Abstract

In the field of polarimetry, ferroelectric liquid crystal cells are mostly used as bistable polarization rotators suitable to analyze crossed polarizations. This paper shows that, provided such a cell is used at its nominal wavelength and correctly driven, its behavior is close to that of a tunable half-wave plate, and it can be used with much benefit in lightweight imaging polarimetric setups. A partial Stokes polarimeter using a single digital video camera and a single ferroelectric liquid crystal modulator is designed and implemented for linear polarization analysis. Polarization azimuthal angle and degree of linear polarization are available at 150 frames per second with a good accuracy.

© 2010 Optical Society of America

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References

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2009 (1)

K. Fujita, Y. Itoh, and T. Mukai, “Development of simultaneous imaging polarimeter for asteroids,” Adv. Space. Res. 43, 325–327 (2009).
[CrossRef]

2008 (4)

J. S. Harchanko, L. Pezzaniti, D. Chenault, and G. Eades, “Comparing a MWIR and LWIR polarimetric imager for surface swimmer detection,” Proc. SPIE 6972, 697211 (2008).
[CrossRef]

D. A. Lavigne, M. Breton, M. Pichette, V. Larochelle, and J.-R. Simard, “Evaluation of active and passive polarimetric electro-optic imagery for civilian and military targets discrimination,” Proc. SPIE 6972, 69720X (2008).
[CrossRef]

A. Jaulin, L. Bigué, and P. Ambs, “High-speed degree-of-polarization imaging with a ferroelectric liquid-crystal modulator,” Opt. Eng. 47, 033201 (2008).
[CrossRef]

A. Jaulin and L. Bigué, “High speed partial stokes imaging using a ferroelectric liquid crystal modulator,” J. Eur. Opt. Soc. Rap. Pub. 3, 08019 (2008).
[CrossRef]

2007 (4)

L. Bigué and N. Cheney, “High-speed portable polarimeter using a ferroelectric liquid crystal modulator,” Proc. SPIE 6682, 668205 (2007).
[CrossRef]

E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous stokes imaging polarimeter,” Proc. SPIE 6682, 668215 (2007).
[CrossRef]

A. Jaulin and L. Bigué, “High speed linear polarization evaluation with a single light modulator,” in EOS 3rd Topical Meeting on Advanced Imaging Techniques (European Optical Society, 2007).

A. Gruev, A. Ortu, N. Lazarus, J. van der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express 15, 4994–5007 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

2004 (2)

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

F. A. Sadjadi and C. S. L. Chun, “Remote sensing using passive infrared stokes parameters,” Opt. Eng. 43, 2283–2291 (2004).
[CrossRef]

2003 (1)

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

2000 (2)

1999 (2)

A. M. Gandorfer, “Ferroelectric retarders as an alternative to piezoelastic modulators for use in solar stokes vector polarimetry,” Opt. Eng. 38, 1402–1408 (1999).
[CrossRef]

J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Lett. 24, 64–66 (1999).
[CrossRef]

1998 (1)

1997 (1)

L. B. Wolff, T. A. Mancini, P. Pouliquen, and A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Robot. Autom. 13, 195–203 (1997).
[CrossRef]

1996 (1)

1995 (1)

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M.Bass, ed. (McGraw-Hill, 1995).

1991 (1)

K.-F. Reinhart, L. Dorfmüller, K. Marx, and T. Matszczyk, “Addressing of ferroelectric liquid crystal matrices and electrooptical characterization,” Ferroelectrics 113, 405–417 (1991).
[CrossRef]

1986 (1)

1981 (1)

1977 (1)

R. Walraven, “Polarization imagery,” Proc. SPIE 112, 164–167(1977).

1852 (1)

G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Camb. Philos. Soc. 9, 339–416 (1852).

Alouini, M.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

Ambs, P.

A. Jaulin, L. Bigué, and P. Ambs, “High-speed degree-of-polarization imaging with a ferroelectric liquid-crystal modulator,” Opt. Eng. 47, 033201 (2008).
[CrossRef]

Andreou, A. G.

