## Gradual loss of polarization in light scattered from rough surfaces: Electromagnetic prediction |

Optics Express, Vol. 18, Issue 15, pp. 15832-15843 (2010)

http://dx.doi.org/10.1364/OE.18.015832

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### Abstract

Electromagnetic theory is used to calculate the gradual loss of polarization in light scattering from surface roughness. The receiver aperture is taken into account by means of a multiscale spatial averaging process. The polarization degrees are connected with the structural parameters of surfaces.

© 2010 OSA

## 1. Introduction

4. M. E. Knotts and K. A. O’Donnell, “Multiple scattering by deep perturbed gratings,” J. Opt. Soc. Am. A **11**(11), 2837–2843 (1994). [CrossRef]

5. E. Wolf, “Unified theory of Coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**(5-6), 263–267 (2003). [CrossRef]

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 **16**(14), 10372–10383 (2008). [CrossRef] [PubMed]

**ρ**:with α the normalization coefficient:and E

_{S}, E

_{P}the real polarization modes, <>

_{t}an average process over time parameter t. Here mutual coherence is a time constant due to the value of optical frequencies in the visible regime, in regard to detector band-passes. Notice that Eq. (1) is for a single realization of the process E

_{SorP}(

**ρ**,t) under study [7].

5. E. Wolf, “Unified theory of Coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**(5-6), 263–267 (2003). [CrossRef]

11. J. Borky, J. Ellis, and A. Dogariu, “Identifying non-stationarities in random EM fields: are speckles really disturbing?” Opt. Express **16**(19), 14469–14475 (2008). [CrossRef]

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]

11. J. Borky, J. Ellis, and A. Dogariu, “Identifying non-stationarities in random EM fields: are speckles really disturbing?” Opt. Express **16**(19), 14469–14475 (2008). [CrossRef]

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. Multi-scale analysis of polarization

**ρ**

_{1}and

**ρ**

_{2}, that is:

**ρ**

_{1}=

**ρ**

_{2}=

**ρ**), as:

**ρ**, and

*J*the coherence matrix:

## 3. Electromagnetic prediction

12. J. Li, G. Yao, and L. V. Wang, “Degree of polarization in laser speckles from turbid media: implications in tissue optics,” J. Biomed. Opt. **7**(3), 307–312 (2002). [CrossRef] [PubMed]

15. 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]

_{rms}and their correlation length

*L*. A 6 mm-long profile is illuminated with a fully polarized Gaussian beam under normal incidence at 632.8nm wavelength. The profiles are assumed to be engraved in aluminum, of complex optical refractive index 1.39 + j 7.65 at this wavelength.

_{cor}_{S}and A

_{P}of the scattered field are calculated for 10000 scattering angles in the θ angular range (0°− 30°) and regularly sampled with a step of 5e-5rad, that is with δθ = 2.8e-3°. The angle runs from 0.6761° to 29.3239°, so that the specular region (θ = 0°) is avoided. Such an angular resolution permits us to precisely resolve the speckle, which is of characteristic size 1.15e-2° ( = 2e-4 rad≈⌊/3mm = 2.11e-4) in this configuration. The profiles are generated using the spectral method [16]. Profiles are spatially sampled with a step that depends on the correlation length. It is set to 92nm for

*L*= 2μm and to 23nm when

_{cor}*L*= 100nm.

_{cor}## 3. The case of two extreme polarization regimes

_{P}/A

_{S}| and the polarimetric phase delay δ of the polarized speckles for two surfaces of strongly different topography. One surface is specific of the perturbative regime with parameters h

_{rms}= 50 nm and L

_{cor}= 2 μm, which implies a low quadratic slope (s ≈h

_{rms}/L

_{cor}= 2.5%). On the other hand, the second surface has a significant slope (s ≈100%, h

_{rms}= 100 nm and L

_{cor}= 0.1 μm) and scatters the whole incident light.

_{P}/A

_{S}| ≈1); this is no more the case for the high slope surface which shows 2 decades fluctuations for the amplitude ratio (| β

_{max}/ β

_{min}| ≈100). Strong differences also appear when the phase data (Fig. 2) are considered: while the phase term remains close to -π for the perturbative surface, it appears quasi-uniformly distributed within ± π for the other one. These results provide a preliminary signature to identify the scattering regime; in other words, a perturbative surface cannot depolarize light [3] Calculation of the MDOP within the receiver aperture ΔΩ = 1° gives MDOP = 1 for the low slope surface, and MDOP = 0.3 for the other one.

## 4. Scan of surface parameters

_{cor}= 100nm) were considered. The roughness values h

_{rms}vary from 1nm to 100nm, so that the slope is in the range (1%-100%). For each surface 350 data points were calculated in the angular range (10°-11°) and plotted on the Poincaré sphere. Results are given in Figs. 5 and 6 . As predicted, low slope surfaces slighlty spread the polarization location over the sphere, which indicates that full polarization is globally maintained for these surfaces and similar to the incident polarization (linear 45°).

