Abstract

Rotatable retarder fixed polarizer (RRFP) Stokes polarimeters, which employ uniformly spaced angles over 180° or 360°, are most commonly used to detect the state of polarization (SOP) of an electromagnetic (EM) wave. The misalignment error of the retarder is one of the major error sources. We suppose that the misalignment errors of the retarder obey a uniform normal distribution and are independent of each other. Then, we derive analytically the covariance matrices of the measurement errors. Based on the covariance matrices derived, we can conclude that 1) the measurement errors are independent of the incident intensity s0, but seriously depend on the Stokes parameters (s1, s2, s3) and the retardance of the retarder δ; 2) for any mean incident SOP, the optimal initial angle and retardance to minimize the measurement error both can be achieved; 3) when N = 5, 10, 12, the initial orienting angle could be used as an added degree of freedom to strengthen the immunity of RRFP Stokes polarimeters to the misalignment error. Finally, a series of simulations are performed to verify these theoretical results.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
  5. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. H. Dong, P. Shum, Y. D. Gong, and Q. Z. Sun, “Measurement errors induced by retardance deviation in a rotatable retarder fixed polarizer Stokes polarimeter,” Opt. Eng. 51(3), 033001 (2012).
    [Crossref]
  13. 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]
  14. D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. L. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeter optimization,” in Proceedings of Polarization: Measurement, Analysis, and Remote Sensing III, D. B. Chenault, eds., Proc. SPIE 4133 (2000).

2013 (1)

H. Dong, M. Tang, and Y. D. Gong, “Noise properties of uniformly-rotating RRFP Stokes polarimeters,” Opt. Express 21(8), 9674–9690 (2013).
[Crossref] [PubMed]

2012 (1)

H. Dong, P. Shum, Y. D. Gong, and Q. Z. Sun, “Measurement errors induced by retardance deviation in a rotatable retarder fixed polarizer Stokes polarimeter,” Opt. Eng. 51(3), 033001 (2012).
[Crossref]

2011 (1)

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

2009 (1)

F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. 34(5), 647–649 (2009).
[Crossref] [PubMed]

2006 (2)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).
[Crossref] [PubMed]

J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45(22), 5497–5503 (2006).
[Crossref] [PubMed]

2004 (1)

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

2002 (2)

J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minization of systematic error,” Appl. Opt. 41(4), 619–630 (2002).
[PubMed]

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]

2000 (1)

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(11), 802–804 (2000).
[Crossref] [PubMed]

1992 (1)

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

1981 (1)

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Aebersold, F.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Athay, R. G.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Beckers, J. M.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Brandt, J. C.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Bruner, E. C.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Chapman, R. D.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Chenault, D. B.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).
[Crossref] [PubMed]

Cheng, C. C.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

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(11), 802–804 (2000).
[Crossref] [PubMed]

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(11), 802–804 (2000).
[Crossref] [PubMed]

Dong, H.

H. Dong, M. Tang, and Y. D. Gong, “Noise properties of uniformly-rotating RRFP Stokes polarimeters,” Opt. Express 21(8), 9674–9690 (2013).
[Crossref] [PubMed]

H. Dong, P. Shum, Y. D. Gong, and Q. Z. Sun, “Measurement errors induced by retardance deviation in a rotatable retarder fixed polarizer Stokes polarimeter,” Opt. Eng. 51(3), 033001 (2012).
[Crossref]

Egger, U.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Feller, A.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Gandorfer, A. M.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Ghosh, N.

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

Gisler, D.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Goldstein, D. H.

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

Goldstein, D. L.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).
[Crossref] [PubMed]

Gong, Y. D.

H. Dong, M. Tang, and Y. D. Gong, “Noise properties of uniformly-rotating RRFP Stokes polarimeters,” Opt. Express 21(8), 9674–9690 (2013).
[Crossref] [PubMed]

H. Dong, P. Shum, Y. D. Gong, and Q. Z. Sun, “Measurement errors induced by retardance deviation in a rotatable retarder fixed polarizer Stokes polarimeter,” Opt. Eng. 51(3), 033001 (2012).
[Crossref]

Goudail, F.

