Abstract

We study theoretically the angular dependence of the correlation functions of the scattering amplitudes occurring in the interaction of a beam of polarized light with a one-dimensional, perfectly conducting rough surface. For an ensemble of surface-profile functions that are realizations of a stationary stochastic process, a necessary condition for the exact scattering amplitudes to be correlated is established: the projection on the mean surface of the difference between the incident wave vectors must equal that of the scattered wave vectors. The exact expressions for the amplitudes of the scattered plane waves constituting the far field are derived from Green’s second integral theorem. By numerical simulations, the dependence of these amplitude correlation functions on the angle of incidence and on the incident and the scattered polarization states is computed for Gaussian surfaces producing enhanced backscattering. Results are presented for the complex correlation functions of p- and s-polarized scattering amplitudes. However, it is argued that, for an incident field polarized at +45°, the single- and the multiple-scattering contributions to the amplitude correlation functions are clearly separated if the scattered field is resolved into −45° and +45° polarized plane waves. In this case the real part of the correlation function of the −45° scattering amplitudes displays two peaks of large angular width. The maxima occur for angles of incidence at which the correlation between the +45° polarized scattering amplitudes has sharp peaks of angular width approximately equal to λ/a, where the surface correlation length a may be interpreted as an estimate for the mean distance between successive scattering points on the surface. One of these maxima occurs when the correlated amplitudes are identical, and the other occurs when the role of the incoming and the outgoing directions is interchanged between the two scattering amplitudes, which are then related by the reciprocity condition. The coherent addition of the amplitudes arising from multiple-scattering processes and those of their time-reversed partners in directions close to the retroreflection direction is demonstrated in the calculations of the correlation functions.

© 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. M.-J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
    [Crossref]
  4. 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]
  5. M. E. Knotts, K. A. O’Donnell, “Anomalous scattering from a perturbed grating,” Opt. Lett. 15, 1485–1487 (1990).
    [Crossref] [PubMed]
  6. K. A. O’Donnell, M. E. Knotts, “The polarization dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
    [Crossref]
  7. 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]
  8. Ya-Qiu Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
    [Crossref]
  9. P. Tran, A. A. Maradudin, V. Celli, “Backscattering enhancement from a dielectric surface,” J. Opt. Soc. Am. B 8, 1526–1530 (1991).
    [Crossref]
  10. A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
    [Crossref]
  11. 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]
  12. 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]
  13. D. Léger, E. Mathieu, J. C. Perrin, “Optical surface roughness determination using speckle correlation techniques,” Appl. Opt. 14, 872–877 (1975).
    [Crossref]
  14. 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]
  15. There is a typographical error in Eq. (14) of Ref. 14: the correlation length Tin the argument of the exponential should be squared.
  16. S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave propagation through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
    [Crossref] [PubMed]
  17. I. Freund, M. Rosenbluh, “Memory effect in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
    [Crossref] [PubMed]
  18. 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]
  19. 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]
  20. I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wavepropagation through random media,” Phys. Rev. B 39, 12403–12406 (1989).
    [Crossref]
  21. R. Berkovits, M. Kaveh, “Time-reversed memory effects,” Phys. Rev. B 41, 2635–2638 (1990).
    [Crossref]
  22. R. Berkovits, M. Kaveh, “The vector memory effect for waves,” Europhys. Lett. 13, 97–101 (1990).
    [Crossref]
  23. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 269–278.
  24. D. Maystre, O. Mata-Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
    [Crossref]
  25. W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).
  26. E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
    [Crossref]
  27. See, for example, A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.
  28. A. A. Maradudin, T. Michel, “The transverse correlation correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
    [Crossref]
  29. G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,” IEEE Trans. Antennas Propag. AP-30, 1135–1143 (1982).
    [Crossref]
  30. G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,” IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
    [Crossref]

1992 (1)

1991 (3)

1990 (7)

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]

Ya-Qiu Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[Crossref]

M.-J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional surfaces,” J. Opt. Soc. Am. A 7, 569–577 (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]

A. A. Maradudin, T. Michel, “The transverse correlation correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[Crossref]

1989 (4)

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]

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, “Geometric scaling of the optical memory effect in coherent-wavepropagation through random media,” Phys. Rev. B 39, 12403–12406 (1989).
[Crossref]

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]

1988 (2)

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

I. Freund, M. Rosenbluh, “Memory effect in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (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]

1985 (1)

G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,” IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
[Crossref]

1983 (1)

D. Maystre, O. Mata-Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[Crossref]

1982 (2)

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[Crossref]

G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,” IEEE Trans. Antennas Propag. AP-30, 1135–1143 (1982).
[Crossref]

1976 (1)

1975 (2)

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]

D. Léger, E. Mathieu, J. C. Perrin, “Optical surface roughness determination using speckle correlation techniques,” Appl. Opt. 14, 872–877 (1975).
[Crossref]

1956 (1)

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Berkovits, R.