## 5. Gradual transition and the MDOP function

_{0}= 10.5°, so that we studied the function MDOP (θ

_{0}= 10.5°, ΔΩ). In what follows 3 values are considered for θ

_{0}, that are 10°, 10.5° and 11°. In all cases the maximum aperture is 1° apart from this average angle θ

_{0}.

_{0}, ΔΩ<1°) functions are plotted in Fig. 7 versus the coherence areas or speckle grains; each coherence area is about 5 data points so that 40 coherence areas are enough to cover a 0.5° aperture, which is enough to approach the asymptotic behavior. The first figures just confirm that perturbative surfaces keep full polarization whatever the angular aperture lower than 1°. When the slope increases, we first observe, for a given average θ

_{0}angle, fluctuations of polarization versus the aperture; hence the curves are not monotonic and one may measure a slight “re-polarization” when increasing the aperture, and this local re-polarization is specific of the θ

_{0}angle.

_{cor}= 100nm.

^{n}+ c = 1 for a unity DOP value at the origin, so, the fit parameter b can be deduced from a,c and n. Results are given in Fig. 8 and show a quasi perfect agreement between MDOP* and the fit f, since the two curves are superimposed whatever the surface.

*a*,

*c*and

*n*are given versus the surface slopes, and parameter c is compared to MDOP(1°). As predicted, the asymptotic MDOP* is quasi-identical to this parameter. Moreover, the power parameter (n) rapidly reaches a stationary value of the order of 1.2.

## 6. DOP cartography versus slope and height

_{cor}= 0.1μm, which reduces our MDOP prediction to short correlation surfaces. For this reason the correlation length is now scanned with values in the range 0.1μm - 3μm. Results are plotted in Fig. 10 , and consider the asymptotic MDOP value (ΔΩ = 1°) versus both h

_{rms}and L

_{cor}. We observe at a constant roughness (vertical line) that the DOP increases up to 1 when the correlation length varies from 0.1 to 3 μm. Then at a constant correlation length (horizontal line), the DOP falls to zero when the roughness increases. Lastly, at a constant slope (oblique line) given by the L

_{cor}/h

_{rms}ratio, we notice in a first approximation that the MDOP remains quasi-constant; this last result emphasizes the key role of surface slope in the MDOP prediction.

## 7. Conclusion

## References and links

1. | J. W. Goodman, |

2. | L. Mandel, and E. Wolf, eds., |

3. | J. W. Goodman, |

4. | M. E. Knotts and K. A. O’Donnell, “Multiple scattering by deep perturbed gratings,” J. Opt. Soc. Am. A |

5. | E. Wolf, “Unified theory of Coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

6. | C. Brosseau, “Polarization and Coherence Optics: Historical Perspective, Status and Future Directions,” presented at the Frontiers in Optics, 2008. |

7. | E. Wolf, ed., |

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

9. | J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” 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. | J. Borky, J. Ellis, and A. Dogariu, “Identifying non-stationarities in random EM fields: are speckles really disturbing?” Opt. Express |

12. | J. Li, G. Yao, and L. V. Wang, “Degree of polarization in laser speckles from turbid media: implications in tissue optics,” J. Biomed. Opt. |

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

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

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

16. | L. Tsang, J. A. Kong, and K.-H. Ding, |

**OCIS Codes**

(030.5770) Coherence and statistical optics : Roughness

(260.5430) Physical optics : Polarization

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: May 18, 2010

Manuscript Accepted: June 24, 2010

Published: July 12, 2010

**Citation**

Myriam Zerrad, Jacques Sorrentini, Gabriel Soriano, and Claude Amra, "Gradual loss of polarization in light scattered from rough surfaces: Electromagnetic prediction," Opt. Express **18**, 15832-15843 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15832

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### References

- J. W. Goodman, Statistical Optics (Wiley Classic Library, 1985).
- L. Mandel, and E. Wolf, eds., Optical Coherence and Quantum Optics (Cambridge University Press 1995).
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).
- M. E. Knotts and K. A. O’Donnell, “Multiple scattering by deep perturbed gratings,” J. Opt. Soc. Am. A 11(11), 2837–2843 (1994). [CrossRef]
- E. Wolf, “Unified theory of Coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
- C. Brosseau, “Polarization and Coherence Optics: Historical Perspective, Status and Future Directions,” presented at the Frontiers in Optics, 2008.
- E. Wolf, ed., Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
- P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13(16), 6051–6060 (2005). [CrossRef] [PubMed]
- J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004). [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. Express 16(14), 10372–10383 (2008). [CrossRef] [PubMed]
- J. Broky, J. Ellis, and A. Dogariu, “Identifying non-stationarities in random EM fields: are speckles really disturbing?” Opt. Express 16(19), 14469–14475 (2008). [CrossRef]
- J. Li, G. Yao, and L. V. Wang, “Degree of polarization in laser speckles from turbid media: implications in tissue optics,” J. Biomed. Opt. 7(3), 307–312 (2002). [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]
- D. Colton, and R. Kress, Integral Equations methods in Scattering Theory (New-York, 1983).
- 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]
- L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of electromagnetic waves: numerical simulations, Wiley series in remote sensing (Wiley-Interscience, 2001).

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