## Optimum input states of polarization for Mueller matrix measurement in a system having finite polarization-dependent loss or gain

Optics Express, Vol. 17, Issue 25, pp. 23044-23057 (2009)

http://dx.doi.org/10.1364/OE.17.023044

Acrobat PDF (333 KB)

### Abstract

We present the theoretical and simulation results of the relationship between three input states of polarization (SOP) and the Mueller matrix measurement error in an optical system having birefringence and finite polarization-dependent loss or gain (PDL/G). By using the condition number as the criterion, it can be theoretically demonstrated that the three input SOPs should be equally-spaced on the Poincaré sphere and centered on the reversed PDL/G vector to achieve better measurement accuracy in a single test. Further, an upper bound of the mean of the Mueller matrix measurement error is derived when the measurement errors of output Stokes parameters independently and identically follow the ideal Gaussian distribution. This upper bound also shows that the statistically best Mueller matrix measurement accuracy can be obtained when the three input SOPs have the same relationship mentioned above. Simulation results confirm the validity of the theoretical findings.

© 2009 OSA

## 1. Introduction

^{3}E) seriously depends on the choice of the three input states of polarization (SOP) [1

1. H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express **17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

^{3}E is statistically achieved when the three input SOPs are coplanar with an angle of 120° between any two of them in Stokes space [1

1. H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express **17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

^{3}E than other input SOPs [1

1. H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express **17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

2. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. **142**(1-3), 119–125 (
1997). [CrossRef]

3. K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. **12**(4), 650–657 (
1994). [CrossRef]

2. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. **142**(1-3), 119–125 (
1997). [CrossRef]

5. B. Huttner and N. Gisin, “Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,” Opt. Lett. **22**(8), 504–506 (
1997). [CrossRef] [PubMed]

6. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express **14**(12), 5067–5072 (
2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067. [CrossRef] [PubMed]

7. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. **274**(1), 116–123 (
2007). [CrossRef]

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

^{3}E. By calculating the minimum of this CN, the relationship among the three optimum input SOPs, which can lead to a smaller M

^{3}E in a single test, is clearly presented. Secondly, in Sections 3 and 4, the statistical relationship between M

^{3}E and the three input SOPs is investigated under the same two preconditions mentioned above. Finally, some simulation results are used to verify the theoretical findings.

## 2. Optimization using CN as the criterion

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

^{3}E, which is depicted by the matrix norm

8. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. **34**(6), 1651–1655 (
1995). [CrossRef]

9. W. C. Waterhouse, “Do symmetric problems have symmetric solutions?” Am. Math. Mon. **90**(6), 378–387 (
1983). [CrossRef]

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

*D*is the value of PDL/G ;

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

10. H. Dong, J. Q. Zhou, M. Yan, P. Shum, L. Ma, Y. D. Gong, and C. Q. Wu, “Quasi-monochromatic fiber depolarizer and its application to polarization-dependent loss measurement,” Opt. Lett. **31**(7), 876–878 (
2006). [CrossRef] [PubMed]

7. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. **274**(1), 116–123 (
2007). [CrossRef]

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

11. H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. **19**(1), 27–29 (
2007). [CrossRef]

12. E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE **6120**, 61200Q (
2005). [CrossRef]

7. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. **274**(1), 116–123 (
2007). [CrossRef]

13. H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. **282**(18), 3671–3675 (
2009). [CrossRef]

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

^{3}E will dramatically increase when the value of PDL/G is up to tens of dB. Therefore, when the system under test has such a big PDL/G, its Mueller matrix cannot be accurately measured by using only three inputs in a single test.

^{3}E depends on not only the CN, but also on the noise realization. When many tests can be performed, the mean of M

^{3}E should be investigated. To carry out such an investigation, we must know the statistical properties of the Stokes parameter measurement errors in advance. In this paper, we assume that all Stokes parameter measurement errors independently and identically follow the Gaussian distribution

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

## 3. Statistical properties of Δ | M ˜ |

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

## 4. Upper bound of 〈 ‖ Δ M ˜ ‖ 〉

*D*has a finite value, from Eq. (14) of Ref [1

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

**17**(15), 13017–13030 (
2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed]

**F**, which are

^{3}E will dramatically increase when the value of PDL/G is up to tens of dB. Therefore, when the system under test has such a big PDL/G, the mean of the Mueller matrix also cannot be accurately measured.

