## Enpolarization of light by scattering media |

Optics Express, Vol. 19, Issue 22, pp. 21313-21320 (2011)

http://dx.doi.org/10.1364/OE.19.021313

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### Abstract

The polarization of a coherent depolarized incident light beam passing through a scattering medium is investigated at the speckle scale. The polarization of the scattered far field at each direction and the probability density function of the degree of polarization are calculated and show an excellent agreement with experimental data. It is demonstrated that complex media may confer high degree of local polarization (0.75 DOP average) to the incident unpolarized light.

© 2011 OSA

## 1. Introduction

14. 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]

14. 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]

13. 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]

## 2. Repolarization by a scattering process: principles

### 2.1 **The incident unpolarized field**

*M*=

*(*ν

*, and*

_{uv})*r*is the spatial coordinate. In the plane z = z

_{0}(Fig. 1 ), this field is written as:where

_{0}, no temporal correlation exists between the Transverse Electric (TE or s) and the Transverse Magnetic (TM or p) modes, so that the complex modes correlation follows:with bars denoting the complex conjugation. In this relation, e

_{S}(t) and e

_{P}(t) are normalized as:

*ω*of

_{0}and matches the quasi-monochromatic condition: Δ

*ω/ω*<<1. Moreover this beam illuminates a scattering medium whose linear response is not frequency-dependent within the spectral domain, in order to preserve temporal coherence. In relation (2a) the correlation μ represents the non-diagonal term of the coherency matrix as defined in [22]

_{0}### 2.2 The scattered field

*E*

^{sc}scattered in the far field at one direction at infinity as E

^{sc}≈M E, that is:where the scattering coefficients

*(*ν

*of the Jones matrix M are for the v-polarized scattered waves resulting from a u-polarized illumination. Notice here that relation (3) is for the field scattered at a particular direction at infinity in the far field; that is, the scattering coefficients ν*

_{uv})*and the Jones matrix M are direction dependent. Notice also in this relation the absence of wave packet to take into account the propagation from z*

_{UV}_{1}to z

_{2}; indeed we know from the stationary phase theorem [4] that in the far field at infinity, the wave packet (whose first-order approximation gives the Fresnel formalism) that describes the exact field at a particular direction can be reduced to a single Fourier component of the packet, here characterized by the (ν

_{uv}) coefficients. In other words, the scattered field is described by a plane wave in the far field.

### 2.3 Polarization parameters

^{SC}of the scattered field

*(*ν

*coefficients are independent in the general case of arbitrary scattering media, Eq. (8) ensures that the temporal correlation*

_{uv})28. K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media **11**(1), 1–30 (2001). [CrossRef]

_{UV}≈0) in the incidence plane; with these samples the temporal correlation remains zero (μ

^{SC}= 0) and the scattered light remains unpolarized (dop

^{SC}= 0) if the polarization ratio is unity (β = 1). On the other hand, in the general case of arbitrary samples, the presence of cross-scattering coefficients will make the temporal correlation and the dop

^{SC}not to be zero. So, even though the illumination beam is perfectly unpolarized, relation (8) shows that the scattered light can be partially or totally polarized in the far field depending on the scattering samples and the space direction.

## 3. Comparison of experiment and numerical calculation

### 3.1 Numerical calculation

*(*ν

*is obtained via the Fourier Transform of a random phasor matrix [29]. Here, the non-zero domain is a square of 2*

_{uv})^{7}points length within a square of 2

^{10}points length. Figure 2(a) shows the spatial repartition of the local dop

^{SC}of the scattered far field at infinity in a plane perpendicular to propagation. Depending on the space location (or direction), the dop varies from 0 to 1. Therefore it is different from that of the incident light, which was zero at any location. Such result is in agreement with the prediction of relation (8) given in the preceding section.

^{2}law, and the resulting spatial average of local dop is found to be:

14. 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]

### 3.2 Experiment

_{2}sample often used for calibration in scattering apparatuses. This means that the sample scatters all the incident light and that its angular pattern follows a lambertian law (cosθ curve, with θ the scattering angle). Moreover, previous experiments [13

13. 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]

_{2}.

## 4. Conclusion

^{2}pdf probability local dop function.

9. M. Mujat and A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. **28**(22), 2153–2155 (2003). [CrossRef] [PubMed]

## References and links

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

2. | C. Brosseau, |

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

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

5. | D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A |

6. | E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. |

7. | O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. |

8. | X. Y. Du and D. M. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express |

9. | M. Mujat and A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. |

10. | C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, “Partial polarization of light induced by random defects at surfaces or bulks,” Opt. Express |

11. | L. H. Jin, M. Kasahara, B. Gelloz, and K. Takizawa, “Polarization properties of scattered light from macrorough surfaces,” Opt. Lett. |

12. | S. M. F. Nee and T. W. Nee, “Polarization of transmission scattering simulated by using a multiple-facets model,” J. Opt. Soc. Am. A |

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

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

15. | I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. |

16. | D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

17. | B. J. DeBoo, J. M. Sasian, and R. A. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt. |

18. | T. W. Nee, S. M. F. Nee, D. M. Yang, and A. Chiou, “Optical scattering depolarization in a biomedium with anisotropic biomolecules,” J. Opt. Soc. Am. A |

