Abstract

We study a selective light scattering elimination procedure in the case of highly scattering rough surfaces. Contrary to the case of low scattering levels, the elimination parameters are shown to depend on the sample microstructure and to present rapid variations with the scattering angle. On the other hand, when the slope of the surface is moderated, we show that this parameters present smoother variations and little dependence to the microstructure, even when the roughness is high. These results allow an important selective reduction of the scattered light, with a basic experimental mounting and an analytical determination of the elimination parameters. Such selective scattering reduction is demonstrated by simulations and experiments and applied to the imaging of an object situated under a highly rough surface.

© 2009 Optical Society of America

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References

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  1. M. Saillard, P. Vincent, and G. Micolau, "Reconstruction of buried objects surrounded by small inhomogeneities," Inv. Problems 16, 1195-1208 (2000).
    [CrossRef]
  2. W. Chew and Y. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
    [CrossRef]
  3. D. Ausserré and M. Valignat, "Wide-field optical imaging of surface nanostructures," Nano Lett. 6, 1384-1388 (2006).
    [CrossRef] [PubMed]
  4. C. Dunsby and P. French, "Techniques for depth-resolved imaging through turbid media including coherencegated imaging," J. Phys. D: Appl. Phys. 36, R207-R227 (2003).
    [CrossRef]
  5. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
    [CrossRef]
  6. C. Amra, C. Grèzes-Besset, and L. Bruel, "Comparison of surface and bulk scattering in optical multilayers," Appl. Opt. 32, 5492-5503 (1993).
    [CrossRef] [PubMed]
  7. O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005).
    [CrossRef] [PubMed]
  8. C. Amra, C. Deumie, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 854-864 (2005).
    [CrossRef]
  9. G. Georges, C. Deumié, and C. Amra, "Selective probing and imaging in random media based on the elimination of polarized scattering," Opt. Express 15, 9804-9816 (2007).
    [CrossRef] [PubMed]
  10. G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
    [CrossRef]
  11. C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
    [CrossRef]
  12. E. Popov and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
    [CrossRef]
  13. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1037 (1996).
    [CrossRef]
  14. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon Press New York, 1999).
  16. C. Amra, "Light scattering from multilayer optics. I. Tools of investigation," J. Opt. Soc. Am. A 11197-210 (1994).
    [CrossRef]
  17. A. Voronovich, Wave Scattering from Rough Surfaces (Springer, 1994).
  18. I. Ohlidal and K. Navratil, "Analysis of the basic statistical properties of randomly rough curved surfaces by shearing interferometry," Appl. Opt. 24, 2690-2695 (1985).
    [CrossRef] [PubMed]
  19. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Norwood, MA, Artech House, Inc., 1987).
  20. R. Petit, "Diffraction d’une onde plane par un reseau metallique," Rev. Opt. 45, 353-370 (1966).
  21. G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Acad. Sci. 268, 1060-1063 (1969).
  22. H. Giovannini and C. Amra, "Scattering-reduction effect with overcoated rough surfaces: theory and experiment," Appl. Opt. 36, 5574-5579 (1997).
    [CrossRef] [PubMed]
  23. H. Giovannini, M. Saillard, and A. Sentenac, "Numerical study of scattering from rough inhomogeneous films," J. Opt. Soc. Am. A 15, 1182-1191 (1998).
    [CrossRef]
  24. M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  25. L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008).
    [CrossRef]
  26. R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

2008 (3)

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
[CrossRef]

L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008).
[CrossRef]

2007 (1)

2006 (2)

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
[CrossRef]

D. Ausserré and M. Valignat, "Wide-field optical imaging of surface nanostructures," Nano Lett. 6, 1384-1388 (2006).
[CrossRef] [PubMed]

2005 (2)

C. Amra, C. Deumie, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 854-864 (2005).
[CrossRef]

O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005).
[CrossRef] [PubMed]

2003 (1)

C. Dunsby and P. French, "Techniques for depth-resolved imaging through turbid media including coherencegated imaging," J. Phys. D: Appl. Phys. 36, R207-R227 (2003).
[CrossRef]

2000 (2)

M. Saillard, P. Vincent, and G. Micolau, "Reconstruction of buried objects surrounded by small inhomogeneities," Inv. Problems 16, 1195-1208 (2000).
[CrossRef]

E. Popov and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

1998 (1)

1997 (1)

1996 (2)

1994 (1)

1993 (1)

1990 (1)

W. Chew and Y. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef]

1985 (1)

1969 (1)

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Acad. Sci. 268, 1060-1063 (1969).

