Abstract

The final product of polarimetric measurements is often such polarimetric parameters as degree of polarization (DOP), angle of polarization (AOP) and ellipticity (EOP). Since these parameters are nonlinear functions of the Stokes vector, it is difficult to derive closed-form expressions of their variances. We derive approximate but accurate expressions of the estimation variances of DOP, AOP, and EOP in the presence of both additive and Poisson noise for optimal spherical design-based Stokes polarimeters. These original closed-from expressions provide a clear insight into the physical parameters that govern the estimation precision of each polarimetric parameter. They are validated through optical experiments on a real-world polarimeter. These expressions are important for designing and sizing polarimeters or polarimetric imagers aimed at different types of applications, and for assessing their performance.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  6. J. Naghizadeh-Khouei and D. Clarke, “On the statistical behaviour of the position angle of linear polarization,” Astronomy and Astrophysics 274, 968–974 (1993).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

2017 (1)

2016 (1)

2015 (1)

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Physical Review Letters 115, 263901 (2015).
[Crossref]

2014 (2)

M. Kupinski, R. Chipman, and E. Clarkson, “Relating the statistics of the angle of linear polarization to measurement uncertainty of the Stokes vector,” Optical Engineering 53, 113108 (2014).
[Crossref]

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

2013 (1)

M. Gecevičius, M. Beresna, and P. G. Kazansky, “Polarization sensitive camera by femtosecond laser nanostructuring,” Optics Letters 38, 4096–4099 (2013).
[Crossref]

2012 (1)

G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Applied Optics 51, 5302–5309 (2012).
[Crossref] [PubMed]

2010 (3)

2009 (1)

2008 (1)

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive poisson noises,” Journal of the European Optical Society Rapid Publications 3, 08002 (2008).
[Crossref]

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, 5453–5469 (2006).
[Crossref] [PubMed]

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” App. Opt. 45, 4062–4068 (2006).
[Crossref]

H. Shao, Y. He, W. Li, and H. Ma, “Polarization-degree imaging contrast in turbid media: a quantitative study,” Applied Optics 45, 4491–4496 (2006).
[Crossref] [PubMed]

2000 (2)

1996 (1)

R. H. Hardin and N. J. A. Sloane, “Mclaren’s improved snub cube and other new spherical designs in three dimensions,” Discrete and Computational Geometry 15, 429–441 (1996).
[Crossref]

1993 (1)

J. Naghizadeh-Khouei and D. Clarke, “On the statistical behaviour of the position angle of linear polarization,” Astronomy and Astrophysics 274, 968–974 (1993).

1990 (1)

Y. Mimura, “A construction of spherical 2-design,” Graphs and Combinatorics 6, 369–372 (1990).
[Crossref]

1977 (1)

P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geometriae Dedicata 6, 363–388 (1977).
[Crossref]

Aiello, A.

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Physical Review Letters 115, 263901 (2015).
[Crossref]

Alouini, M.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive poisson noises,” Journal of the European Optical Society Rapid Publications 3, 08002 (2008).
[Crossref]

Anna, G.

G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Applied Optics 51, 5302–5309 (2012).
[Crossref] [PubMed]

Balakrishnan, K.

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

Bénière, A.

F. Goudail and A. Bénière, “Estimation of the degree of linear polarization and of the angle of polariztion in the presence of different sources of noise,” Appl. Opt. 49, 683–693 (2010).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive poisson noises,” Journal of the European Optical Society Rapid Publications 3, 08002 (2008).
[Crossref]

Beresna, M.

M. Gecevičius, M. Beresna, and P. G. Kazansky, “Polarization sensitive camera by femtosecond laser nanostructuring,” Optics Letters 38, 4096–4099 (2013).
[Crossref]

Bermak, A.

X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging invisible spectrum,” Optics Express 18, 17776–17787 (2010).
[Crossref]

Boffety, M.

Boussaid, F.

X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging invisible spectrum,” Optics Express 18, 17776–17787 (2010).
[Crossref]

Brock, N.

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

Campos, J.

Chenault, D. B.

Chigrinov, V. G.

X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging invisible spectrum,” Optics Express 18, 17776–17787 (2010).
[Crossref]

Chipman, R.

