## Mueller matrix measurements and modeling pertaining to Spectralon white reflectance standards |

Optics Express, Vol. 20, Issue 14, pp. 15045-15053 (2012)

http://dx.doi.org/10.1364/OE.20.015045

Acrobat PDF (2294 KB)

### Abstract

The full Mueller matrix for a Spectralon white reflectance standard was measured in the incidence plane, to obtain the polarization state of the scattered light for different angles of illumination. The experimental setup was a Mueller matrix ellipsometer, by which measurements were performed for scattering angles measured relative to the normal of the Spectralon surface from −90° to 90° sampled at every 2.5° for an illumination wavelength of 532 nm. Previously, the polarization of light scattered from Spectralon white reflectance standards was measured only for four of the elements of the Muller matrix. As in previous investigations, the reflection properties of the Spectralon white reflectance standard was found to be close to those of a Lambertian surface for small scattering and illumination angles. At large scattering and illumination angles, all elements of the Mueller matrix were found to deviate from those of a Lambertian surface. A simple empirical model with only two parameters, was developed, and used to simulate the measured results with fairly good accuracy.

© 2012 OSA

## Introduction

1. “A guide to diffuse reflectance coatings & materials;” http://www.prolite.co.uk/File/coatings materials documentation.php.

2. “Reflectance standards product sheet 8.pdf” http://www.labsphere.com/data/userFiles/.

4. D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of Spectralon illuminated by coherent light,” Appl. Opt. **38**(30), 6350–6356 (1999). [CrossRef] [PubMed]

6. D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. **37**(18), 3996–3999 (1998). [CrossRef] [PubMed]

7. K. J. Voss and H. Zhang, “Bidirectional reflectance of dry and submerged Labsphere Spectralon plaque,” Appl. Opt. **45**(30), 7924–7927 (2006). [CrossRef] [PubMed]

8. G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE **5570**, 492–502 (2004). [CrossRef]

9. AA. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. **50**(16), 2431–2442 (2011). [CrossRef] [PubMed]

11. B. T. McGuckin, D. A. Haner, R. T. Menzies, C. Esproles, and A. M. Brothers, “Directional reflectance characterization facility and measurement methodology,” Appl. Opt. **35**(24), 4827–4834 (1996). [CrossRef] [PubMed]

12. M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res. **112**(C5), 5026–5039 (2007). [CrossRef]

13. G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. **6**(4), 741–746 (1967). [CrossRef] [PubMed]

14. G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. **241**(4-6), 255–261 (2004). [CrossRef]

15. G. W. Kattawar and D. J. Gray, “Mueller matrix imaging of targets in turbid media: effect of the volume scattering function,” Appl. Opt. **42**(36), 7225–7230 (2003). [CrossRef] [PubMed]

16. P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt. **48**(2), 250–260 (2009). [CrossRef] [PubMed]

*M*

_{22}and

*M*

_{44}to be less influenced by the interface than the other elements. Tyo et al. [17

17. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. **45**(22), 5453–5469 (2006). [CrossRef] [PubMed]

18. F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. **47**(7), 073604 (2008). [CrossRef]

## Theory for Mueller matrix of a diffuser

4. D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of Spectralon illuminated by coherent light,” Appl. Opt. **38**(30), 6350–6356 (1999). [CrossRef] [PubMed]

9. AA. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. **50**(16), 2431–2442 (2011). [CrossRef] [PubMed]

18. F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. **47**(7), 073604 (2008). [CrossRef]

*s*and

*p*polarized components. The notation used here for the Stokes vector and Mueller matrix follows the definition given by Hauge et al. [19

19. P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. **96**(1-3), 81–107 (1980). [CrossRef]

*s*and

*p*polarized states of the incident light, and analyzing

*s*and

*p*states of the scattered light, are included as a subset of the current measurements of the full Mueller matrix. However, the normalization issues for an absolute BRDF measurement remain as complicated as described in e.g. Bhandari et al. [9