1966 (1)

R. Petit, "Diffraction d’une onde plane par un reseau metallique," Rev. Opt. 45, 353-370 (1966).

Amra, C.

L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008).
[CrossRef]

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
[CrossRef]

G. Georges, C. Deumié, and C. Amra, "Selective probing and imaging in random media based on the elimination of polarized scattering," Opt. Express 15, 9804-9816 (2007).
[CrossRef] [PubMed]

O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005).
[CrossRef] [PubMed]

C. Amra, C. Deumie, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 854-864 (2005).
[CrossRef]

H. Giovannini and C. Amra, "Scattering-reduction effect with overcoated rough surfaces: theory and experiment," Appl. Opt. 36, 5574-5579 (1997).
[CrossRef] [PubMed]

C. Amra, "Light scattering from multilayer optics. I. Tools of investigation," J. Opt. Soc. Am. A 11197-210 (1994).
[CrossRef]

C. Amra, C. Grèzes-Besset, and L. Bruel, "Comparison of surface and bulk scattering in optical multilayers," Appl. Opt. 32, 5492-5503 (1993).
[CrossRef] [PubMed]

Arnaud, L.

L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008).
[CrossRef]

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

Ausserré, D.

D. Ausserré and M. Valignat, "Wide-field optical imaging of surface nanostructures," Nano Lett. 6, 1384-1388 (2006).
[CrossRef] [PubMed]

Bruel, L.

Cadilhac, M.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Acad. Sci. 268, 1060-1063 (1969).

Cerutti-Maori, G.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Acad. Sci. 268, 1060-1063 (1969).

Chazallet, F.

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

Chew, W.

W. Chew and Y. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef]

de Rosny, J.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
[CrossRef]

Derode, A.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
[CrossRef]

Deumie, C.

C. Amra, C. Deumie, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 854-864 (2005).
[CrossRef]

Deumié, C.

L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008).
[CrossRef]

C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
[CrossRef]

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

G. Georges, C. Deumié, and C. Amra, "Selective probing and imaging in random media based on the elimination of polarized scattering," Opt. Express 15, 9804-9816 (2007).
[CrossRef] [PubMed]

O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005).
[CrossRef] [PubMed]

Dunsby, C.

C. Dunsby and P. French, "Techniques for depth-resolved imaging through turbid media including coherencegated imaging," J. Phys. D: Appl. Phys. 36, R207-R227 (2003).
[CrossRef]

Fink, M.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
[CrossRef]

French, P.

C. Dunsby and P. French, "Techniques for depth-resolved imaging through turbid media including coherencegated imaging," J. Phys. D: Appl. Phys. 36, R207-R227 (2003).
[CrossRef]

Georges, G.

L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008).
[CrossRef]

C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
[CrossRef]

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

G. Georges, C. Deumié, and C. Amra, "Selective probing and imaging in random media based on the elimination of polarized scattering," Opt. Express 15, 9804-9816 (2007).
[CrossRef] [PubMed]

Gilbert, O.

O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005).
[CrossRef] [PubMed]

C. Amra, C. Deumie, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 854-864 (2005).
[CrossRef]

Giovannini, H.

Grèzes-Besset, C.

Le Neindre, N.

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

Lerosey, G.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
[CrossRef]

Li, L.

Micolau, G.

M. Saillard, P. Vincent, and G. Micolau, "Reconstruction of buried objects surrounded by small inhomogeneities," Inv. Problems 16, 1195-1208 (2000).
[CrossRef]

Navratil, K.

Neviere, M.

Ohlidal, I.

Petit, R.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Acad. Sci. 268, 1060-1063 (1969).

R. Petit, "Diffraction d’une onde plane par un reseau metallique," Rev. Opt. 45, 353-370 (1966).

Popov, E.

Saillard, M.