L. B. Wolff, T. A. Mancini, P. Pouliquen, and A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Robot. Autom. 13, 195–203 (1997).
[CrossRef]

Artal, P.

Bigué, L.

A. Jaulin, L. Bigué, and P. Ambs, “High-speed degree-of-polarization imaging with a ferroelectric liquid-crystal modulator,” Opt. Eng. 47, 033201 (2008).
[CrossRef]

A. Jaulin and L. Bigué, “High speed partial stokes imaging using a ferroelectric liquid crystal modulator,” J. Eur. Opt. Soc. Rap. Pub. 3, 08019 (2008).
[CrossRef]

A. Jaulin and L. Bigué, “High speed linear polarization evaluation with a single light modulator,” in EOS 3rd Topical Meeting on Advanced Imaging Techniques (European Optical Society, 2007).

L. Bigué and N. Cheney, “High-speed portable polarimeter using a ferroelectric liquid crystal modulator,” Proc. SPIE 6682, 668205 (2007).
[CrossRef]

Brandt, R.

E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous stokes imaging polarimeter,” Proc. SPIE 6682, 668215 (2007).
[CrossRef]

Breton, M.

D. A. Lavigne, M. Breton, M. Pichette, V. Larochelle, and J.-R. Simard, “Evaluation of active and passive polarimetric electro-optic imagery for civilian and military targets discrimination,” Proc. SPIE 6972, 69720X (2008).
[CrossRef]

Bueno, J. M.

Chenault, D.

J. S. Harchanko, L. Pezzaniti, D. Chenault, and G. Eades, “Comparing a MWIR and LWIR polarimetric imager for surface swimmer detection,” Proc. SPIE 6972, 697211 (2008).
[CrossRef]

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

Cheney, N.

L. Bigué and N. Cheney, “High-speed portable polarimeter using a ferroelectric liquid crystal modulator,” Proc. SPIE 6682, 668205 (2007).
[CrossRef]

Chipman, R. A.

Chun, C. S. L.

F. A. Sadjadi and C. S. L. Chun, “Remote sensing using passive infrared stokes parameters,” Opt. Eng. 43, 2283–2291 (2004).
[CrossRef]

Clémenceau, P.

P. Clémenceau, A. Dogariu, and J. Stryewski, “Polarization active imaging,” Proc. SPIE 4035, 401–409 (2000).
[CrossRef]

de Leon, E.

E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous stokes imaging polarimeter,” Proc. SPIE 6682, 668215 (2007).
[CrossRef]

Dereniak, E. L.

Descour, M. R.

Dogariu, A.

P. Clémenceau, A. Dogariu, and J. Stryewski, “Polarization active imaging,” Proc. SPIE 4035, 401–409 (2000).
[CrossRef]

Dolfi, D.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

Dorfmüller, L.

K.-F. Reinhart, L. Dorfmüller, K. Marx, and T. Matszczyk, “Addressing of ferroelectric liquid crystal matrices and electrooptical characterization,” Ferroelectrics 113, 405–417 (1991).
[CrossRef]

Eades, G.

J. S. Harchanko, L. Pezzaniti, D. Chenault, and G. Eades, “Comparing a MWIR and LWIR polarimetric imager for surface swimmer detection,” Proc. SPIE 6972, 697211 (2008).
[CrossRef]

Engheta, N.

Fujita, K.

K. Fujita, Y. Itoh, and T. Mukai, “Development of simultaneous imaging polarimeter for asteroids,” Adv. Space. Res. 43, 325–327 (2009).
[CrossRef]

Gandorfer, A. M.

A. M. Gandorfer, “Ferroelectric retarders as an alternative to piezoelastic modulators for use in solar stokes vector polarimetry,” Opt. Eng. 38, 1402–1408 (1999).
[CrossRef]

Goldstein, D.

Goudail, F.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

Grisard, A.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

Gruev, A.

Harchanko, J. S.

J. S. Harchanko, L. Pezzaniti, D. Chenault, and G. Eades, “Comparing a MWIR and LWIR polarimetric imager for surface swimmer detection,” Proc. SPIE 6972, 697211 (2008).
[CrossRef]

Itoh, Y.

