Abstract

We propose and study a numerical procedure for the reconstruction of surface profiles from far-field scattering data. The algorithm, based on wave-front-matching principles, is used to reconstruct one-dimensional surface profiles from amplitude scattering data calculated by using rigorous techniques. The study is complemented by the development of a sampling strategy and considerations of the tolerance of the algorithm to noise in the data.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. DeSanto, G. Brown, “Analytical techniques for multiple scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23, pp. 2–62.
  2. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).
  3. K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
    [CrossRef]
  4. M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
    [CrossRef]
  5. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  6. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  7. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 29.
  8. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  9. A. Mendoza-Suárez, E. R. Méndez, “Light scattering by reentrant fractal surfaces,” Appl. Opt. 36, 3521–3531 (1997).
    [CrossRef]
  10. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  11. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 61–95 (1987).
  12. P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a randomly rough 2-D metallic surface,” Opt. Commun. 110, 269–273 (1994).
    [CrossRef]
  13. P. Tran, “Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interactions,” Waves Random Media 7, 295–302 (1997).
    [CrossRef]
  14. K. Pak, L. Tsang, J. T. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough sur-faces with the sparse-matrix canonical grid method,” J. Opt. Soc. Am. A 14, 1515–1529 (1997).
    [CrossRef]
  15. F. Chen, “Computer simulation of wave scattering from three-dimensional conducting random surfaces,” Int. J. Remote Sens. 21, 777–790 (2000).
    [CrossRef]
  16. P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
    [CrossRef]
  17. W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
    [CrossRef]
  18. P. J. Chandley, “Determination of the probability density function of height on a rough surface from far-field coherent light scattering,” Opt. Quantum Electron. 11, 413–418 (1979).
    [CrossRef]
  19. J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
    [CrossRef]
  20. J. C. Stover, S. A. Serati, C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
    [CrossRef]
  21. V. Malyshkin, S. Simeonov, A. R. McGurn, A. A. Maradudin, “Determination of surface profile statistics from electromagnetic scattering data,” Opt. Lett. 22, 58–60 (1997).
    [CrossRef] [PubMed]
  22. R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 1. Gaussian spectrum,” Radio Sci. 33, 821–834 (1998).
    [CrossRef]
  23. R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 2. Pierson–Moskowitz spectrum,” Radio Sci. 33, 835–843 (1998).
    [CrossRef]
  24. R. J. Wombell, J. A. DeSanto, “Reconstruction of rough-surface profiles with the Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
    [CrossRef]
  25. R. J. Wombell, J. A. DeSanto, “The reconstruction of shallow rough-surfaces profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
    [CrossRef]
  26. J. C. Quartel, C. J. R. Sheppard, “Surface reconstruc-tion using an algorithm based on confocal imaging,” J. Mod. Opt. 43, 469–486 (1996).
    [CrossRef]
  27. J. C. Quartel, C. J. R. Sheppard, “A surface reconstruction algorithm based on confocal interferometric profiling,” J. Mod. Opt. 43, 591–605 (1996).
    [CrossRef]
  28. K. Harada, A. Noguchi, “Reconstruction of two dimensional rough surface with Gaussian beam illumination,” IEICE Trans. Electron. E79-C, 1345–1349 (1996).
  29. N. Destouches, C. Guérin, M. Lequime, H. Giovaninni, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
    [CrossRef]
  30. E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
    [CrossRef]
  31. 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]
  32. E. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 485.
  33. C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  34. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).
  35. T. Wilson, ed., Confocal Microscopy (Academic, New York, 1990).
  36. D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
    [CrossRef]
  37. T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
    [CrossRef]
  38. J. F. Aguilar, M. Lera, C. J. R. Sheppard, “Imaging of spheres and surface profiling by confocal microscopy,” Appl. Opt. 39, 4621–4628 (2000).
    [CrossRef]
  39. I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of optical signatures of materials,” Appl. Phys. Lett. 41, 604–606 (1982).
    [CrossRef]
  40. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 550.
  41. J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984), pp. 342, 485.