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]

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wavepropagation through random media,” Phys. Rev. B 39, 12403–12406 (1989).
[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]

Brown, G. S.

G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,” IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
[Crossref]

G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,” IEEE Trans. Antennas Propag. AP-30, 1135–1143 (1982).
[Crossref]

Celli, V.

Chen, J. S.

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

Dainty, J. C.

Feng, S.

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]

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]

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

Freund, I.

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wavepropagation through random media,” Phys. Rev. B 39, 12403–12406 (1989).
[Crossref]

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

Friberg, A. T.

Ishimaru, A.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 269–278.

Jin, Ya-Qiu

Ya-Qiu Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[Crossref]

Kane, C.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave propagation 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.

Lax, M.

Ya-Qiu Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[Crossref]

Lee, P. A.

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

Léger, D.

Liszka, E. G.

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[Crossref]

Maradudin, A. A.

P. Tran, A. A. Maradudin, V. Celli, “Backscattering enhancement from a dielectric surface,” J. Opt. Soc. Am. B 8, 1526–1530 (1991).
[Crossref]

A. A. Maradudin, T. Michel, “The transverse correlation correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[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]

See, for example, A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Mata-Mendez, O.

D. Maystre, O. Mata-Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[Crossref]

Mathieu, E.

Maystre, D.

D. Maystre, O. Mata-Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[Crossref]

McCoy, J. J.

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[Crossref]

McGurn, A. R.

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]

Meecham, W. C.

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Méndez, E. R.

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]

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]

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]

See, for example, A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

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]

A. A. Maradudin, T. Michel, “The transverse correlation correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[Crossref]

See, for example, A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Michel, T. R.

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.

Roger, A.

D. Maystre, O. Mata-Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[Crossref]

Rosenbluh, M.

I. Freund, M. Rosenbluh, R. Berkovits, “Geometric scaling of the optical memory effect in coherent-wavepropagation through random media,” Phys. Rev. B 39, 12403–12406 (1989).
[Crossref]

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

Sant, A. J.

Stone, A. D.

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

Tran, P.

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]

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. (1)

Europhys. Lett. (1)

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

IEEE Trans. Antennas Propag. (2)

G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,” IEEE Trans. Antennas Propag. AP-30, 1135–1143 (1982).
[Crossref]

G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,” IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
[Crossref]

J. Acoust. Soc. Am. (1)

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

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

J. Rat. Mech. Anal. (1)

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

J. Stat. Phys. (1)

A. A. Maradudin, T. Michel, “The transverse correlation correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[Crossref]

Opt. Acta (2)

D. Maystre, O. Mata-Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[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. (2)

Phys. Rev. B (5)

Ya-Qiu Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[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]

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, “Geometric scaling of the optical memory effect in coherent-wavepropagation through random media,” Phys. Rev. B 39, 12403–12406 (1989).
[Crossref]

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

Phys. Rev. Lett. (2)

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

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

Waves Random Media (1)

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

Other (3)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 269–278.

There is a typographical error in Eq. (14) of Ref. 14: the correlation length Tin the argument of the exponential should be squared.

See, for example, A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

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Figures (7)

Fig. 1
Fig. 1

Dependence of the real (solid curves) and the imaginary (curves with open circles) parts of the four complex amplitude correlation functions Γ++pp, Γ++ss, Γ++ps, and Γ++sp on θi2 for θi1 = −10° and θs1 = 30°. The one-dimensional profile of the randomly rough, perfectly conducting surface is characterized by δ/a = 0.6. The wavelength of the incident light is λ = a/3, the half-width of the Gaussian beam is w = 7λ, and the length of the segment of surface considered is L = 6w. Nx = 400 discretization points were used, and the estimates of the correlation functions were obtained by averaging over an ensemble of Np = 2000 independent surface realizations.

Fig. 2
Fig. 2

Two complex amplitude correlation functions, Γ++++ and Γ++−−, computed from the results of Fig. 1. The solid curves represent the real parts, and the curves marked with open circles represent the imaginary parts.

Fig. 3
Fig. 3

Dependence of the two complex amplitude correlation functions Γ++++ and Γ++−− on the transverse correlation length a of the surface profile. The values of the parameters kept constant are θi1 = 0° θs1 = 30°, δ/a = 0.6, and Nx = 400. Top row: a = λ, w/λ = 7, L/w = 6, Np = 1000; middle row: a/λ = 3, w/λ = 7, L/w = 6, Np = 2300; bottom row: a/λ = 6, w/λ = 12, L/w = 4, Np = 1400. The solid curves represent the real parts, and the curves marked with open circles represent the imaginary parts.

