Abstract

An optical procedure is presented to measure at the speckle size and with high accuracy, the polarization degree of patterns scattered by disordered media. Whole mappings of polarization ratio, polarimetric phase and polarization degree are pointed out. Scattered clouds are emphasized on the Poincaré sphere, and are completed by probability density functions of the polarization degree. A special care is attributed to the accuracy of data. The set-up provides additional signatures of scattering media.

© 2014 Optical Society of America

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References

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  1. E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge University, 1995)
  2. C. Brosseau, Fundamentals of Polarized Light - A Statistical Approach (Wiley, 1998)
  3. J. W. Goodman, Statistical Optics (Wiley- Interscience, 2000).
  4. E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor and Francis Group, 2006)
  5. E. Wolf, Theory of coherence and polarization of light (Cambridge University, ed. 2007)
  6. R. Barakat, “Polarization entropy transfer and relative polarization entropy,” Opt. Commun. 123(4-6), 443–448 (1996).
    [CrossRef]
  7. P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25(11), 2749–2757 (2008).
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  8. H. D. Noble and R. A. Chipman, “Mueller matrix roots algorithm and computational considerations,” Opt. Express 20(1), 17–31 (2012).
    [CrossRef] [PubMed]
  9. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition for direction reversal: application to samples measured in reflection and backscattering,” Opt. Express 19(15), 14348–14353 (2011).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007)
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    [CrossRef] [PubMed]
  14. J. Sorrentini, M. Zerrad, G. Soriano, and C. Amra, “Enpolarization of light by scattering media,” Opt. Express 19(22), 21313–21320 (2011).
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  16. 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(3), 2787–2794 (2013).
    [CrossRef] [PubMed]
  17. G. Soriano, M. Zerrad, and C. Amra, “Mapping the coherence time of far-field speckle scattered by disordered media,” Opt. Express 21(20), 24191–24200 (2013).
    [CrossRef] [PubMed]
  18. B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12(20), 4941–4958 (2004).
    [CrossRef] [PubMed]
  19. O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17(18), 15623–15634 (2009).
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    [CrossRef] [PubMed]

2013 (2)

2012 (3)

2011 (3)

2010 (3)

2009 (4)

2008 (1)

2004 (2)

1996 (1)

R. Barakat, “Polarization entropy transfer and relative polarization entropy,” Opt. Commun. 123(4-6), 443–448 (1996).
[CrossRef]

Alouini, M.

Amra, C.

Angelsky, O. V.

Arce-Diego, J. L.

Barakat, R.

R. Barakat, “Polarization entropy transfer and relative polarization entropy,” Opt. Commun. 123(4-6), 443–448 (1996).
[CrossRef]

Bénière, A.

Broky, J.

Chipman, R.

Chipman, R. A.

De Martino, A.

DeBoo, B.

Dogariu, A.

Ellis, J.

Fade, J.

Ghabbach, A.

Gorodyns’ka, N. V.

Gorsky, M. P.

Goudail, F.

Hamel, C.

Hanson, S. G.

Hatit, S. B.

Ibrahim, B. H.

Luis, A.

Noble, H. D.

Ortega-Quijano, N.

Ponomarenko, S.

Pouget, L.

Réfrégier, P.

Sasian, J.

Soriano, G.

Sorrentini, J.

Wolf, E.

Yura, H. T.

Zenkova, C. Yu.

Zerrad, M.

Appl. Opt. (3)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

R. Barakat, “Polarization entropy transfer and relative polarization entropy,” Opt. Commun. 123(4-6), 443–448 (1996).
[CrossRef]

Opt. Express (10)

J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010).
[CrossRef] [PubMed]

J. Broky and A. Dogariu, “Correlations of polarization in random electro-magnetic fields,” Opt. Express 19(17), 15711–15719 (2011).
[CrossRef] [PubMed]

M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, “Gradual loss of polarization in light scattered from rough surfaces: Electromagnetic prediction,” Opt. Express 18(15), 15832–15843 (2010).
[CrossRef] [PubMed]

J. Sorrentini, M. Zerrad, G. Soriano, and C. Amra, “Enpolarization of light by scattering media,” Opt. Express 19(22), 21313–21320 (2011).
[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(3), 2787–2794 (2013).
[CrossRef] [PubMed]

G. Soriano, M. Zerrad, and C. Amra, “Mapping the coherence time of far-field speckle scattered by disordered media,” Opt. Express 21(20), 24191–24200 (2013).
[CrossRef] [PubMed]

B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12(20), 4941–4958 (2004).
[CrossRef] [PubMed]

O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17(18), 15623–15634 (2009).
[CrossRef] [PubMed]

H. D. Noble and R. A. Chipman, “Mueller matrix roots algorithm and computational considerations,” Opt. Express 20(1), 17–31 (2012).
[CrossRef] [PubMed]

N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition for direction reversal: application to samples measured in reflection and backscattering,” Opt. Express 19(15), 14348–14353 (2011).
[CrossRef] [PubMed]

Opt. Lett. (3)

Other (6)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007)

E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge University, 1995)

C. Brosseau, Fundamentals of Polarized Light - A Statistical Approach (Wiley, 1998)

J. W. Goodman, Statistical Optics (Wiley- Interscience, 2000).

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor and Francis Group, 2006)

E. Wolf, Theory of coherence and polarization of light (Cambridge University, ed. 2007)

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Figures (16)

Fig. 1
Fig. 1

Basic polarization set-up for specular optics.

