Abstract

We present an approach to spectropolarimetry that requires neither moving parts nor time dependent modulation, and that offers the prospect of achieving high sensitivity. The technique applies equally well, in principle, in the optical, UV, or IR. The concept, which is one of those generically known as channeled polarimetry, is to encode the polarization information at each wavelength along the spatial dimension of a two-dimensional data array using static, robust optical components. A single 2D data frame contains the full polarization information and can be configured to measure either two or all of the Stokes polarization parameters. By acquiring full polarimetric information in a single observation, we simplify polarimetry of transient sources and in situations where the instrument and target are in relative motion. The robustness and simplicity of the approach, coupled with its potential for high sensitivity, and applicability over a wide wavelength range, is likely to prove useful for applications in challenging environments such as space.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  5. F. Snik, T. Karalidi, and C. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48, 1337–1346(2009).
    [CrossRef]
  6. G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  15. M. W. Kudenov, L. Pezzaniti, E. L. Dereniak, and G. R. Gerhart, “Prismatic imaging polarimeter calibration for the infrared spectral region,” Opt. Express 16, 13720 (2008).
    [CrossRef]
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  17. E. H. Geyer, K. Jockers, N. N. Kiselev, and G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to Comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
    [CrossRef]
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  19. C. Pernechele, E. Giro, and D. Fantinel, “Device for optical linear polarization measurements with a single exposure,” Proc. SPIE  4843, 156–163 (2003).
  20. T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
    [CrossRef]
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    [CrossRef]
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2011 (1)

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

2010 (1)

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

2009 (3)

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

F. Snik, T. Karalidi, and C. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48, 1337–1346(2009).
[CrossRef]

2008 (3)

T. Wakayama, Y. Otani, and N. Umeda, “One-shot birefringence dispersion measurement based on channeled spectrum technique,” Opt. Commun. 281, 3668–3672 (2008).
[CrossRef]

J. Howard, “Snapshot-imaging motional stark effect polarimetry,” Plasma Phys. Control Fusion 50, 125003 (2008).

M. W. Kudenov, L. Pezzaniti, E. L. Dereniak, and G. R. Gerhart, “Prismatic imaging polarimeter calibration for the infrared spectral region,” Opt. Express 16, 13720 (2008).
[CrossRef]

2007 (1)

2006 (1)

R. W. Oka and N. Saito, “Snapshot complete imaging polarimeter using savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

2004 (1)

M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. 21, 2244–2249 (2004).
[CrossRef]

2003 (2)

K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11, 1510–1519 (2003).
[CrossRef]

C. Pernechele, E. Giro, and D. Fantinel, “Device for optical linear polarization measurements with a single exposure,” Proc. SPIE  4843, 156–163 (2003).

2001 (1)

T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
[CrossRef]

1999 (1)

1997 (1)

E. Oliva, “Wedged double Wollaston, a device for single shot polarimetric measurements,” Astrophys. Space Sci. 123, 589–592 (1997).

1996 (1)

E. H. Geyer, K. Jockers, N. N. Kiselev, and G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to Comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

1974 (1)

K. Nordseick, “A simple polarimetric system for the lick observatory image-tube scanner,” Publ. Astron. Soc. Pac. 86, 324–329 (1974).
[CrossRef]

1972 (1)

K. Serkowski, “A polarimetric method of measuring radial velocities,” Publ. Astron. Soc. Pac. 84, 649–651 (1972).
[CrossRef]

1967 (1)

Baleine, E.

M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. 21, 2244–2249 (2004).
[CrossRef]

Beckers, J. M.

Bevington, P.

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. (McGraw-Hill, 2002).

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences, 1st ed. (McGraw-Hill, 1968).

Boulbry, B.

Brown, T. M.

T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
[CrossRef]

Burrows, A.

T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
[CrossRef]

Charbonneau, D.

T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
[CrossRef]

Chen, F.

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

Chernova, G. P.

E. H. Geyer, K. Jockers, N. N. Kiselev, and G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to Comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

DasSarma, P.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

DasSarma, S.

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

de Boer, J.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

Dereniak, E. L.

Diamantopoulou, R.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

Dogariu, A.

M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. 21, 2244–2249 (2004).
[CrossRef]

Fantinel, D.

C. Pernechele, E. Giro, and D. Fantinel, “Device for optical linear polarization measurements with a single exposure,” Proc. SPIE  4843, 156–163 (2003).

Gerhart, G. R.