Fig. 4
Fig. 4

Two complex amplitude correlation functions, Γ++++ and Γ++−− for the same parameters as in Fig. 3 (a/λ = 3) but for δ/a = 0.3 and Np = 1800. The solid curves represent the real parts, the curve marked with open circles represents the imaginary part of Γ++−− and the dotted-dashed curve represents the imaginary part of Γ++++.

Fig. 5
Fig. 5

Two complex amplitude correlation functions, Γ++++ and Γ++−−, for the same parameters as in Fig. 3; that is, a/λ = 3, w/λ = 7, and L/w = 6 but θs1 = 0°. Top row: θi1 = 15°, Np = 2000; middle row: θi1 = 5°, Np = 2000; bottom row: θi1 = 0°, Np = 1300. The solid curves represent the real parts, and the curves marked with open circles represent the imaginary parts.

Fig. 6
Fig. 6

Dependence of the two complex amplitude correlation functions Γ++++ and Γ++−− on the transverse correlation length a of the surface profile. The values of the parameters kept constant are θi1 = −θs1 = 10°, δ/a = 0.6, and Nx = 400. Top row: a = λ, w/λ = 7, L/w = 5, Np = 2000; middle row: a/λ = 3, w/λ = 7, L/w = 5, Np = 1300; bottom row: a/λ = 6, w/λ = 12, L/w = 4, Np = 1800. The solid curves represent the real parts, and the curves marked with open circles represent the imaginary parts.

Fig. 7
Fig. 7

Two complex amplitude correlation functions, Γ++++ and Γ++−−, for a/λ = 3 as in Fig. 5 (θi1 = 0°) but for δ/a = 0.3 and Np = 1200. The solid curves represent the real parts, and the curves marked with open circles represent the imaginary parts.

Equations (29)