Fig. 2
Fig. 2

Minimum (η ≈π- top figure) and maximum (η ≈0 – bottom figure) intensity of the direct beam at the system output, with grey levels on a logarithmic scale. Data are normalized to the integration times. The area of investigation is 1.5mm x 2mm.

Fig. 3
Fig. 3

Cartography of first extracted parameters δ and α.

Fig. 4
Fig. 4

Mapping of the polarization degree (right figure) and associated normalized distance (left figure).

Fig. 5
Fig. 5

Polarization curves of a few pixels chosen with different maximum grey levels. Logarithmic coordinates.

Fig. 6
Fig. 6

Polarization states on the Poincaré sphere (right figure- with a zoom on the top), plotted for the square region of the left figure.

Fig. 7
Fig. 7

a: dop histograms measured within the investigation area, weighted (middle figure) or not (top figure) by grey levels (see text). b: calculation of a dop histogram ((see text)) of full polarized pattern in the presence of Gaussian noise with 10% root mean square.

Fig. 8
Fig. 8

Histogram of the distance function versus the grey levels. The region of investigation is given at the right, top. The distance provides a criteria for the quality of the LMS procedure (see text).

Fig. 9
Fig. 9

Basic polarization set-up for analysis of the scattering pattern.

Fig. 10
Fig. 10

Minimum (η = π- left figure) and maximum (η = 0- right figure) intensity of the scattering pattern at the system output, with the grey levels in logarithmic scale. Data are normalized to the integration time. The area of investigation is 3mmx3.5mm.

Fig. 11
Fig. 11

mapping of the parameters α and δ for the surface scattering pattern.

Fig. 12
Fig. 12

mapping the dop (right figure) of the surface scattering pattern and its associated distance (left figure).

Fig. 13
Fig. 13

complete polarization curves (bottom figures) measured for the pixels identified in the top figure across the frontiers of a grain size.

Fig. 14
Fig. 14

Polarization states on the Poincaré sphere for different areas within the speckle pattern.

Fig. 15
Fig. 15

dop histograms of the scattering pattern, weighted (bottom) or not (top) by the energy levels.

Fig. 16
Fig. 16

Distance function of the scattering pattern versus the grey levels. The region of investigation is given at the right, top. The distance provides a criteria for the quality of the LMS procedure (see text related to Fig. 8).

Equations (27)

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E'( t )=cosϕ E S ( t )exp( j η S )+sinϕ E P ( t )exp( j η P )
V=K{ cos 2 ϕ | E S ( t ) | 2 + sin 2 ϕ | E P ( t ) | 2 } +2cosϕsinϕeal{ E S ( t ) E P * ( t ) exp( jη ) }
V( ϕ,η )=K cos 2 ϕ | E S ( t ) | 2 { 1+ β 2 tan 2 ϕ+2β| μ |tanϕcos( δ+η ) }
DO P 2 =14 β ( 1+β ) 2 ( 1 | μ | 2 )
V( ϕ,η )=γ( ϕ ){ 1+α( ϕ )cos( δ+η ) }
γ( ϕ )=K cos 2 ϕ | E S ( t ) | 2 { 1+ β 2 tan 2 ϕ }
α( ϕ )= 2β| μ |tanϕ 1+ β 2 tan 2 ϕ
V k =V( ϕ, η k )=γ( ϕ ){ 1+α( ϕ )cos( δ+ η k ) }
V k k = 1 N k V k =γ( ϕ )
M k ( ϕ )= V k V k =1+α( ϕ )cos( δ+ η k )
N k ( ϕ )= M k ( ϕ )1=α( ϕ )cos( δ+ η k )
F( α,δ )= 1 N k [ N ' k α( ϕ )cos( δ+ η k ) ] 2
α= 2 N [ k N ' k ( ϕ )cos( η k ) ] 2 + [ k N ' k ( ϕ )sin( η k ) ] 2 >0
cosδ= 2 N 1 α k N ' k ( ϕ )cos( η k )
sinδ= 2 N 1 α k N ' k ( ϕ )sin( η k )
| μ |= q α ' q u q q u q 2 =f( β ) with u q = 2βtan ϕ q 1+ β 2 tan 2 ϕ q
| μ |= q α ' q tg ϕ q 1 β 2 tan 2 ϕ q ( 1+ β 2 tan 2 ϕ q ) 2 2β q α ' q tan 2 ϕ q 1 β 2 tan 2 ϕ q ( 1+ β 2 tan 2 ϕ q ) 3 =g( β )
V( ϕ=0 )= V 0 = γ 0 =K | E S ( t ) | 2
V 0 η = γ 0 =K | E S ( t ) | 2
V( ϕ=45° ) η = V 45 η = γ 45 ( 1+ β 2 )= 1 2 K | E S ( t ) | 2 ( 1+ β 2 )
β= 1+2 V 45 V 0
| μ |= α 45 1+ β 2 2β
V ij ( ϕ, η k )= V ij ( ϕ )= γ ij ( ϕ )[ 1+ α ij ( ϕ )cos( δ ij + η k ) ]
α ij ( ϕ )=2 β ij | μ ij | tanϕ 1+ β ij 2 tan 2 ϕ
γ ij ( ϕ )=K cos 2 ϕ | E S,ij ( t ) | 2 { 1+ β ij 2 tan 2 ϕ }
V ij ( ϕ=45° )K | E S,ij | 2 [ 1+cos( η k ) ]
D ij = σ m = 1 M ij,k k 1 N k { M ij,k [ 1+ α ij cos( δ ij + η k ) ] }

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