M. Kupinski, R. Chipman, and E. Clarkson, “Relating the statistics of the angle of linear polarization to measurement uncertainty of the Stokes vector,” Optical Engineering 53, 113108 (2014).
[Crossref]

Clarke, D.

J. Naghizadeh-Khouei and D. Clarke, “On the statistical behaviour of the position angle of linear polarization,” Astronomy and Astrophysics 274, 968–974 (1993).

Clarkson, E.

M. Kupinski, R. Chipman, and E. Clarkson, “Relating the statistics of the angle of linear polarization to measurement uncertainty of the Stokes vector,” Optical Engineering 53, 113108 (2014).
[Crossref]

Delsarte, P.

P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geometriae Dedicata 6, 363–388 (1977).
[Crossref]

Dereniak, E. L.

Descour, M. R.

Dolfi, D.

G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Applied Optics 51, 5302–5309 (2012).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive poisson noises,” Journal of the European Optical Society Rapid Publications 3, 08002 (2008).
[Crossref]

Dupont, J.

Favaro, A.

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Physical Review Letters 115, 263901 (2015).
[Crossref]

Foreman, M. R.

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Physical Review Letters 115, 263901 (2015).
[Crossref]

Gecevicius, M.

M. Gecevičius, M. Beresna, and P. G. Kazansky, “Polarization sensitive camera by femtosecond laser nanostructuring,” Optics Letters 38, 4096–4099 (2013).
[Crossref]

Goethals, J. M.

P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geometriae Dedicata 6, 363–388 (1977).
[Crossref]

Goldstein, D.

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

Goldstein, D. L.

Gorria, P.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” App. Opt. 45, 4062–4068 (2006).
[Crossref]

Goudail, F.

Hamaoui, M.

Hardin, R. H.

R. H. Hardin and N. J. A. Sloane, “Mclaren’s improved snub cube and other new spherical designs in three dimensions,” Discrete and Computational Geometry 15, 429–441 (1996).
[Crossref]

He, Y.

H. Shao, Y. He, W. Li, and H. Ma, “Polarization-degree imaging contrast in turbid media: a quantitative study,” Applied Optics 45, 4491–4496 (2006).
[Crossref] [PubMed]

Hsu, W.-L.

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

Ibn-Elhaj, M.

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

Iemmi, C.

Kazansky, P. G.

M. Gecevičius, M. Beresna, and P. G. Kazansky, “Polarization sensitive camera by femtosecond laser nanostructuring,” Optics Letters 38, 4096–4099 (2013).
[Crossref]

Kemme, S. A.

Kupinski, M.

M. Kupinski, R. Chipman, and E. Clarkson, “Relating the statistics of the angle of linear polarization to measurement uncertainty of the Stokes vector,” Optical Engineering 53, 113108 (2014).
[Crossref]

Li, W.

H. Shao, Y. He, W. Li, and H. Ma, “Polarization-degree imaging contrast in turbid media: a quantitative study,” Applied Optics 45, 4491–4496 (2006).
[Crossref] [PubMed]

Lizana, A.

Ma, H.

H. Shao, Y. He, W. Li, and H. Ma, “Polarization-degree imaging contrast in turbid media: a quantitative study,” Applied Optics 45, 4491–4496 (2006).
[Crossref] [PubMed]

Meriaudeau, F.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” App. Opt. 45, 4062–4068 (2006).
[Crossref]

Mimura, Y.

Y. Mimura, “A construction of spherical 2-design,” Graphs and Combinatorics 6, 369–372 (1990).
[Crossref]

Morel, O.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” App. Opt. 45, 4062–4068 (2006).
[Crossref]

Myhre, G.

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

Naghizadeh-Khouei, J.

J. Naghizadeh-Khouei and D. Clarke, “On the statistical behaviour of the position angle of linear polarization,” Astronomy and Astrophysics 274, 968–974 (1993).

Papoulis, A.

A. Papoulis, Probability, random variables and stochastic processes (Mc Graw-Hill, 1991).

Pau, S.

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

Peinado, A.

Phipps, G. S.

Sabatke, D. S.

Sauer, H.

G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Applied Optics 51, 5302–5309 (2012).
[Crossref] [PubMed]

Seidel, J. J.

