## Experimental validation of Mueller matrix differential decomposition |

Optics Express, Vol. 20, Issue 2, pp. 1151-1163 (2012)

http://dx.doi.org/10.1364/OE.20.001151

Acrobat PDF (1245 KB)

### Abstract

Mueller matrix differential decomposition is a novel method for retrieving the polarimetric properties of general depolarizing anisotropic media [N. Ortega-Quijano and J. L. Arce-Diego, Opt. Lett. 36, 1942 (2011), R. Ossikovski, Opt. Lett. 36, 2330 (2011)]. The method has been verified for Mueller matrices available in the literature. We experimentally validate the decomposition for five different experimental setups with different commutation properties and controlled optical parameters, comparing the differential decomposition with the forward and reverse polar decompositions. The results enable to verify the method and to highlight its advantages for certain experimental applications of high interest.

© 2012 OSA

## 1. Introduction

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

5. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**(5), 1106–1113 (1996). [CrossRef]

7. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. **29**(19), 2234–2236 (2004). [CrossRef] [PubMed]

10. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. **41**(10), 1903–1915 (1994). [CrossRef]

11. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A **26**(5), 1109–1118 (2009). [CrossRef] [PubMed]

12. C. Fallet, A. Pierangelo, R. Ossikovski, and A. De Martino, “Experimental validation of the symmetric decomposition of Mueller matrices,” Opt. Express **18**(2), 831–842 (2010). [CrossRef] [PubMed]

13. N. Ortega-Quijano, F. Fanjul-Vélez, I. Salas-García, and J. L. Arce-Diego, “Comparative study of optical activity in chiral biological media by polar decomposition and differential Mueller matrices analysis,” Proc. SPIE **7906**, 790612 (2011). [CrossRef]

17. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. **36**(12), 2330–2332 (2011). [CrossRef] [PubMed]

14. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. **36**(10), 1942–1944 (2011). [CrossRef] [PubMed]

15. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. **36**(13), 2429–2431 (2011). [CrossRef] [PubMed]

17. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. **36**(12), 2330–2332 (2011). [CrossRef] [PubMed]

18. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus,” J. Opt. Soc. Am. **68**(12), 1756–1767 (1978). [CrossRef]

## 2. Experimental

### 2.1 Polarimetric setup

19. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. **43**(14), 2824–2832 (2004). [CrossRef] [PubMed]

20. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. **38**(16), 3490–3502 (1999). [CrossRef] [PubMed]

### 2.2 Samples

#### 2.2.1 Glucose and milk dilution in distilled water

21. M. D. Waterworth, B. J. Tarte, A. J. Joblin, T. van Doorn, and H. E. Niesler, “Optical transmission properties of homogenised milk used as a phantom material in visible wavelength imaging,” Australas. Phys. Eng. Sci. Med. **18**(1), 39–44 (1995). [PubMed]

23. R. R. Ansari, S. Böckle, and L. Rovati, “New optical scheme for a polarimetric-based glucose sensor,” J. Biomed. Opt. **9**(1), 103–115 (2004). [CrossRef] [PubMed]

#### 2.2.2 Milk dilution with a linear polarizer at the bottom or at the top

#### 2.2.3 Milk dilution with a linear polarizer inside the container

### 2.3 Method

14. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. **36**(10), 1942–1944 (2011). [CrossRef] [PubMed]

17. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. **36**(12), 2330–2332 (2011). [CrossRef] [PubMed]

14. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. **36**(10), 1942–1944 (2011). [CrossRef] [PubMed]

18. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus,” J. Opt. Soc. Am. **68**(12), 1756–1767 (1978). [CrossRef]

5. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**(5), 1106–1113 (1996). [CrossRef]

## 3. Results

### 3.1 Validation for commutative distributed effects

24. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express **14**(1), 190–202 (2006). [CrossRef] [PubMed]

25. J. S. Maier, S. A. Walker, S. Fantini, M. A. Franceschini, and E. Gratton, “Possible correlation between blood glucose concentration and the reduced scattering coefficient of tissues in the near infrared,” Opt. Lett. **19**(24), 2062–2064 (1994). [CrossRef] [PubMed]

26. J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, M. Essenpreis, G. Schmelzeisen-Redeker, and D. Böcker, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. **22**(3), 190–192 (1997). [CrossRef] [PubMed]

27. A. N. Bashkatov, E. A. Genina, Y. P. Sinichkin, N. A. Lakodina, V. I. Kochubey, and V. V. Tuchin, “Estimation of glucose diffusion coefficient in scleral tissue,” Proc. SPIE **4001**, 345–355 (2000). [CrossRef]

### 3.2 Comparison for non-commutative sequential effects

7. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. **29**(19), 2234–2236 (2004). [CrossRef] [PubMed]

8. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. **32**(6), 689–691 (2007). [CrossRef] [PubMed]

8. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. **32**(6), 689–691 (2007). [CrossRef] [PubMed]

### 3.3 Comparison for non-commutative distributed-like effects

28. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues,” Opt. Commun. **283**(6), 1200–1208 (2010). [CrossRef]

## 4. Conclusions

## Acknowledgments

*ScatteroMueller*contract.

