Abstract

A division-of-focal-plane polarimeter based on a dichroic dye and liquid crystal polymer guest-host system is presented. Two Stokes polarimeters are demonstrated: a linear Stokes and the first ever Full-Stokes division-of-focal-plane polarimeter. The fabrication, packaging, and characterization of the systems are presented. Finally, optimized polarimeter designs are discussed for future works.

© 2012 OSA

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References

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  1. J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt.35(11), 1855–1870 (1996).
    [CrossRef] [PubMed]
  2. K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express16(26), 21339–21354 (2008).
    [CrossRef] [PubMed]
  3. C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
    [CrossRef] [PubMed]
  4. J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, Pixelated Phase-Mask Dynamic Interferometers, W. Osten, ed. (Springer Berlin Heidelberg, 2006), pp. 640–647.
  5. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt.44(32), 6861–6868 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. B. E. Bayer, “Color imaging array,” U.S. Patent 3,971,065 (1976).
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    [CrossRef]
  10. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express18(18), 19087–19094 (2010).
    [CrossRef] [PubMed]
  11. V. Gruev, A. Ortu, N. Lazarus, J. Van der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express15(8), 4994–5007 (2007).
    [CrossRef] [PubMed]
  12. A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J.2(6), 566–576 (2002).
    [CrossRef]
  13. G. Myhre, A. Sayyad, and S. Pau, “Patterned color liquid crystal polymer polarizers,” Opt. Express18(26), 27777–27786 (2010).
    [CrossRef] [PubMed]
  14. G. Myhre and S. Pau, “Imaging capability of patterned liquid crystals,” Appl. Opt.48(32), 6152–6158 (2009).
    [CrossRef] [PubMed]
  15. C. F. LaCasse, R. A. Chipman, and J. S. Tyo, “Band limited data reconstruction in modulated polarimeters,” Opt. Express19(16), 14976–14989 (2011).
    [CrossRef] [PubMed]
  16. R. A. Chipman, “Polarized Light and Polarimetry” (University of Arizona, 2010).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. K. A. Bachman, J. J. Peltzer, P. D. Flammer, T. E. Furtak, R. T. Collins, and R. E. Hollingsworth, “Spiral plasmonic nanoantennas as circular polarization transmission filters,” Opt. Express20(2), 1308–1319 (2012).
    [CrossRef] [PubMed]
  21. X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging invisible spectrum,” Opt. Express18(17), 17776–17787 (2010).
    [CrossRef] [PubMed]
  22. H. Arwin, R. Magnusson, J. Landin, and K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Philos. Mag.92(12), 1583–1599 (2012).
    [CrossRef]
  23. D. H. Goldstein, “Polarization properties of Scarabaeidae,” Appl. Opt.45(30), 7944–7950 (2006).
    [CrossRef] [PubMed]
  24. 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]
  25. S. K. Gao and V. Gruev, “Bilinear and bicubic interpolation methods for division of focal plane polarimeters,” Opt. Express19(27), 26161–26173 (2011).
    [CrossRef] [PubMed]
  26. D. A. LeMaster and S. C. Cain, “Multichannel blind deconvolution of polarimetric imagery,” J. Opt. Soc. Am. A25(9), 2170–2176 (2008).
    [CrossRef] [PubMed]

2012 (2)

H. Arwin, R. Magnusson, J. Landin, and K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Philos. Mag.92(12), 1583–1599 (2012).
[CrossRef]

K. A. Bachman, J. J. Peltzer, P. D. Flammer, T. E. Furtak, R. T. Collins, and R. E. Hollingsworth, “Spiral plasmonic nanoantennas as circular polarization transmission filters,” Opt. Express20(2), 1308–1319 (2012).
[CrossRef] [PubMed]

2011 (2)

2010 (3)

2009 (1)

2008 (2)

2007 (1)

2006 (3)

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]

D. H. Goldstein, “Polarization properties of Scarabaeidae,” Appl. Opt.45(30), 7944–7950 (2006).
[CrossRef] [PubMed]

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

2005 (1)

2002 (1)

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J.2(6), 566–576 (2002).
[CrossRef]

2000 (1)

1999 (1)

1998 (2)

J.-H. Kim, S. Kumar, and S.-D. Lee, “Alignment of liquid crystals on polyimide films exposed to ultraviolet light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics57(5), 5644–5650 (1998).
[CrossRef]

M. Nishikawa, B. Taheri, and J. L. West, “Mechanism of unidirectional liquid-crystal alignment on polyimides with linearly polarized ultraviolet light exposure,” Appl. Phys. Lett.72(19), 2403–2405 (1998).
[CrossRef]

1996 (1)

1995 (1)

J. L. Pezzaniti and R. A. Chipman, “Mueller Matrix Imaging Polarimetry,” Opt. Eng.34(6), 1558–1568 (1995).
[CrossRef]

Andreou, A. G.

