## Light enpolarization by disordered media under partial polarized illumination: The role of cross-scattering coefficients |

Optics Express, Vol. 21, Issue 3, pp. 2787-2794 (2013)

http://dx.doi.org/10.1364/OE.21.002787

Acrobat PDF (1635 KB)

### Abstract

We show how disordered media allow to increase the local degree of polarization (DOP) of an arbitrary (partial) polarized incident beam. The role of cross-scattering coefficients is emphasized, together with the probability density functions (PDF) of the scattering DOP. The average DOP of scattering is calculated versus the incident illumination DOP.

© 2013 OSA

## 1. Introduction

6. S. G. Hanson and H. T. Yura, “Statistics of spatially integrated speckle intensity difference,” J. Opt. Soc. Am. A **26**(2), 371–375 (2009). [CrossRef] [PubMed]

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

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

10. 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). [CrossRef] [PubMed]

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

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

12. P. Réfrégier, M. Zerrad, and C. Amra, “Coherence and polarization properties in speckle of totally depolarized light scattered by totally depolarizing media,” Opt. Lett. **37**(11), 2055–2057 (2012). [CrossRef] [PubMed]

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

12. P. Réfrégier, M. Zerrad, and C. Amra, “Coherence and polarization properties in speckle of totally depolarized light scattered by totally depolarizing media,” Opt. Lett. **37**(11), 2055–2057 (2012). [CrossRef] [PubMed]

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

^{2}, with PDF the probability density function. Furthermore an analytical demonstration followed in [12

12. P. Réfrégier, M. Zerrad, and C. Amra, “Coherence and polarization properties in speckle of totally depolarized light scattered by totally depolarizing media,” Opt. Lett. **37**(11), 2055–2057 (2012). [CrossRef] [PubMed]

13. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. **31**(6), 688–690 (2006). [CrossRef] [PubMed]

14. C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. **32**(28), 5492–5503 (1993). [CrossRef] [PubMed]

- ➢ The first process lies in the arbitrary value of the energy ratio of the scattered polarization modes, due to the relative amplitude of the scattering coefficients on each polarization axis.
- ➢ The second process is less common and is related to an increase of mutual correlation between the polarization modes of scattering, which results from a linear combination of the incident modes on each polarization axis. This last property is specific of the scattering process and enforces itself with the value of cross-scattering coefficients; it will be further discussed throughout the next sections.

**19**(22), 21313–21320 (2011). [CrossRef] [PubMed]

**37**(11), 2055–2057 (2012). [CrossRef] [PubMed]

^{2}PDF law.

**19**(22), 21313–21320 (2011). [CrossRef] [PubMed]

**37**(11), 2055–2057 (2012). [CrossRef] [PubMed]

## 2. Principles of enpolarization

### 2.1. Incident partial polarization

_{0S}and E

_{0P}the polarization modes. These modes are analytical signals and the polarization properties can be analyzed through the complex coherency matrix J

_{0}given by [5]:where

_{0}) can be directly calculated from the determinant (det) and the trace (tr) of the matrix, that is [5]:Equation (2) can also be written versus the two parameters that are the mutual coherence μ

_{0}(or correlation) and the polarization ratio β

_{0}of the incident field:with:Hence the correlation factor μ and the polarization ratio β control the DOP value. Notice that these parameters vary with the choice of the axis while the DOP remains invariant for a unitary transformation [15

15. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express **13**(16), 6051–6060 (2005). [CrossRef] [PubMed]

_{0}, β

_{0}) of parameters may allow to keep a constant DOP with the condition:However this parameter range is strongly reduced with the DOP value, which means that an arbitrary DOP cannot be reached with an arbitrary polarization ratio (or correlation).

### 2.2. The scattered field and the cross-scattering coefficients

16. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Correlation matrix of a completely polarized, statistically stationary electromagnetic field,” Opt. Lett. **29**(13), 1536–1538 (2004). [CrossRef] [PubMed]

_{uv}) allows to write the polarization modes (E

_{S}, E

_{P}) of the scattered field E versus the polarization modes (E

_{0S}, E

_{0P}) of the incident field E

_{0}, with scalar terms ν

_{UV}that are the scattering coefficients:Notice that the scattering coefficients are here assumed to be static since the sample is not in motion. Moreover and following perturbative theories [14

14. C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. **32**(28), 5492–5503 (1993). [CrossRef] [PubMed]

_{S}or E

_{P}) is the result of a combination of the two incident polarization modes (E

_{0S}or E

_{0P}), due to the presence of cross-scattering coefficients (ν

_{SP}and ν

_{PS}). In other words on each polarization axis (S or P) the scattering process performs a linear combination of the two initial random variables E

_{0S}and E

_{0P}, and the result is another couple of random variables E

_{S}and E

_{P}with new statistics and correlation. This remark directly announces why mutual coherence (and therefore the DOP) can be increased by a scattering process.