## References and links

1. | D. Goldstein, |

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

3. | S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.) |

4. | S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE |

5. | S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

6. | J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.) |

7. | J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. |

8. | R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. |

9. | M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. Rapid Publ. |

10. | R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. |

11. | R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A |

12. | C. Fallet, A. Pierangelo, R. Ossikovski, and A. De Martino, “Experimental validation of the symmetric decomposition of Mueller matrices,” Opt. Express |

13. | N. Ortega-Quijano, F. Fanjul-Vélez, I. Salas-García, and J. L. Arce-Diego, “Comparative study of optical activity in chiral biological media by polar decomposition and differential Mueller matrices analysis,” Proc. SPIE |

14. | N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. |

15. | N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. |

16. | 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 |

17. | R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. |

18. | R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus,” J. Opt. Soc. Am. |

19. | B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. |

20. | E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. |

21. | M. D. Waterworth, B. J. Tarte, A. J. Joblin, T. van Doorn, and H. E. Niesler, “Optical transmission properties of homogenised milk used as a phantom material in visible wavelength imaging,” Australas. Phys. Eng. Sci. Med. |

22. | M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, |

23. | R. R. Ansari, S. Böckle, and L. Rovati, “New optical scheme for a polarimetric-based glucose sensor,” J. Biomed. Opt. |

24. | S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express |

25. | J. S. Maier, S. A. Walker, S. Fantini, M. A. Franceschini, and E. Gratton, “Possible correlation between blood glucose concentration and the reduced scattering coefficient of tissues in the near infrared,” Opt. Lett. |

26. | J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, M. Essenpreis, G. Schmelzeisen-Redeker, and D. Böcker, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. |

27. | A. N. Bashkatov, E. A. Genina, Y. P. Sinichkin, N. A. Lakodina, V. I. Kochubey, and V. V. Tuchin, “Estimation of glucose diffusion coefficient in scleral tissue,” Proc. SPIE |

28. | N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues,” Opt. Commun. |

**OCIS Codes**

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(260.2130) Physical optics : Ellipsometry and polarimetry

(260.5430) Physical optics : Polarization

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: October 14, 2011

Manuscript Accepted: November 17, 2011

Published: January 4, 2012

**Citation**

Noé Ortega-Quijano, Bicher Haj-Ibrahim, Enric García-Caurel, José Luis Arce-Diego, and Razvigor Ossikovski, "Experimental validation of Mueller matrix differential decomposition," Opt. Express **20**, 1151-1163 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1151

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### References

- D. Goldstein, Polarized Light (Marcel Dekker, 2003).
- R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them,” Phys. Status Solidi (A)205(4), 720–727 (2008). [CrossRef]
- S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.)75, 26–36 (1986).
- S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE1166, 177–185 (1989).
- S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A13(5), 1106–1113 (1996). [CrossRef]
- J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttg.)76, 67–71 (1987).
- J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett.29(19), 2234–2236 (2004). [CrossRef] [PubMed]
- R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett.32(6), 689–691 (2007). [CrossRef] [PubMed]
- M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. Rapid Publ.2, 070181–070187 (2007).
- R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt.41(10), 1903–1915 (1994). [CrossRef]
- R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A26(5), 1109–1118 (2009). [CrossRef] [PubMed]
- C. Fallet, A. Pierangelo, R. Ossikovski, and A. De Martino, “Experimental validation of the symmetric decomposition of Mueller matrices,” Opt. Express18(2), 831–842 (2010). [CrossRef] [PubMed]
- N. Ortega-Quijano, F. Fanjul-Vélez, I. Salas-García, and J. L. Arce-Diego, “Comparative study of optical activity in chiral biological media by polar decomposition and differential Mueller matrices analysis,” Proc. SPIE7906, 790612 (2011). [CrossRef]
- N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett.36(10), 1942–1944 (2011). [CrossRef] [PubMed]
- N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett.36(13), 2429–2431 (2011). [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. Express19(15), 14348–14353 (2011). [CrossRef] [PubMed]
- R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett.36(12), 2330–2332 (2011). [CrossRef] [PubMed]
- R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus,” J. Opt. Soc. Am.68(12), 1756–1767 (1978). [CrossRef]
- B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt.43(14), 2824–2832 (2004). [CrossRef] [PubMed]
- E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt.38(16), 3490–3502 (1999). [CrossRef] [PubMed]
- M. D. Waterworth, B. J. Tarte, A. J. Joblin, T. van Doorn, and H. E. Niesler, “Optical transmission properties of homogenised milk used as a phantom material in visible wavelength imaging,” Australas. Phys. Eng. Sci. Med.18(1), 39–44 (1995). [PubMed]
- M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).
- R. R. Ansari, S. Böckle, and L. Rovati, “New optical scheme for a polarimetric-based glucose sensor,” J. Biomed. Opt.9(1), 103–115 (2004). [CrossRef] [PubMed]
- S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express14(1), 190–202 (2006). [CrossRef] [PubMed]
- J. S. Maier, S. A. Walker, S. Fantini, M. A. Franceschini, and E. Gratton, “Possible correlation between blood glucose concentration and the reduced scattering coefficient of tissues in the near infrared,” Opt. Lett.19(24), 2062–2064 (1994). [CrossRef] [PubMed]
- J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, M. Essenpreis, G. Schmelzeisen-Redeker, and D. Böcker, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett.22(3), 190–192 (1997). [CrossRef] [PubMed]
- A. N. Bashkatov, E. A. Genina, Y. P. Sinichkin, N. A. Lakodina, V. I. Kochubey, and V. V. Tuchin, “Estimation of glucose diffusion coefficient in scleral tissue,” Proc. SPIE4001, 345–355 (2000). [CrossRef]
- N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues,” Opt. Commun.283(6), 1200–1208 (2010). [CrossRef]

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