Abstract

A study of the statistics of the polarization properties of one-dimensional randomly rough surfaces is presented. Based on the assumption that the s and p components of the electric-field vector constitute correlated circular complex Gaussian processes, some first-order statistical properties of the polarization of scattered fields are first established. In particular, results are presented for the probability density function of the Stokes parameters and their correlations. For each realization of the surface the random Mueller matrix elements associated with the surface may be determined from measurements of the Stokes parameters of the scattered light. Choosing a +45° linear polarization for the incident field, one is then able to study the statistics of the Mueller matrix elements. Numerical and experimental data on the statistics of the Mueller matrix elements are presented and compared with theoretical results. Finally, the usefulness and the significance of the results are illustrated with some examples.

© 1995 Optical Society of America

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References

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  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]
  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]
  3. 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]
  4. 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]
  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. 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]
  7. 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]
  8. 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]
  9. M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
    [CrossRef]
  10. N. C. Bruce, A. J. Sant, J. C. Dainty, “The Mueller matrix for rough surface scattering using the Kirchhoff approximation,” Opt. Commun. 88, 471–484 (1992).
    [CrossRef]
  11. M. E. Knotts, K. A. O’Donnell, “Backscattering enhancement from a conducting surface with isotropic roughness,” Opt. Commun. 99, 1–6 (1993).
    [CrossRef]
  12. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser and Stellar Speckle, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 9–75.
  13. A. F. Fercher, P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
    [CrossRef]
  14. P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
    [CrossRef]
  15. R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
    [CrossRef]
  16. R. Barakat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
    [CrossRef]
  17. F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
    [CrossRef]
  18. I. Freund, “Polarization correlations in two-dimensional random media,” Waves Random Media 1, 245–263 (1991).
    [CrossRef]
  19. D. Eliyahu, M. Rosenbluh, I. Freund, “Angular intensity and polarization dependence of diffuse transmission through random media,” J. Opt. Soc. Am. A 10, 477–491 (1993).
    [CrossRef]
  20. D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
    [CrossRef]
  21. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 108.
  22. N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Math. Stat. 34, 152–177 (1962).
    [CrossRef]
  23. Ref. 21, p. 137.
  24. P. W. Johnson, R. W. Christie, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  25. R. García-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep.194, 351–359 (1990).
    [CrossRef]
  26. A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A (Utrecht) 207, 302–314 (1994).
    [CrossRef]
  27. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  28. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged edition (Academic, New York, 1980), p. 1039.

1994 (1)

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A (Utrecht) 207, 302–314 (1994).
[CrossRef]

1993 (4)

1992 (2)

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]

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

1991 (2)

1990 (2)

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]

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 (2)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

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]

1987 (3)

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]

R. Barakat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
[CrossRef]

1985 (2)

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]

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

1984 (1)

1981 (1)

A. F. Fercher, P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[CrossRef]

1978 (1)

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

1972 (1)

P. W. Johnson, R. W. Christie, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1962 (1)

N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Math. Stat. 34, 152–177 (1962).
[CrossRef]

Asakura, T.

Barakat, R.

R. Barakat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
[CrossRef]

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

Bruce, N. C.

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

Christie, R. W.

P. W. Johnson, R. W. Christie, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Dainty, J. C.

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

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]

Eliyahu, D.

Fercher, A. F.

P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
[CrossRef]

A. F. Fercher, P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[CrossRef]

Freund, I.

Friberg, A. T.

García-Molina, R.

R. García-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.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser and Stellar Speckle, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 9–75.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 108.

Goodman, N. R.

N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Math. Stat. 34, 152–177 (1962).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged edition (Academic, New York, 1980), p. 1039.

Gray, P. F.

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

Johnson, P. W.

P. W. Johnson, R. W. Christie, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kim, M. J.

Knotts, M. E.

Leskova, T. A.

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

MacKintosh, F. C.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A (Utrecht) 207, 302–314 (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]

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

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. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A (Utrecht) 207, 302–314 (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]

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]

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.

Nieto-Vesperinas, M.

O’ Donnell, K. A.

O’Donnell, K. A.

Pine, D. J.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Rosenbluh, M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged edition (Academic, New York, 1980), p. 1039.

Sant, A. J.

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

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]

Soto-Crespo, J. M.

Steeger, P. F.

P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
[CrossRef]

A. F. Fercher, P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[CrossRef]

Weitz, D. A.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Zhu, J. X.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Zocha, K.