P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geometriae Dedicata 6, 363–388 (1977).
[Crossref]

Shao, H.

H. Shao, Y. He, W. Li, and H. Ma, “Polarization-degree imaging contrast in turbid media: a quantitative study,” Applied Optics 45, 4491–4496 (2006).
[Crossref] [PubMed]

Shaw, J. A.

Sloane, N. J. A.

R. H. Hardin and N. J. A. Sloane, “Mclaren’s improved snub cube and other new spherical designs in three dimensions,” Discrete and Computational Geometry 15, 429–441 (1996).
[Crossref]

Stolz, C.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” App. Opt. 45, 4062–4068 (2006).
[Crossref]

Sweatt, W. C.

Tyo, J. S.

Vidal, J.

Zhao, X.

X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging invisible spectrum,” Optics Express 18, 17776–17787 (2010).
[Crossref]

App. Opt. (1)

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” App. Opt. 45, 4062–4068 (2006).
[Crossref]

Appl. Opt. (3)

Applied Optics (2)

H. Shao, Y. He, W. Li, and H. Ma, “Polarization-degree imaging contrast in turbid media: a quantitative study,” Applied Optics 45, 4491–4496 (2006).
[Crossref] [PubMed]

G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Applied Optics 51, 5302–5309 (2012).
[Crossref] [PubMed]

Astronomy and Astrophysics (1)

J. Naghizadeh-Khouei and D. Clarke, “On the statistical behaviour of the position angle of linear polarization,” Astronomy and Astrophysics 274, 968–974 (1993).

Discrete and Computational Geometry (1)

R. H. Hardin and N. J. A. Sloane, “Mclaren’s improved snub cube and other new spherical designs in three dimensions,” Discrete and Computational Geometry 15, 429–441 (1996).
[Crossref]

Geometriae Dedicata (1)

P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geometriae Dedicata 6, 363–388 (1977).
[Crossref]

Graphs and Combinatorics (1)

Y. Mimura, “A construction of spherical 2-design,” Graphs and Combinatorics 6, 369–372 (1990).
[Crossref]

J. Opt. Soc. Am. A (1)

Journal of the European Optical Society Rapid Publications (1)

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive poisson noises,” Journal of the European Optical Society Rapid Publications 3, 08002 (2008).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Optical Engineering (1)

M. Kupinski, R. Chipman, and E. Clarkson, “Relating the statistics of the angle of linear polarization to measurement uncertainty of the Stokes vector,” Optical Engineering 53, 113108 (2014).
[Crossref]

Optics Express (2)

X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging invisible spectrum,” Optics Express 18, 17776–17787 (2010).
[Crossref]

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Optics Express 22, 3063–3074 (2014).
[Crossref] [PubMed]

Optics Letters (1)

M. Gecevičius, M. Beresna, and P. G. Kazansky, “Polarization sensitive camera by femtosecond laser nanostructuring,” Optics Letters 38, 4096–4099 (2013).
[Crossref]

Physical Review Letters (1)

M. R. Foreman, A. Favaro, and A. Aiello, “Optimal frames for polarization state reconstruction,” Physical Review Letters 115, 263901 (2015).
[Crossref]

Other (3)

A. Papoulis, Probability, random variables and stochastic processes (Mc Graw-Hill, 1991).

Wikipedia, “Delta method,” https://en.wikipedia.org/w/index.php?title=Delta_method&oldid=841130321

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

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Figures (5)