Germer, T. A.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

B. Boulbry, J. C. Ramella-Roman, and T. A. Germer, “Improved method for calibrating a Stokes polarimeter,” Appl. Opt. 46, 8533–8541 (2007).
[CrossRef]

Geyer, E. H.

E. H. Geyer, K. Jockers, N. N. Kiselev, and G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to Comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

Gilliland, R. L.

T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
[CrossRef]

Giro, E.

C. Pernechele, E. Giro, and D. Fantinel, “Device for optical linear polarization measurements with a single exposure,” Proc. SPIE  4843, 156–163 (2003).

Goldstein, D.

D. Goldstein, Polarized Light3rd ed. (CRC Press, 2011).

Hannemann, S.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

Hasekamp, O. P.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

Hough, J. H.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Howard, J.

J. Howard, “Snapshot-imaging motional stark effect polarimetry,” Plasma Phys. Control Fusion 50, 125003 (2008).

Jockers, K.

E. H. Geyer, K. Jockers, N. N. Kiselev, and G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to Comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

Kaneko, T.

Karalidi, T.

Kato, T.

Keller, C.

F. Snik, T. Karalidi, and C. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48, 1337–1346(2009).
[CrossRef]

F. Snik and C. Keller, “Astronomical polarimetry,” in Planets, Stars and Stellar Systems (Springer, 2012).

Keller, C. U.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Kiselev, N. N.

E. H. Geyer, K. Jockers, N. N. Kiselev, and G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to Comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

Kolokolova, L.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Kudenov, M. W.

Laan, E. C.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Macchetto, F.

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Macchetto, F. D.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

Manset, N.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Martin, W.

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

Moon, S. G.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Mujat, M.

M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. 21, 2244–2249 (2004).
[CrossRef]

Navarro, R.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Nordseick, K.

K. Nordseick, “A simple polarimetric system for the lick observatory image-tube scanner,” Publ. Astron. Soc. Pac. 86, 324–329 (1974).
[CrossRef]

Noyes, R. W.

T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
[CrossRef]

Oka, K.

Oka, R. W.

R. W. Oka and N. Saito, “Snapshot complete imaging polarimeter using savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

Oliva, E.

E. Oliva, “Wedged double Wollaston, a device for single shot polarimetric measurements,” Astrophys. Space Sci. 123, 589–592 (1997).

Otani, Y.

T. Wakayama, Y. Otani, and N. Umeda, “One-shot birefringence dispersion measurement based on channeled spectrum technique,” Opt. Commun. 281, 3668–3672 (2008).
[CrossRef]

Pernechele, C.

C. Pernechele, E. Giro, and D. Fantinel, “Device for optical linear polarization measurements with a single exposure,” Proc. SPIE  4843, 156–163 (2003).

Pezzaniti, L.

Ramella-Roman, J. C.

Reid, I.

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Reid, I. N.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

Rietjens, J. H. H.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Robb, F.

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Robb, F. T.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

Robinson, D.

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. (McGraw-Hill, 2002).

Saito, N.

R. W. Oka and N. Saito, “Snapshot complete imaging polarimeter using savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

Serkowski, K.

K. Serkowski, “A polarimetric method of measuring radial velocities,” Publ. Astron. Soc. Pac. 84, 649–651 (1972).
[CrossRef]

Smit, J. M.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Snik, F.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

F. Snik, T. Karalidi, and C. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48, 1337–1346(2009).
[CrossRef]

F. Snik and C. Keller, “Astronomical polarimetry,” in Planets, Stars and Stellar Systems (Springer, 2012).

Sparks, W. B.

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Stam, D. M.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Ter Horst, R.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Umeda, N.

T. Wakayama, Y. Otani, and N. Umeda, “One-shot birefringence dispersion measurement based on channeled spectrum technique,” Opt. Commun. 281, 3668–3672 (2008).
[CrossRef]

van Harten, G.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Verlaan, A. L.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Vliegenthart, W. A.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

Voors, R.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Wakayama, T.

T. Wakayama, Y. Otani, and N. Umeda, “One-shot birefringence dispersion measurement based on channeled spectrum technique,” Opt. Commun. 281, 3668–3672 (2008).
[CrossRef]

Wielinga, K.