F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. 34(5), 647–649 (2009).
[Crossref] [PubMed]

Gurman, J. B.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Hagenbuch, S.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Henze, W.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Hyder, C. L.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

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(11), 802–804 (2000).
[Crossref] [PubMed]

Kenney, P. J.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Michalitsianos, A. G.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

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(11), 802–804 (2000).
[Crossref] [PubMed]

Povel, H. P.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Rehse, R. A.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

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(11), 802–804 (2000).
[Crossref] [PubMed]

Schoolman, S. A.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Shaw, J. A.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).
[Crossref] [PubMed]

Shine, R. A.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Shum, P.

H. Dong, P. Shum, Y. D. Gong, and Q. Z. Sun, “Measurement errors induced by retardance deviation in a rotatable retarder fixed polarizer Stokes polarimeter,” Opt. Eng. 51(3), 033001 (2012).
[Crossref]

Steiner, P.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Stenflo, J. O.

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Sun, Q. Z.

H. Dong, P. Shum, Y. D. Gong, and Q. Z. Sun, “Measurement errors induced by retardance deviation in a rotatable retarder fixed polarizer Stokes polarimeter,” Opt. Eng. 51(3), 033001 (2012).
[Crossref]

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(11), 802–804 (2000).
[Crossref] [PubMed]

Tandberg-Hanssen, E.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

Tang, M.

H. Dong, M. Tang, and Y. D. Gong, “Noise properties of uniformly-rotating RRFP Stokes polarimeters,” Opt. Express 21(8), 9674–9690 (2013).
[Crossref] [PubMed]

Tyo, J. S.

J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45(22), 5497–5503 (2006).
[Crossref] [PubMed]

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).
[Crossref] [PubMed]

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]

J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minization of systematic error,” Appl. Opt. 41(4), 619–630 (2002).
[PubMed]

Vitkin, I. A.

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

Wei, H.

J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45(22), 5497–5503 (2006).
[Crossref] [PubMed]

Woodgate, B. E.

E. Tandberg-Hanssen, C. C. Cheng, B. E. Woodgate, J. C. Brandt, R. D. Chapman, P. J. Kenney, A. G. Michalitsianos, R. A. Shine, R. G. Athay, J. M. Beckers, E. C. Bruner, R. A. Rehse, S. A. Schoolman, J. B. Gurman, C. L. Hyder, and W. Henze, “Solar maximum mission experiment: Ultraviolet spectroscopy and polarimetry on the solar maximum mission,” Adv. Space Res. 1(13), 275–283 (1981).
[Crossref]

A&A. (1)

A. M. Gandorfer, H. P. Povel, P. Steiner, F. Aebersold, U. Egger, A. Feller, D. Gisler, S. Hagenbuch, and J. O. Stenflo, “Solar polarimetry in the near UV with the Zurich Imaging Polarimeter ZIMPOL II,” A&A. 422(2), 703–708 (2004).
[Crossref]

Adv. Space Res. (1)

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Figures (7)

Fig. 1
Fig. 1 The schematic structure of a RRFP Stokes polarimeter.
Fig. 2
Fig. 2 The relationships between C1, C2, C3, WS1, WS2 and the retardance δ for the case of employing N (N = 9, 11, 13, 15, 17, 18, 19…∞) uniformly spaced angles over 360°.
Fig. 3
Fig. 3 The relationship between WS and the initial orientation angle θ1 when N = 5 or 10, 7 or 14, 12, 16.
Fig. 4
Fig. 4 The coefficients and WSs for N = 5 or 10, 7 or 14, 12, 16 at the respective optimal initial angle when the mean incident SOP is ( 1,1/ 2 ,1/ 3 ,1/ 6 ) T .
Fig. 5
Fig. 5 The comparison of WSs for N (N = 10, 12, 14, 16) uniformly spaced angles over between180 °and 360 °.
Fig. 6
Fig. 6 Simulation and theoretical results of i 2 ,i=0,1,2,3 at the SOP ( 1,1/ 3 ,1/ 3 ,1/ 3 ) T and the initial angle θ 1 = 9 with σ = 0.05. The real line “-” represents the simulation result while the star “*” denotes theoretical result.
Fig. 7
Fig. 7 Simulation and theoretical results of i 2 ,i=0,1,2,3 at the SOP ( 1,1/ 2 ,1/ 3 ,1/ 6 ) T and the initial angle θ 1 = 13 with σ = 0.02. The real line “-” represents the simulation result while the star “*” denotes theoretical result.