Fig. 1
Fig. 1 Theoretical (solid line) and estimated (marker) variance of DOP estimator P ^ in the presence of AWGN (red) and PSN (blue) as a function of the true value of P. The measurement structure is an octahedron. In the presence of AWGN, SNR   g=48864 and in the presence of PSN, S 0 = 24432, so that the theoretical values of var   [ P ^ ] in the presence of AWGN and PSN are equal for P = 0.
Fig. 2
Fig. 2 Theoretical and estimated variance of DOP estimator P ^ (first row), AOP estimator α ^ (second row) and EOP estimator ε ^ (third row) as a function of the true value of α with ε = 20 (left column) and of the true value of ε with α = 29 (right column) in the presence of PSN with three kinds of measurement matrices: regular tetrahedron W R (tetra), octahedron (octa), and cube. S 0 = 24432 and P = 0.9.
Fig. 3
Fig. 3 Scheme of experimental setup.
Fig. 4
Fig. 4 Poincaré sphere representation of two sequences of input polarization states generated by PSG in the experiment. These red circles (forming meridian) indicate a class of polarization states with the same α = 29 and varying e p s i l o n ranging from -45   to 45   with step 5   . These blue circles (forming parallel) represent another class of polarization states with the same ε = 20 and varying α ranging from -45   to 45   with step 5   .
Fig. 5
Fig. 5 Experimental and theoretical variance of DOP estimator P ^ (first row), AOP estimator α ^ (second row) and EOP estimator ε ^ (third row) as a function of the true value of α (left column) and of ε(right column) in the presence of Poisson noise with two kind of measurement matrices: regular tetrahedron W R (tetra) and octahedron (octa). The theoretical values have been computed from the experimentally estimated values of P, α, l o n, and S 0, averaged over the 10,000 measurements.

Equations (29)

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

S = S 0 [ 1 P cos  2 α cos  2 ε P sin  2 α cos  2 ε P sin  2 ε ]
I = W S
S ^ = W + I
W + = ( W T W ) 1 W T
I = W S + ν S ^ = S + W + ν
Γ S ^ = σ 2 ( W T W ) 1
EWV = trace [ Γ S ^ ]
Γ S ^ = 4 N σ 2 [ 1 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3 ]
Γ i j   S ^ = k = 0 3 S k n = 1 N W i n + W j n + W n k
W R = 1 2 [ 1 1   3 1   3 1   3 1 1   3 1   3 1   3 1 1   3 1   3 1   3 1 1   3 1   3 1   3 ]
Γ S ^ = 1 2 [ S 0 S 1 S 2 S 3 S 1 3 S 0 3 S 3 3 S 2 S 2 3 S 3 3 S 0 3 S 1 S 3 3 S 2 3 S 1 3 S 0 ]
Γ S ^ = 2 N [ S 0 S 1 S 2 S 3 S 1 3 S 0 0 0 S 2 0 3 S 0 0 S 3 0 0 3 S 0 ]
P ^ = S ^ 1 + S ^ 2 + S ^ 3 S ^ 0
α ^ = { 1 2 arctan  ( S ^ 2 S ^ 1 ) S ^ 1 0 , π 2 × sgn ( S ^ 2 ) + 1 2 arctan  ( S ^ 2 S ^ 1 ) S ^ 1 < 0
ε ^ = 1 2 arctan  ( S ^ 3 S ^ 1 2 + S ^ 2 2 )
y f ( X ) var [ y ] [ f ( X ) ] T Γ X f ( X )
P ^ = 1 P S 0 2 [ P 2 S 0 , S 1 , S 2 , S 3 ] T
α ^ = 1 2 [ 0 , S 2 S 1 2 + S 2 2 , S 1 S 1 2 + S 2 2 , 0 ] T
ε ^ = 1 2 P 2 S 0 2 S 1 2 + S 2 2 [ 0 , S 1 S 3 , S 2 S 3 , ( S 1 2 + S 2 2 ) ] T
var  [ P ^ ] = 4 N 1 S N R g ( 3 + P 2 )
var  [ α ^ ] = 3 N 1 S N R g 1 P 2 cos   2 2 ε
var  [ ε ^ ] = 3 N 1 S N R g 1 P 2
SNR g = ( S 0 σ ) 2
var  [ P ^ ] = 1 2 S 0 ( 3 P 2 + 3 3 P sin   4 α sin   4 ε cos   2 ε 2 )
var  [ α ^ ] = 1 8 S 0 ( 3 P 2 cos   2 2 ε 3 sin   4 α sin   4 ε 2 P cos   3 2 ε )
var  [ ε ^ ] = 1 8 S 0 ( 3 P 2 + 3 sin   4 α sin   3 2 ε 3 sin   4 α sin   4 ε cos   2 ε P )
var  [ P ^ ] = 4 N 1 2 S 0 [ 3 P 2 ]
var  [ α ^ ] = 3 N 1 2 S 0 1 P 2 cos   2 2 ε
var  [ ε ^ ] = 3 N 1 2 S 0 1 P 2

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