Abstract

Data processing for sequential in time polarimeters based on the Data Reduction Matrix technique yield polarization artifacts in the presence of time varying signals. To overcome these artifacts, polarimeters are designed to operate at higher and higher speeds. In this paper we describe a band limited reconstruction algorithm that allows the measurement and processing of temporally varying Stokes parameters without artifacts. An example polarimeter consisting of a rotating retarder and polarizer is considered, and conventional processing methods are compared to a band limited reconstruction algorithm for the example polarimeter. We demonstrate that a significant reduction in error is possible using these methods.

© 2011 OSA

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    [CrossRef] [PubMed]
  2. L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166.
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    [CrossRef] [PubMed]
  4. M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
    [CrossRef]
  5. D. B. Chenault and J. L. Pezzaniti, “Polarization imaging through scattering media,”(SPIE, Bellingham, WA, 2000), pp. 124 – 133.
  6. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” in “ACM SIGGRAPH ASIA 2008 courses,”(ACM, New York, NY, USA, 2008), SIGGRAPH Asia ’08, pp. 71:1–71:15
  7. V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “Proc. SPIE 6682 ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2007), p. 668206.
    [CrossRef]
  8. D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoeleastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. 46, 8428–8445 (2007).
    [CrossRef] [PubMed]
  9. K. Sassen, “Polarization in LIDAR,” in “LIDAR: Range-resolved optical remote sensing of the atmosphere ,” ,C. Weitkamp, ed. (Springer, 2005), pp. 19–42.
  10. D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.
  11. R. A. Chipman, “Polarimetry,” in “Handbook of Optics ,” , M. Bass, ed. (McGraw-Hill, 2009), 3rd ed.
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    [CrossRef]
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    [CrossRef]
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  19. L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.
  20. M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in “Proceedings of SPIE vol. 3754, Polarization Measurement, Analysis, and Remote Sensing II ,” , D. H. Goldstein and D. B. Chenault, eds. (SPIE, Bellingham, WA, 1999), pp. 50–54.
  21. B. M. Ratliff, C. F. Lacasse, and J. S. Tyo, “Quantifying ifov error and compensating its effects in dofp polarimeters,” Opt. Express 17, 9112 – 9125 (2009).
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  22. D. S. Sabatke, M. R. Descour, E. 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]
  23. J. S. Tyo and T. S. Turner, “Variable retardance, Fourier transform imaging spectropolarimeters for visible spectrum remote sensing,” Appl. Opt. 40, 1450–1458 (2001).
    [CrossRef]
  24. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651–1655 (1995).
    [CrossRef]
  25. J. S. Tyo, “Design of optimal polarimeters: maximization of SNR and minimization of systematic errors,” Appl. Opt. 41, 619–630 (2002).
    [CrossRef] [PubMed]
  26. F. Goudail and A. Beniere, “Estimation precision of the linear degree of polarization and of the angle of polarization in the presence of different types of noises,” Appl. Opt. 49, 683–693 (2010).
    [CrossRef] [PubMed]
  27. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  28. R. A. Chipman, “Polarimetric impulse response,” in “Proc. SPIE col. 1317: Polarimetery: radar, Infrared, Visible, Ultraviolet, and X-Ray ,”(SPIE, Bellingham, WA, 1990), pp. 223 – 241.
  29. J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I: formulation and example,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
    [CrossRef]

2010 (1)

2009 (2)

2007 (1)

2006 (1)

2003 (1)

2002 (2)

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566–576 (2002).
[CrossRef]

J. S. Tyo, “Design of optimal polarimeters: maximization of SNR and minimization of systematic errors,” Appl. Opt. 41, 619–630 (2002).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

1999 (1)

1997 (1)

M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

1996 (1)

1995 (1)

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

1990 (1)

1989 (1)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

1988 (1)

Ambirajan, A.

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

Andreou, A. G.

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566–576 (2002).
[CrossRef]

Azzam, R. M. A.

Beniere, A.

Bigué, L.

L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.

Cairns, B.

Chenault, D. B.

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

D. B. Chenault and J. L. Pezzaniti, “Polarization imaging through scattering media,”(SPIE, Bellingham, WA, 2000), pp. 124 – 133.

J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” in “Proceedigns of SPIE vol. 5888: Polarization Science and Remote Sensing II ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2005), p. 5888OV.

Cheng, L. J.

L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166.

Chipman, R. A.

D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoeleastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. 46, 8428–8445 (2007).
[CrossRef] [PubMed]

J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I: formulation and example,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
[CrossRef]

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

R. A. Chipman, “Polarimetric impulse response,” in “Proc. SPIE col. 1317: Polarimetery: radar, Infrared, Visible, Ultraviolet, and X-Ray ,”(SPIE, Bellingham, WA, 1990), pp. 223 – 241.