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J.2(6), 566–576 (2002).
[CrossRef]

Arwin, H.

H. Arwin, R. Magnusson, J. Landin, and K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Philos. Mag.92(12), 1583–1599 (2012).
[CrossRef]

Bachman, K. A.

Bermak, A.

Boussaid, F.

Brock, N.

Cain, S. C.

Chenault, D. B.

Chigrinov, V. G.

Chipman, R. A.

Collins, R. T.

Deguzman, P. C.

Dereniak, E. L.

Descour, M. R.

Elsner, A. E.

Engheta, N.

Flammer, P. D.

Furtak, T. E.

Gao, S. K.

Goldstein, D. H.

Goldstein, D. L.

Gruev, V.

Hayes, J.

Hollingsworth, R. E.

Hsu, I. J.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

Järrendahl, K.

H. Arwin, R. Magnusson, J. Landin, and K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Philos. Mag.92(12), 1583–1599 (2012).
[CrossRef]

Jones, M. W.

Kalayjian, Z. K.

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J.2(6), 566–576 (2002).
[CrossRef]

Kemme, S. A.

Kiang, Y.-W.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

Kim, J.-H.

J.-H. Kim, S. Kumar, and S.-D. Lee, “Alignment of liquid crystals on polyimide films exposed to ultraviolet light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics57(5), 5644–5650 (1998).
[CrossRef]

Kumar, S.

J.-H. Kim, S. Kumar, and S.-D. Lee, “Alignment of liquid crystals on polyimide films exposed to ultraviolet light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics57(5), 5644–5650 (1998).
[CrossRef]

LaCasse, C. F.

Landin, J.

H. Arwin, R. Magnusson, J. Landin, and K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Philos. Mag.92(12), 1583–1599 (2012).
[CrossRef]

Lazarus, N.

Lee, S.-D.

J.-H. Kim, S. Kumar, and S.-D. Lee, “Alignment of liquid crystals on polyimide films exposed to ultraviolet light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics57(5), 5644–5650 (1998).
[CrossRef]

LeMaster, D. A.

Lu, C.-W.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

Lu, L.-S.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

Magnusson, R.

H. Arwin, R. Magnusson, J. Landin, and K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Philos. Mag.92(12), 1583–1599 (2012).
[CrossRef]

Meier, J. T.

Millerd, J.

Myhre, G.

Nishikawa, M.

M. Nishikawa, B. Taheri, and J. L. West, “Mechanism of unidirectional liquid-crystal alignment on polyimides with linearly polarized ultraviolet light exposure,” Appl. Phys. Lett.72(19), 2403–2405 (1998).
[CrossRef]

Nordin, G. P.

North-Morris, M.

Novak, M.

Ortu, A.

Pau, S.

Peltzer, J. J.

Perkins, R.

Pezzaniti, J. L.

J. L. Pezzaniti and R. A. Chipman, “Mueller Matrix Imaging Polarimetry,” Opt. Eng.34(6), 1558–1568 (1995).
[CrossRef]

Phipps, G. S.

Pugh, E. N.

Rowe, M. P.

Sabatke, D. S.

Sayyad, A.

Shaw, J. A.

Sun, C.-W.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

Sweatt, W. C.

Taheri, B.

M. Nishikawa, B. Taheri, and J. L. West, “Mechanism of unidirectional liquid-crystal alignment on polyimides with linearly polarized ultraviolet light exposure,” Appl. Phys. Lett.72(19), 2403–2405 (1998).
[CrossRef]

Tsai, M.-T.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

Twietmeyer, K. M.

Tyo, J. S.

Van der Spiegel, J.

VanNasdale, D.

Wang, Y.-M.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

West, J. L.

M. Nishikawa, B. Taheri, and J. L. West, “Mechanism of unidirectional liquid-crystal alignment on polyimides with linearly polarized ultraviolet light exposure,” Appl. Phys. Lett.72(19), 2403–2405 (1998).
[CrossRef]

Wu, C.-C.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

Wyant, J.