M. Saillard, P. Vincent, and G. Micolau, "Reconstruction of buried objects surrounded by small inhomogeneities," Inv. Problems 16, 1195-1208 (2000).
[CrossRef]

H. Giovannini, M. Saillard, and A. Sentenac, "Numerical study of scattering from rough inhomogeneous films," J. Opt. Soc. Am. A 15, 1182-1191 (1998).
[CrossRef]

Sentenac, A.

Siozade, L.

C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
[CrossRef]

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

Tourin, A.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
[CrossRef]

Valignat, M.

D. Ausserré and M. Valignat, "Wide-field optical imaging of surface nanostructures," Nano Lett. 6, 1384-1388 (2006).
[CrossRef] [PubMed]

Vincent, P.

M. Saillard, P. Vincent, and G. Micolau, "Reconstruction of buried objects surrounded by small inhomogeneities," Inv. Problems 16, 1195-1208 (2000).
[CrossRef]

Wang, Y.

W. Chew and Y. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef]

Zerrad, M.

G. Georges, L. Arnaud, L. Siozade, N. Le Neindre, F. Chazallet, M. Zerrad, C. Deumié, and C. Amra, "From angle-resolved ellipsometry of light scattering to imaging in random media," Appl. Opt. 47, 257-265 (2008).
[CrossRef]

C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, "Time reversal of wideband microwaves," Appl. Phys. Lett. 88, 101 (2006).
[CrossRef]

C. R. Acad. Sci. (1)

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Acad. Sci. 268, 1060-1063 (1969).

IEEE Trans. Med. Imaging (1)

W. Chew and Y. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990).
[CrossRef]

Inv. Problems (1)

M. Saillard, P. Vincent, and G. Micolau, "Reconstruction of buried objects surrounded by small inhomogeneities," Inv. Problems 16, 1195-1208 (2000).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Phys. D: Appl. Phys. (1)

C. Dunsby and P. French, "Techniques for depth-resolved imaging through turbid media including coherencegated imaging," J. Phys. D: Appl. Phys. 36, R207-R227 (2003).
[CrossRef]

Nano Lett. (1)

D. Ausserré and M. Valignat, "Wide-field optical imaging of surface nanostructures," Nano Lett. 6, 1384-1388 (2006).
[CrossRef] [PubMed]

Opt. Commun. (1)

L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281,1739-1744 (2008).
[CrossRef]

Opt. Express (4)

O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005).
[CrossRef] [PubMed]

G. Georges, C. Deumié, and C. Amra, "Selective probing and imaging in random media based on the elimination of polarized scattering," Opt. Express 15, 9804-9816 (2007).
[CrossRef] [PubMed]

C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express 16, 372-383 (2008).
[CrossRef]

C. Amra, C. Deumie, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 854-864 (2005).
[CrossRef]

Rev. Opt. (1)

R. Petit, "Diffraction d’une onde plane par un reseau metallique," Rev. Opt. 45, 353-370 (1966).

Other (5)

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

A. Voronovich, Wave Scattering from Rough Surfaces (Springer, 1994).

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon Press New York, 1999).

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Norwood, MA, Artech House, Inc., 1987).

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Figures (13)

Fig. 1.
Fig. 1.

Definition of the axes and description of the polarization of the scattered field.

Fig. 2.
Fig. 2.

Basic principles of the mounting used for selective reduction of the scatterd light.

Fig. 3.
Fig. 3.

Local reflection on the rough surface.

Fig. 4.
Fig. 4.

Ellipsometric parameters ∣Ap ∣/∣As ∣ and d of the scattered field by a rough surface with a high slope (w = 100%, i.e. 45°) for a roughness of 100 nm. The parameters are presented as a function of the scattering angle, the incidence angle is 50°, the incident field is linearly polarized at 45° of the s direction, the wavelength is 632.8 nm and the sample is dielectric with a refractive index equal to 1.5. Calculations are given for perturbative (first order) and rigorous (differential) theories.

Fig. 5.
Fig. 5.

Ellipsometric parameters ∣Ap ∣/∣As ∣ and δ of the field scattered by a rough surface with low slope (w = 2%, i.e. 1.15°) and a roughness of 100 nm. Same conditions and medium properties as for Fig. 4.

Fig. 6.
Fig. 6.

s and p intensity scattered by a rough surface with moderated slope (w = 15%) and a roughness of 500 nm. Same conditions and medium properties as for Fig. 4.

