Abstract

Intensity contrast in a fully developed speckle pattern resulting from the elastic scattering of a partially polarized light from a strongly scattering medium is theoretically and numerically studied. Simple expressions are derived when the illumination bandwidth is much smaller or larger than the chromatic length of the scattering medium.

© 2015 Optical Society of America

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References

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  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  2. J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Co, 2007).
  3. L. Mandel, “Intensity fluctuations of partially polarized light,” Proceedings of the Physical Society 81, 1104 (1963).
    [Crossref]
  4. I. A. Stegun and M. Abramowitz, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  5. J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.
  6. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [Crossref]
  7. G. Soriano, M. Zerrad, and C. Amra, “Enpolarization and depolarization of light scattered from chromatic complex media,” Opt. Express 22, 12603–12613 (2014).
    [Crossref] [PubMed]
  8. J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Let. 34, 2429–2431 (2009).
    [Crossref]
  9. M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Light enpolarization by disordered media under partial polarized illumination: The role of cross-scattering coefficients,” Opt. Express 21, 2787–2794 (2013).
    [Crossref] [PubMed]
  10. M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Comportement polarimétrique de la lumière diffuse : rôle des coefficients croisés de diffusion,” Assemblée Générale du GDR Ondes (2013).
  11. M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Repolarisation de la lumière par des milieux désordonnés,” JNOG, Paris (2013).
  12. W. H. Press, Numerical Recipes in Fortran 77: the Art of Scientific Computing, vol. 1 (Cambridge University, 1992).
  13. L. Mandel, “Concept of cross-spectral purity in coherence theory,” J. Opt. Soc. Am. 51, 1342–1350 (1961).
    [Crossref]
  14. J. J. Gil, “Polarimetric characterization of light and media,” The Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [Crossref]
  15. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [Crossref]

2014 (1)

2013 (1)

2009 (1)

J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Let. 34, 2429–2431 (2009).
[Crossref]

2007 (1)

J. J. Gil, “Polarimetric characterization of light and media,” The Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

1987 (1)

1976 (1)

1963 (1)

L. Mandel, “Intensity fluctuations of partially polarized light,” Proceedings of the Physical Society 81, 1104 (1963).
[Crossref]

1961 (1)

Abramowitz, M.

I. A. Stegun and M. Abramowitz, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Amra, C.

G. Soriano, M. Zerrad, and C. Amra, “Enpolarization and depolarization of light scattered from chromatic complex media,” Opt. Express 22, 12603–12613 (2014).
[Crossref] [PubMed]

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Light enpolarization by disordered media under partial polarized illumination: The role of cross-scattering coefficients,” Opt. Express 21, 2787–2794 (2013).
[Crossref] [PubMed]

J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Let. 34, 2429–2431 (2009).
[Crossref]

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Repolarisation de la lumière par des milieux désordonnés,” JNOG, Paris (2013).

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Comportement polarimétrique de la lumière diffuse : rôle des coefficients croisés de diffusion,” Assemblée Générale du GDR Ondes (2013).

Dainty, J.

J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.

Ennos, A.

J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.

Françon, M.

J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.

Ghabbach, A.

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Light enpolarization by disordered media under partial polarized illumination: The role of cross-scattering coefficients,” Opt. Express 21, 2787–2794 (2013).
[Crossref] [PubMed]

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Repolarisation de la lumière par des milieux désordonnés,” JNOG, Paris (2013).

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Comportement polarimétrique de la lumière diffuse : rôle des coefficients croisés de diffusion,” Assemblée Générale du GDR Ondes (2013).

Gil, J. J.

J. J. Gil, “Polarimetric characterization of light and media,” The Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

Goodman, J.

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Co, 2007).

J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.

Goodman, J. W.

Kim, K.

Mandel, L.

McKechnie, T.

J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.

Parry, G.

J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.

Press, W. H.

W. H. Press, Numerical Recipes in Fortran 77: the Art of Scientific Computing, vol. 1 (Cambridge University, 1992).

Soriano, G.

G. Soriano, M. Zerrad, and C. Amra, “Enpolarization and depolarization of light scattered from chromatic complex media,” Opt. Express 22, 12603–12613 (2014).
[Crossref] [PubMed]

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Light enpolarization by disordered media under partial polarized illumination: The role of cross-scattering coefficients,” Opt. Express 21, 2787–2794 (2013).
[Crossref] [PubMed]

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Comportement polarimétrique de la lumière diffuse : rôle des coefficients croisés de diffusion,” Assemblée Générale du GDR Ondes (2013).

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Repolarisation de la lumière par des milieux désordonnés,” JNOG, Paris (2013).

Sorrentini, J.

J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Let. 34, 2429–2431 (2009).
[Crossref]

Stegun, I. A.

I. A. Stegun and M. Abramowitz, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Wolf, E.

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Zerrad, M.

G. Soriano, M. Zerrad, and C. Amra, “Enpolarization and depolarization of light scattered from chromatic complex media,” Opt. Express 22, 12603–12613 (2014).
[Crossref] [PubMed]

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Light enpolarization by disordered media under partial polarized illumination: The role of cross-scattering coefficients,” Opt. Express 21, 2787–2794 (2013).
[Crossref] [PubMed]

J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Let. 34, 2429–2431 (2009).
[Crossref]

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Repolarisation de la lumière par des milieux désordonnés,” JNOG, Paris (2013).

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Comportement polarimétrique de la lumière diffuse : rôle des coefficients croisés de diffusion,” Assemblée Générale du GDR Ondes (2013).

