## Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain

Optics Express, Vol. 17, Issue 15, pp. 13017-13030 (2009)

http://dx.doi.org/10.1364/OE.17.013017

Acrobat PDF (304 KB)

### Abstract

For the measurement of Mueller matrix in an optical system with birefringence and small polarization-dependent loss or gain (PDL/G), we theoretically derive the statistical relationship between the Mueller matrix measurement error and three input states of polarization (SOP). Based on this theoretical relation and simulation results, it can be concluded that the three input SOPs, that are coplanar with an angle of 120° between any two of them in Stokes space, can be considered as a substitute for the best input SOPs which can statistically lead to the minimum measurement error. This conclusion is valid when the PDL/G of the optical system under test is less than 0.35dB.

© 2009 OSA

## 1. Introduction

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

2. J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE **5268**, 19–26 (2004). [CrossRef]

**M**, or equivalently, a

**J**. Polarization properties of an optical system, such as PDL/G vector, polarization mode dispersion (PMD) vector and so on, can be extracted from

**M**or

**J**using appropriate algorithms [3

3. R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. **16**(7), 1285–1294 (1998). [CrossRef]

6. B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. **4**(9), 1066–1069 (1992). [CrossRef]

7. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. **31**(31), 6676–6683 (1992). [CrossRef] [PubMed]

8. S.-M. F. Nee, “Depolarization and principal mueller matrix measured by null ellipsometry,” Appl. Opt. **40**(28), 4933–4939 (2001). [CrossRef]

9. B. J. Howell, “Measurement of the polarization effect of an instrument using partially polarized light,” Appl. Opt. **18**(6), 809–812 (1979). [CrossRef] [PubMed]

10. D. B. Chenault, R. A. Chipman, K. M. Johnson, and D. Doroski, “Infrared linear diattenuation and birefringence spectra of ferroelectric liquid crystals,” Opt. Lett. **17**(6), 447–449 (1992). [CrossRef] [PubMed]

4. 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]

12. R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. **37**(2), 110–112 (1947). [CrossRef]

13. 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]

12. R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. **37**(2), 110–112 (1947). [CrossRef]

13. 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, 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]

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

17. Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. **46**(30), 7354–7364 (2007). [CrossRef] [PubMed]

18. S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A **20**(8), 1651–1657 (2003). [CrossRef]

19. D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A **7**(4), 693–700 (1990). [CrossRef]

20. H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. **47**(6), 065007 (2008). [CrossRef]

## 2. Some properties of Mueller-Jones matrix

21. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. **53**(3), 317–323 (1963). [CrossRef]

22. R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. **38**(3), 159–161 (1981). [CrossRef]

**M**. Based on this definition and the property of Lorentz transformation [22

22. R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. **38**(3), 159–161 (1981). [CrossRef]

**I**stands for the identity matrix.

23. 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]

23. 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]

23. 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]

*ϕ*denotes the rotation angle in Stokes space;

**13**(5), 1106–1113 (1996). [CrossRef]

*α*is the angle between

**274**(1), 116–123 (2007). [CrossRef]

## 3. Measurement approach and measurement error

**274**(1), 116–123 (2007). [CrossRef]

**274**(1), 116–123 (2007). [CrossRef]

**F**. Equations (12) and (14) are the starting point of the theoretical analysis.

*α*,

*β*and

*γ*are the angles between

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

*α*,

*β*and

*γ*, but also on the relative relationship between three input SOPs and the PDL/G vector, namely

*α*,

*β*and

*γ*if

## 5. Upper limit of 〈 ‖ Δ M ˜ ‖ 〉

**I**. Then, the complex Mueller matrix

**13**(5), 1106–1113 (1996). [CrossRef]

**F**, which are

*α*,

*β*and

*γ*. From the calculation, we can find that

## 6. Simulation results

*D*, show that the theoretical upper limit (ULimit) is valid when

*D*is less than 0.35 dB,

*D*, have been shown in Fig. 6(a). However, to achieve these minimums, the PDL/G vectors must be known before the measurement. If the PDL/G vector is unknown,

## 7. Conclusion

## Acknowledgements

## References and Links

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

2. | J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE |

3. | R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. |

4. | 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 |

5. | B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission though optical devices,” IEEE Photon. Technol. Lett. |

6. | B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. |

7. | D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. |

8. | S.-M. F. Nee, “Depolarization and principal mueller matrix measured by null ellipsometry,” Appl. Opt. |

9. | B. J. Howell, “Measurement of the polarization effect of an instrument using partially polarized light,” Appl. Opt. |

10. | D. B. Chenault, R. A. Chipman, K. M. Johnson, and D. Doroski, “Infrared linear diattenuation and birefringence spectra of ferroelectric liquid crystals,” Opt. Lett. |

11. | H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted). |

12. | R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. |

13. | 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. |

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

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

16. | J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. |

17. | Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. |

18. | S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A |

19. | D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A |

20. | H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. |

21. | R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. |

22. | R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. |

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

**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:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 7, 2009

Revised Manuscript: June 21, 2009

Manuscript Accepted: June 23, 2009

Published: July 15, 2009

**Citation**

H. Dong, Y. D. Gong, Varghese Paulose, P. Shum, and Malini 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**, 13017-13030 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017

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### References

- N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997). [CrossRef]
- J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004). [CrossRef]
- R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998). [CrossRef]
- 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]
- B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission though optical devices,” IEEE Photon. Technol. Lett. 4(5), 451–454 (1992). [CrossRef]
- B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4(9), 1066–1069 (1992). [CrossRef]
- D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992). [CrossRef] [PubMed]
- S.-M. F. Nee, “Depolarization and principal mueller matrix measured by null ellipsometry,” Appl. Opt. 40(28), 4933–4939 (2001). [CrossRef]
- B. J. Howell, “Measurement of the polarization effect of an instrument using partially polarized light,” Appl. Opt. 18(6), 809–812 (1979). [CrossRef] [PubMed]
- D. B. Chenault, R. A. Chipman, K. M. Johnson, and D. Doroski, “Infrared linear diattenuation and birefringence spectra of ferroelectric liquid crystals,” Opt. Lett. 17(6), 447–449 (1992). [CrossRef] [PubMed]
- H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).
- R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. 37(2), 110–112 (1947). [CrossRef]
- 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]
- A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34(6), 1656–1658 (1995). [CrossRef]
- J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002). [CrossRef] [PubMed]
- Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. 46(30), 7354–7364 (2007). [CrossRef] [PubMed]
- S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20(8), 1651–1657 (2003). [CrossRef]
- D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7(4), 693–700 (1990). [CrossRef]
- H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008). [CrossRef]
- R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53(3), 317–323 (1963). [CrossRef]
- R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981). [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]

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