K. Fujita, Y. Itoh, and T. Mukai, “Development of simultaneous imaging polarimeter for asteroids,” Adv. Space. Res. 43, 325–327 (2009).
[CrossRef]

Jaulin, A.

A. Jaulin, L. Bigué, and P. Ambs, “High-speed degree-of-polarization imaging with a ferroelectric liquid-crystal modulator,” Opt. Eng. 47, 033201 (2008).
[CrossRef]

A. Jaulin and L. Bigué, “High speed partial stokes imaging using a ferroelectric liquid crystal modulator,” J. Eur. Opt. Soc. Rap. Pub. 3, 08019 (2008).
[CrossRef]

A. Jaulin and L. Bigué, “High speed linear polarization evaluation with a single light modulator,” in EOS 3rd Topical Meeting on Advanced Imaging Techniques (European Optical Society, 2007).

Kemme, S. A.

Lallier, E.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

Larochelle, V.

D. A. Lavigne, M. Breton, M. Pichette, V. Larochelle, and J.-R. Simard, “Evaluation of active and passive polarimetric electro-optic imagery for civilian and military targets discrimination,” Proc. SPIE 6972, 69720X (2008).
[CrossRef]

Lavigne, D. A.

D. A. Lavigne, M. Breton, M. Pichette, V. Larochelle, and J.-R. Simard, “Evaluation of active and passive polarimetric electro-optic imagery for civilian and military targets discrimination,” Proc. SPIE 6972, 69720X (2008).
[CrossRef]

Lazarus, N.

Lu, S.-Y.

Mancini, T. A.

L. B. Wolff, T. A. Mancini, P. Pouliquen, and A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Robot. Autom. 13, 195–203 (1997).
[CrossRef]

Marx, K.

K.-F. Reinhart, L. Dorfmüller, K. Marx, and T. Matszczyk, “Addressing of ferroelectric liquid crystal matrices and electrooptical characterization,” Ferroelectrics 113, 405–417 (1991).
[CrossRef]

Matszczyk, T.

K.-F. Reinhart, L. Dorfmüller, K. Marx, and T. Matszczyk, “Addressing of ferroelectric liquid crystal matrices and electrooptical characterization,” Ferroelectrics 113, 405–417 (1991).
[CrossRef]

Mukai, T.

K. Fujita, Y. Itoh, and T. Mukai, “Development of simultaneous imaging polarimeter for asteroids,” Adv. Space. Res. 43, 325–327 (2009).
[CrossRef]

Ortu, A.

Pezzaniti, L.

J. S. Harchanko, L. Pezzaniti, D. Chenault, and G. Eades, “Comparing a MWIR and LWIR polarimetric imager for surface swimmer detection,” Proc. SPIE 6972, 697211 (2008).
[CrossRef]

Phenis, A.

E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous stokes imaging polarimeter,” Proc. SPIE 6682, 668215 (2007).
[CrossRef]

Phipps, G. S.

Pichette, M.

D. A. Lavigne, M. Breton, M. Pichette, V. Larochelle, and J.-R. Simard, “Evaluation of active and passive polarimetric electro-optic imagery for civilian and military targets discrimination,” Proc. SPIE 6972, 69720X (2008).
[CrossRef]

Pouliquen, P.

L. B. Wolff, T. A. Mancini, P. Pouliquen, and A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Robot. Autom. 13, 195–203 (1997).
[CrossRef]

Réfrégier, P.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

Reinhart, K.-F.

K.-F. Reinhart, L. Dorfmüller, K. Marx, and T. Matszczyk, “Addressing of ferroelectric liquid crystal matrices and electrooptical characterization,” Ferroelectrics 113, 405–417 (1991).
[CrossRef]

Sabatke, D. S.

Sadjadi, F. A.

F. A. Sadjadi and C. S. L. Chun, “Remote sensing using passive infrared stokes parameters,” Opt. Eng. 43, 2283–2291 (2004).
[CrossRef]

Shaw, J. A.

Simard, J.-R.

D. A. Lavigne, M. Breton, M. Pichette, V. Larochelle, and J.-R. Simard, “Evaluation of active and passive polarimetric electro-optic imagery for civilian and military targets discrimination,” Proc. SPIE 6972, 69720X (2008).
[CrossRef]

Solomon, J.

