Abstract

The polarization analysis of light is typically carried out using modulation schemes. The light of an unknown polarization state is passed through a set of known modulation optics, and a detector is used to measure the total intensity passing the system. The modulation optics is modified several times, and, with the aid of several such measurements, the unknown polarization state of the light can be inferred. How to find the optimal demodulation process has been investigated in the past. However, since the modulation matrix has to be measured for a given instrument and the optical elements can present problems of repeatability, some uncertainty is present in the elements of the modulation matrix or covariances between these elements. We analyze in detail this issue, presenting analytical formulas for calculating the covariance matrix produced by the propagation of such uncertainties on the demodulation matrix, on the inferred Stokes parameters, and on the efficiency of the modulation process. We demonstrate that even if the covariance matrix of the modulation matrix is diagonal, the covariance matrix of the demodulation matrix is in general nondiagonal because matrix inversion is a nonlinear operation. This propagates through the demodulation process and induces correlations on the inferred Stokes parameters.

© 2008 Optical Society of America

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  1. C. R. Givens and A. Kostinski, “Physical significance of experimental Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
    [CrossRef]
  2. E. Landi Degl'Innocenti and J. C. del Toro Iniesta, “Physical significance of experimental Mueller matrices,” J. Opt. Soc. Am. A 15, 533-537 (1998).
    [CrossRef]
  3. H. P. Povel, H. Aebersold, and J. O. Stenflo, “Charge-coupled-device image sensor as a demodulator in a 2-D polarimeter with a piezoelastic modulator,” Appl. Opt. 29, 1186-1190 (1990).
  4. H. P. Povel, “Imaging Stokes polarimetry with modulators and charge coupled-device image sensors,” Opt. Eng. 34, 1870-1878 (1995).
  5. M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. III. Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).
  6. J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).
  7. V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.
  8. A. López Ariste, J. Rayrole, and M. Semel, “First results from THEMIS spectropolarimetric mode,” Astron. Astrophys. Suppl. Ser. 142, 137-148 (2000).
    [CrossRef]
  9. J. C. del Toro Iniesta and M. Collados, “Optimum modulation and demodulation matrices for solar polarimetry,” Appl. Opt. 39, 1637-1642 (2000).
  10. M. Collados, “High resolution spectropolarimetry and magnetography, in Third Advances in Solar Physics Euroconference: Magnetic Fields and Oscillations, Vol. 184 of Astronomical Society of the Pacific Conference Series, B. Schmieder, A. Hofmann, and J. Staude, eds. (Astronomical Society of the Pacific, 1999), p. 3.
  11. E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394-395 (1920).
  12. R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406-413 (1955).
  13. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804(2000).
    [CrossRef]
  14. Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. 46, 7354-7364 (2007).
    [CrossRef]
  15. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519-1528 (1978).
  16. J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
    [CrossRef]
  17. S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20, 1651-1657 (2003).
    [CrossRef]
  18. S.-M. F. Nee, “Errors of Mueller matrix measurements with a partially polarized light source,” Appl. Opt. 45, 6497-6506(2006).
    [CrossRef]
  19. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25, 1198-1200 (2000).
    [CrossRef]
  20. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise and minimization of systematic error,” Appl. Opt. 41, 619-630 (2002).
    [CrossRef]
  21. M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 41, 2488-2493 (2002).
    [CrossRef]
  22. D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 6, 693-700 (1990).
  23. G. J. Hahn and S. S. Shapiro, Statistical Models in Engineering (Wiley, 1967).
  24. M. Lefebvre, R. K. Keeler, R. Sobie, and J. White, “Propagation of errors for matrix inversion,” Nucl. Instrum. Methods Phys. Res. A 451, 520 (2000).
  25. E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).
  26. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1655 (1995).
  27. V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).
  28. D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

2007 (1)

2006 (2)

J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

S.-M. F. Nee, “Errors of Mueller matrix measurements with a partially polarized light source,” Appl. Opt. 45, 6497-6506(2006).
[CrossRef]

2003 (1)

2002 (2)

2000 (5)

1999 (1)

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

1998 (1)

1997 (1)

J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).