_{SP}and ν

_{PS}) that allow the linear combination to occur on each polarization axis. Moreover, the strength of these phenomena increase in average with the ratios ν

_{PS}/ν

_{SS}and ν

_{SP}/ν

_{PP}. Inversely, specular processes (ie: processes that do not involve optical cross-coefficients) such as reflection or transmission do not modify the modes correlation (μ = μ

_{0}) and can only change the DOP via the modification of the energy ratio β on the axis. In a similar way slightly disordered media cannot modify the correlation because perturbative theories [14

14. C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. **32**(28), 5492–5503 (1993). [CrossRef] [PubMed]

### 2.3. Polarization degree of the scattered field

### 2.4. Relationship between input and output DOP

_{0}. The results are the following: Relations (17-18) now provide a way to calculate the scattering polarization parameters β and μ versus those (β

_{0}and μ

_{0}) of the incident beam. From these values one can extract the DOP of the scattered light following relation (15), a procedure that we use in section 3 devoted to numerical calculation.

_{SP}= ν

_{PS}= 0), relations (17-18) are reduced to:so that correlation is not modified and the DOP can only be changed via the energy ratio β.

## 3. Numerical results on light enpolarization

### 3.1. The scattering model

_{uv}in the far field is assumed to be the Fourier Transform of an exponential function exp[jϕ

_{uv}(x,y)

_{]}, with ϕ

_{uv}(x,y) a random phase uniformly distributed within [0;2π]. With this model the scattering coefficients are independent numbers and the speckle is fully developed. Such phasor formalism has been used in many situations with success [19]; also, this model was recently used to predict enpolarization effects and revealed large agreement with experiment, provided that the media are highly disordered and the incident beam fully depolarized [11

**19**(22), 21313–21320 (2011). [CrossRef] [PubMed]

^{2}, and is included within another square area of surface S

_{0}= L

_{0}

^{2}, with L < L

_{0}. Therefore the intrinsic speckle resolution or grain size is given in the Fourier plane by δσ = 1/L, while the spectral step follows the Shannon/Niquist criteria: Δσ = 1/L

_{0}< 1/L. The result is that L

_{0}/L is the number of data points within the speckle size, and must be chosen as an integer M < N, with N

^{2}the total number of data points. Hence the speckle patterns that follow are plotted versus the vector spatial frequency

**σ**= (σ

_{x}, σ

_{y}), and were calculated with M = 16 and N = 1024.

**ρ**) within a receiver. Indeed the spatial frequency can also be written [14

**32**(28), 5492–5503 (1993). [CrossRef] [PubMed]

### 3.2. The DOP statistics of the scattered field

_{0}of the incident light increases from 0 to 1. Here the input DOP

_{0}was calculated with zero correlation (μ

_{0}= 0) so that its value was controlled by the polarization ratio β

_{0}.

_{0}. We first observe at low and medium DOP

_{0}values, that from one speckle grain to another, the scattered DOP may take arbitrary values within the range [0;1]. This proves that disordered media may locally increase the polarization of scattering, in regard to that of the incident beam. Also, as predicted the scattered light remains fully polarized (DOP = 1) when the incident beam is fully polarized (DOP

_{0}= 1).

_{0}is increased from 0 to 1. For the first value (DOP

_{0}= 0) which corresponds to a fully unpolarized illumination, we observe that the data points spread over the whole disk, which means that the scattered light is not unpolarized in average and can take arbitrary DOP values within the range [0;1]; in other words, the disordered medium may arbitrarily increase the incident polarization degree from one speckle grain to another. Then when the incident DOP

_{0}is increased, the data points get closer to the circle and finally vanish within the disk for fully polarized incident light; this last result recalls that full polarized illumination creates fully polarized scattering.

_{0}. The resulting variations are plotted in Fig. 4 . The 3u

^{2}law plotted in red dashed line is recalled for the case of totally unpolarized incident light [11

**19**(22), 21313–21320 (2011). [CrossRef] [PubMed]

**37**(11), 2055–2057 (2012). [CrossRef] [PubMed]

_{0}. This means that for all incident DOP

_{0}the most probable situation for scattering is full polarization, while the less probable is unpolarized scattering. Therefore the scattered light will be highly polarized in average. Notice that the scattered DOP naturally tends towards a Dirac function around DOP = 1 when the incident light is fully polarized.

