Abstract

Dual photoelastic modulator polarimeter systems are widely used for the measurement of light beam polarization, most often described by Stokes vectors, that carry information about an interrogated sample. The sample polarization properties can be described more thoroughly through its Mueller matrix, which can be derived from judiciously chosen input polarization Stokes vectors and correspondingly measured output Stokes vectors. However, several sources of error complicate the construction of a Mueller matrix from the measured Stokes vectors. Here we present a general formalism to examine these sources of error and their effects on the derived Mueller matrix, and identify the optimal input polarization states to minimize their effects in a dual photoelastic modulator polarimeter configuration. The input Stokes vector states leading to the most robust Mueller matrix determination are shown to form Platonic solids in the Poincaré sphere space; we also identify the optimal 3D orientation of these solids for error minimization.

© 2012 OSA

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References

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  1. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc.9, 399–416 (1852).
  2. E. Collett, Field Guide to Polarization (SPIE Press, 2005).
    [CrossRef]
  3. H. Poincaré, Théorie mathématique de la lumière (Gauthiers-Villars, 1892).
    [PubMed]
  4. H. Mueller, “Memorandum on the polarization optics of the photoelastic shutter,” Report No. 2 of the OSRD project OEMsr-576, (1943).
  5. N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
    [CrossRef]
  6. D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt.9, 213–220 (2004).
    [CrossRef] [PubMed]
  7. X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt.11, 041105 (2006).
    [CrossRef] [PubMed]
  8. S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, 2nd ed. (Prentice-Hall, 1989).
  9. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 1,” Opt. Eng.34, 1651–1655 (1995).
    [CrossRef]
  10. E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004).
    [CrossRef]
  11. P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23095–23103 (2010).
    [CrossRef] [PubMed]
  12. A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004).
    [CrossRef]
  13. A. D. Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett.28, 616–618 (2003).
    [CrossRef] [PubMed]
  14. 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, 802–804 (2000).
    [CrossRef]
  15. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng.41, 965–972 (2002).
    [CrossRef]
  16. M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt.41, 2488–2493 (2002).
    [CrossRef] [PubMed]
  17. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express16, 11589–11603 (2008).
    [CrossRef] [PubMed]
  18. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt.41, 619–630 (2002).
    [CrossRef] [PubMed]
  19. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett.25, 1198–1200 (2000).
    [CrossRef]
  20. I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express16, 2091–2108 (2008).
    [CrossRef] [PubMed]
  21. J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt.8, 807–814 (2006).
    [CrossRef]
  22. G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins University Press, 1996).
  23. W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys.103, 043104 (2008).
    [CrossRef]
  24. G. H. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Numer. Anal.10, 413–432 (1973).
    [CrossRef]
  25. J. Dattorro, Convex Optimization & Euclidean Distance Geometry (Meboo Publishing USA, 2005).
    [PubMed]
  26. J. Stewart, Calculus Early Transcendentals, 6th ed. (Thompson Brooks/Cole, 2008).
  27. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 2,” Opt. Eng.34, 1656–1658 (1995).
    [CrossRef]
  28. R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A5, 681–689 (1988).
    [CrossRef]
  29. M. Atiyah and P. Sutcliffe, “Polyhedra in physics, chemistry and geometry,” Milan J. Math.71, 33–58 (2003).
    [CrossRef]
  30. A. P. Arya, Introduction to Classical Mechanics, 2nd ed. (Prentice-Hall, 1998).

2010

2009

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

2008

2006

X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt.11, 041105 (2006).
[CrossRef] [PubMed]

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

2004

E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004).
[CrossRef]

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004).
[CrossRef]

D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt.9, 213–220 (2004).
[CrossRef] [PubMed]

2003

2002

2000

1995

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 2,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 1,” Opt. Eng.34, 1651–1655 (1995).
[CrossRef]

1988

1973

G. H. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Numer. Anal.10, 413–432 (1973).
[CrossRef]

1852

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc.9, 399–416 (1852).

Aas, L. M. S.

Aïnouz, S.

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

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 1,” Opt. Eng.34, 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 2,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

Arya, A. P.

A. P. Arya, Introduction to Classical Mechanics, 2nd ed. (Prentice-Hall, 1998).

Atiyah, M.

M. Atiyah and P. Sutcliffe, “Polyhedra in physics, chemistry and geometry,” Milan J. Math.71, 33–58 (2003).
[CrossRef]

Azzam, R. M. A.