2001 (3)

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

N. Destouches, C. Guérin, M. Lequime, H. Giovaninni, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

2000 (2)

J. F. Aguilar, M. Lera, C. J. R. Sheppard, “Imaging of spheres and surface profiling by confocal microscopy,” Appl. Opt. 39, 4621–4628 (2000).
[CrossRef]

F. Chen, “Computer simulation of wave scattering from three-dimensional conducting random surfaces,” Int. J. Remote Sens. 21, 777–790 (2000).
[CrossRef]

1998 (2)

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 1. Gaussian spectrum,” Radio Sci. 33, 821–834 (1998).
[CrossRef]

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 2. Pierson–Moskowitz spectrum,” Radio Sci. 33, 835–843 (1998).
[CrossRef]

1997 (4)

1996 (3)

J. C. Quartel, C. J. R. Sheppard, “Surface reconstruc-tion using an algorithm based on confocal imaging,” J. Mod. Opt. 43, 469–486 (1996).
[CrossRef]

J. C. Quartel, C. J. R. Sheppard, “A surface reconstruction algorithm based on confocal interferometric profiling,” J. Mod. Opt. 43, 591–605 (1996).
[CrossRef]

K. Harada, A. Noguchi, “Reconstruction of two dimensional rough surface with Gaussian beam illumination,” IEICE Trans. Electron. E79-C, 1345–1349 (1996).

1994 (1)

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a randomly rough 2-D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

1992 (1)

1991 (2)

R. J. Wombell, J. A. DeSanto, “Reconstruction of rough-surface profiles with the Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
[CrossRef]

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

1990 (1)

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

1989 (1)

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

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 (1)

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 61–95 (1987).

1985 (1)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

1984 (1)

J. C. Stover, S. A. Serati, C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

1982 (2)

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of optical signatures of materials,” Appl. Phys. Lett. 41, 604–606 (1982).
[CrossRef]

1979 (2)

P. J. Chandley, “Determination of the probability density function of height on a rough surface from far-field coherent light scattering,” Opt. Quantum Electron. 11, 413–418 (1979).
[CrossRef]

J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
[CrossRef]

1977 (2)

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

1976 (1)

P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Aguilar, J. F.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 29.

Bennett, J. M.

Born, E.

E. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 485.

Brown, G.

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

Brown, G. S.

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 1. Gaussian spectrum,” Radio Sci. 33, 821–834 (1998).
[CrossRef]

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 2. Pierson–Moskowitz spectrum,” Radio Sci. 33, 835–843 (1998).
[CrossRef]

Celli, V.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Chandley, P. J.

P. J. Chandley, “Determination of the probability density function of height on a rough surface from far-field coherent light scattering,” Opt. Quantum Electron. 11, 413–418 (1979).
[CrossRef]

P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

Chen, F.

F. Chen, “Computer simulation of wave scattering from three-dimensional conducting random surfaces,” Int. J. Remote Sens. 21, 777–790 (2000).
[CrossRef]

Chew, W. C.

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Cox, I. J.

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of optical signatures of materials,” Appl. Phys. Lett. 41, 604–606 (1982).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984), pp. 342, 485.

DeSanto, J. A.

R. J. Wombell, J. A. DeSanto, “Reconstruction of rough-surface profiles with the Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
[CrossRef]

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

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

Destouches, N.

N. Destouches, C. Guérin, M. Lequime, H. Giovaninni, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Elson, J. M.

Giovaninni, H.

N. Destouches, C. Guérin, M. Lequime, H. Giovaninni, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 550.

Guérin, C.

N. Destouches, C. Guérin, M. Lequime, H. Giovaninni, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Guillespie, C. H.

J. C. Stover, S. A. Serati, C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Hamilton, D. K.

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of optical signatures of materials,” Appl. Phys. Lett. 41, 604–606 (1982).
[CrossRef]

Harada, K.

K. Harada, A. Noguchi, “Reconstruction of two dimensional rough surface with Gaussian beam illumination,” IEICE Trans. Electron. E79-C, 1345–1349 (1996).