Tables (1)

Tables Icon

Table 1 Optimal Initial Angle and Retardance for the Incident SOP ( 1,1/ 2 ,1/ 3 ,1/ 6 ) T

Equations (26)

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I =W S
W= 1 2 ( 1 cos 2 2 θ 1 +cosδ sin 2 2 θ 1 sin 2 ( δ/2 )sin4 θ 1 sinδsin2 θ 1 1 cos 2 2 θ 2 +cosδ sin 2 2 θ 2 sin 2 ( δ/2 )sin4 θ 2 sinδsin2 θ 2 1 cos 2 2 θ N +cosδ sin 2 2 θ N sin 2 ( δ/2 )sin4 θ N sinδsin2 θ N )
Δ I =( W ' W ) S =ΔW S
Δ W ij =W ' ij W ij = ξ i [ W ij ( θ i + ξ i ) W ij ( θ i ) ξ i ] ξ i W ij (θ) θ | θ i
ΔW=( 0 sin4 θ 1 ( cosδ1 ) ξ 1 cos4 θ 1 ( cosδ1 ) ξ 1 cos2 θ 1 sinδ ξ 1 0 sin4 θ 2 ( cosδ1 ) ξ 2 cos4 θ 2 ( cosδ1 ) ξ 2 cos2 θ 2 sinδ ξ 2 0 sin4 θ N ( cosδ1 ) ξ N cos4 θ N ( cosδ1 ) ξ N cos2 θ N sinδ ξ N )
Δ I =Q ξ
Q ij ={ s 1 sin4 θ i (cosδ1) s 2 cos4 θ i (cosδ1) s 3 cos2 θ i sinδ,i=j; 0,ij.
= W + Δ I = ( W T W ) 1 W T Q ξ
W + = ( W T W ) -1 W T
Γ = T =( 0 2 0 1 0 2 0 3 1 0 1 2 1 2 1 2 2 0 2 1 2 2 2 3 3 0 3 1 3 2 3 2 )= ( W T W ) 1 W T Q ξ ξ T Q T W ( W T W ) 1
Γ = σ 2 ( W T W ) 1 Λ ( W T W ) 1 = σ 2 Γ
Λ= W T Q ( W T Q ) T
Γ= ( W T W ) 1 Λ ( W T W ) 1
( W T W ) 1 = 4 N ( 3+2cosδ+3 cos 2 δ ( 1cosδ ) 2 4( 1+cosδ ) ( 1cosδ ) 2 0 0 4( 1+cosδ ) ( 1cosδ ) 2 8 ( 1cosδ ) 2 0 0 0 0 8 ( 1cosδ ) 2 0 0 0 0 2 sin 2 δ )
{ Λ 11 = N 8 [ s 1 2 ( cosδ1 ) 2 + s 2 2 ( cosδ1 ) 2 + s 3 2 sin 2 δ ] Λ 12 = Λ 21 = N 16 s 1 2 sin 2 δ( 1cosδ )+ N 16 s 2 2 sin 2 δ( 1cosδ ) + N 32 s 3 2 ( 3+cosδ ) sin 2 δ Λ 13 = Λ 31 = Λ 34 = Λ 43 =0 Λ 14 = Λ 41 = N 8 s 1 s 3 sin 2 δ( cosδ1 ) Λ 22 = N 128 s 1 2 ( cosδ1 ) 2 ( 5+6cosδ+5 cos 2 δ )+ N 128 s 2 2 ( cosδ1 ) 2 ( 7+2cosδ+7 cos 2 δ ) + N 64 s 3 2 sin 2 δ( 5+2cosδ+ cos 2 δ ) Λ 23 = Λ 32 = N 32 s 1 s 2 ( cosδ1 ) 3 sin 2 δ 2 Λ 24 = Λ 42 = N 16 s 1 s 2 sin 4 δ Λ 33 = 3N 32 s 1 2 ( cosδ1 ) 2 sin 4 δ 2 + N 32 s 2 2 ( cosδ1 ) 2 sin 4 δ 2 + N 16 s 3 2 sin 2 δ sin 4 δ 2 Λ 44 = N 16 s 1 2 ( cosδ1 ) 2 sin 2 δ+ N 16 s 2 2 ( cosδ1 ) 2 sin 2 δ+ N 32 s 3 2 sin 4 δ
{ Γ 11 = 4 N s 1 2 ( 1+ cos 2 δ )+ 8 N s 2 2 ( 1+cosδ+ cos 2 δ ) 2 N s 3 2 ( 5 cos 3 δ+7 cos 2 δ+3cosδ+1 ) ( cosδ1 ) Γ 22 = 8 N s 1 2 + 24 N s 2 2 + 16 N s 3 2 cot 2 δ 2 Γ 33 = 24 N s 1 2 + 8 N s 2 2 + 16 N s 3 2 cot 2 δ 2 Γ 44 = 4 N s 1 2 tan 2 δ 2 + 4 N s 2 2 tan 2 δ 2 + 2 N s 3 2
i=0 3 i 2 =Tr( Γ )= 4 σ 2 N ( C 1 s 1 2 + C 2 s 2 2 + C 3 s 3 2 + C 4 s 1 s 2 + C 5 s 2 s 3 + C 6 s 1 s 3 )
{ C 1 = cos 4 δ+7 cos 2 +2cosδ10 cos 2 δ1 , C 2 = 2 cos 4 δ+2 cos 3 δ+7 cos 2 δ11 cos 2 δ1 , C 3 = 5 cos 4 δ+12 cos 3 δ+25 cos 2 δ+36cosδ+18 2( cos 2 δ1 ) , C 4 = C 5 = C 6 =0
PD= s 1 2 + s 2 2 + s 3 2 s 0
WS= s 1 ¯ 2 C 1 + s 2 ¯ 2 C 2 + s 3 ¯ 2 C 3 + s 1 ¯ s 2 ¯ C 4 + s 2 ¯ s 3 ¯ C 5 + s 1 ¯ s 3 ¯ C 6
W S 1 = C 1 2 + C 2 3 + C 3 6
W S 2 = 1 3 ( C 1 + C 2 + C 3 )
{ Γ 11 = 16 N s 1 2 ( 1+ cos 2 δ )+ 32 N s 2 2 ( cos 2 δ+cosδ+1 ) 8 N s 3 2 ( 15 16 1cosδ +12cosδ+5 cos 2 δ ) 16 N s 1 s 3 sin10 θ 1 sinδ( 5 4 1cosδ +3cosδ )+ 16 N s 2 s 3 cos10 θ 1 sinδ( 5 4 1cosδ +3cosδ ) Γ 22 = 32 N s 1 2 + 96 N s 2 2 + 64 N s 3 2 1+cosδ 1cosδ 64 N s 2 s 3 cos10 θ 1 sinδ 1cosδ + 64 N s 1 s 3 sin10 θ 1 sinδ 1cosδ Γ 33 = 96 N s 1 2 + 32 N s 2 2 + 64 N s 3 2 cot 2 δ 2 + 64 N s 2 s 3 cos10 θ 1 cot δ 2 64 N s 1 s 3 sin 10 θ 1 cot δ 2 Γ 44 = 16 N s 1 2 1cosδ 1+cosδ + 16 N s 2 2 1cosδ 1+cosδ + 8 N s 3 2 16 N s 1 s 3 sin10 θ 1 sinδ 1+cosδ + 16 N s 2 s 3 cos10 θ 1 sinδ 1+cosδ