R. A. Chipman, “Polarimetry,” in “Handbook of Optics ,” , M. Bass, ed. (McGraw-Hill, 2009), 3rd ed.

Creusere, C. D.

V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “Proc. SPIE 6682 ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2007), p. 668206.
[CrossRef]

Dank, J. A.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

Davis, A.

Dereniak, E.

Descour, M. R.

Diner, D. J.

Elminyawi, I. M.

El-Saba, A. M.

Engheta, N.

Foulonneau, A.

L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.

Gendre, L.

L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.

Goldstein, D. H.

Goudail, F.

Gutt, G.

Hamilton, M.

L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166.

Hancock, B.

Howe, J. D.

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in “Proceedings of SPIE vol. 3754, Polarization Measurement, Analysis, and Remote Sensing II ,” , D. H. Goldstein and D. B. Chenault, eds. (SPIE, Bellingham, WA, 1999), pp. 50–54.

Illing, R. M. E.

R. M. E. Illing, “High-speed fieldable imaging stokes vector polarimeter,” in “Proceedigns of SPIE vol. 5888: Polarization Science and Remote Sensing II ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2005), p. 58880X.

Kalayjian, Z. K.

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566–576 (2002).
[CrossRef]

Kaneko, T.

Kato, T.

Kemme, S. A.

LaCasse, C. F.

Look, D. C.

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

Mahoney, C.

L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166.

McGuire, J. P.

Mudge, J. D.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

Narasimhan, S. G.

Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” in “ACM SIGGRAPH ASIA 2008 courses,”(ACM, New York, NY, USA, 2008), SIGGRAPH Asia ’08, pp. 71:1–71:15

Nayar, S. K.

Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” in “ACM SIGGRAPH ASIA 2008 courses,”(ACM, New York, NY, USA, 2008), SIGGRAPH Asia ’08, pp. 71:1–71:15

Oka, K.

Pezzaniti, J. L.

J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” in “Proceedigns of SPIE vol. 5888: Polarization Science and Remote Sensing II ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2005), p. 5888OV.

D. B. Chenault and J. L. Pezzaniti, “Polarization imaging through scattering media,”(SPIE, Bellingham, WA, 2000), pp. 124 – 133.

Phenis, A. M.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

Phipps, G. S.

Pugh, E. N.

Ratliff, B. M.

Reyes, G.

L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166.

Rowe, M. P.

Sabatke, D. S.

Sassen, K.

K. Sassen, “Polarization in LIDAR,” in “LIDAR: Range-resolved optical remote sensing of the atmosphere ,” ,C. Weitkamp, ed. (Springer, 2005), pp. 19–42.

Schechner, Y. Y.

Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” in “ACM SIGGRAPH ASIA 2008 courses,”(ACM, New York, NY, USA, 2008), SIGGRAPH Asia ’08, pp. 71:1–71:15

Shaw, J. A.

Silverman, M. P.

M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Smith, M. H.

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in “Proceedings of SPIE vol. 3754, Polarization Measurement, Analysis, and Remote Sensing II ,” , D. H. Goldstein and D. B. Chenault, eds. (SPIE, Bellingham, WA, 1999), pp. 50–54.

Strange, W.

M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Stryjewski, J. S.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

Sweatt, W. C.

Thilak, V.

V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “Proc. SPIE 6682 ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2007), p. 668206.
[CrossRef]

Tietjen, A. B.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

Turner, T. S.

Tyler, D. W.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

Tyo, J. S.

Virgen, M.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

Voelz, D. G.

V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “Proc. SPIE 6682 ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2007), p. 668206.
[CrossRef]

Woodruff, J. B.

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in “Proceedings of SPIE vol. 3754, Polarization Measurement, Analysis, and Remote Sensing II ,” , D. H. Goldstein and D. B. Chenault, eds. (SPIE, Bellingham, WA, 1999), pp. 50–54.

Appl. Opt. (6)

IEEE Sens. J. (1)

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566–576 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Opt. Eng. (2)

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

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

Opt. Express (2)

Opt. Lett. (3)

Other (12)

R. A. Chipman, “Polarimetric impulse response,” in “Proc. SPIE col. 1317: Polarimetery: radar, Infrared, Visible, Ultraviolet, and X-Ray ,”(SPIE, Bellingham, WA, 1990), pp. 223 – 241.

R. M. E. Illing, “High-speed fieldable imaging stokes vector polarimeter,” in “Proceedigns of SPIE vol. 5888: Polarization Science and Remote Sensing II ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2005), p. 58880X.

L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in “Proceedings of SPIE vol. 3754, Polarization Measurement, Analysis, and Remote Sensing II ,” , D. H. Goldstein and D. B. Chenault, eds. (SPIE, Bellingham, WA, 1999), pp. 50–54.