Jackson, D. R.

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

Johnson, J. T.

Knotts, M. E.

Lequime, M.

N. Destouches, C. Guérin, M. Lequime, H. Giovaninni, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Lera, M.

Malyshkin, V.

Maradudin, A. A.

V. Malyshkin, S. Simeonov, A. R. McGurn, A. A. Maradudin, “Determination of surface profile statistics from electromagnetic scattering data,” Opt. Lett. 22, 58–60 (1997).
[CrossRef] [PubMed]

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a randomly rough 2-D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

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

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Marchand, R. T.

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 2. Pierson–Moskowitz spectrum,” Radio Sci. 33, 835–843 (1998).
[CrossRef]

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 1. Gaussian spectrum,” Radio Sci. 33, 821–834 (1998).
[CrossRef]

McGurn, A. R.

V. Malyshkin, S. Simeonov, A. R. McGurn, A. A. Maradudin, “Determination of surface profile statistics from electromagnetic scattering data,” Opt. Lett. 22, 58–60 (1997).
[CrossRef] [PubMed]

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

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Méndez, E. R.

A. Mendoza-Suárez, E. R. Méndez, “Light scattering by reentrant fractal surfaces,” Appl. Opt. 36, 3521–3531 (1997).
[CrossRef]

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

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 61–95 (1987).

Mendoza-Suárez, A.

Michel, T.

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

Michel, T. R.

Noguchi, A.

K. Harada, A. Noguchi, “Reconstruction of two dimensional rough surface with Gaussian beam illumination,” IEICE Trans. Electron. E79-C, 1345–1349 (1996).

O’Donnell, K. A.

T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
[CrossRef]

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 61–95 (1987).

Ogilvy, J. A.

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

Pak, K.

Quartel, J. C.

J. C. Quartel, C. J. R. Sheppard, “Surface reconstruc-tion using an algorithm based on confocal imaging,” J. Mod. Opt. 43, 469–486 (1996).
[CrossRef]

J. C. Quartel, C. J. R. Sheppard, “A surface reconstruction algorithm based on confocal interferometric profiling,” J. Mod. Opt. 43, 591–605 (1996).
[CrossRef]

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Saillard, M.

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

Sentenac, A.

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

Serati, S. A.

J. C. Stover, S. A. Serati, C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Sheppard, C. J. R.

J. F. Aguilar, M. Lera, C. J. R. Sheppard, “Imaging of spheres and surface profiling by confocal microscopy,” Appl. Opt. 39, 4621–4628 (2000).
[CrossRef]

J. C. Quartel, C. J. R. Sheppard, “A surface reconstruction algorithm based on confocal interferometric profiling,” J. Mod. Opt. 43, 591–605 (1996).
[CrossRef]

J. C. Quartel, C. J. R. Sheppard, “Surface reconstruc-tion using an algorithm based on confocal imaging,” J. Mod. Opt. 43, 469–486 (1996).
[CrossRef]

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of optical signatures of materials,” Appl. Phys. Lett. 41, 604–606 (1982).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Simeonov, S.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 29.

Stover, J. C.

J. C. Stover, S. A. Serati, C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Thorsos, E. I.

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[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]

Tran, P.

P. Tran, “Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interactions,” Waves Random Media 7, 295–302 (1997).
[CrossRef]

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a randomly rough 2-D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

Tsang, L.

Voronovich, G.

G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

Warnick, K. F.

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Welford, W. T.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Wilson, T.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Wolf, E.

E. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 485.

Wombell, R. J.

R. J. Wombell, J. A. DeSanto, “Reconstruction of rough-surface profiles with the Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
[CrossRef]

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

Ann. Phys. (N.Y.) (1)

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

Appl. Opt. (2)

Appl. Phys. Lett. (1)

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of optical signatures of materials,” Appl. Phys. Lett. 41, 604–606 (1982).
[CrossRef]

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

IEICE Trans. Electron. (1)

K. Harada, A. Noguchi, “Reconstruction of two dimensional rough surface with Gaussian beam illumination,” IEICE Trans. Electron. E79-C, 1345–1349 (1996).

