. 1
Fig. 1

Geometry used for the Kirchhoff calculation. It can be seen that, for a constant dx, ds 1 is smaller than ds 2 so that in this second section the sampling of the incident field on the surface is worse. Also shown are the definitions of angles β, θinc, and θsc.

Fig. 2
Fig. 2

Geometry of the sample surface with a square groove.

Fig. 3
Fig. 3

Graphs of scattered intensity versus scatter angle for grooves of 0.5λ depth and 4λ width. Top left, 0° incidence; top right, 20° incidence; bottom left, 40° incidence; bottom right, 50° incidence. Continuous curves, integral equation results; open circles, the Kirchhoff single-scatter results; crosses, the Kirchhoff double-scatter results.

Fig. 4
Fig. 4

Same as Fig. 3 but for a groove depth of 1λ.

Fig. 5
Fig. 5

Same as Fig. 3 but for a groove depth of 1.75λ.

Fig. 6
Fig. 6

Same as Fig. 3 except the continuous curves represent the integral equation results and the filled squares represent the Kirchhoff total (single plus double) results.

Fig. 7
Fig. 7

Same as Fig. 4 but for a groove depth of 1λ.

Fig. 8
Fig. 8

Same as Fig. 4 but for a groove depth of 1.75λ.

Fig. 9
Fig. 9

Geometry of the sample with a line.

Fig. 10
Fig. 10

Graphs of scattered intensity versus scatter angle for a line of height 1λ and width 8λ. Top left, 0° incidence; top right, 20° incidence; bottom left, 40° incidence; bottom right, 60° incidence. Continuous curves, the integral equation results; filled squares, the Kirchhoff total scatter results.

Fig. 11
Fig. 11

Graphs of scattered intensity versus scatter angle for multiple grooves of width 8λ and depth 0.5λ. The graphs on the top line are for an incident angle of 20° and those on the bottom line are for an incident angle of 40°. The graphs at left are for a surface with two grooves and a total length of 20λ, the curves on the right are for a surface with four grooves and a total length of 36λ. Continuous curves, the integral equation results; filled squares, the Kirchhoff total scatter results.

Tables (3)

Tables Icon

Table 1 Conservation of Energy for a Square Groove of Width 4λ

Tables Icon

Table 2 Conservation of Energy for a Line of Width 8λ and Height 1λ

Tables Icon

Table 3 Conservation of Energy for Multiple Square Grooves of Width 8λ and Depth 1λ

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Φsc=14πSΦSx, zH01krn-H01krΦSx, znds,
ΦSx, z=1+RΦincx, z,
ΦSx, zn=i1-Rkinc·nΦincx, z
Φsc=14πSΦincx, z1+Rksc·nH11kr-H01kri1-Rkinc·nds,
H01krn=ksc·nH11kr.
kinc=k sin θincx-k cos θincz, ksc=k sin θscx+k cos θscz, n=-sin βx+cos βz, ds=dxcos β, tan β=dhxdx=mx,
Φsc=14πSΦincx, z1+Rmxsin θsc-cos θscH11kr-H01kri1-R×mxsin θinc+cos θincdx.
n=-sin βx+cos βz=-dzdSx+dxdSz.
Φsc=14πSΦincx, z1+R-dz sin θsc+dx cos θscH11kr+H01kri1-R×dz sin θinc+dx cos θinc,
Φsc=-14πSΦincx, z1+Rsin θscH11kr-H01kri1-Rsin θincdz-14πSΦincx, z1+Rcos θscH11kr+H01kri1-Rcos θincdx.
Φsc1x2, z2=-14πSΦincx1, z11+R1sin θ12×H11kr12-H01kr12i1-R1sin θincdz1-14πSΦincx1, z11+R1cos θ12H11kr12+H01kr12i1-R1cos θincdx1,
Φsc2=-14πSΦsc1x2, z21+R2sin θscH11kr-H01kri1-R2sin θ12dz2-14πS×Φsc1x2, z21+R2cos θscH11kr+H01kri1-R2cos θ12dx2,
Siθinc=1if point i is illuminated by the incident field0if point i is not illuminated by the incident field, Siθsc=1if point i is visible from the detector position0if point i is not visible from the detector position. Siθji=1if point i is visible from point j0if point i is not visible from point j.
Φincx, z-14πH01krΦSx, znds=Φx, zifx, z is above the surface0ifx, z is below the surface.
E=-90°90° |Φscθsc|2dθscEinc
E=-90°90° |Φsc2θsc|2dθscEinc.
E=-90°90° |Φscθsc+Φsc2θsc|2dθscEinc,
E=-90°90° |Φθsc|2dθscEinc.

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