Stokes, G.

G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Camb. Philos. Soc. 9, 339–416 (1852).

Stryewski, J.

P. Clémenceau, A. Dogariu, and J. Stryewski, “Polarization active imaging,” Proc. SPIE 4035, 401–409 (2000).
[CrossRef]

Sweatt, W. C.

Tyo, J. S.

van der Spiegel, J.

Virgen, M.

E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous stokes imaging polarimeter,” Proc. SPIE 6682, 668215 (2007).
[CrossRef]

Walraven, R.

R. Walraven, “Polarization imagery,” Proc. SPIE 112, 164–167(1977).

Williams, M. W.

Wolff, L. B.

L. B. Wolff, T. A. Mancini, P. Pouliquen, and A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Robot. Autom. 13, 195–203 (1997).
[CrossRef]

Adv. Space. Res. (1)

K. Fujita, Y. Itoh, and T. Mukai, “Development of simultaneous imaging polarimeter for asteroids,” Adv. Space. Res. 43, 325–327 (2009).
[CrossRef]

Appl. Opt. (4)

Ferroelectrics (1)

K.-F. Reinhart, L. Dorfmüller, K. Marx, and T. Matszczyk, “Addressing of ferroelectric liquid crystal matrices and electrooptical characterization,” Ferroelectrics 113, 405–417 (1991).
[CrossRef]

IEEE Trans. Robot. Autom. (1)

L. B. Wolff, T. A. Mancini, P. Pouliquen, and A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Robot. Autom. 13, 195–203 (1997).
[CrossRef]

J. Eur. Opt. Soc. Rap. Pub. (1)

A. Jaulin and L. Bigué, “High speed partial stokes imaging using a ferroelectric liquid crystal modulator,” J. Eur. Opt. Soc. Rap. Pub. 3, 08019 (2008).
[CrossRef]

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

Opt. Eng. (3)

A. Jaulin, L. Bigué, and P. Ambs, “High-speed degree-of-polarization imaging with a ferroelectric liquid-crystal modulator,” Opt. Eng. 47, 033201 (2008).
[CrossRef]

F. A. Sadjadi and C. S. L. Chun, “Remote sensing using passive infrared stokes parameters,” Opt. Eng. 43, 2283–2291 (2004).
[CrossRef]

A. M. Gandorfer, “Ferroelectric retarders as an alternative to piezoelastic modulators for use in solar stokes vector polarimetry,” Opt. Eng. 38, 1402–1408 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (7)

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[CrossRef]

L. Bigué and N. Cheney, “High-speed portable polarimeter using a ferroelectric liquid crystal modulator,” Proc. SPIE 6682, 668205 (2007).
[CrossRef]

D. A. Lavigne, M. Breton, M. Pichette, V. Larochelle, and J.-R. Simard, “Evaluation of active and passive polarimetric electro-optic imagery for civilian and military targets discrimination,” Proc. SPIE 6972, 69720X (2008).
[CrossRef]

E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous stokes imaging polarimeter,” Proc. SPIE 6682, 668215 (2007).
[CrossRef]

R. Walraven, “Polarization imagery,” Proc. SPIE 112, 164–167(1977).

J. S. Harchanko, L. Pezzaniti, D. Chenault, and G. Eades, “Comparing a MWIR and LWIR polarimetric imager for surface swimmer detection,” Proc. SPIE 6972, 697211 (2008).
[CrossRef]

P. Clémenceau, A. Dogariu, and J. Stryewski, “Polarization active imaging,” Proc. SPIE 4035, 401–409 (2000).
[CrossRef]

Trans. Camb. Philos. Soc. (1)

G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Camb. Philos. Soc. 9, 339–416 (1852).

Other (4)

URL:http://www.bnonlinear.com/products/polarRotators/polarRotators.htm.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M.Bass, ed. (McGraw-Hill, 1995).

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

A. Jaulin and L. Bigué, “High speed linear polarization evaluation with a single light modulator,” in EOS 3rd Topical Meeting on Advanced Imaging Techniques (European Optical Society, 2007).