_{0}, β

_{0}) that control the incident DOP

_{0}. This average strictly increases with the incident DOP

_{0}, so that the minimum value (DOP = 0.75) is obtained for totally unpolarized light. Therefore the scattering DOP is higher than 0.75 whatever the incident polarization (0 < DOP

_{0}< 1), which proves that light is strongly locally enpolarized by the scattering medium. These results are completed by those of Fig. 6 where the ratio of the scattered DOP to the incident DOP

_{0}is plotted. This ratio is greater than 1.

_{0}) and polarization (β

_{0}) parameters. The reason is that the incident DOP

_{0}can be reached with different sets (μ

_{0}, β

_{0}) of parameters leading to different intensity patterns, as given in relation (13); however we have checked that the average of the scattering DOP does not depend on these two parameters, but only on the incident DOP

_{0}. Hence we were allowed to plot in Fig. 7 the variation of scattering DOP versus the incident DOP

_{0}. Notice here that the fact that the average output DOP only depends on the input DOP

_{0}does not prove that all statistics of the output DOP only depend on the input DOP

_{0}.

## 4. Conclusion

_{0}. These phenomena do not violate the entropy principles since scattering is specific of a loss process.

_{0}) and can be used to emphasize new signatures when probing complex media [18]. Applications may concern remote sensing and biophotonics, defense, cosmetics and lightening.

## Acknowledgments

## Referencesand links

1. | L. Mandel and E. Wolf, |

2. | C. Brosseau, |

3. | J. W. Goodman, |

4. | E. Jakeman and K. D. Ridley, |

5. | E. Wolf, |

6. | S. G. Hanson and H. T. Yura, “Statistics of spatially integrated speckle intensity difference,” J. Opt. Soc. Am. A |

7. | M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, “Gradual loss of polarization in light scattered from rough surfaces: electromagnetic prediction,” Opt. Express |

8. | J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express |

9. | R. Barakat, “Polarization entropy transfer and relative polarization entropy,” Opt. Commun. |

10. | P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A |

11. | J. Sorrentini, M. Zerrad, G. Soriano, and C. Amra, “Enpolarization of light by scattering media,” Opt. Express |

12. | P. Réfrégier, M. Zerrad, and C. Amra, “Coherence and polarization properties in speckle of totally depolarized light scattered by totally depolarizing media,” Opt. Lett. |

13. | F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. |

14. | C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. |

15. | P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express |

16. | J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Correlation matrix of a completely polarized, statistically stationary electromagnetic field,” Opt. Lett. |

17. | D. L. Colton and R. Kress, |

18. | L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, |

19. | J. W. Goodman, |

20. | J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(260.5430) Physical optics : Polarization

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: November 8, 2012

Revised Manuscript: January 1, 2013

Manuscript Accepted: January 4, 2013

Published: January 29, 2013

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2787

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- C. Brosseau, Fundamentals of Polarized Light - A Statistical Optics 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, Introduction to the Theory of coherence and polarization of light, Cambridge University Press, 2007).
- S. G. Hanson and H. T. Yura, “Statistics of spatially integrated speckle intensity difference,” J. Opt. Soc. Am. A26(2), 371–375 (2009). [CrossRef] [PubMed]
- M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, “Gradual loss of polarization in light scattered from rough surfaces: electromagnetic prediction,” Opt. Express18(15), 15832–15843 (2010). [CrossRef] [PubMed]
- J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express18(19), 20105–20113 (2010). [CrossRef] [PubMed]
- R. Barakat, “Polarization entropy transfer and relative polarization entropy,” Opt. Commun.123(4-6), 443–448 (1996). [CrossRef]
- P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A25(11), 2749–2757 (2008). [CrossRef] [PubMed]
- J. Sorrentini, M. Zerrad, G. Soriano, and C. Amra, “Enpolarization of light by scattering media,” Opt. Express19(22), 21313–21320 (2011). [CrossRef] [PubMed]
- P. Réfrégier, M. Zerrad, and C. Amra, “Coherence and polarization properties in speckle of totally depolarized light scattered by totally depolarizing media,” Opt. Lett.37(11), 2055–2057 (2012). [CrossRef] [PubMed]
- F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett.31(6), 688–690 (2006). [CrossRef] [PubMed]
- C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt.32(28), 5492–5503 (1993). [CrossRef] [PubMed]
- P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express13(16), 6051–6060 (2005). [CrossRef] [PubMed]
- J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Correlation matrix of a completely polarized, statistically stationary electromagnetic field,” Opt. Lett.29(13), 1536–1538 (2004). [CrossRef] [PubMed]
- D. L. Colton and R. Kress, Integral Equations methods in Scattering Theory (Wiley, 1983).
- L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of electromagnetic waves: numerical simulations (Wiley-Interscience, 2001).
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).
- J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett.34(16), 2429–2431 (2009). [CrossRef] [PubMed]

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