Abstract

In this paper we present the analysis, optimization and implementation of several Stokes polarimeter configurations based on a set-up including two variable retarders. The polarimeter analysis is based on the Mueller-Stokes formalism, and as a consequence, it is suitable to deal with depolarized light. Complete Stokes polarimeters are optimized by minimizing the amplification of simulated errors into the final solution. Different indicators useful to achieve this aim, as the condition number or the equally weighted variance, are compared in this paper. Moreover, some of the optimized polarimeters are experimentally implemented and it is studied the influence of small deviations from the theoretical ones on the amplification of the Stokes component error. In addition, the benefit of using incomplete polarimeters, when detecting specific ranges of states of polarization, is discussed.

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  1. M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).
  2. K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express 16(26), 21339–21354 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-26-21339 .
    [CrossRef] [PubMed]
  3. J. L. November and L. M. Wilkins, “The Liquid Crystal Polarimeter for solid-state imaging of solar vector magnetic fields,” Proc. SPIE 2265, 210–221 (1992).
    [CrossRef]
  4. A. Márquez, I. Moreno, C. Iemmi, A. Lizana, J. Campos, and M. J. Yzuel, “Mueller-Stokes characterization and optimization of a liquid crystal on silicon display showing depolarization,” Opt. Express 16, 1669–1685 (2008), http://www.opticsinfobase.org/ oe/ abstract.cfm ?uri=oe-16-3-1669.
  5. S. Firdous and M. Ikram, “Stokes Polarimetry for the Characterization of Bio-Materials using Liquid Crystal Variable Retarders,” Proc. of the SPIE-OSA Biomedical Optics 6632, 66320F–1 66320F–13 (2007).
  6. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., McGraw-Hill, (New York, Chapter 22, 1995).
  7. 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]
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    [CrossRef]
  12. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. 2(3), 216–222 (2000).
    [CrossRef]
  13. A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003).
    [CrossRef] [PubMed]
  14. J. Zallat, S. Aïnourz, and M. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006).
    [CrossRef]
  15. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. 34(5), 647–649 (2009).
    [CrossRef] [PubMed]
  16. D. Lara and C. Paterson, “Stokes polarimeter optimization in the presence of shot and Gaussian noise,” Opt. Express 17(23), 21240–21249 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-23-21240 .
    [CrossRef] [PubMed]
  17. V. L. Gamiz and J. F. Belsher, “Performance limits of a four-channel polarimeter in the presence of detection noise: Non-Ideal polarimeter,” Opt. Eng. 41, 973–980 (2002).
    [CrossRef]
  18. P. Taylor, Theory and Applications of Numerical Analysis, (Academic Press, London, 1974).
  19. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31(7), 488–493 (1941).
    [CrossRef]
  20. R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1977).
  21. D. Goldstein, Polarized Light, (Marcel Dekker, NY, 2003).
  22. G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for mathematical computations, (Prentice-Hall, New Jersey, 1977).
  23. E. Compain, S. Poirier, and B. Drevillón, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999).
    [CrossRef]
  24. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25(16), 1198–1200 (2000).
    [CrossRef]
  25. S. R. Davis, R. Uberna, and R. A. Herke, “Retardance sweep polarimeter and method,” United State Patent, patent US 6744509 B” (2004).

2009 (2)

2008 (2)

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express 16(26), 21339–21354 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-26-21339 .
[CrossRef] [PubMed]

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

2006 (1)

J. Zallat, S. Aïnourz, and M. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006).
[CrossRef]

2004 (1)

E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films 455–456, 120–123 (2004).
[CrossRef]

2003 (1)

2002 (1)

V. L. Gamiz and J. F. Belsher, “Performance limits of a four-channel polarimeter in the presence of detection noise: Non-Ideal polarimeter,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

2000 (3)

1999 (1)

1995 (1)

1992 (1)

J. L. November and L. M. Wilkins, “The Liquid Crystal Polarimeter for solid-state imaging of solar vector magnetic fields,” Proc. SPIE 2265, 210–221 (1992).
[CrossRef]

1941 (1)

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31(7), 488–493 (1941).
[CrossRef]

Aïnourz, S.