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Figures (16)

Fig. 1
Fig. 1

Polarimeter scheme and photo. The modulator is composed of an FLC cell and a linear polarizer.

Fig. 2
Fig. 2

Setup for partial Mueller characterization of the FLC device. Microscope objective (MO), Pin Hole (PH), Lenses ( L 1 and L 2 ), static linear polarizers oriented at 0 ° ( P 1 and P 2 ), rotating half-wave plate ( λ / 2 ), ferroelectric liquid crystal cell (FLC).

Fig. 3
Fig. 3

Blue continuous line: average measured intensities (a) I h , (b) I 45 , (c) I v , and (d) I 135 of five acquisitions versus FLC control voltage for each input Stokes vectors S h , S 45 , S v , and S 135 . Red dashed line: intensities theoretically measured with an ideal half-wave plate whose orientation varies according to Fig. 4.

Fig. 4
Fig. 4

FLC orientation angle estimation (average of five acquisitions).

Fig. 5
Fig. 5

Variation of intensity (a) I h + I v and (b) zoom at a specific voltage range of the curve I h + I v .

Fig. 6
Fig. 6

(a) Bipolar control signal, (b) composite control signal, (c) FLC cell responses to bipolar and composite signals at specific amplitudes of 5 V , and (d) 0.1 V for S h polarization input.

Fig. 7
Fig. 7

Condition number for ideal and experimental systems. The red square denotes our actual configuration.

Fig. 8
Fig. 8

(a) Estimated angle in practice and theoretical angle of the polarization of the light emerging from a rotating half-wave plate. (b) Angle estimation error.

Fig. 9
Fig. 9

Estimation of the DOLP of the light emerging from a rotating half-wave plate.

Fig. 10
Fig. 10

Setup for imaging validation: microscope objective (MO), pinhole (PH), lens ( L 1 ) quarter-wave plate ( λ / 4 ), ferroelectric liquid crystal cell (FLC), static linear polarizers oriented at 0 ° (P).

Fig. 11
Fig. 11

Test sample used for imaging validation: a microscope plate with three strips of linear polarizer.

Fig. 12
Fig. 12

Stokes vector estimation, at a frame rate of 150 fps , for a test sample oriented at 0 ° . (a) s 0 , (b) | s 1 | / s 0 , and (c) | s 2 | / s 0 . Polarizer orientations are mentioned in (b).

Fig. 13
Fig. 13

Angle estimation, at a frame rate of 150 fps , for different rotation angles of the test sample: (a) 0 ° , (b) 40 ° , (c) 90 ° , and (d) 130 ° . (e) Color bar. The reference position at 0 ° is arbitrarily defined so that one polarizer strip is actually aligned with the modulator axis. Polarizers orientations are mentioned in (a).

Fig. 14
Fig. 14

DOLP estimation, at a frame rate of 150 fps , for different rotation angles of the test sample: (a) 0 ° , (b) 40 ° , (c) 90 ° , and (d) 130 ° . (e) Color bar. Polarizers orientations are mentioned in (a).

Fig. 15
Fig. 15

Regions of interest used to quantify the results, presented on a DOLP picture of a test sample oriented at 0 ° . Polarizers orientations are also mentioned.

Fig. 16
Fig. 16

Spatial average error and standard deviation of the angle and DOLP estimation, at a frame rate of 150 fps , for a test sample oriented at 0 ° . Blue line with dots: polarizer strip oriented at 0 ° . Green line with squares: polarizer strip oriented at 45 ° . Red line with diamonds: polarizer strip oriented at 90 ° .