J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” in “Proceedigns of SPIE vol. 5888: Polarization Science and Remote Sensing II ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2005), p. 5888OV.

L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166.

D. B. Chenault and J. L. Pezzaniti, “Polarization imaging through scattering media,”(SPIE, Bellingham, WA, 2000), pp. 124 – 133.

Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” in “ACM SIGGRAPH ASIA 2008 courses,”(ACM, New York, NY, USA, 2008), SIGGRAPH Asia ’08, pp. 71:1–71:15

V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “Proc. SPIE 6682 ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2007), p. 668206.
[CrossRef]

K. Sassen, “Polarization in LIDAR,” in “LIDAR: Range-resolved optical remote sensing of the atmosphere ,” ,C. Weitkamp, ed. (Springer, 2005), pp. 19–42.

D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

R. A. Chipman, “Polarimetry,” in “Handbook of Optics ,” , M. Bass, ed. (McGraw-Hill, 2009), 3rd ed.

Supplementary Material (1)

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Figures (11)

Fig. 1
Fig. 1

A modulated measurement of the Stokes parameters using a rotating analyzer polarimeter, with each parameter band limited such that the signal can be ideally reconstructed. WB is band width of each parameter, f 0 is the frequency of analyzer rotation, and fs is the detector sampling frequency. The dashed blue line indicates that S 2 is in the quadrature component of the side band. The configuration with maximum allowed bandwidth is shown.

Fig. 2
Fig. 2

Two Stokes parameter input signals with different bandwidths; one satisfies the polarimeter band limit criteria (dashed), while the other has twice the allowed bandwidth for ideal reconstruction (solid).

Fig. 3
Fig. 3

The components of the flux I for an excitation of (a) sinc 2 ( n 10 ) [ 1 0 0 0 ] , (b) sinc 2 ( n 10 ) [ 0 1 3 0 0 ] , (c) sinc 2 ( n 10 ) [ 0 0 1 3 0 ] , and (d) sinc 2 ( n 10 ) [ 0 0 0 1 3 ] . The vertical red lines indicate an example position of a 10 measurement sequence for comparison to the signal feature size.

Fig. 4
Fig. 4

Single frame excerpt from graphic displaying the measured flux as the bandwidth of the input signals increases (Media 1). (a) The measured flux I for S = { 1 , 1 3 , 1 3 , 1 3 } sinc 2 ( n 10 ) in the temporal domain for a rotating retarder polarimeter with a third wave retarder. (b) The Fourier transform of the measured flux, with dashed lines representing imaginary components. The Fourier transform of (a) is given by the sum of all 4 components in (b). The colors in (b) indicate each individual Stokes parameter’s contributions.

Fig. 5
Fig. 5

The results of the homodyne processes shown in the Fourier domain. (a) {A 0(t)I(t)}, (b) {A 1(t)I(t)}, (c) {A 2(t)I(t)}, (d) {A 3(t)I(t)}. Dashed lines indicate imaginary components. In panels (a) and (b) the base band exhibits coupling between the s 0 and the s 1 signals since the modulators A 0 and A 1 for these two signals are not orthogonal.

Fig. 6
Fig. 6

The four components of the estimated Stokes parameters given by Ŝ (f) = {ZA(t)I(t)} prior to low pass filtering with w(t) according to Eq. (20). Also shown in the dotted line is the rectangular low pass filter that ideally reconstructs the correct individual Stokes parameters. The marking γ is an example of self-error, while the marking ε is an example of cross-error. If the low pass filter w(t) does not reject these frequencies outside of the shaded base band artifacts will arise due to these error terms.

Fig. 7
Fig. 7

The four components of the estimated Stokes parameters in the Fourier domain given by {w(t) * ZA(t)I(t)} with w ( n ) = rect ( n 16 ) , a 16 sample rect window. (a) {ŝ 0}, (b) {ŝ 1}, (c) {ŝ 2}, and (d) {ŝ 3}. Differences from a triangle function in the Fourier domain indicate errors in the data reduction method.

Fig. 8
Fig. 8

The time dependent Stokes parameters calculated using a sliding 16 element window compared to the input signal, which is ideally reconstructible a band limited window. (a) s 0, (b) s 1, (c) s 2, and (d) s 3

Fig. 9
Fig. 9

Single frame excerpt from animation (Fig. 4 (Media 1)) displaying the measured flux for a scene with bandwidth given by 2WB , with WB the required bandwidth of the polarimeter for ideal reconstruction. (a) The measured flux I for S = [ 1 1 3 1 3 1 3 ] sinc 2 ( n 5 ) in the temporal domain. (b) The Fourier transform of the contributions from individual Stokes parameters to the measured flux, with dashed lines representing imaginary components. The Fourier transform of (a) is given by the summing the 4 components in (b). The colors in (b) indicate contributions of the individual Stokes parameters from Fig. 3. This figure corresponds to Fig. 4 in section 5.2.