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E ( x 1 , x 3 ω ) inc = ω w 2 π c - π / 2 π / 2 d θ exp [ - ( ω w 2 c ) 2 ( θ - θ i ) 2 ] × [ E p e ^ p ( θ ) inc + E s e ^ s ( θ ) inc ] × exp [ i ω c ( x 1 sin θ - x 3 cos θ ) ] ,             θ i < π 2 ,
e ^ s ( θ ) inc = x ^ 2 ,
e ^ p ( θ ) inc = - x ^ 1 cos θ - x ^ 3 sin θ ,
E ( x 1 , x 3 ω ) sc = i 4 π × - π / 2 π / 2 d θ s [ E p r p ( θ s ) e ^ p ( θ s ) sc + E s r s ( θ s ) e ^ s ( θ s ) sc ] × exp [ i ω c ( x 1 sin θ s + x 3 cos θ s ) ] ,
e ^ s ( θ s ) sc = x ^ 2 ,
e ^ p ( θ s ) sc = x ^ 1 cos θ s - x ^ 3 sin θ s .
e ^ + = 1 2 ( e ^ p + e ^ s ) ,
e ^ - = 1 2 ( e ^ p - e ^ s ) ,
e ^ R = 1 2 ( e ^ p - i e ^ s ) ,
e ^ L = 1 2 ( e ^ p + i e ^ s ) ,
P β ( θ s ) sc = L 2 c 2 64 π 2 ω × e ^ β ( θ s ) sc * · [ E p r p ( θ s ) e ^ p ( θ s ) sc + E s r s ( θ s ) e ^ s ( θ s ) sc ] 2 .
A ( θ s , β θ i , α ) = c 8 π ( L 2 ω P inc ) 1 / 2 [ ( e ^ β * · e ^ p ) ( e ^ p * · e ^ α ) r p ( θ s ) + ( e ^ β * · e ^ s ) ( e ^ s * · e ^ α ) r s ( θ s ) ] .
A ( θ s 1 , β 1 θ i 1 , α 1 ) A * ( θ s 2 , β 2 θ i 2 , α 2 ) = exp { - ( ω w 2 c ) 2 [ ( sin θ i 1 - sin θ s 1 ) - ( sin θ i 2 - sin θ s 2 ) ] 2 cos 2 θ i 1 + cos 2 θ i 2 } × A ( θ s 1 , β 1 θ i 1 , α 1 ) A * ( θ ˜ s 2 , β 2 θ i 2 , α 2 ) ,
sin θ i 1 - sin θ s 1 = sin θ i 2 - sin θ ˜ s 2
( sin θ i 1 - sin θ s 1 ) - ( sin θ i 2 - sin θ s 2 ) λ ( cos 2 θ i 1 + cos 2 θ i 2 ) 1 / 2 π w
Γ α 1 α 2 β 1 β 2 ( θ i 1 , θ i 2 , θ s 1 ) = A ( θ s 1 , β 1 θ i 1 , α 1 ) A * ( θ ˜ s 2 , β 2 θ i 2 , α 2 ) ,
Γ + + - - ( θ i 1 , θ i 2 , θ s 1 ) ( c 8 π ) 2 L 2 ω P inc r p ( 1 ) ( θ s 1 ) r p ( 1 ) * ( θ s 2 ) ,
Γ + + + + ( θ i 1 , θ i 2 , θ s 1 ) ( c 8 π ) 2 L 2 ω P inc r p ( 2 ) ( θ s 1 ) r p ( 2 ) * ( θ s 2 ) .
r α ( θ s ) = - d x 1 exp { - i ω c [ x 1 sin θ s + ζ ( x 1 ) cos θ s ] } j α ( x 1 ) ,             α = p , s .
r α ( θ s 1 θ i 1 ) r β * ( θ s 2 θ i 2 ) = - d x 1 - d x 1 × exp [ - i ( k s 1 x 1 - k s 2 x 1 ) ] × j α ( x 1 ) j β * ( x 1 ) exp { - i [ α ( k s 1 ) ζ ( x 1 ) - α ( k s 2 ) ζ ( x 1 ) ] } .
j α ( x 1 ) j β * ( x 1 ) exp { - i [ α ( k s 1 ) ζ ( x 1 ) - α ( k s 2 ) ζ ( x 1 ) ] } = exp [ i ( k i 1 x 1 - k i 2 x 1 ) ] χ α β [ α ( k s 1 ) , α ( k s 2 ) , x 1 - x 1 ] .
r α ( θ s 1 θ i 1 ) r β * ( θ s 2 θ i 2 ) = δ [ ( k i 1 - k s 1 ) - ( k i 2 - k s 2 ) ] C α β ( k i 1 , k i 2 , k s 1 ) ,
C α β ( k i 1 , k i 2 , k s 1 ) = 2 π - d s exp [ - i s ( k i 1 - k s 1 ) ] χ α β [ α ( k s 1 ) , α ( k s 2 ) , s ]
r α ( θ s 1 ) r β * ( θ s 2 ) = 1 4 π ( ω w c ) 2 1 α ( k i 1 ) α ( k i 2 ) × - d k 1 - d k 2 exp [ - ( ω w 2 c ) 2 ( k 1 - k i 1 ) 2 α 2 ( k i 1 ) ] × exp [ - ( ω w 2 c ) 2 ( k 2 - k i 2 ) 2 α 2 ( k i 2 ) ] × δ [ ( k 1 - k s 1 ) - ( k 2 - k s 2 ) ] C α β ( k 1 , k 2 , k s 1 ) .
r α ( θ s 1 ) r β * ( θ s 2 ) = ω c w 2 π exp { - ( ω w / 2 c ) 2 [ ( k i 1 - k s 1 ) - ( k i 2 - k s 2 ) ] 2 / [ α 2 ( k i 1 ) + α 2 ( k i 2 ) ] } [ α 2 ( k i 1 ) + α 2 ( k i 2 ) ] 1 / 2 C α β ( k i 1 , k i 2 , k s 1 ) .
r α ( θ s 1 ) r β * ( θ s 2 ) = exp { - ( ω w 2 c ) 2 × [ ( sin θ i 1 - sin θ s 1 ) - ( sin θ i 2 - sin θ s 2 ) ] 2 cos 2 θ i 1 + cos 2 θ i 2 } × r α ( θ s 1 ) r β * ( θ ˜ s 2 ) ,
sin θ i 1 - sin θ s 1 = sin θ i 2 - sin θ ˜ s 2 .
A ( θ s 1 , β 1 θ i 1 , α 1 ) A * ( θ s 2 , β 2 θ i 2 , α 2 ) = exp { - ( ω w 2 c ) 2 [ ( sin θ i 1 - sin θ s 1 ) - ( sin θ i 2 - sin θ s 2 ) ] 2 cos 2 θ i 1 + cos 2 θ i 2 } × A ( θ s 1 , β 1 θ i 1 , α 1 ) A * ( θ ˜ s 2 , β 2 θ i 2 , α 2 ) ,
( cos 2 θ i 1 + cos 2 θ i 2 ) 1 / 2 A ( θ s 1 , β 1 θ i 1 , α 1 ) A * ( θ s 2 , β 2 θ i 2 , α 2 ) = ( cos 2 θ i 1 + cos 2 θ s 2 ) 1 / 2 A ( θ s 1 , β 1 θ i 1 , α 1 ) × A * ( - θ i 2 , β 2 - θ s 2 , α 2 ) = ( cos 2 θ s 1 + cos 2 θ i 2 ) 1 / 2 A ( - θ i 1 , β 1 - θ s 1 , α 1 ) × A * ( θ s 2 , β 2 θ i 2 , α 2 ) .

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