J. Opt. Soc. Am. (2)

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

Opt. Express (2)

Opt. Let. (1)

J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Let. 34, 2429–2431 (2009).
[Crossref]

Proceedings of the Physical Society (1)

L. Mandel, “Intensity fluctuations of partially polarized light,” Proceedings of the Physical Society 81, 1104 (1963).
[Crossref]

The Eur. Phys. J. Appl. Phys. (1)

J. J. Gil, “Polarimetric characterization of light and media,” The Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

Other (7)

I. A. Stegun and M. Abramowitz, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

J. Dainty, A. Ennos, M. Françon, J. Goodman, T. McKechnie, and G. Parry, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” Topics in Applied Physics, vol. 9 (Springer, 1975), pp. 9–75.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Co, 2007).

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Comportement polarimétrique de la lumière diffuse : rôle des coefficients croisés de diffusion,” Assemblée Générale du GDR Ondes (2013).

M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, “Repolarisation de la lumière par des milieux désordonnés,” JNOG, Paris (2013).

W. H. Press, Numerical Recipes in Fortran 77: the Art of Scientific Computing, vol. 1 (Cambridge University, 1992).

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Figures (3)

Fig. 1
Fig. 1 Probability density function of the normalized scattered intensity I/I〉 for (a) β0 = 1 and five real values of μ0 between 0 and 1, (b) μ0 = 3/4 and five values of β0 between 1 and 100.
Fig. 2
Fig. 2 (a) Scattered intensity probability density functions for unpolarized incident light and six values of the ratio R between 1/7 and 7. (b) Scattered intensity contrast σI/〈I〉 against the R ratio for six real values of the incident DoP P0 between 0 and 1.
Fig. 3
Fig. 3 (a) Number N = (σI/〈I〉)−2 against the R ratio for six real values of the incident DoP P0 between 0 and 1. (b) Parameters a and b of the linear regression of N = f(R) for large values of the ratio R against the incident DoP P0.

Equations (23)

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G [ k , θ ] ( x 0 ) = x k 1 e x / θ Γ ( k ) θ k .
I = | E S | 2 ¯ + | E P | 2 ¯ β = | E P | 2 ¯ | E S | 2 ¯ μ = E S * E P ¯ | E S | 2 ¯ | E P | 2 ¯
P = 1 4 β ( 1 + β ) 2 ( 1 | μ | 2 ) .
[ E S E P ] = [ Σ S S Σ S P Σ P S Σ P P ] [ E S 0 E P 0 ]
I = | Σ S S | 2 + | Σ P S | 2 + β 0 ( | Σ S P | 2 + | Σ P P | 2 ) + 2 β 0 e { μ 0 ( Σ S S * Σ S P + Σ P S * Σ P P ) } 1 + β 0 I 0 .
I = | Σ S S + β 0 e i δ 0 Σ S P | 2 + | Σ P S + β 0 e i δ 0 Σ P P | 2 1 + β 0 I 0
p I = G [ 2 , 2 σ 2 I 0 ]
p I = 1 ( 1 β 0 ) 2 { β 0 2 G [ 2 , 2 σ 2 β 0 I 0 1 + β 0 ] + G [ 2 , 2 σ 2 I 0 1 + β 0 ] } + 2 β 0 ( 1 β 0 ) ( 1 β 0 ) 2 { β 0 G [ 1 , 2 σ 2 β 0 I 0 1 + β 0 ] G [ 1 , 2 σ 2 I 0 1 + β 0 ] } .
I = | γ 1 | 2 + | γ 2 | 2 + | γ 3 | 2 + | γ 4 | 2
γ 1 = ( cos θ Σ S S + cos φ β 0 e i δ 0 Σ S P ) I 0 / ( 1 + β 0 )
γ 2 = ( sin θ Σ S S sin φ β 0 e i δ 0 Σ S P ) I 0 / ( 1 + β 0 )
γ 3 = ( cos θ Σ P S + cos φ β 0 e i δ 0 Σ P P ) I 0 / ( 1 + β 0 )
γ 4 = ( sin θ Σ P S sin φ β 0 e i δ 0 Σ P P ) I 0 / ( 1 + β 0 )
cos ( θ + φ ) = | μ 0 | sin 2 θ = β 0 sin 2 φ .
I = 4 σ 2 I 0 σ I = I 1 2 β 0 ( 1 + β 0 ) 2 ( 1 | μ 0 | 2 )
σ I I = 1 + P 0 2 4 .
E X * E Y ¯ = E ˜ X ( ν ) * E ˜ Y ( ν ) d ν
= ( Σ X S ( ν ) * Σ Y S ( ν ) | E ˜ S 0 ( ν ) | 2 + Σ X S ( ν ) * Σ Y P ( ν ) E ˜ S 0 ( ν ) * E ˜ P 0 ( ν ) ) d ν + ( Σ X P ( ν ) * Σ Y S ( ν ) E ˜ P 0 ( ν ) * E ˜ S 0 ( ν ) + Σ X P ( ν ) * Σ Y P ( ν ) | E ˜ P 0 ( ν ) | 2 ) d ν .
E ˜ X 0 ( ν ) * E ˜ Y 0 ( ν ) = E X 0 * E Y 0 ¯ g ( ν ) g ( ν ) d ν = 1
E X * E Y ¯ = C X S Y S + β 0 μ 0 C X S Y P + β 0 μ 0 * C X P Y S + β 0 C X P Y P 1 + β 0 I 0
C A B C D = Σ A B ( ν ) * Σ C D ( ν ) g ( ν ) d ν
R = Δ ν i Δ ν Σ
σ I I ~ 1 a R + b .

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