1995 (2)

H. P. Povel, “Imaging Stokes polarimetry with modulators and charge coupled-device image sensors,” Opt. Eng. 34, 1870-1878 (1995).

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1655 (1995).

1993 (2)

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. III. Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

C. R. Givens and A. Kostinski, “Physical significance of experimental Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

1990 (2)

H. P. Povel, H. Aebersold, and J. O. Stenflo, “Charge-coupled-device image sensor as a demodulator in a 2-D polarimeter with a piezoelastic modulator,” Appl. Opt. 29, 1186-1190 (1990).

D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 6, 693-700 (1990).

1978 (1)

1955 (1)

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406-413 (1955).

1920 (1)

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394-395 (1920).

Aebersold, H.

Ahmad, J. E.

Aïnouz, S.

J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Almeida, J. Sánchez

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1655 (1995).

Ariste, A. López

A. López Ariste, J. Rayrole, and M. Semel, “First results from THEMIS spectropolarimetric mode,” Astron. Astrophys. Suppl. Ser. 142, 137-148 (2000).
[CrossRef]

Bellot Rubio, L. R.

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

Carter, B. D.

J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).

Chambellan, C. W.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Chipman, R. A.

D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 6, 693-700 (1990).

Cobo, B. Ruiz

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

Collados, M.

J. C. del Toro Iniesta and M. Collados, “Optimum modulation and demodulation matrices for solar polarimetry,” Appl. Opt. 39, 1637-1642 (2000).

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

M. Collados, “High resolution spectropolarimetry and magnetography, in Third Advances in Solar Physics Euroconference: Magnetic Fields and Oscillations, Vol. 184 of Astronomical Society of the Pacific Conference Series, B. Schmieder, A. Hofmann, and J. Staude, eds. (Astronomical Society of the Pacific, 1999), p. 3.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Collier Cameron, A.

J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).

Cruz-Lopez, A.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Degl'Innocenti, E. Landi

del Toro Iniesta, J. C.

Dereniak, E. L.

D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804(2000).
[CrossRef]

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Descour, M. R.

D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804(2000).
[CrossRef]

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Diaz, J.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Donati, J.-F.

J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. III. Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

Dunn, R. B.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Elmore, D. F.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Feeney, O.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

García, J. P.

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Givens, C. R.

C. R. Givens and A. Kostinski, “Physical significance of experimental Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Goldstein, D. H.

D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 6, 693-700 (1990).

González, V.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Hahn, G. J.

G. J. Hahn and S. S. Shapiro, Statistical Models in Engineering (Wiley, 1967).

Hamilton, T.

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Hauge, P. S.

Hidalgo, I. Rodríguez

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

Hull, H. K.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Joven, E.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Keeler, R. K.

M. Lefebvre, R. K. Keeler, R. Sobie, and J. White, “Propagation of errors for matrix inversion,” Nucl. Instrum. Methods Phys. Res. A 451, 520 (2000).

Kemme, S. A.

D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804(2000).
[CrossRef]

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Kostinski, A.

C. R. Givens and A. Kostinski, “Physical significance of experimental Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Leach, T. W.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Lefebvre, M.

M. Lefebvre, R. K. Keeler, R. Sobie, and J. White, “Propagation of errors for matrix inversion,” Nucl. Instrum. Methods Phys. Res. A 451, 520 (2000).

Lites, B. W.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Locke, M. R.

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1655 (1995).

Manescau, A.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Moore, E. H.

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394-395 (1920).

Nee, S.-M. F.

Paez, E.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Penrose, R.

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406-413 (1955).

Phipps, G. S.

D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804(2000).
[CrossRef]

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Pillet, V. Martínez

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Povel, H. P.

