## Mueller matrix differential decomposition for direction reversal: application to samples measured in reflection and backscattering |

Optics Express, Vol. 19, Issue 15, pp. 14348-14353 (2011)

http://dx.doi.org/10.1364/OE.19.014348

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### Abstract

Mueller matrix differential decomposition is a novel method for analyzing the polarimetric properties of optical samples. It is performed through an eigenanalysis of the Mueller matrix and the subsequent decomposition of the corresponding differential Mueller matrix into the complete set of 16 differential matrices which characterize depolarizing anisotropic media. The method has been proposed so far only for measurements in transmission configuration. In this work the method is extended to the backward direction. The modifications of the differential matrices according to the reference system are discussed. The method is successfully applied to Mueller matrices measured in reflection and backscattering.

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## 1. Introduction

1. 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. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. **36** (in press). [PubMed]

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

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

4. 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, 790612-6 (2011). [CrossRef]

## 2. Mueller Matrix Differential Decomposition for Direction Reversal

**M**bywhere the non-zero diagonal elements of

**m**according to the general expression given in Eq. (5) enables to obtain the 16 parameters associated with each optical effect [3

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

**36**(10), 1942–1944 (2011). [CrossRef] [PubMed]

*z*, i.e. the optical path undergone by the measured photons. The determination of

*z*is not readily achievable in many applications. In general, for those situations in which

*z*is unknown, the optical path weighted differential matrix

6. N. C. Pistoni, “Simplified approach to the Jones calculus in retracing optical circuits,” Appl. Opt. **34**(34), 7870–7876 (1995). [CrossRef] [PubMed]

*α*is measured counterclockwise from the

*x*axis. For the specific example considered, the observed azimuth angle is positive when the measurement is performed in transmission, but it remarkably takes the opposite sign when measuring in the backward direction as a result of the reference and sign convention (Fig. 1). Therefore, it is self-evident that the measurement configuration has to be taken into account for the correct analysis of the measured polarimetric properties of the sample. The introduction of these considerations into the basic differential Mueller matrices entails a change of sign of the basic types of optical behavior for the linear ± 45° direction. This result is analogous to the conclusion obtained for differential Jones matrices [6

6. N. C. Pistoni, “Simplified approach to the Jones calculus in retracing optical circuits,” Appl. Opt. **34**(34), 7870–7876 (1995). [CrossRef] [PubMed]

## 3. Application to experimental Mueller matrices

**36**(10), 1942–1944 (2011). [CrossRef] [PubMed]

7. F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. **282**(5), 692–704 (2009). [CrossRef]

8. 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]

9. 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]

7. F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. **282**(5), 692–704 (2009). [CrossRef]

_{2}. The measurement was also performed in reflection [10

10. M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. **25**(20), 3616–3622 (1986). [CrossRef] [PubMed]

7. F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. **282**(5), 692–704 (2009). [CrossRef]

11. 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]

11. 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]

## 4. Conclusion

## Acknowledgments

## References and links

1. | 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. |

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

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

4. | 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 |

5. | D. S. Kliger, J. W. Lewis, and C. E. Randall, |

6. | N. C. Pistoni, “Simplified approach to the Jones calculus in retracing optical circuits,” Appl. Opt. |

7. | F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. |

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

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

10. | M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. |

11. | 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 |

**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: May 3, 2011

Revised Manuscript: May 31, 2011

Manuscript Accepted: May 31, 2011

Published: July 12, 2011

**Citation**

Noé Ortega-Quijano and José Luis Arce-Diego, "Mueller matrix differential decomposition for direction reversal: application to samples measured in reflection and backscattering," Opt. Express **19**, 14348-14353 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14348

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### References

- 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]
- N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36 (in press). [PubMed]
- N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36(10), 1942–1944 (2011). [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. SPIE 7906, 790612, 790612-6 (2011). [CrossRef]
- D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
- N. C. Pistoni, “Simplified approach to the Jones calculus in retracing optical circuits,” Appl. Opt. 34(34), 7870–7876 (1995). [CrossRef] [PubMed]
- F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009). [CrossRef]
- 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]
- 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]
- M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25(20), 3616–3622 (1986). [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. Express 14(1), 190–202 (2006). [CrossRef] [PubMed]

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