Ann. Math. Stat. (1)

N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Math. Stat. 34, 152–177 (1962).
[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]

J. Opt. Soc. Am. A (9)

P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
[CrossRef]

R. Barakat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
[CrossRef]

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]

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]

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]

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]

D. Eliyahu, M. Rosenbluh, I. Freund, “Angular intensity and polarization dependence of diffuse transmission through random media,” J. Opt. Soc. Am. A 10, 477–491 (1993).
[CrossRef]

M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
[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]

Opt. Acta (3)

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

A. F. Fercher, P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[CrossRef]

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

Opt. Commun. (3)

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

M. E. Knotts, K. A. O’Donnell, “Backscattering enhancement from a conducting surface with isotropic roughness,” Opt. Commun. 99, 1–6 (1993).
[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]

Phys. Rev. B (3)

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]

P. W. Johnson, R. W. Christie, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Phys. Rev. E (1)

D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
[CrossRef]

Physica A (Utrecht) (1)

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A (Utrecht) 207, 302–314 (1994).
[CrossRef]

Waves Random Media (1)

I. Freund, “Polarization correlations in two-dimensional random media,” Waves Random Media 1, 245–263 (1991).
[CrossRef]

Other (5)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser and Stellar Speckle, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 9–75.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 108.

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

Ref. 21, p. 137.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged edition (Academic, New York, 1980), p. 1039.

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Figures (11)

Fig. 1
Fig. 1

Scattering geometry. A one-dimensional surface is illuminated by a plane wave with wave vector kinc. The angles of incidence and scattering are denoted by θ0 and θs, respectively. The state of polarization of the incident light is a linear combination of s and p states.

Fig. 2
Fig. 2

Average Mueller matrix elements. The curves show the results of averaging over 10,000 realizations of the numerical experiment. The symbols show the estimated averages in the backscattering direction from the scaled CCD data, comprising approximately 9000 independent samples (see Section 6): S11(◊), S12(○), S33(●), S34(□). The statistical parameters of the surface are δ = 0.85 μm and a = 2.1 μm.

Fig. 3
Fig. 3

Histograms of the four Mueller matrix elements obtained with data from the numerical simulation. The angle of incidence is θ0 = 0°, and the angle of scattering is θs = 0°. The solid curves were calculated from expressions (24) and (27) with the use of the sample mean quantities.

Fig. 4
Fig. 4

Second-order moments of the four Mueller matrix elements: (a) calculated from the numerical simulation, (b) calculated from the first-order moments and relations (26) and (29).

Fig. 5
Fig. 5

Correlations between the Mueller matrix elements calculated from the numerical simulation.

Fig. 6
Fig. 6

Correlations between the Mueller matrix elements calculated from the first-order moments and relation (19).

Fig. 7
Fig. 7

Schematic diagram of the experimental setup employed for the study of the speckle statistics.

Fig. 8
Fig. 8

Histograms of the four Mueller matrix elements obtained with the CCD data. The angle of incidence is θ0 = 0°, and the angle of scattering is θs = 0°. The solid curves were calculated from expressions (24) and (27) with the use of the sample mean quantities.

Fig. 9
Fig. 9

Average values of the scattering cross sections σss and σpp.

Fig. 10
Fig. 10

Correlation coefficient between the scattering cross sections σss and σpp calculated from the numerical simulation (thin curves) and from the average Mueller matrix elements and Eq. (35) (thick curves).

Fig. 11
Fig. 11

Correlation coefficient between the scattering cross sections σRR and σRL calculated from the average Mueller matrix elements and Eq. (38).

Equations (38)