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

Appl. Opt. (3)

Astrophys. J. (1)

T. M. Brown, D. Charbonneau, R. L. Gilliland, R. W. Noyes, and A. Burrows, “Hubble Space Telescope time-series photometry of the transiting planet of HD 209458,” Astrophys. J. 552, 699–709 (2001).
[CrossRef]

Astrophys. Space Sci. (2)

E. H. Geyer, K. Jockers, N. N. Kiselev, and G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to Comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

E. Oliva, “Wedged double Wollaston, a device for single shot polarimetric measurements,” Astrophys. Space Sci. 123, 589–592 (1997).

J. Opt. Soc. Am. (1)

M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. 21, 2244–2249 (2004).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

W. B. Sparks, J. H. Hough, L. Kolokolova, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, I. N. Reid, F. D. Macchetto, and W. Martin, “Circular polarization in scattered light as a possible biomarker,” J. Quant. Spectrosc. Radiat. Transfer 110, 1771–1779 (2009).
[CrossRef]

Opt. Commun. (1)

T. Wakayama, Y. Otani, and N. Umeda, “One-shot birefringence dispersion measurement based on channeled spectrum technique,” Opt. Commun. 281, 3668–3672 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Plasma Phys. Control Fusion (1)

J. Howard, “Snapshot-imaging motional stark effect polarimetry,” Plasma Phys. Control Fusion 50, 125003 (2008).

Proc. Natl. Acad. Sci. USA (1)

W. B. Sparks, J. H. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. Robb, N. Manset, L. Kolokolova, I. Reid, F. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” Proc. Natl. Acad. Sci. USA 106, 7816–7821 (2009).
[CrossRef]

Proc. SPIE (4)

R. W. Oka and N. Saito, “Snapshot complete imaging polarimeter using savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

G. van Harten, F. Snik, J. H. H. Rietjens, J. M. Smit, J. de Boer, R. Diamantopoulou, O. P. Hasekamp, D. M. Stam, C. U. Keller, E. C. Laan, A. L. Verlaan, W. A. Vliegenthart, R. Ter Horst, R. Navarro, K. Wielinga, S. Hannemann, S. G. Moon, and R. Voors, “Prototyping for the Spectropolarimeter for Planetary EXploration (SPEX): calibration and sky measurements,” Proc. SPIE  8160, 81600Z (2011).

F. Snik, J. H. H. Rietjens, G. van Harten, D. M. Stam, C. U. Keller, J. M. Smit, E. C. Laan, A. L. Verlaan, R. Ter Horst, R. Navarro, K. Wielinga, S. G. Moon, and R. Voors, “SPEX: the spectropolarimeter for planetary exploration,” Proc. SPIE  7731, 77311B (2010).

C. Pernechele, E. Giro, and D. Fantinel, “Device for optical linear polarization measurements with a single exposure,” Proc. SPIE  4843, 156–163 (2003).

Publ. Astron. Soc. Pac. (2)

K. Serkowski, “A polarimetric method of measuring radial velocities,” Publ. Astron. Soc. Pac. 84, 649–651 (1972).
[CrossRef]

K. Nordseick, “A simple polarimetric system for the lick observatory image-tube scanner,” Publ. Astron. Soc. Pac. 86, 324–329 (1974).
[CrossRef]

Other (4)

F. Snik and C. Keller, “Astronomical polarimetry,” in Planets, Stars and Stellar Systems (Springer, 2012).

D. Goldstein, Polarized Light3rd ed. (CRC Press, 2011).

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences, 1st ed. (McGraw-Hill, 1968).

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. (McGraw-Hill, 2002).

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Figures (11)

Fig. 1.
Fig. 1.

Illustration of the optical bench layout. Light enters a spatially elongated slit, passes through a birefringent wedge or wedges and a polarization analyzer, and then enters a conventional long-slit spectrograph. A quarter-wave retarder (not shown) may be inserted before the wedge.

Fig. 2.
Fig. 2.

Concept—the polarization optics imprint an amplitude modulation on the dimension orthogonal to the dispersion direction of the spectrograph. Typically, this direction corresponds to the spatial dimension along the slit.

Fig. 3.
Fig. 3.

Illustration of the compound birefringent wedges. A single compound optic may be used for linear polarimetry (left) and double (both) for full Stokes. The fast axes run at ±45° in the left optic and horizontal and vertical in the right optic, and the slit direction is horizontal or vertical.

Fig. 4.
Fig. 4.