Chipman, R. A.

Collett, E.

E. Collett, Field Guide to Polarization (SPIE Press, 2005).
[CrossRef]

Côté, D.

D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt.9, 213–220 (2004).
[CrossRef] [PubMed]

Dattorro, J.

J. Dattorro, Convex Optimization & Euclidean Distance Geometry (Meboo Publishing USA, 2005).
[PubMed]

Dereniak, E. L.

Descour, M. R.

Drévillon, B.

E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004).
[CrossRef]

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004).
[CrossRef]

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

Ellingsen, P. G.

Elminyawi, I. M.

El-Saba, A. M.

Friedberg, S. H.

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, 2nd ed. (Prentice-Hall, 1989).

Garcia-Caurel, E.

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004).
[CrossRef]

E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004).
[CrossRef]

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

Ghosh, N.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

Golub, G. H.

G. H. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Numer. Anal.10, 413–432 (1973).
[CrossRef]

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins University Press, 1996).

Guan, W.

W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys.103, 043104 (2008).
[CrossRef]

Guo, X.

X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt.11, 041105 (2006).
[CrossRef] [PubMed]

Hoover, B. G.

Insel, A. J.

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, 2nd ed. (Prentice-Hall, 1989).

Jones, G. A.

W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys.103, 043104 (2008).
[CrossRef]

Kemme, S. A.

Kildemo, M.

Kim, Y.-K.

Laude, B.

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004).
[CrossRef]

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

Letnes, P. A.

Li, R.-K.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

Li, S.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

Liu, Y.

W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys.103, 043104 (2008).
[CrossRef]

Loan, C. F. V.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins University Press, 1996).

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 2,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 1,” Opt. Eng.34, 1651–1655 (1995).
[CrossRef]

Martino, A. D.

E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004).
[CrossRef]

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004).
[CrossRef]

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

Mueller, H.

H. Mueller, “Memorandum on the polarization optics of the photoelastic shutter,” Report No. 2 of the OSRD project OEMsr-576, (1943).

Nerbø, I. S.

Pereyra, V.

G. H. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Numer. Anal.10, 413–432 (1973).
[CrossRef]

Phipps, G. S.

Poincaré, H.

H. Poincaré, Théorie mathématique de la lumière (Gauthiers-Villars, 1892).
[PubMed]

Sabatke, D. S.

Savenkov, S. N.

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng.41, 965–972 (2002).
[CrossRef]

Shen, T. H.

W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys.103, 043104 (2008).
[CrossRef]

Smith, M. H.

Spence, L. E.

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, 2nd ed. (Prentice-Hall, 1989).

Stewart, J.

J. Stewart, Calculus Early Transcendentals, 6th ed. (Thompson Brooks/Cole, 2008).

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc.9, 399–416 (1852).

Stoll, M. P.

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

Sutcliffe, P.

M. Atiyah and P. Sutcliffe, “Polyhedra in physics, chemistry and geometry,” Milan J. Math.71, 33–58 (2003).
[CrossRef]

Sweatt, W. C.

Twietmeyer, K. M.

Tyo, J. S.

Vaughn, I. J.

Vitkin, I. A.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt.11, 041105 (2006).
[CrossRef] [PubMed]

D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt.9, 213–220 (2004).
[CrossRef] [PubMed]

Weisel, R. D.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

Wilson, B. C.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

Wood, M. F. G.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt.11, 041105 (2006).
[CrossRef] [PubMed]

Zallat, J.

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

Appl. Opt.

J. Appl. Phys.

W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys.103, 043104 (2008).
[CrossRef]

J. Biomed. Opt.

D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt.9, 213–220 (2004).
[CrossRef] [PubMed]

X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt.11, 041105 (2006).
[CrossRef] [PubMed]

J. Biophoton.

N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

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

J. Opt. Soc. Am. A

Milan J. Math.

M. Atiyah and P. Sutcliffe, “Polyhedra in physics, chemistry and geometry,” Milan J. Math.71, 33–58 (2003).
[CrossRef]

Opt. Eng.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 2,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng.41, 965–972 (2002).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part 1,” Opt. Eng.34, 1651–1655 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

SIAM J. Numer. Anal.

G. H. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Numer. Anal.10, 413–432 (1973).
[CrossRef]

Thin Solid Films

E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004).
[CrossRef]

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004).
[CrossRef]

Trans. Cambridge Phil. Soc.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc.9, 399–416 (1852).