Yang, C. C.

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

York, T.

Zhao, X.

Zhao, Y.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

M. Nishikawa, B. Taheri, and J. L. West, “Mechanism of unidirectional liquid-crystal alignment on polyimides with linearly polarized ultraviolet light exposure,” Appl. Phys. Lett.72(19), 2403–2405 (1998).
[CrossRef]

IEEE Sens. J. (1)

A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J.2(6), 566–576 (2002).
[CrossRef]

J. Biomed. Opt. (1)

C.-W. Sun, Y.-M. Wang, L.-S. Lu, C.-W. Lu, I. J. Hsu, M.-T. Tsai, C. C. Yang, Y.-W. Kiang, and C.-C. Wu, “Myocardial tissue characterization based on a polarization-sensitive optical coherence tomography system with an ultrashort pulsed laser,” J. Biomed. Opt.11(5), 054016 (2006).
[CrossRef] [PubMed]

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

Opt. Eng. (1)

J. L. Pezzaniti and R. A. Chipman, “Mueller Matrix Imaging Polarimetry,” Opt. Eng.34(6), 1558–1568 (1995).
[CrossRef]

Opt. Express (8)

K. A. Bachman, J. J. Peltzer, P. D. Flammer, T. E. Furtak, R. T. Collins, and R. E. Hollingsworth, “Spiral plasmonic nanoantennas as circular polarization transmission filters,” Opt. Express20(2), 1308–1319 (2012).
[CrossRef] [PubMed]

X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging invisible spectrum,” Opt. Express18(17), 17776–17787 (2010).
[CrossRef] [PubMed]

V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express18(18), 19087–19094 (2010).
[CrossRef] [PubMed]

V. Gruev, A. Ortu, N. Lazarus, J. Van der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express15(8), 4994–5007 (2007).
[CrossRef] [PubMed]

G. Myhre, A. Sayyad, and S. Pau, “Patterned color liquid crystal polymer polarizers,” Opt. Express18(26), 27777–27786 (2010).
[CrossRef] [PubMed]

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express16(26), 21339–21354 (2008).
[CrossRef] [PubMed]

S. K. Gao and V. Gruev, “Bilinear and bicubic interpolation methods for division of focal plane polarimeters,” Opt. Express19(27), 26161–26173 (2011).
[CrossRef] [PubMed]

C. F. LaCasse, R. A. Chipman, and J. S. Tyo, “Band limited data reconstruction in modulated polarimeters,” Opt. Express19(16), 14976–14989 (2011).
[CrossRef] [PubMed]

Opt. Lett. (1)

Philos. Mag. (1)

H. Arwin, R. Magnusson, J. Landin, and K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Philos. Mag.92(12), 1583–1599 (2012).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

J.-H. Kim, S. Kumar, and S.-D. Lee, “Alignment of liquid crystals on polyimide films exposed to ultraviolet light,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics57(5), 5644–5650 (1998).
[CrossRef]

Other (4)

R. A. Chipman, “Polarized Light and Polarimetry” (University of Arizona, 2010).

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, Pixelated Phase-Mask Dynamic Interferometers, W. Osten, ed. (Springer Berlin Heidelberg, 2006), pp. 640–647.

B. E. Bayer, “Color imaging array,” U.S. Patent 3,971,065 (1976).

D. H. Goldstein, Polarized Light (CRC Press, 2011).

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Figures (17)

Fig. 1
Fig. 1

(a) A linear polarizer focal plane array comprised of 0°, 45°, 90°, and 135° linear polarizers. (b) Each different polarizer orientation transmits a differing polarization state and that intensity is measured by the individual pixel.

Fig. 2
Fig. 2

(a) Alignment marks defined on the bare borosilicate wafer using a chrome lift off process. (b) Alignment marks on the second mask index the chrome mask 7.4 μm in each direction. (c) The majority of the mask is comprised of 7.4 μm boxes that cover every fourth pixel. Each exposure through the mask defines a quarter of the macro pixel.

Fig. 3
Fig. 3

(a) A micrograph of a completed FPA shows the four orientations in a macro pixel. (b) The sensor with the aligned and affixed FPA. (c) The sensor is replaced in the SBIG Camera with the modified version. (d) The SBIG ST-2000XM Camera with a Nikon 50 mm F-mount lens.