Fig. 10
Fig. 10

Shown are the four components of the estimated Stokes parameters reconstructed according to Eq. (20) with w ˜ ( f ) = rect ( f 0.2 f s ) (ideal band limited window), estimating a signal that has a twice the bandwidth required for error free reconstruction. (a) {ŝ 0}, (b) {ŝ 1}, (c) {ŝ 2}, and (d) {ŝ 3}. The total signal for the estimate is calculated by summing each of the signals in the corresponding panel, so for example any contribution to the s 0 estimate in panel (a) from the other three Stokes parameters results in reconstruction artifacts. This figure corespondents to Fig. 7 in section 5.2.

Fig. 11
Fig. 11

The time dependent Stokes parameters calculated using the time limited rect window and the band limited sinc window to be compared with the the input signal in Fig. 2. This figure corresponds to Fig. 8 in section 5.2.

Equations (24)

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S = [ s 0 s 1 s 2 s 3 ] T = [ I x + I y I x I y I 45 I 135 I L I R ] T .
I ( t ) = A ( t ) T S ( t ) = [ A 0 ( t ) A 1 ( t ) A 2 ( t ) A 3 ( t ) ] S ( t ) ,
I k = A k T S = A 0 k s 0 + A 1 k s 1 + A 2 k s 2 + A 3 k s 3 ,
I = [ I 1 I K ] = [ A 1 T A K T ] S = WS .
Ŝ = W 1 I ,
W p 1 = ( W T W ) 1 W T
A ( t ) = 1 2 [ 1 cos ( 4 π f 0 t ) sin ( 4 π f 0 t ) 0 ] T ,
I ( t ) = 1 2 ( s 0 ( t ) + cos ( 4 π f 0 t ) s 1 ( t ) + sin ( 4 π f 0 t ) s 2 ( t ) ) .
I ˜ ( f ) = s ˜ A 0 ( f ) * s ˜ 0 ( f ) + s ˜ A 1 ( f ) * s ˜ 1 ( f ) + s ˜ A 2 ( f ) * s ˜ 2 ( f ) + s ˜ A 3 ( f ) * s ˜ 3 ( f ) ,
I ˜ ( f ) = 1 2 ( s ˜ 0 ( f ) + 1 2 ( s ˜ 1 ( f 2 f 0 ) + s ˜ 1 ( f + 2 f 0 ) ) + 1 2 j ( s ˜ 2 ( f 2 f 0 ) s ˜ 2 ( f + 2 f 0 ) ) ) .
W B 2 f 0 ,
f 0 f s 6 ,
I ( x , y , t , λ ) = ( A ( x , y , t , λ ) T ( h ( x , y , t , λ ) * S ( x , y , t , λ ) ) ) * d * ( x , y , t , λ ) ,
f ( x ) * g ( x ) = f ( α ) h ( x , α ) d α
f ( x ) * g ( x ) = f ( α ) h ( x α ) d α ,
Z 1 = ( W T W ) 1 = ( k = 1 N A k A K T ) 1 .
Z i j = A i ( t ) A j ( t ) d t ,
Z i j ( t ) = w ( t ) * A i ( t ) A j ( t ) = w ( t t 0 ) A i ( t 0 ) A j ( t 0 ) d t 0 ,
W T I = W T WS .
W p 1 W T I w ( x , y , t , λ ) * Z 1 ( x , y , t , λ ) A ( x , y , t , λ ) I ( x , y , t , λ ) .
Ŝ ( x , y , t , λ ) = W 1 { I ( x , y , t , λ ) } = w ( x , y , t , λ ) * Z 1 A ( x , y , t , λ ) I ( x , y , t , λ ) .
A ( x , y , t , λ ) = A ( t ) = [ A 0 ( t ) A 1 ( t ) A 2 ( t ) A 3 ( t ) ] = 1 2 [ 1 1 + cos δ 2 + 1 cos δ 2 cos 8 π f 0 t ( 1 cos δ ) 2 sin 8 π f 0 t sin δ sin 4 π f 0 t ] .
Z = ( t = 0 t = 1 f 0 A ( t ) A T ( t ) d t ) = [ 1 4 cos ( δ ) + 1 8 0 0 cos ( δ ) + 1 8 3 cos 2 δ + 2 cos δ + 3 32 0 0 0 0 cos 2 δ 2 cos δ + 1 32 0 0 0 0 sin 2 δ 8 ] .
S = sinc 2 ( t 10 t 0 ) [ 1 1 3 1 3 1 3 ] T ,

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