H. P. Povel, “Imaging Stokes polarimetry with modulators and charge coupled-device image sensors,” Opt. Eng. 34, 1870-1878 (1995).

H. P. Povel, H. Aebersold, and J. O. Stenflo, “Charge-coupled-device image sensor as a demodulator in a 2-D polarimeter with a piezoelastic modulator,” Appl. Opt. 29, 1186-1190 (1990).

Rayrole, J.

A. López Ariste, J. Rayrole, and M. Semel, “First results from THEMIS spectropolarimetric mode,” Astron. Astrophys. Suppl. Ser. 142, 137-148 (2000).
[CrossRef]

Rees, D. E.

J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. III. Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

Sabatke, D. S.

D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804(2000).
[CrossRef]

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Sánchez, V.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Sass, D.

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Scharmer, G.

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Schuenke, J. A.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Semel, M.

A. López Ariste, J. Rayrole, and M. Semel, “First results from THEMIS spectropolarimetric mode,” Astron. Astrophys. Suppl. Ser. 142, 137-148 (2000).
[CrossRef]

J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. III. Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

Shapiro, S. S.

G. J. Hahn and S. S. Shapiro, Statistical Models in Engineering (Wiley, 1967).

Skumanich, A. P.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Smith, M. H.

Sobie, R.

M. Lefebvre, R. K. Keeler, R. Sobie, and J. White, “Propagation of errors for matrix inversion,” Nucl. Instrum. Methods Phys. Res. A 451, 520 (2000).

Soltau, D.

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

Stenflo, J. O.

Stoll, M. P.

J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Streander, K. V.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Sweatt, W. C.

D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804(2000).
[CrossRef]

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

Takakura, Y.

Tomczyk, S.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

Tyo, J. S.

White, J.

M. Lefebvre, R. K. Keeler, R. Sobie, and J. White, “Propagation of errors for matrix inversion,” Nucl. Instrum. Methods Phys. Res. A 451, 520 (2000).

Zallat, J.

J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Appl. Opt. (6)

Astron. Astrophys. (1)

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. III. Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

Astron. Astrophys. Suppl. Ser. (1)

A. López Ariste, J. Rayrole, and M. Semel, “First results from THEMIS spectropolarimetric mode,” Astron. Astrophys. Suppl. Ser. 142, 137-148 (2000).
[CrossRef]

Astron. Gess. Abstract Ser. (1)

V. Martínez Pillet, M. Collados, L. R. Bellot Rubio, I. Rodríguez Hidalgo, B. Ruiz Cobo, and D. Soltau, “The Tenerife Infrared Polarimeter,” Astron. Gess. Abstract Ser. 15, 141 (1999).

Bull. Am. Math. Soc. (1)

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394-395 (1920).

J. Mod. Opt. (1)

C. R. Givens and A. Kostinski, “Physical significance of experimental Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Mon. Not. R. Astron. Soc. (1)

J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron, “Spectropolarimetric observations of active stars,” Mon. Not. R. Astron. Soc. 291, 658-682 (1997).

Nucl. Instrum. Methods Phys. Res. A (1)

M. Lefebvre, R. K. Keeler, R. Sobie, and J. White, “Propagation of errors for matrix inversion,” Nucl. Instrum. Methods Phys. Res. A 451, 520 (2000).

Opt. Eng. (1)

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1655 (1995).

Opt. Lett. (2)

Proc. Cambridge Philos. Soc. (1)

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406-413 (1955).

Other (6)

V. Martínez Pillet, M. Collados, J. Sánchez Almeida, V. González, A. Cruz-Lopez, A. Manescau, E. Joven, E. Paez, J. Diaz, O. Feeney, V. Sánchez, G. Scharmer, and D. Soltau, “LPSP & TIP: full Stokes polarimeters for the Canary Islands Observatories,” in High Resolution Solar Physics: Theory, Observations, and Techniques, Vol. 183 of Astronomical Society of the Pacific Conference Series, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds. (Astronomical Society of the Pacific, 1999), p. 264.