Equations (22)

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DOLP = s 1 2 + s 2 2 / s 0 ,
tan ( 2 ψ ) = s 2 s 1 .
S out = M . S in .
M Pol ( β ) = 1 2 [ 1 cos ( 2 β ) sin ( 2 β ) 0 cos ( 2 β ) cos 2 ( 2 β ) cos ( 2 β ) sin ( 2 β ) 0 sin ( 2 β ) cos ( 2 β ) sin ( 2 β ) sin 2 ( 2 β ) 0 0 0 0 0 ] ,
M λ 2 ( θ ) = [ 1 0 0 0 0 cos ( 4 θ ) sin ( 4 θ ) 0 0 sin ( 4 θ ) cos ( 4 θ ) 0 0 0 0 1 ] .
I ( θ ) = s out 0 ( θ ) = 1 2 [ s in 0 + s in 1 . cos ( 4 θ ) + s in 2 . sin ( 4 θ ) ] .
[ I 1 I 2 I 3 ] = A [ s in 0 s in 1 s in 2 ] ,
A = 1 2 [ 1 cos ( 4 θ 1 ) sin ( 4 θ 1 ) 1 cos ( 4 θ 2 ) sin ( 4 θ 2 ) 1 cos ( 4 θ 3 ) sin ( 4 θ 3 ) ] .
[ s in 0 s in 1 s in 2 ] = A 1 [ I 1 I 2 I 3 ] .
M Δ = [ 1 0 0 0 0 d 0 0 0 0 d 0 0 0 0 d ] ,
M λ 2 ( θ ) = [ 1 0 0 0 0 d cos ( 4 θ ) d sin ( 4 θ ) 0 0 d sin ( 4 θ ) d cos ( 4 θ ) 0 0 0 0 d ] ,
I ( θ ) = s out 0 ( θ ) = 1 2 [ s in 0 + s in 1 . d . cos ( 4 θ ) + s in 2 . d . sin ( 4 θ ) ] ,
A = 1 2 [ 1 d cos ( 4 θ 1 ) d sin ( 4 θ 1 ) 1 d cos ( 4 θ 2 ) d sin ( 4 θ 2 ) 1 d cos ( 4 θ 3 ) d sin ( 4 θ 3 ) ] .
{ S h , S v , S 45 , S 135 } = { [ 1 1 0 0 ] , [ 1 1 0 0 ] , [ 1 0 1 0 ] , [ 1 0 1 0 ] } .
{ I h = 1 2 [ 1 + d cos ( 4 θ ) ] for S h in input I v = 1 2 [ 1 d cos ( 4 θ ) ] for S v in input I 45 = 1 2 [ 1 + d sin ( 4 θ ) ] for S 45 in input I 135 = 1 2 [ 1 d sin ( 4 θ ) ] for S 135 in input ,
{ θ = 1 4 arctan ( I 45 I 135 I h I v ) for I h I v θ = + 22.5 ° for I h = I v and I 135 = 0 θ = 22.5 ° for I h = I v and I 45 = 0 ,
d = ( I h I v ) 2 + ( I 45 I 135 ) 2 .
I ( V ) = s out 0 ( V ) = 1 2 [ s in 0 . A ( V ) + s in 1 . B ( V ) + s in 2 . C ( V ) + s in 3 . D ( V ) ] .
{ I h ( V ) = 1 2 [ A ( V ) + B ( V ) ] for S h in input I v ( V ) = 1 2 [ A ( V ) B ( V ) ] for S v in input I 45 ( V ) = 1 2 [ A ( V ) + C ( V ) ] for S 45 in input I 135 ( V ) = 1 2 [ A ( V ) C ( V ) ] for S 135 in input .
A = 1 2 [ I h ( V 1 ) + I v ( V 1 ) + I 45 ( V 1 ) + I 135 ( V 1 ) 2 I h ( V 1 ) I v ( V 1 ) I 45 ( V 1 ) I 135 ( V 1 ) I h ( V 2 ) + I v ( V 2 ) + I 45 ( V 2 ) + I 135 ( V 2 ) 2 I h ( V 2 ) I v ( V 2 ) I 45 ( V 2 ) I 135 ( V 2 ) I h ( V 3 ) + I v ( V 3 ) + I 45 ( V 3 ) + I 135 ( V 3 ) 2 I h ( V 3 ) I v ( V 3 ) I 45 ( V 3 ) I 135 ( V 3 ) ] .
A 1 = [ 0.7743 0.2453 0.6094 0.7140 0.1844 0.5375 1.4477 1.8313 0.3789 ] .
A 1 = [ 1 0 1 1 0 1 1 2 1 ] .

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