Fig. 7.
Fig. 7.

s and p phases and their difference for a rough glass surface with a 100 nm roughness, a 10% slope, and the same parameters as for Fig. 4 otherwise. Both phases are strongly correlated, which leads to few oscillations for their difference δ.

Fig. 8.
Fig. 8.

Ap ∣/∣As ∣ for several roughness, in the case of a rough surface with a 15% slope. Normal incidence, wavelength of 632.8 nm, refractive index equal to 1.5.

Fig. 9.
Fig. 9.

Reduction of light scattered by a surface with 15% slope and 500 nm roughness: simulation by the differential method. Top: intensities before and after reduction. Bottom: reduction procedure’s parameters obtained by the local reflections model. The ratio between the total scattered intensity (integrated between -90° and +90°) after and before the reduction procedure is 2.9 · 10-4. The incident field is polarized linearly at 45° of the s direction, has an incidence of 50° and a wavelength of 632.8 nm. The parameters of the surface are identical to those of Fig. 6.

Fig. 10.
Fig. 10.

Cylindrical inclusions with refractive index 2.0, situated in a medium with refractive index 1.5, under a rough interface. Total height: 10 μm, total length: 55.8 μm.

Fig. 11.
Fig. 11.

(a): Intensity scattered by cylindrical inclusions of Fig. 10, without rough interface, before and after reduction of the scattered light using the parameters of Fig. 9. The attenuation coefficient of the whole scattered light is 0.12 (compared to 0.00029 for surface scattered light reduction presented Fig. 9). (b): Intensity scattered by the cylindrical inclusions and the rough interface of Fig. 10, before and after reduction of the light scattered by the interface. The remaining light is comparable to the remaining light in the absence of rough interface (cf. (a)).

Fig. 12.
Fig. 12.

Scattering from a highly rough surface, before (a) and after (b) application of the light scattering reduction parameters. The attenuation coefficient is around 10-3. The aperture and exposure time of the camera are identical for both images, but the intensity of the incident beam was reduced by a factor 102 for image (a) to ovoid overexposure.

Fig. 13.
Fig. 13.

Rough surface with heterogeneous object situated underneath: (a) without reduction of the scattered light, (b) with reduction of the light scattered by the interface. Same aperture and exposure time for (a) and (b), but the intensity of the incident beam is reduced by a factor 10 for image (a) to avoid overexposure.

Equations (29)

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E = ( A s + A p ) exp ( i k · ρ )
A s = A s exp ( i δ s ) e s
A p = A p exp ( i δ p ) e p ,
δ = δ p δ s .
A ' = cos ψ * ( A s + tan ψ * A p exp ( i δ * ) )
f ( A ) = cos ψ * ( A s + ζ A p ) ,
ζ = tan ψ * exp ( i δ * ) .
tan ψ * = A s A p
δ * = δ + π
A = A 1 + A 2 + A 12 ,
R q 2 = 1 S S ( h ( r ) h 0 ) 2 d r ,
Γ ( r ) = 1 S S h ( r ' ) h ( r ' + r ) d r ' .
w 2 = 1 S S grad ( h ( r ) ) 2 d r .
θ = 2 θ local θ i
θ local = θ + θ i 2 .
A p = r p A 0 p
A s = r s A 0 s ,
r p = β 1 β 2 / n 2 β 1 + β 2 / n 2
r s = β 1 β 2 β 1 + β 2 .
β 1 = k cos ( θ i + θ 2 )
β 2 = k n 2 sin 2 ( θ i + θ 2 ) .
A p A s = β 1 + β 2 β 1 β 2 n 2 β 1 β 2 n 2 β 1 + β 2 A 0 p A 0 s ,
δ = arg ( r p A 0 p r s A 0 s ) .
θ i + θ 2 = arctan ( n ) .
θ step = 2 arctan ( n ) θ i .
I ( σ ) = C ( σ , σ 0 ) γ ( σ σ 0 )
γ ( σ ) = 4 π S h ˜ ( σ ) 2 ,
θ b = arcsin ( n 2 ( n 2 sin 2 θ i ) n 2 + ( n 4 1 ) sin 2 θ i ) .
I annul cos 2 ψ * ( A s 2 + tan 2 ψ * A p 2 + 2 A s A p tan ψ * cos ( δ + δ * ) ) .

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