Example theoretical data frames for each of the configurations discussed in the text when viewed with 100% polarized light. Each panel has x running horizontally and wavelength vertically, increasing up. Parameters correspond to 2 mm in x of a 3° quartz wedge set, running from 450 to 750 nm. Top row shows 100% Stokes Q, middle row 100% Stokes U, and bottom row 100% Stokes V. Left to right, in the notation of Tables 13, the configurations are qw, qww, wW, wW, and wwWW. Note that if the quarter-wave retarder was omitted in the first two columns, U and V, which show no sensitivity with the quarter-wave retarder, would be interchanged. For the first two columns, the analyzer angle is set at 0° and for the remaining three at 45° (see text).

Fig. 5.
Fig. 5.

Actual spectra obtained with a preliminary test optical bench: (upper) a spectrum obtained with unpolarized light, (center) a spectrum with one quartz birefringent wedge and 100% linearly polarized light, and (lower) a spectrum obtained with two quartz birefringent wedges reversed as in the manner of the compound optics with 100% linearly polarized light.

Fig. 6.
Fig. 6.

The uncertainties on normalized Stokes parameters for double wedge pair configuration, Subsection 4.B.3. Blue is Stokes q, green is u, and red is v. The vertical lines indicate the positions of the minima for q and v, the horizontal lines indicates 1/Ntot (dotted), and the analytically determined minimum 1.24× higher (dashed).

Fig. 7.
Fig. 7.

The uncertainties on normalized Stokes parameters for double wedge pair configuration, Subsection 4.B.3, with miscentering of order 1/32× the spatial distance of one wavelength of retardance. Blue is Stokes q, green is u, and red is v. The vertical lines indicate the positions of the analyzer angles that have no formal covariance, which is independent of the miscentering. Smooth lines through the simulated data are the analytic solutions ignoring covariance terms, while the plus signs are the results of Monte Carlo simulations. The horizontal lines are as in the previous figure.

Fig. 8.
Fig. 8.

Laboratory optical bench layout as implemented. Light enters an integrating sphere, and illuminates a diffusing screen on exit. It then passes through calibration or sample components, before entering the long-slit spectrograph with its associated polarization components, as described in the text. The upper panel shows the actual optical bench with baffles and boxes removed for visibility. The integrating sphere is to the left, and the camera to the right.

Fig. 9.
Fig. 9.

Example data frames for each of the configurations discussed in the text when viewed with 100% polarized light, obtained in the laboratory. The rows and columns correspond to those shown in Fig. 4.

Fig. 10.
Fig. 10.

Retrieved polarization curves for a BG18 colored glass filter presented orthogonal to the beam, blue, and at an angle tilted by 30° to orthogonal, green, observed using the qww configuration. At right angles, we expect no polarization, and inclined at 30°, approximately 6.3%, consistent with the least-squares retrieval. The black curves show arbitrarily normalized throughputs for the two configurations (solid, orthogonal and dotted, inclined) derived from the data, serving to illustrate that we also obtain full Stokes I spectroscopy using these methods.

Fig. 11.
Fig. 11.

Retrieved circular polarization for a pair of polarizing cinema 3D glasses, expected to exhibit 100% Stokes V left and right circularly polarized light for the left and right eyes, measured using the wwWW configuration. The retrieval is consistent with expectations. For completeness, and to illustrate that we obtain full Stokes polarimetry from a single data frame, the dashed lines show the retrieved degree of linear polarization.

Tables (3)

Tables Icon

Table 1. Coefficients of Stokes Parameters for Different Wedge Configurations

Tables Icon

Table 2. Error Estimates for Normalized Stokes Parameters for Different Wedge Configurations

Tables Icon

Table 3. Error Estimates for Unnormalized Stokes Parameters for Different Wedge Configurations