## 5. Simulation results

## 6. Conclusion

^{3}E in a system having birefringence and a finite PDL/G. Firstly, by using the CN as the criterion, it has been demonstrated that the three optimum input SOPs should be equally-spaced and centred on the reversed PDL/G vector for achieving a smaller M

^{3}E in a single test. Secondly, the statistical relationship, which is expressed as an upper bound of the mean of M

^{3}E, has been derived when SOP measurement errors follow the same Gaussian distribution. This upper bound also tells us that the minimum M

^{3}E will be statistically achieved when the three input SOPs are equally-spaced and centred on the reversed PDL/G vector. Finally, the simulation results confirm the validity of the proposed conclusion.

14. M. Reimer and D. Yevick, “Least-squares analysis of the Mueller matrix,” Opt. Lett. **31**(16), 2399–2401 (
2006). [CrossRef] [PubMed]

- 1) The PDL/G vector can be measured in the first test using three “not-too-bad” input SOPs, for example, the three inputs we suggested in Ref [1
**17**(15), 13017–13030 ( 2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017. [CrossRef] [PubMed] - 2) The second test is carried out using three input SOPs optimized using the relative relationship shown in Section 4 and the PDL/G vector measured in the first test. Then a more accurate PDL/G vector can be obtained;
- 3) The third test is carried out based on the knowledge of the more accurate PDL/G to result in a further more accurate PDL/G vector;
- 4) The measurement is repeated in this iterative way. Then the final averaged measurement result will be statistically the best.

^{3}E and the three input SOPs will not be as same as the one we obtained in this paper. It will be polarimeter-dependent. Further analysis will be presented in other papers.

## Acknowledgements

## References and Links

1. | H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express |

2. | N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. |

3. | K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. |

4. | E. Lichtmann, “Performance degradation due to polarization dependent gain and loss in lightwave systems with optical amplifiers,” IEEE Photon. Technol. Lett. |

5. | B. Huttner and N. Gisin, “Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,” Opt. Lett. |

6. | H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express |

7. | H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. |

8. | A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. |

9. | W. C. Waterhouse, “Do symmetric problems have symmetric solutions?” Am. Math. Mon. |

10. | H. Dong, J. Q. Zhou, M. Yan, P. Shum, L. Ma, Y. D. Gong, and C. Q. Wu, “Quasi-monochromatic fiber depolarizer and its application to polarization-dependent loss measurement,” Opt. Lett. |

11. | H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. |

12. | E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE |

13. | H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. |

14. | M. Reimer and D. Yevick, “Least-squares analysis of the Mueller matrix,” Opt. Lett. |

**OCIS Codes**

(060.2300) Fiber optics and optical communications : Fiber measurements

(060.2310) Fiber optics and optical communications : Fiber optics

(260.3090) Physical optics : Infrared, far

(260.5430) Physical optics : Polarization

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 31, 2009

Revised Manuscript: November 9, 2009

Manuscript Accepted: November 9, 2009

Published: December 2, 2009

**Citation**

H. Dong, Y. D. Gong, Varghese Paulose, P. Shum, and Malini Olivo, "Optimum input states of polarization for Mueller matrix measurement in a system having finite polarization-dependent loss or gain," Opt. Express **17**, 23044-23057 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23044

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### References

- H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express 17(15), 13017–13030 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017 . [CrossRef] [PubMed]
- N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997). [CrossRef]
- K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994). [CrossRef]
- E. Lichtmann, “Performance degradation due to polarization dependent gain and loss in lightwave systems with optical amplifiers,” IEEE Photon. Technol. Lett. 5, 1969–1970 (1993).
- B. Huttner and N. Gisin, “Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,” Opt. Lett. 22(8), 504–506 (1997). [CrossRef] [PubMed]
- H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 . [CrossRef] [PubMed]
- H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007). [CrossRef]
- A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995). [CrossRef]
- W. C. Waterhouse, “Do symmetric problems have symmetric solutions?” Am. Math. Mon. 90(6), 378–387 (1983). [CrossRef]
- H. Dong, J. Q. Zhou, M. Yan, P. Shum, L. Ma, Y. D. Gong, and C. Q. Wu, “Quasi-monochromatic fiber depolarizer and its application to polarization-dependent loss measurement,” Opt. Lett. 31(7), 876–878 (2006). [CrossRef] [PubMed]
- H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007). [CrossRef]
- E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005). [CrossRef]
- H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009). [CrossRef]
- M. Reimer and D. Yevick, “Least-squares analysis of the Mueller matrix,” Opt. Lett. 31(16), 2399–2401 (2006). [CrossRef] [PubMed]

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