9. AA. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. **50**(16), 2431–2442 (2011). [CrossRef] [PubMed]

20. R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi **205**(4), 720–727 (2008). [CrossRef]

*a, b,*and

*c*in Eq. (1) are positive and less than 1. On the other hand, a non-depolarizing Mueller matrix can be made up of the basic building blocks of the so-called isotropic dichroic retarder, commonly encountered in specular reflection from an isotropic surface [21

21. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A **25**(2), 473–482 (2008). [CrossRef] [PubMed]

*ψ*and

*Δ*are the commonly defined ellipsometric parameters

21. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A **25**(2), 473–482 (2008). [CrossRef] [PubMed]

*R*

_{p}and

*R*

_{s}are the reflection coefficients for the

*s*and

*p*components, respectively.

8. G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE **5570**, 492–502 (2004). [CrossRef]

**50**(16), 2431–2442 (2011). [CrossRef] [PubMed]

*p*and

*s*directions were used in the characterization, and hence only the elements

*M*

_{11},

*M*

_{12},

*M*

_{21}, and

*M*

_{22}were measured. All other elements were disregarded, but it was noted that the Mueller matrix must be of the form:where

*M*

_{11}for small illumination and scattering angles is much larger than either of |

*M*

_{12}|, |

*M*

_{21}|, and |

*M*

_{22}|. However, |

*M*

_{12}|, |

*M*

_{21}|, and |

*M*

_{22}| would increase significantly for large scattering and illumination angles. In this work, the aim is to understand the significance of the Mueller matrix elements denoted by question marks in Eq. (3) for a Spectralon diffuser used as a reference standard. To that end, the full Mueller matrix pertaining to four different illuminations angles

*θ*

_{0}= 0°, 30°, 45°, and 60° was measured for scattering angles

*θ*between 0° and 180° for every 2.5°. Earlier measurements showed that speckle effects from laser light would influence BRDF measurements [10, 11

11. B. T. McGuckin, D. A. Haner, R. T. Menzies, C. Esproles, and A. M. Brothers, “Directional reflectance characterization facility and measurement methodology,” Appl. Opt. **35**(24), 4827–4834 (1996). [CrossRef] [PubMed]

*M*

_{11}, which depends both on the illumination angle

*θ*

_{0}and the scattering angle

*θ*.

## Results and discussion

*p*(or degree of purity), which is obtained from the full Mueller matrix as follows [22

22. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) **33**(2), 185–189 (1986). [CrossRef]

*θ),*the depolarization index is seen to be very small (regardless of the value of the illumination angle (

*θ*

_{0})), but as

*θ*and

*θ*

_{0}increase, p increases, as would be expected since then many of the elements of the Mueller matrix differ from zero. Note that all other common forms of the linear depolarization index previously reported [4

4. D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of Spectralon illuminated by coherent light,” Appl. Opt. **38**(30), 6350–6356 (1999). [CrossRef] [PubMed]

**50**(16), 2431–2442 (2011). [CrossRef] [PubMed]

*M*

_{11}element (representing the scattering phase function) is shown for the same four illumination angles as in the left-hand side of the figure. g. For a Lambertian surface,

*M*

_{11}would be independent of the illumination angle

*θ*

_{0}, and thus only depend on the scattering angle

*θ*, as indicated by the black curve on the right-hand side of Fig. 1.

*θ*

_{0}= 0°, 30°, 45°, and 60° together with simulated results (red curves) based on a simple empirical model to be described below. Many of the elements appear insignificant compared to the noise level in the measurement. A close inspection of the Mueller matrices in Fig. 2 and 3 shows that the most significant elements can be summarized as:where each quantity

*m*

_{ij}represents a normalized Mueller matrix element. The elements

*m*

_{13},

*m*

_{14},

*m*

_{31}, and

*m*

_{41}are all constant (within the noise level) for all illumination and scattering angles and less than 0.05.