Int. J. Remote Sens. (1)

F. Chen, “Computer simulation of wave scattering from three-dimensional conducting random surfaces,” Int. J. Remote Sens. 21, 777–790 (2000).
[CrossRef]

Inverse Probl. (1)

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

J. Acoust. Soc. Am. (2)

E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[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. (2)

J. C. Quartel, C. J. R. Sheppard, “Surface reconstruc-tion using an algorithm based on confocal imaging,” J. Mod. Opt. 43, 469–486 (1996).
[CrossRef]

J. C. Quartel, C. J. R. Sheppard, “A surface reconstruction algorithm based on confocal interferometric profiling,” J. Mod. Opt. 43, 591–605 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (2)

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

Opt. Commun. (3)

N. Destouches, C. Guérin, M. Lequime, H. Giovaninni, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 61–95 (1987).

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a randomly rough 2-D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[CrossRef]

Opt. Eng. (1)

J. C. Stover, S. A. Serati, C. H. Guillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (3)

P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

P. J. Chandley, “Determination of the probability density function of height on a rough surface from far-field coherent light scattering,” Opt. Quantum Electron. 11, 413–418 (1979).
[CrossRef]

Phys. Rev. B (1)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Radio Sci. (2)

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 1. Gaussian spectrum,” Radio Sci. 33, 821–834 (1998).
[CrossRef]

R. T. Marchand, G. S. Brown, “Inferring rough surface parameters from average scattering data using approximate scattering models. 2. Pierson–Moskowitz spectrum,” Radio Sci. 33, 835–843 (1998).
[CrossRef]

Waves Random Media (3)

K. F. Warnick, W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

P. Tran, “Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interactions,” Waves Random Media 7, 295–302 (1997).
[CrossRef]

Other (9)

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

G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

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

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963), p. 29.

E. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 485.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

T. Wilson, ed., Confocal Microscopy (Academic, New York, 1990).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 550.

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984), pp. 342, 485.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Illustration of the reflection of a focused beam: (a) mirror in focus and (b) mirror out of focus.

Fig. 3
Fig. 3

Illustration of the effect of sampling on the incident field. The plots represent the intensity of the incident field in reversed contrast: (a) 21 angles of incidence, (b) 17 angles of incidence, (c) intensity resulting from the product of the corresponding field distributions shown in (a) and (b). The cutoff angle is θ0m=80°.

Fig. 4
Fig. 4

Realization of a Gaussian random process with correlation length a=2λ and standard deviation of heights δ=λ/2. The first and second derivatives are shown in the lower plot.

Fig. 5
Fig. 5

Angular spectrum Rp(q|k) as a function of q and k corresponding to the surface profile shown in Fig. 4.

Fig. 6
Fig. 6

Function Fp,s(wm)(ξ, η) when the random profile shown in Fig. 4 is illuminated with either (a) p-polarized or (b) s-polarized light. The solid curves in reversed contrast show the original profile.

Fig. 7
Fig. 7

Surface-profile reconstruction using the data shown in Fig. 6. The solid curve represents the actual profile used in the scattering calculations, while the circles represent estimated points of the profile.

Fig. 8
Fig. 8

Confocal signal Fp,s(conf)(ξ, η) when the random profile shown in Fig. 4 is illuminated with either (a) p-polarized or (b) s-polarized light. The solid curves in reversed contrast show the original profile.

Fig. 9
Fig. 9

Illustration of the effects of sampling of the incident and scattered fields on the function Fs-p(wm)(ξ, η) for the cases of (a) 61 angles of incidence and 61 angles of scattering, (b) 31 angles of incidence and 31 angles of scattering, and (c) 21 angles of incidence and 17 angles of scattering.

Fig. 10
Fig. 10

Illustration of the function Fp(wm)(ξ, η) for the case of (a) a silver surface and (b) a glass surface. The random profile that gave rise to the corresponding scattering amplitudes is denoted by the solid curves in reversed contrast.