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

I=0.5(I+Qcosϕ+Usinϕ).
I=0.5(IQcosϕUsinϕ).
I=(II)/(I+I)=qcosϕ+usinϕ.
I=0.5I(1+pcos(ϕ2ψ)),I=0.5I(1pcos(ϕ2ψ)),I=pcos(ϕ2ψ),
I=0.5(I+Qcos2ϕ+Usin2ϕ)
I=qcos2ϕ+usin2ϕ.
Iobs=Iic+Qqc+Uuc+Vvc
I=(II)/(I+I)=q·qc+u·uc+v·vc
IQ=0.5(I+QcosϕVsinϕ).
IU=0.5(I+Ucosϕ+Vsinϕ).
I=0.5(I+Q(cosϕcos2θ+sinϕsin2ϕsin2θ)+Ucos2ϕsin2θ+V(cosϕsin2ϕsin2θsinϕcos2θ)),
I=0.5(IQ(cosϕcos2θ+sinϕsin2ϕsin2θ)Ucos2ϕsin2θV(cosϕsin2ϕsin2θsinϕcos2θ)),
I=q(cosϕcos2θ+sinϕsin2ϕsin2θ)+ucos2ϕsin2θ+v(cosϕsin2ϕsin2θsinϕcos2θ).
I=0.5(I+Q(cosϕcos2θ+sinϕsin(ζ2ϕ)sin2θ)+Ucos(ζ2ϕ)sin2θ+V(cosϕsin(ζ2ϕ)sin2θsinϕcos2θ)),
I=0.5(IQ(cosϕcos2θ+sinϕsin(ζ2ϕ)sin2θ)Ucos(ζ2ϕ)sin2θV(cosϕsin(ζ2ϕ)sin2θsinϕcos2θ)),
I=qcosϕcos2θ+sinϕsin(ζ2ϕ)sin2θ)ucos(ζ2ϕ)sin2θv(cosϕsin(ζ2ϕ)sin2θsinϕcos2θ).
I=0.5(I+Q(cos2ϕcos2θ+sin2ϕsin4ϕsin2θ)+Ucos4ϕsin2θ+V(cos2ϕsin2ϕsin2θsin2ϕcos2θ)),
I=0.5(IQ(cos2ϕcos2θ+sin2ϕsin4ϕsin2θ)Ucos4ϕsin2θV(cos2ϕsin2ϕsin2θsin2ϕcos2θ)),
I=q(cos2ϕcos2θ+sin2ϕsin4ϕsin2θ)ucos4ϕsin2θv(cos2ϕsin4ϕsin2θsin2ϕcos2θ).
I=0.5(I+Q(cos2ϕcos2θ+sin2ϕsin(2a+4ϕ)sin2θ)+Ucos(2a+4ϕ)sin2θ+V(cos2ϕsin(2a+4ϕ)sin2θsin2ϕcos2θ)),
I=0.5(IQ(cos2ϕcos2θ+sin2ϕsin(2a+4ϕ)sin2θ)Ucos(2a+4ϕ)sin2θV(cos2ϕsin(2a+4ϕ)sin2θsin2ϕcos2θ)),
I=q(cos2ϕcos2θ+sin2ϕsin(2a+4ϕ)sin2θ)ucos(2a+4ϕ)sin2θv(cos2ϕsin(2a+4ϕ)sin2θsin2ϕcos2θ).
y(xi)=aic(xi)+bqc(xi)+cuc(xi)+dvc(xi),
χ2=i=1Nϵi2/σi2=i=1N1σi2[yiy(xi)]2=i=1N1σi2[yiaic(xi)bqc(xi)cuc(xi)dvc(xi)]2,
χ2a=0=21σi2ic(yiaicbqccucdvc),χ2b=0=21σi2qc(yiaicbqccucdvc),χ2c=0=21σi2uc(yiaicbqccucdvc),χ2d=0=21σi2vc(yiaicbqccucdvc).
B(1σi2ic21σi2icqc1σi2icuc1σi2icvc1σi2icqc1σi2qc21σi2qcuc1σi2qcvc1σi2icuc1σi2qcuc1σi2uc21σi2ucvc1σi2icvc1σi2qcvc1σi2ucvc1σi2vc2),
sy(1σi2icyi,1σi2qcyi,1σi2ucyi,1σi2vcyi),
sy=B·(abcd).
a=(a,b,c,d)=B1·sy.
σai2=Bii1,
B1σ2(ic2icqcicucicqcqc2qcucicucqcucuc2).
B=14σ2(nxcos(2πxi/X)sin(2πxi/X)cos(2πxi/X)cos2(2πxi/X)cos(2πxi/X)sin(2πxi/X)sin(2πxi/X)cos(2πxi/X)sin(2πxi/X)sin2(2πxi/X)).
B=nx8σ2(200010001),
B1=4σ2nx(100020002).
Ia=(Ntot2σ2)(4σ2nx)=2Ntotnx=2y,Qb=(4nx)yicos(2πxi/X),Uc=(4nx)yisin(2πxi/X).
σ(I)σa=2Ntotnx,σ(Q)σb=22Ntotnx,σ(U)σc=22Ntotnx.
σ(q)=σ(u)=2Ntot.
σ(p)=2Ntot.

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