J. Zallat, S. Aïnourz, and M. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006).
[CrossRef]

Anastasiadou, M.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Belsher, J. F.

V. L. Gamiz and J. F. Belsher, “Performance limits of a four-channel polarimeter in the presence of detection noise: Non-Ideal polarimeter,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

Bueno, J. M.

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. 2(3), 216–222 (2000).
[CrossRef]

Chipman, R. A.

Clement, D.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Cohen, H. A. S.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Compain, E.

De Martino, A.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films 455–456, 120–123 (2004).
[CrossRef]

A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003).
[CrossRef] [PubMed]

Dereniak, E. L.

Descour, M. R.

Dou, R.

Drevillón, B.

Drévillon, B.

E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films 455–456, 120–123 (2004).
[CrossRef]

A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003).
[CrossRef] [PubMed]

Dreyfuss, J.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Elsner, A. E.

Gamiz, V. L.

V. L. Gamiz and J. F. Belsher, “Performance limits of a four-channel polarimeter in the presence of detection noise: Non-Ideal polarimeter,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

Garcia-Caurel, E.

E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films 455–456, 120–123 (2004).
[CrossRef]

A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003).
[CrossRef] [PubMed]

Giles, M. K.

Goudail, F.

Huynh, B.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Jones, R. C.

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31(7), 488–493 (1941).
[CrossRef]

Kemme, S. A.

Kim, Y. K.

Lara, D.

Laude, B.

Laude-Boulesteix, B.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Liège, F.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Nazac, A.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

November, J. L.

J. L. November and L. M. Wilkins, “The Liquid Crystal Polarimeter for solid-state imaging of solar vector magnetic fields,” Proc. SPIE 2265, 210–221 (1992).
[CrossRef]

Paterson, C.

Phipps, G. S.

Poirier, S.

Quang, N.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Sabatke, D. S.

Schwartz, L.

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Stoll, M.

J. Zallat, S. Aïnourz, and M. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006).
[CrossRef]

Sweatt, W. C.

Twietmeyer, K. M.

Tyo, J. S.

VanNasdale, D.

Wilkins, L. M.

J. L. November and L. M. Wilkins, “The Liquid Crystal Polarimeter for solid-state imaging of solar vector magnetic fields,” Proc. SPIE 2265, 210–221 (1992).
[CrossRef]

Zallat, J.

J. Zallat, S. Aïnourz, and M. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006).
[CrossRef]

Zhao, Y.

Appl. Opt. (1)

J. Opt. A, Pure Appl. Opt. (2)

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. 2(3), 216–222 (2000).
[CrossRef]

J. Zallat, S. Aïnourz, and M. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006).
[CrossRef]

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

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31(7), 488–493 (1941).
[CrossRef]

Opt. Eng. (1)

V. L. Gamiz and J. F. Belsher, “Performance limits of a four-channel polarimeter in the presence of detection noise: Non-Ideal polarimeter,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Status Solidi (1)

M. Anastasiadou, A. De Martino, D. Clement, F. Liège, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. A. S. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi 5(5), (2008).

Proc. SPIE (1)

J. L. November and L. M. Wilkins, “The Liquid Crystal Polarimeter for solid-state imaging of solar vector magnetic fields,” Proc. SPIE 2265, 210–221 (1992).
[CrossRef]

Thin Solid Films (1)

E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films 455–456, 120–123 (2004).
[CrossRef]

Other (10)

A. Márquez, I. Moreno, C. Iemmi, A. Lizana, J. Campos, and M. J. Yzuel, “Mueller-Stokes characterization and optimization of a liquid crystal on silicon display showing depolarization,” Opt. Express 16, 1669–1685 (2008), http://www.opticsinfobase.org/ oe/ abstract.cfm ?uri=oe-16-3-1669.

S. Firdous and M. Ikram, “Stokes Polarimetry for the Characterization of Bio-Materials using Liquid Crystal Variable Retarders,” Proc. of the SPIE-OSA Biomedical Optics 6632, 66320F–1 66320F–13 (2007).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., McGraw-Hill, (New York, Chapter 22, 1995).