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

= [ S 11 S 12 0 0 S 12 S 11 0 0 0 0 S 33 S 34 0 0 S 34 S 33 ] ,
V inc = ( I inc Q inc U inc V inc ) ,
V sc = ( I sc Q sc U sc V sc ) .
V sc = V inc .
I sc = | E p | 2 + | E s | 2 , Q sc = | E p | 2 | E s | 2 , U sc = E p E s * + E p * E s , V sc = ι ( E p * E s E p E s * ) ,
P = ( Q sc 2 + U sc 2 + V sc 2 ) 1 / 2 I sc .
V + = ( 1 0 1 0 ) ,
V sc = ( I sc Q sc U sc V sc ) = ( S 11 S 12 S 33 S 34 ) .
P ( E s r , E s i , E p r , E p i ) = 1 π 2 d exp { 1 d [ j 11 | E s | 2 + j 22 | E p | 2 2 Re ( j 12 E s * E p ) ] } ,
J = [ j 11 j 12 j 21 j 22 ] = [ E p * E p E s * E p E p * E s E s * E s ] ,
d = j 11 j 22 j 12 j 21 .
j 11 = I sc + Q sc 2 , j 22 = I sc Q sc 2 , j 12 = j 21 * = U sc + V sc 2 ,
d = 1 4 ( I sc 2 Q sc 2 U sc 2 V sc 2 ) = I sc 2 4 ( 1 P 2 ) .
P ( E s r , E s i , E p r , E p i ) = 1 π 2 d exp ( 1 d { I sc 2 ( | E p | 2 + | E s | 2 ) + Q sc 2 ( | E s | 2 | E p | 2 ) Re [ ( U sc + V sc ) E s * E p ] } ) .
I sc = | E p | 2 + | E s | 2 , 0 < I sc < , Q sc = | E p | 2 | E s | 2 , < Q sc < , U sc = E p E s * + E p * E s , < U sc < , V sc = ι ( E p * E s E p E s * ) , < V sc < ,
P I sc ( i sc ) = 1 P I sc { exp [ 2 i sc ( 1 + P ) I sc ] exp [ 2 i sc ( 1 + P ) I sc ] } .
P Y ( y ) = 1 [ I sc 2 ( 1 P 2 ) + Y 2 ] 1 / 2 × exp ( 1 2 d { y Y | y | [ I sc 2 ( 1 P 2 ) + Y 2 ] 1 / 2 } ) ,
E l E m * E n E o * = E l E m * E n E o * + E l E o * E m * E n ,
Y Z = 2 Y Z , Y Z ,
= [ S 11 S 12 0 0 S 12 S 11 0 0 0 0 S 33 S 34 0 0 S 34 S 33 ] .
S 11 = I sc , S 12 = Q sc , S 33 = U sc , S 34 = V sc .
P = ( S 12 2 + S 33 2 + S 34 2 ) 1 / 2 S 11 ,
d = S 11 2 4 ( 1 P 2 ) .
P s 11 ( s 11 ) = 1 P S 11 { exp [ 2 s 11 ( 1 + P ) S 11 ] exp [ 2 s 11 ( 1 P ) S 11 ] } ,
S 11 n = 1 P s 11 { n ! [ ( 1 + P M ) S 11 2 ] n + 1 n ! [ ( 1 P ) S 11 2 ] n + 1 } .
S 11 2 = S 11 2 2 ( 3 + P 2 ) .
P Y ( y ) = 1 [ s 11 2 ( 1 P 2 ) + Y 2 ] 1 / 2 × exp ( 2 2 d { y Y | y | [ S 11 2 ( 1 P 2 ) + Y 2 ] 1 / 2 } ) ,
Y n = n ! 2 n + 1 [ S 11 2 ( 1 P 2 ) + Y 2 ] 1 / 2 × ( { [ S 11 2 ( 1 P 2 ) + Y 2 ] 1 / 2 + Y 2 } n + 1 + ( 1 ) n { [ S 11 2 ( 1 P 2 ) + Y 2 ] 1 / 2 Y } n + 1 ) .
Y 2 = S 11 2 2 ( 1 P 2 ) + 2 Y 2 .
S 11 = I p + I s , S 12 = I p I s , S 33 = I + I , S 34 = I L I R ,
S 11 = 2500 , S 12 = 361 , S 33 = 626 , S 34 = 666 .
V p = ( 1 1 0 0 ) , V s = ( 1 1 0 0 )
σ p p = S 11 + S 12 , σ s s = S 11 S 12 ,
ρ ( X , Y ) = X Y X Y [ ( X 2 X 2 ) ( Y 2 Y 2 ) ] 1 / 2 .
ρ ( σ p p , σ p p ) = S 11 2 P 2 S 12 2 S 11 2 S 12 2 .
σ p p n σ s s m σ p p n σ s s m = ( 4 d S 11 2 S 12 2 ) n × F 2 1 ( n , m + 1 ; 1 ; 1 S 11 2 S 12 2 4 d ) ,
σ R R = 1 2 ( S 11 + S 33 ) , σ R L = 1 2 ( S 11 S 33 ) .
ρ ( σ R R , σ R L ) = S 11 2 P 2 S 33 2 S 11 2 S 33 2 .

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