19. | L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A |

20. | A. C. Maggs and V. Rossetto, “Writhing photons and Berry phases in polarized multiple scattering,” Phys. Rev. Lett. |

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

22. | J. W. Goodman, |

23. | C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A |

24. | L. Arnaud, G. Georges, J. Sorrentini, M. Zerrad, C. Deumié, and C. Amra, “An enhanced contrast to detect bulk objects under arbitrary rough surfaces,” Opt. Express |

25. | D. Colton and R. Kress, |

26. | C. Macaskill and B. J. Kachoyan, “Iterative approach for the numerical simulation of scattering from one- and two-dimensional rough surfaces,” Appl. Opt. |

27. | G. Soriano and M. Saillard, “Scattering of electromagnetic waves from two-dimensional rough surfaces with an impedance approximation,” J. Opt. Soc. Am. A |

28. | K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media |

29. | J. W. Goodman, |

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

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.6140) Coherence and statistical optics : Speckle

(030.6600) Coherence and statistical optics : Statistical optics

(260.2130) Physical optics : Ellipsometry and polarimetry

(260.5430) Physical optics : Polarization

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: July 25, 2011

Revised Manuscript: September 10, 2011

Manuscript Accepted: September 19, 2011

Published: October 12, 2011

**Citation**

J. Sorrentini, M. Zerrad, G. Soriano, and C. Amra, "Enpolarization of light by scattering media," Opt. Express **19**, 21313-21320 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21313

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### References

- E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor and Francis Group, 2006).
- C. Brosseau, Fundamentals of Polarized Light—A Statistical Approach (Wiley, 1998).
- R. Barakat, “Polarization entropy transfer and relative polarization entropy,” Opt. Commun.123(4-6), 443–448 (1996). [CrossRef]
- E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (C. Cambridge University Press, 1995).
- D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A11(5), 1641–1643 (1994). [CrossRef]
- E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett.28(13), 1078–1080 (2003). [CrossRef] [PubMed]
- O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.233(4-6), 225–230 (2004). [CrossRef]
- X. Y. Du and D. M. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express17(6), 4257–4262 (2009). [CrossRef] [PubMed]
- M. Mujat and A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett.28(22), 2153–2155 (2003). [CrossRef] [PubMed]
- C. Amra, M. Zerrad, L. Siozade, G. Georges, and C. Deumié, “Partial polarization of light induced by random defects at surfaces or bulks,” Opt. Express16(14), 10372–10383 (2008). [CrossRef] [PubMed]
- L. H. Jin, M. Kasahara, B. Gelloz, and K. Takizawa, “Polarization properties of scattered light from macrorough surfaces,” Opt. Lett.35(4), 595–597 (2010). [CrossRef] [PubMed]
- S. M. F. Nee and T. W. Nee, “Polarization of transmission scattering simulated by using a multiple-facets model,” J. Opt. Soc. Am. A21(9), 1635–1644 (2004). [CrossRef] [PubMed]
- 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]
- 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]
- I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett.104(22), 223904 (2010). [CrossRef] [PubMed]
- D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics49(2), 1767–1770 (1994). [CrossRef] [PubMed]
- B. J. DeBoo, J. M. Sasian, and R. A. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt.44(26), 5434–5445 (2005). [CrossRef] [PubMed]
- T. W. Nee, S. M. F. Nee, D. M. Yang, and A. Chiou, “Optical scattering depolarization in a biomedium with anisotropic biomolecules,” J. Opt. Soc. Am. A26(5), 1101–1108 (2009). [CrossRef] [PubMed]
- L. F. Rojas-Ochoa, D. Lacoste, R. Lenke, P. Schurtenberger, and F. Scheffold, “Depolarization of backscattered linearly polarized light,” J. Opt. Soc. Am. A21(9), 1799–1804 (2004). [CrossRef] [PubMed]
- A. C. Maggs and V. Rossetto, “Writhing photons and Berry phases in polarized multiple scattering,” Phys. Rev. Lett.87(25), 253901 (2001). [CrossRef] [PubMed]
- J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express18(19), 20105–20113 (2010). [CrossRef] [PubMed]
- J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).
- C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A10(2), 365–374 (1993). [CrossRef]
- L. Arnaud, G. Georges, J. Sorrentini, M. Zerrad, C. Deumié, and C. Amra, “An enhanced contrast to detect bulk objects under arbitrary rough surfaces,” Opt. Express17(7), 5758–5773 (2009). [CrossRef] [PubMed]
- D. Colton and R. Kress, Integral Equations in Scattering Theory (Elsevier, 1983).
- C. Macaskill and B. J. Kachoyan, “Iterative approach for the numerical simulation of scattering from one- and two-dimensional rough surfaces,” Appl. Opt.32(15), 2839–2847 (1993). [CrossRef] [PubMed]
- G. Soriano and M. Saillard, “Scattering of electromagnetic waves from two-dimensional rough surfaces with an impedance approximation,” J. Opt. Soc. Am. A18(1), 124–133 (2001). [CrossRef] [PubMed]
- K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media11(1), 1–30 (2001). [CrossRef]
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).
- 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]

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