*M*

_{11}see Fig. 1), increases with the scattering angle

*θ*, and also with the illumination angle

*θ*

_{0}. In particular, the normalized diagonal elements

*m*

_{22},

*m*

_{33}, and

*m*

_{44}increase to 0.5 for

*θ*

_{0}

*=*60° and

*θ =*85°, indicating that the Spectralon sample to a significant degree behaves like the partial diffuser described by Eq. (1), and hence rather different from a Lambertian diffuser, which is described by Eq. (1) with

*θ*

_{0}(less than 30°) and scattering angles less than |80°|, the Spectralon diffuser behaves to a large extent as a Lambertian diffuser for which each normalized diagonal element

*m*

_{22},

*m*

_{33}, or

*m*

_{44}is less than 0.05.

*m*

_{12},

*m*

_{21},

*m*

_{43}, or

*m*

_{34}is smaller than the diagonal elements by a factor between 2 and 3, but is still significant, particularly at large illumination angles

*θ*

_{0}(see Fig. 2 and 3). A comparison with Eq. (2) shows that the non-vanishing values of the elements

*m*

_{43}and

*m*

_{34}indicate the presence of a dichroic retarder component, i.e. scattering where the polarization of the scattered light is not scrambled or lost. As for an isotropic dichroic retarder, there appears to be significant symmetry between elements

*m*

_{34}and

*m*

_{43}(i.e.

*m*

_{34}≈-

*m*

_{43}) as well as between

*m*

_{12}and

*m*

_{21}.

*m*

_{23},

*m*

_{32},

*m*

_{42}, and

*m*

_{24}(see Fig. 2 and 3), which one also would expect for a rotated dichroic retarder, may be explained by light scattering from surface roughness of low spatial frequency. Furthermore, an apparent symmetry may be observed with

*m*

_{24}≈

*m*

_{42},

*m*

_{24}≈-

*m*

_{42}and

*m*

_{34}≈-

*m*

_{43}. As a result, a simplified Mueller matrix for the Spectralon diffuser in the plane of incidence is proposed to be as Eq. (5).

*θ*

_{0}, a simple empirical model with only one parameter [see Eqs. (6) and (7) below] can be derived that gives results that agree fairly well with the measured Muller matrices for all four illumination angles

*θ*

_{0}and scattering angles

*θ*between −80° and 80°. This empirical model can be split into the sum of two matrices, where the first matrix is that of an ideal isotropic diffuser or Lambertian surface, and the second matrix represents a non-depolarizing surface [19

19. P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. **96**(1-3), 81–107 (1980). [CrossRef]

22. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) **33**(2), 185–189 (1986). [CrossRef]

*p*, the ideal isotropic diffuser or Lambertian reflector, represented by the first matrix, and its correction, represented by the second matrix.

*θ*is the scattering angle,

*θ*

_{0}is the illumination angle,

*κ*

_{1}= 0.67,

*κ*

_{2}= 0.17,

*κ*

_{3}= 0.45,

*κ*

_{4}= 0.04, and

*κ*

_{5}= 0.03. Our empirical model in Eqs. (6) and (7) and the values for

*κ*

_{1},

*κ*

_{2},

*κ*

_{3,}

*κ*

_{4}and

*κ*

_{5}were found by a best fit to the measured results. Figure 1 shows that the empirical model give results for the depolarization index that agree fairly well with those calculated from the measured Muller matrix using Eq. (6) for scattering angles

*θ*between −80° and 80°. For all four illumination angles

*θ*

_{0}= 0°, 30°, 45°, and 60°, the absolute value of the error between normalized simulated and normalized measured Muller matrix elements are less than 0.03. Since the error between the Mueller matrix elements obtained from the simple empirical model in Eqs. (6) and (7) and the measured Muller matrix elements is small, this model provides a simple way to obtain an estimate of all Mueller matrix for different illumination angles

*θ*

_{0}and scattering angles

*θ*between −80° and 80°.

