Abstract

Experimental results are presented for the angular correlation functions of far-field intensity scattered by a conducting, one-dimensionally rough surface that produces backscattering enhancement. A detailed study of the special case of equal incident and scattering angles is presented, in which it is found that the intensity correlation functions exhibit two distinct and equal maxima, both of which imply perfect correlation within experimental accuracies. One of these is an autocorrelation peak, and the second peak arises from the cross correlation between two distinct intensities related by a reciprocity condition. It is found that, if the rough surface is illuminated by a +45° polarization state, the angular correlation functions of scattered intensity polarized at −45° have broad structures that are interpreted as arising from single-scattering processes. The scattered intensity polarized at +45° has quite different correlation functions whose properties are attributed to multiple-scattering processes; this interpretation is based on generalizations of arguments presented in previous studies of backscattering enhancement. In support of these conclusions, rigorous theoretical results for the angular correlation functions of scattered amplitude are presented.

© 1992 Optical Society of America

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References

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  1. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  2. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  3. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [CrossRef] [PubMed]
  4. T. Michel, A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from a non-Gaussian random metal surface,” J. Opt. Soc. Am. B 6, 2438–2446 (1989).
    [CrossRef]
  5. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  6. A. J. Sant, J. C. Dainty, M. J. Kim, “Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers,” Opt. Lett. 14, 1183–1185 (1989).
    [CrossRef] [PubMed]
  7. 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]
  8. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  9. J. S. Chen, A. Ishimaru, “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]
  10. M. E. Knotts, K. A. O’Donnell, “Anomalous scattering from a perturbed grating,” Opt. Lett. 15, 1485–1487 (1990).
    [CrossRef] [PubMed]
  11. J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  12. A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Med. 1, 21–34 (1991).
    [CrossRef]
  13. 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]
  14. A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Med. 3, S91–S107 (1991).
    [CrossRef]
  15. E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Med. 3, S165–S190 (1991).
    [CrossRef]
  16. K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
    [CrossRef]
  17. 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]
  18. See, for example, J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), and references therein.
    [CrossRef]
  19. H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
    [CrossRef]
  20. D. Léger, J. C. Perrin, “Real-time measurement of surface roughness by correlation of speckle patterns,”J. Opt. Soc. Am. 66, 1210–1217 (1976).
    [CrossRef]
  21. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  22. S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
    [CrossRef] [PubMed]
  23. R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
    [CrossRef]
  24. L. Wang, S. Feng, “Correlations and fluctuations in reflection coefficients for coherent wave propagation in disordered scattering media,” Phys. Rev. B 40, 8284–8289 (1989).
    [CrossRef]
  25. I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
    [CrossRef] [PubMed]
  26. I. Freund, M. Rosenbluh, R. Berkovits, “Geometrical scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
    [CrossRef]
  27. R. Berkovits, M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
    [CrossRef]
  28. R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
    [CrossRef]
  29. T. R. Michel, K. A. O’Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
    [CrossRef]
  30. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  31. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
  32. D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
    [CrossRef]
  33. See Eq. (17) of Ref. 20.
  34. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).
  35. G. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
    [CrossRef]
  36. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  37. R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
    [CrossRef]

1992 (2)

1991 (6)

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

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

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Med. 3, S91–S107 (1991).
[CrossRef]

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Med. 3, S165–S190 (1991).
[CrossRef]

K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
[CrossRef]

1990 (7)

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]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

J. S. Chen, A. Ishimaru, “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]

M. E. Knotts, K. A. O’Donnell, “Anomalous scattering from a perturbed grating,” Opt. Lett. 15, 1485–1487 (1990).
[CrossRef] [PubMed]

R. Berkovits, M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

1989 (7)

1988 (2)

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

1987 (2)

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

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

1984 (1)

G. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
[CrossRef]

1978 (1)

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

1976 (1)

1975 (1)

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

1955 (1)

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

Beckmann, P.

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

Berkovits, R.

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

R. Berkovits, M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, “Geometrical scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

Brown, G.

G. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
[CrossRef]

Bruce, N. 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]

Chen, J. S.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Med. 3, S91–S107 (1991).
[CrossRef]

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

J. S. Chen, A. Ishimaru, “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.

Feng, S.

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[CrossRef]

L. Wang, S. Feng, “Correlations and fluctuations in reflection coefficients for coherent wave propagation in disordered scattering media,” Phys. Rev. B 40, 8284–8289 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Freund, I.

I. Freund, M. Rosenbluh, R. Berkovits, “Geometrical scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Friberg, A. T.

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

Garcia-Molina, R.

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Goodman, J. W.