M. Collados, “High resolution spectropolarimetry and magnetography, in Third Advances in Solar Physics Euroconference: Magnetic Fields and Oscillations, Vol. 184 of Astronomical Society of the Pacific Conference Series, B. Schmieder, A. Hofmann, and J. Staude, eds. (Astronomical Society of the Pacific, 1999), p. 3.

H. P. Povel, “Imaging Stokes polarimetry with modulators and charge coupled-device image sensors,” Opt. Eng. 34, 1870-1878 (1995).

E. L. Dereniak, D. S. Sabatke, M. R. Locke, M. R. Descour, W. C. Sweatt, J. P. García, D. Sass, T. Hamilton, S. A. Kemme, and G. S. Phipps, “Design and optimization of a complete Stokes polarimeter for the MWIR,” (Office of Scientific & Technical Information, 2000).

G. J. Hahn and S. S. Shapiro, Statistical Models in Engineering (Wiley, 1967).

D. F. Elmore, B. W. Lites, S. Tomczyk, A. P. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, and H. K. Hull, “The advanced Stokes polarimeter: a new instrument for solar magnetic field research,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 22-24 (1992).

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Figures (3)

Fig. 1
Fig. 1

Distribution of the D 00 matrix element of the TIP demodulation matrix when the modulation matrix elements are known with the uncertainty indicated in each panel. Note that the distribution is centered at the value obtained from applying Eq. (7), but a dispersion is present around this value.

Fig. 2
Fig. 2

Distribution of the inferred Stokes parameters when no uncertainty is assumed in the measurement of the I out vector but only in the measurement of the TIP modulation matrix, whose uncertainty is σ = 2.5 × 10 3 . The input Stokes vector was S in = ( 1 , 10 3 , 10 3 , 10 3 ) . The distributions are close to Gaussian centered on the original values. The standard deviation of the inferred value is given in each panel. Note that the errors are roughly similar or slightly smaller than the uncertainty in the modulation matrix. Note also that correlations between the elements of the inferred vector are also expected and are produced by the nondiagonal covariance matrix of the demodulation matrix.

Fig. 3
Fig. 3

Left panel: two-dimensional distribution obtained from the Monte Carlo simulation for S 0 in and S 3 in using TIP’s modulation scheme, i.e., the inferred Stokes I and V, respectively. The plot shows the presence of a small correlation between the two results, a direct consequence of the appearance of correlations during the inversion process of the modulation matrix. Right panel: the same result but for the ASP case and for S 0 in and S 1 in , i.e., the inferred Stokes I and Q. Note that the correlation between both is even larger, as indicated by the respective covariance matrices given by Eqs. (35, 39).

Equations (43)