## Conclusion

## Acknowledgment

## References and links

1. | “A guide to diffuse reflectance coatings & materials;” http://www.prolite.co.uk/File/coatings materials documentation.php. |

2. | “Reflectance standards product sheet 8.pdf” http://www.labsphere.com/data/userFiles/. |

3. | E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. |

4. | D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of Spectralon illuminated by coherent light,” Appl. Opt. |

5. | B. Gordon, “Integrating sphere diffuse reflectance technology for use with UV-Visible spectroscopy;” Tech. Note: 51450, Thermo Fisher Scientific, WI, USA. |

6. | D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. |

7. | K. J. Voss and H. Zhang, “Bidirectional reflectance of dry and submerged Labsphere Spectralon plaque,” Appl. Opt. |

8. | G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE |

9. | AA. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. |

10. | G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE |

11. | B. T. McGuckin, D. A. Haner, R. T. Menzies, C. Esproles, and A. M. Brothers, “Directional reflectance characterization facility and measurement methodology,” Appl. Opt. |

12. | M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res. |

13. | G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. |

14. | G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. |

15. | G. W. Kattawar and D. J. Gray, “Mueller matrix imaging of targets in turbid media: effect of the volume scattering function,” Appl. Opt. |

16. | P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt. |

17. | J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. |

18. | F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. |

19. | P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. |

20. | R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi |

21. | R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A |

22. | J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) |

23. | F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) |

**OCIS Codes**

(290.1350) Scattering : Backscattering

(290.1990) Scattering : Diffusion

(290.5820) Scattering : Scattering measurements

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: April 19, 2012

Manuscript Accepted: May 23, 2012

Published: June 20, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Øyvind Svensen, Morten Kildemo, Jerome Maria, Jakob J. Stamnes, and Øyvind Frette, "Mueller matrix measurements and modeling pertaining to Spectralon white reflectance standards," Opt. Express **20**, 15045-15053 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15045

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### References

- “A guide to diffuse reflectance coatings & materials;” http://www.prolite.co.uk/File/coatings materials documentation.php .
- “Reflectance standards product sheet 8.pdf” http://www.labsphere.com/data/userFiles/ .
- E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc.17, 1078–1091 (2000).
- D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of Spectralon illuminated by coherent light,” Appl. Opt.38(30), 6350–6356 (1999). [CrossRef] [PubMed]
- B. Gordon, “Integrating sphere diffuse reflectance technology for use with UV-Visible spectroscopy;” Tech. Note: 51450, Thermo Fisher Scientific, WI, USA.
- D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt.37(18), 3996–3999 (1998). [CrossRef] [PubMed]
- K. J. Voss and H. Zhang, “Bidirectional reflectance of dry and submerged Labsphere Spectralon plaque,” Appl. Opt.45(30), 7924–7927 (2006). [CrossRef] [PubMed]
- G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE5570, 492–502 (2004). [CrossRef]
- AA. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt.50(16), 2431–2442 (2011). [CrossRef] [PubMed]
- G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE588, 588203 (2005).
- B. T. McGuckin, D. A. Haner, R. T. Menzies, C. Esproles, and A. M. Brothers, “Directional reflectance characterization facility and measurement methodology,” Appl. Opt.35(24), 4827–4834 (1996). [CrossRef] [PubMed]
- M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res.112(C5), 5026–5039 (2007). [CrossRef]
- G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt.6(4), 741–746 (1967). [CrossRef] [PubMed]
- G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun.241(4-6), 255–261 (2004). [CrossRef]
- G. W. Kattawar and D. J. Gray, “Mueller matrix imaging of targets in turbid media: effect of the volume scattering function,” Appl. Opt.42(36), 7225–7230 (2003). [CrossRef] [PubMed]
- P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt.48(2), 250–260 (2009). [CrossRef] [PubMed]
- J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt.45(22), 5453–5469 (2006). [CrossRef] [PubMed]
- F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng.47(7), 073604 (2008). [CrossRef]
- P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci.96(1-3), 81–107 (1980). [CrossRef]
- R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi205(4), 720–727 (2008). [CrossRef]
- R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A25(2), 473–482 (2008). [CrossRef] [PubMed]
- J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.)33(2), 185–189 (1986). [CrossRef]
- F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.)29, 34–38 (1996).

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