See, for example, J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), and references therein.
[CrossRef]

Gray, P. F.

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Ishimaru, A.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Med. 3, S91–S107 (1991).
[CrossRef]

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

J. S. Chen, A. Ishimaru, “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 Med. 3, S165–S190 (1991).
[CrossRef]

Kane, C.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Kaveh, M.

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

R. Berkovits, M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[CrossRef]

Kim, M. J.

Knotts, M. E.

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Léger, D.

Leskova, T. A.

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Maradudin, A. A.

Maystre, D.

Méndez, E. R.

Michel, T.

Michel, T. R.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Pedersen, H. M.

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Perrin, J. C.

Phu, P.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Med. 3, S91–S107 (1991).
[CrossRef]

Rosenbluh, M.

I. Freund, M. Rosenbluh, R. Berkovits, “Geometrical scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Saillard, M.

Sanchez-Gil, J. A.

Sant, A. J.

Saxon, D. S.

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

Soto-Crespo, J. M.

Spizzichino, A.

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

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Thorsos, E. I.

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Med. 3, S165–S190 (1991).
[CrossRef]

Wang, L.

L. Wang, S. Feng, “Correlations and fluctuations in reflection coefficients for coherent wave propagation in disordered scattering media,” Phys. Rev. B 40, 8284–8289 (1989).
[CrossRef]

Yoshitomi, K.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Med. 3, S91–S107 (1991).
[CrossRef]

Europhys. Lett. (1)

R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. S. Chen, A. Ishimaru, “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. 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. (1)

J. Opt. Soc. Am. A (8)

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

Opt. Acta (2)

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Opt. Commun. (1)

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Opt. Lett. (3)

Phys. Rep. (1)

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Phys. Rev. (1)

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

Phys. Rev. B (4)

R. Berkovits, M. Kaveh, S. Feng, “Memory effect of waves in disordered systems: a real-space approach,” Phys. Rev. B 40, 737–740 (1989).
[CrossRef]

L. Wang, S. Feng, “Correlations and fluctuations in reflection coefficients for coherent wave propagation in disordered scattering media,” Phys. Rev. B 40, 8284–8289 (1989).
[CrossRef]

I. Freund, M. Rosenbluh, R. Berkovits, “Geometrical scaling of the optical memory effect in coherent-wave propagation through random media,” Phys. Rev. B 39, 12,403–12,406 (1989).
[CrossRef]

R. Berkovits, M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
[CrossRef]

Phys. Rev. Lett. (2)

I. Freund, M. Rosenbluh, S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Radio Sci. (1)

G. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
[CrossRef]

Waves Random Med. (3)

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

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Med. 3, S91–S107 (1991).
[CrossRef]

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Med. 3, S165–S190 (1991).
[CrossRef]

Other (6)

See, for example, J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), and references therein.
[CrossRef]

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

See Eq. (17) of Ref. 20.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

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

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Figures (11)

Fig. 1
Fig. 1

Scattering of light by a one-dimensionally rough surface. The incident and scattering angles θi and θs are positive as shown.

Fig. 2
Fig. 2

Scattering experiment used to measure the angular correlation functions of intensity scattered from the rough surface. The incident beam was reflected by a series of mirrors (M’s), passed through a polarizer (P) that produced a +45° polarization state, and was then incident upon the rough surface. One may rotate the surface about the vertical axis to determine the incident angle θi and rotate the detector arm out of the plane of the figure to determine the detected scattering angle θs. A slit of 150-μm width (not shown) was mounted immediately in front of the field lens.

Fig. 3
Fig. 3

Mean scattered intensities 〈I(θi, θi)〉 (circles) and 〈I+(θi, θi)〉 (squares) obtained by direct averaging of the data.

Fig. 4
Fig. 4

Measured correlation functions of −45°-polarized and +45°-polarized scattered intensities Γ−−(θi1, θi1, θi2, θi2) and Γ++(θi1, θi1, θi2, θi2) for θi1 = 25.0°. The autocorrelation and reciprocal peaks are present at θi2 = 25.0° and θi2 = −25.0°, respectively.

Fig. 5
Fig. 5

Measured correlation functions of −45°-polarized and +45°-polarized scattered intensities Γ−−(θi1, θi1, θi2, θi2) and Γ++(θi1, θi1, θi2, θi2) for θi1 = 15.0°. The autocorrelation and reciprocal peaks are present at θi2 = 15.0° and θi2 = −15.0° respectively, and some overlap may now be seen between the two peaks of Γ−−.