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S = ( I , Q , U , V ) T ,
S out = M S in .
I j out = [ M j ] 00 S 0 in + [ M j ] 01 S 1 in + [ M j ] 02 S 2 in + [ M j ] 03 S 3 in ,
I out = O S in ,
S in = D I out ,
ϵ i = ( n j = 1 n D i j 2 ) 1 / 2 .
D = ( O T O ) 1 O T .
D α β = D α β ( O i j ) .
cov ( D α β , D a b ) = i j k l D α β O i j D a b O k l cov ( O i j , O k l ) .
cov ( D α β , D a b ) = i j D α β O i j D a b O i j σ ( O ) i j 2 .
D α β O i j = D α i D j β .
cov ( D α β , D a b ) = i j k l D α i D j β D a k D l b cov ( O i j , O k l ) .
cov ( D α β , D a b ) = i j D α i D j β D a i D j b σ ( O ) i j 2 .
D α β O i j = A α j 1 δ β i + k O β k A α k 1 O i j .
A α k 1 O i j = m n A α k 1 A m n A m n O i j .
A m n O i j = O i n δ m j + O j m δ n j .
A α β 1 O i j = - A α i 1 A j β 1 .
D α β O i j = A α j 1 δ β i A α j 1 n D n β O i n D α i D j β .
n D n β O i n = n O i n D n β = δ i β ,
cov ( D α β , D a b ) = i j k l cov ( O i j , O k l ) × [ A α j 1 δ β i A α j 1 n D n β O i n D α i D j β ] × [ A a k - 1 δ b k A a k - 1 m D m b O k m D a k D l b ] .
cov ( S i in , S j in ) = α β a b I i in D α β I j in D a b cov ( D α β , D a b ) + k l S i in I k out S j in I l out cov ( I k out , I l out ) .
cov ( S i in , S j in ) = α β I α out I β out cov ( D i α , D j β ) + k l D i k D j l cov ( I k out , I l out ) .
cov ( ϵ α , ϵ β ) = i j k l ϵ α O i j ϵ β O k l cov ( O i j , O k l ) .
cov ( ϵ α , ϵ β ) = i j ϵ α O i j ϵ β O i j σ ( O ) i j 2 .
ϵ α O i j = p q ϵ α D p q D p q O i j ,
ϵ α D p q = D α q δ p α n 1 / 2 ( l D α l 2 ) 3 / 2 .
ϵ α O i j = n 1 / 2 ( l D α l 2 ) 3 / 2 q D α q D α q O i j .
σ ( O ) i j 2 = σ 2 .
O ^ i = O + γ i σ ( O ) ,
cov ( D i j , D k l ) = E [ D i j D k l ] E [ D i j ] E [ D k l ] ,
O TIP = [ 1 0.47 0.68 0.48 1 0.91 0.19 0.13 1 0.11 0.57 0.72 1 0.68 0.27 0.58 ] .
D TIP = [ 0.19 0.31 0.18 0.31 0.31 0.66 0.03 0.37 0.72 0.24 0.67 0.29 0.36 0.34 0.60 0.61 ] .
σ ( D TIP ) = [ 1.14 1.08 1.19 1.07 1.81 1.71 1.88 1.70 2.34 2.22 2.43 2.20 2.19 2.07 2.27 2.06 ] × 10 3 .
ϵ TIP = ( 0.969 , 0.612 , 0.473 , 0.506 ) , σ ( ϵ TIP ) = ( 1.38 , 1.26 , 1.25 , 1.27 ) × 10 3 ,
cov ( S in ) = [ 1.66 0.21 0.01 0.76 0.21 4.17 0.14 0.59 0.01 0.14 6.99 0.32 0.76 0.59 0.32 6.09 ] × 10 6 .
V = [ 0.986 0.042 0.012 0.160 0.002 0.963 0.018 0.268 0.057 0.125 0.913 0.385 0.156 0.234 0.408 0.868 ] .
O ASP = [ 1 0.77 0.41 0.36 1 0.06 0.41 0.86 1 0.06 0.41 0.86 1 0.77 0.41 0.36 1 0.77 0.41 0.36 1 0.06 0.41 0.86 1 0.06 0.41 0.86 1 0.77 0.41 0.36 ] .
ϵ ASP = ( 0.760 , 0.415 , 0.410 , 0.659 ) , σ ( ϵ ASP ) = ( 1.37 , 0.94 , 0.88 , 0.88 ) × 10 3 ,
cov ( S in ) = [ 1.35 1.61 0 0 1.61 4.54 0 0 0 0 4.65 0 0 0 0 1.80 ] × 10 6 .
V = [ 0.92 0.39 0 0 0.39 0.92 0 0 0 0 1 0 0 0 0 1 ] .
O IDEAL = [ 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 ] .
ϵ IDEAL = ( 1 , 3 , 3 , 3 ) , σ ( ϵ IDEAL ) = ( 1.02 , 1.02 , 1.02 , 1.02 ) × 10 3 ,
cov ( S in ) = [ 1.04 0 0 0 0 3.13 0 0 0 0 3.13 0 0 0 0 3.13 ] × 10 6 .

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