H. J. Coufal, D. Psaltis, and B. T. Sincerbox, Holographic Data Storage, (Springer-Verlag, Berlin,2000).

P. Taylor, Theory and Applications of Numerical Analysis, (Academic Press, London, 1974).

S. R. Davis, R. Uberna, and R. A. Herke, “Retardance sweep polarimeter and method,” United State Patent, patent US 6744509 B” (2004).

R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1977).

D. Goldstein, Polarized Light, (Marcel Dekker, NY, 2003).

G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for mathematical computations, (Prentice-Hall, New Jersey, 1977).

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, Berlin, 1997).

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Figures (11)

Fig. 1
Fig. 1

Set-up of the LC based polarimeter: a) PSG; b) PSD.

Fig. 2
Fig. 2

CN minimization for: (a) four, (b) six, (c) eight, (d) twelve, (e) twenty and (f) one hundred polarization analyzers. The vertexes of the regular polyhedrons are located upon the surface of the Poincaré sphere.

Fig. 3
Fig. 3

Analysis of: (a) CN; (b) EWV as a function of the polarization analyzers number.

Fig. 4
Fig. 4

Variances of S 0, S 1, S 2, S 3 and ST (Eq. (11) for different values of the rotation angle:a) Regular tetrahedron, b) Cube, c) Regular dodecahedron.

Fig. 5
Fig. 5

Different sets of four polarization analyzers.

Fig. 6
Fig. 6

Variances of the Stokes components (Eq. (11) for the three different non-regular tetrahedrons shown in Fig. 5.

Fig. 7
Fig. 7

Numerical simulations of the variances of S 0, S1 , S2 , S3 and ST (Eq. (11) for 100 different polarimeters obtained from the optimized theoretical one represented in Fig. 2(a). The deviations are obtained by adding to the polarization analyzers a zero mean, uniformly distributed random numbers with amplitudes (a) 0.1; (b) 0.3; (c) 0.5.

Fig. 8
Fig. 8

Simulations of the S0 , S1 , S2 , S3 and ST variances (Eq. (11) for 100 different polarimeters obtained from the optimized theoretical one represented in Fig. 2(e) and by adding to the polarization analyzers a zero mean; uniform distributed random numbers of amplitudes (a) 0.1; (b) 0.3; (c) 0.5.

Fig. 9
Fig. 9

Simulations of the S0 , S1 , S2 , S3 and ST variances (Eq. (11) for 100 different polarimeters obtained from the non-optimized polarimeter represented in Fig. 5(a) and by adding to the polarization analyzers a zero mean; uniform distributed random numbers of amplitudes (a) 0.1; (b) 0.3; (c) 0.5.

Fig. 10
Fig. 10

Experimental set-up for the calibration of the retardance-voltatge look-up table.

Fig. 11
Fig. 11

Azimuth α standard deviation as a function of the number of intensity measurements (sample size) taken with the radiometer by using the optimized and the incomplete polarimeters.

Tables (1)

Tables Icon

Table 1 Azimuth α and Ellipticity ε values corresponding to three different measured SOPs

Equations (15)

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

Sex=MSin
I=AS
S=A1I
S=(ATA)1ATI=A˜1I
S+ΔS=A1(I+ΔI)
S+ΔS=A˜1(I+ΔI)
CN(A)=σmaxσmin
EWV(A)=j=0R11σj2
S=A˜1I=QISi=k=1NqikIk
δSi=[k=1N(SiIk)2δIk2]12=δI[k=1N(SiIk)2]12
δSi2=δI2k=1Nqik2
Spolarimeter=(S0,S1,S2,S3)T=(1,cosφ1,sinφ2sinφ1,cosφ2sinφ1)T
SLP1+LCD=(S0,S1,S2,S3)T=(1,cos22θ+cosφsin22θ,(1cosφ)sin2θcos2θ,sinφsin2θ)T
φ=arctan(S3sin2θS1cos22θ)
Ai0=12(Ii0+Ii90);Ai1=12(Ii0Ii90);Ai2=Ii45Ai0;Ai3=IiCRAi0i=0,...,n

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