Fig. 6
Fig. 6

Measured correlation functions of −45°-polarized and +45°-polarized scattered intensities Γ−− (θi1, θi1, θi2, θi2) and Γ++(θi1, θi1, θi2, θi2) for θi1 = 7.0°, where there is significant interaction of the two peaks of Γ++.

Fig. 7
Fig. 7

Measured correlation functions of −45°-polarized and +45°-polarized scattered intensities Γ−−(θi1, θi1, θi2, θi2) and Γ++(θi1, θi1, θi2, θi2) for θi1 = 3.5°. The central maximum present in Γ++ in Fig. 6 has completely disappeared, and only small autocorrelation and reciprocal peaks are present in Γ++.

Fig. 8
Fig. 8

Measured correlation functions of −45°-polarized and +45°-polarized scattered intensities Γ−−(θi1, θi1, θi2, θi2) and Γ++(θi1, θi1, θi2, θi2) for θi1 = 0.5°, which is the smallest useful angle in the experiment. On comparison with Fig. 7, Γ++ has completely changed form and exhibits a large central maximum, zeros at θi2 = ±3.5°, and significant recorrelation for larger |θi2|.

Fig. 9
Fig. 9

Three-dimensional plot of the correlation function of +45°-polarized scattered intensity Γ++(θi1, θi1, θi2, θi2). The back face at the left is the result at θi1 = 0.5° from Fig. 8, and the suppressed region at θi1 = 3.5° is from Fig. 7. The evolution of the fringelike structures into the forward (θi1 = θi2) and reciprocal (θi1 = θi2) parts of Γ++ is apparent for larger θi1.

Fig. 10
Fig. 10

Pairs of multiple-scattering paths occurring within a valley of a randomly rough surface: (a) a pair of paths for incident and scattered wave vectors ki and ks; (b) the same pair but with incident and scattered wave vectors ki′ and ks′. The contributions to the correlation function of +45°-polarized scattered amplitudes arise from interference between the four distinct combinations of one path of (a) and one of (b).

Fig. 11
Fig. 11

Theoretical calculations of the correlation functions of scattered −45°-polarized and +45°-polarized scattered amplitudes C−−(θi1, θi1, θi2, θi2) and C++(θi1, θi1, θi2, θi2) for θi1 = 0 for a one-dimensionally rough conducting surface. The solid and dashed curves denote, respectively, the real and imaginary parts. The assumed surface statistics are those of a Gaussian process with a Gaussian correlation function [height standard deviation σ = 1.5λ, 1/e correlation half-width a = 3λ, complex refractive index (n + ik) = (0.312 + i7.93)]. These results may be compared with the experimental measurements of Γ−− and Γ++ in Fig. 8.

Equations (14)

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C α β ( θ i 1 , θ s 1 , θ i 2 , θ s 2 ) = A α ( θ i 1 , θ s 1 ) A β * ( θ i 2 , θ s 2 ) ,
θ s 1 - θ s 2 λ / w ,
sin ( θ i 1 ) - sin ( θ s 1 ) = sin ( θ i 2 ) - sin ( θ s 2 )
Γ α β ( θ i 1 , θ s 1 , θ i 2 , θ s 2 ) = Δ I α ( θ i 1 , θ s 1 ) Δ I β ( θ i 2 , θ s 2 ) ,
Γ α β ( θ i 1 , θ s 1 , θ i 2 , θ s 2 ) = A α ( θ i 1 , θ s 1 ) A β * ( θ i 2 , θ s 2 ) 2 ,
Γ + + ( θ i 1 , θ i 1 , θ i 2 , θ i 2 ) = Δ I + ( θ i 1 , θ i 1 ) Δ I + ( θ i 2 , θ i 2 ) ,
Γ - - ( θ i 1 , θ i 1 , θ i 2 , θ i 2 ) = Δ I - ( θ i 1 , θ i 1 ) Δ I - ( θ i 2 , θ i 2 ) ,
I = I + + I - ,
C + + ( θ i , θ s , θ i , θ s ) = A + ( θ i , θ s ) A + * ( θ i , θ s ) .
ϕ A B = k i · r A - k s · r B + k Δ r ,
Δ r = Δ r = r A - r B .
Δ ϕ A B , A B = ½ ( Δ ϕ - Δ ϕ ) , Δ ϕ A B , B A = ½ ( Δ ϕ + Δ ϕ ) , Δ ϕ B A , A B = - ½ ( Δ ϕ + Δ ϕ ) , Δ ϕ B A , B A = - ½ ( Δ ϕ - Δ ϕ ) ,
Δ ϕ = ( k i + k s ) · Δ r ,
Δ ϕ = ( k i + k s ) · Δ r ,

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