{ Γ 11 = 4 N s 1 2 ( cos 2 δ+1 )+ 8 N s 2 2 ( cos 2 δ+cosδ+1 )+ 2 N s 3 2 sin 2 δ( 5 cos 2 δ+2cosδ+1 ) ( cosδ1 ) 2 + 4 N s 2 s 3 cos14 θ 1 sinδ ( cosδ+1 ) 2 cosδ1 4 N s 1 s 3 sin14 θ 1 sinδ ( cosδ+1 ) 2 cosδ1 Γ 22 = 8 N s 1 2 + 24 N s 2 2 + 16 N s 3 2 1+cosδ 1cosδ 16 N s 2 s 3 cos14 θ 1 sinδ 1cosδ + 16 N s 1 s 3 sin14 θ 1 sinδ 1cosδ Γ 33 = 24 N s 1 2 + 8 N s 2 2 + 16 N s 3 2 cot 2 δ 2 + 16 N s 2 s 3 cos14 θ 1 cot δ 2 16 N s 1 s 3 sin14 θ 1 cot δ 2 Γ 44 = 4 N s 1 2 tan 2 δ 2 + 4 N s 2 2 tan 2 δ 2 + 2 N s 3 2
{ Γ 11 = 4 N s 1 2 [ cos12 θ 1 +1 cos 2 δ( cos12 θ 1 1 ) ]+ 4 N s 2 2 [ 2cos12 θ 1 +2cosδ+ cos 2 δ( cos12 θ 1 +2 ) ] 2 N s 3 2 [ 7cos12 θ 1 +15+4cosδ( cos12 θ 1 +3 )+ cos 2 δ( cos12 θ 1 +5 ) 8( cos12 θ 1 +2 ) 1cosδ ] Γ 22 = 8 N s 1 2 + 24 N s 2 2 + 8 N s 3 2 ( cos12 θ 1 +2 ) 1+cosδ 1cosδ Γ 33 = 24 N s 1 2 + 8 N s 2 2 + 8 N s 3 2 ( 2cos12 θ 1 ) cot 2 δ 2 Γ 44 = 2 N s 1 2 ( cos12 θ 1 +2 ) 1cosδ 1+cosδ 2 N s 2 2 ( cos12 θ 1 2 ) 1cosδ 1+cosδ + 2 N s 3 2 + 4 N s 1 s 2 sin12 θ 1 1cosδ 1+cosδ
{ Γ 11 = 2 N s 1 2 [ cos16 θ 1 2+2cos16 θ 1 cosδ+( cos16 θ 1 2 ) cos 2 δ ] + 2 N s 2 2 [ cos16 θ 1 +4+( cos16 θ 1 +2 )cosδ+( cos16 θ 1 +4 ) cos 2 δ ] + 2 N s 3 2 ( 16 1cosδ 1512cosδ5 cos 2 δ ) 4 N s 1 s 2 sin16 θ 1 ( 1+cosδ ) 2 Γ 22 = 8 N s 1 2 ( cos16 θ 1 +1 ) 8 N s 2 2 ( cos16 θ 1 +3 )+ 16 N s 3 2 1+cosδ 1cosδ 16 N s 1 s 2 sin16 θ 1 Γ 33 = 8 N s 1 2 ( 1+cos16 θ 1 )+ 8 N s 2 2 ( 1cos16 θ 1 )+ 8 N s 3 2 sin 2 δ ( cosδ1 ) 2 + 16 N s 1 s 2 sin16 θ 1 Γ 44 = 4 N s 1 2 1cosδ 1+cosδ + 4 N s 2 2 1cosδ 1+cosδ + 2 N s 3 2

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