Fig. 4
Fig. 4

(a) Bulk transmission of the sample with unpolarized, polarized perpendicular, and polarized parallel light incident is shown. Continuous measurements were taken on a spectrometer (lines) and were confirmed with discrete measurements on a polarimeter (circles). (b) The extinction ratio as a function of wavelength. (c) The chemical components of the coated dye doped LCP.

Fig. 5
Fig. 5

Horizontal cut lines are shown for linear diattenuation, linear diattenuation orientation, and depolarization taken at 600 nm. The sample has a slight tilt, resulting in measurements that shift from the center of the pixel on the left to the boundary area between two pixels on the right

Fig. 6
Fig. 6

The DOLP is measured as a function of the fast axis orientation of a 98.96° retarder at 585 nm. The circles mark the measurements and the solid line is the theoretical prediction. The average standard deviation is 5.31%. The images on the right show a 250x250 pixel area captured at different orientations of the retarder.

Fig. 7
Fig. 7

The 1000x1000 pixel image was taken at f/5.6 with a 0.5 second exposure. The average DOLP of the polarizer regions is 0.782 ± 0.111.

Fig. 8
Fig. 8

A 50mm f/5.6 lens with a 0.01 second exposure was used to image a parked car.

Fig. 9
Fig. 9

Each retarder and polarizer orientation combination transmits a differing polarization state. The macro-pixel is comprised of a 0°, 45°, right-hand circular, and 90° polarizers. (left to right)

Fig. 10
Fig. 10

Horizontal cut lines are shown for linear and circular diattenuation at 600 nm. The diattenuation alternates between primarily circular and linear for the two pixels in the cutline.

Fig. 11
Fig. 11

The DOLP and DOCP are measured as a function of the fast axis orientation of an 89.1° retarder at 585 nm. The circles mark the measurements and the solid line the theoretical prediction. The error bars represent one standard deviation in the DOCP or DOLP of the scene. Imbalance design in the measurement space causes large variance in the standard deviation.

Fig. 12
Fig. 12

A 5 second exposure with a 100mm f/11 lens and a 5 nm bandpass filter centered at 580 nm. The image is of a beam chopper with linear polarizers in the outer windows and right circular polarizers in the inner widows.

Fig. 13
Fig. 13

A 10 second exposure with a 100mm f/11 lens and a 5 nm bandpass filter centered at 580 nm. The image is of a Plusiotis optima beetle

Fig. 14
Fig. 14

Polarimeter designs can be illustrated on a Poincaré Sphere. The red dots represent the measurement states of each polarimeter. (a) The linear stokes DofP polarimeter. (b) The full Stokes DoFP polarimeter. (c) A possible optimized full Stokes polarimeter.

Fig. 15
Fig. 15

The diagram shows how spacing between the FPA and CCD results in a periodic artifacts in the Stokes value measurement. The images show how the period of the effect is proportional to the focal length of the system.

Fig. 16
Fig. 16

As the f-number decrease the marginal ray angle increases. This resulting averaging from neighboring pixels decreases the effective extinction ratio and, in the extreme, can alter the polarizer orientation.

Fig. 17
Fig. 17

The f-number is plotted versus the extinction ratio for a 50 mm, 100 mm, and 500 mm lens.

Equations (13)

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

S (x,y)=[ S 0 (x,y) S 1 (x,y) S 2 (x,y) S 3 (x,y) ]=[ I 0° (x,y)+ I 90° (x,y) I 0° (x,y) I 90° (x,y) I 45° (x,y) I 135° (x,y) I RH (x,y) I LH (x,y) ]
θ linear = 1 2 tan 1 S 2 S 1
DOP= S 1 2 + S 2 2 + S 3 2 / S 0
DOLP= S 1 2 + S 2 2 / S 0
DOCP= S 3 / S 0
S out = M sys S in
S out,0 = M 0,0 S in,0 + M 0,1 S in,1 + M 0,2 S in,2 + M 0,3 S in,3 =A S in
I=[ S out,0 1 S out,0 1 S out,0 n ]=[ A 1 A 2 A n ] S in =W S in
S ˙ in = W 1 I
I out,0 0° = W 0,0 + W 0,1 =M S 0°
I out,0 90° = W 0,0 W 0,1 =M S 90°
2 W 0,0 = I out,0 0° + I out,0 90°
2 W 0,1 = I out,0 0° I out,0 90°

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