Fig. 11
Fig. 11

Scattering data Rp(q|k) with multiplicative noise.

Fig. 12
Fig. 12

Function Fp(wm)(ξ, η) obtained by using the noisy scattering data shown in Fig. 11 with 61 angles of incidence and scattering.

Fig. 13
Fig. 13

Function Fs(wm)(ξ, η) for (a) a random profile with a=2λ and δ=λ and (b) a random profile with a=2λ and δ=1.5λ. The seed used for the random-number generator is the same as the one used for the profile of Fig. 4. The surface is illuminated with s-polarized light, and the solid curves in reversed contrast show the profile employed for the scattering calculations.

Equations (34)

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

ψinc(x, z)=ψ0 exp[ikx-iα0(k)z],
ψsc(x, z)=-dq2πRp,s(q|k)exp[iqx+iα0(q)z],
q=ωcsin θs.
Rp,s(q|k)=i2α0(q)-dx{[iqζ(x)-iα0(q)]φ(x)-χ(x)}×exp[-iqx-iα0(q)ζ(x)],
φ(x)=ψ(x, z)|z=ζ(x)+,
χ(x)=-ζ(x)x+zψ(x, z)z=ζ(x)+.
ψinc(x-ζ; z-η)=-dk2πPinc(k)exp[ik(x-ξ)]exp[-iα0(k)(z-η)],
Pinc(k)=rectk2kmax,
ψsc(x, z; φ, η)=-dk2πPinc(k)exp[-ikξ+iα0(k)η]×-dq2πRp,s(q|k)×exp[iqx+iα0(q)z].
Vp,s(ξ, η)=-|ψsc(x, z; ξ, η)+ψref(x-ξ; z-η)|2dx,
ψref(x-ξ; z-η)=-dq2πPref(q)exp[iq(x-ξ)]×exp[iα0(q)(z-η)],
Pref(q)=rectq2qmax
Vp,s(ξ, η)=2I0+2 Re[Up,s(ξ, η)],
2I0=-[|ψsc(x, z; ξ, η)|2+|ψref(x-ξ; z-η)|2]dx,
Up,s(ξ, η)=-ψsc(x, z; ξ, η)ψref*(x-ξ; z-η)dx.
Up,s(ξ, η)=-dk2π-dq2πPinc(k)Pref*(q)Rp,s(q|k)×Φ(q, k|ξ, η),
Φ(q, k|ξ, η)=exp{-i(k-q)ξ+i[α0(k)+α0(q)]η}.
Fp,s(wm)(ξ, η)=2 Re[Up,s(η, η)],
Fp,s(conf)(ξ, η)=|U(ξ, η)|2.
Pinc(k)=rectk2kmaxΔkm=-δ(k-mΔk),
ψinc(x, z; ξ, η)=n-dk2πrectk2kmax×exp[ik(x-ξ-nTinc)-iα0(k)(z-η)]
=nψinc(x, z; ξ+nTinc, η),
Pref(q)=rectq2qmaxΔqm=-δ(q-mΔq),
Ψref(x, z; ξ, η)=nψref(x, z; ξ+nTsc, η),
Ninc>Lkmaxπ,Nsc>Lqmaxπ.
Ninc=Nsc>60.
Rs(n)(q|k)=(-1)nRp(n)(q|k),n>0,
Rp,s(q|k)=Rp,s(1)(q|k)+Rp,s(2)(q|k)+Rp,s(3)(q|k)+ .
Rs-p(q|k)=Rs(q|k)=2[Rs(1)(q|k)+Rs(3)(q|k)+Rs(5)(q|k)+],
P|An|(x)=xσ2exp-x22σ2,x>0,
Pϕ(x)=x2πσϕexp-x22σϕ2.
RN(q|k)=Rp,s(q|k)AN(q|k).
I(q|k)=|Rp,s(q|k)|2,IN(q|k)=|AN(q|k)|2,
IN(q|k)=I(q|k)IN(q|k).

Metrics