Other

E. Collett, Field Guide to Polarization (SPIE Press, 2005).
[CrossRef]

H. Poincaré, Théorie mathématique de la lumière (Gauthiers-Villars, 1892).
[PubMed]

H. Mueller, “Memorandum on the polarization optics of the photoelastic shutter,” Report No. 2 of the OSRD project OEMsr-576, (1943).

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, 2nd ed. (Prentice-Hall, 1989).

J. Dattorro, Convex Optimization & Euclidean Distance Geometry (Meboo Publishing USA, 2005).
[PubMed]

J. Stewart, Calculus Early Transcendentals, 6th ed. (Thompson Brooks/Cole, 2008).

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins University Press, 1996).

A. P. Arya, Introduction to Classical Mechanics, 2nd ed. (Prentice-Hall, 1998).

Supplementary Material (10)

» Media 1: MPEG (2334 KB)     
» Media 2: MPEG (2344 KB)     
» Media 3: MPEG (2176 KB)     
» Media 4: MPEG (2348 KB)     
» Media 5: MPEG (2170 KB)     
» Media 6: MPEG (5032 KB)     
» Media 7: PDF (258 KB)     
» Media 8: PDF (33 KB)     
» Media 9: PDF (41 KB)     
» Media 10: PDF (56 KB)     

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Figures (6)

Fig. 1
Fig. 1

The dual PEM Stokes polarimeter schematic. Panel (a) provides a top view of the system, which is comprised of a movable PSG to illuminate the sample and dual photoelastic modulators followed by a linear polarizer and a photodetector, which form the PSA. The two lock-in amplifiers recover the polarization parameters Q, U, and V using the PEM modulation frequencies as references. Panel (b) depicts the PSA as seen from the photodetector. The fast axis of PEM 1 is at an angle α = 45° above the horizontal (laboratory frame), and the fast axis of PEM 2 is parallel to the optical table. The linear polarizer’s transmission axis is at an angle β = 22.5° above the horizontal.

Fig. 2
Fig. 2

The five Platonic solids inscribed in the Poincaré sphere, where the equator represents linear polarization and the poles represent circular polarization. The three great circles on each sphere represent the areas where Q, U, or V are zero. When the input Stokes vectors form the vertices of any of these shapes, Eqs. (21)(23) are satisfied, and so the error-sensitivity of the determined Mueller matrix will be minimized. In (a) the n = 4 input Stokes vectors form a tetrahedron in polarization space ( Media 1), in (b) where n = 6, they form an octahedron ( Media 2), in (c) where n = 8, they form a cube ( Media 3), in (d) where n = 12, they form an icosahedron ( Media 4), and finally, in (e) where n = 20, they form a dodecahedron ( Media 5).

Fig. 3
Fig. 3

( Media 6). The Poincaré sphere coloured according to the likelihood of phase errors in each region, where red represents a high probability and green represents a low probability. As in Fig. 2, the great circles in black represent the areas where the polarization parameters are 0. As Stokes vectors approach these areas (i.e. they enter the red/yellow zones) they become increasingly prone to phase errors. Here, the cubic configuration of Fig. 2(c) ( Media 3) has been inscribed in the sphere to show that the Stokes vectors forming its vertices are all minimally prone to phase errors.

Fig. 4
Fig. 4

The probability of a simulated phase inversions as a function of q, u, or v. The likelihood of a phase error approaches 0.5 when the polarization parameters are near 0, and it drops quickly to ∼ 0 as said parameters move towards ±1.

Fig. 5
Fig. 5

( Media 7).The Mueller errors resulting from ∼ 104 randomly chosen sets of n Stokes vectors, evenly distributed over 4 ≤ n ≤ 20. Each blue dot represents one simulated Mueller matrix determination, and 〈δM〉 on the vertical axis shows the RMS error in said matrix. On the horizontal plane, n and ||(��in)+|| serve to describe and quantify each configuration. The red dot surrounded by the black circle at the bottom of the plot shows the simulation result corresponding to the optimal cubic configuration in Fig. 3 ( Media 6).

Fig. 6
Fig. 6

The 2-dimensional projections of Fig. 5 ( Media 7). (a) shows the norm of (��in)+ for randomly chosen sets of input states versus n ( Media 8), (b) shows the RMS Mueller error associated with these sets of input states, again versus n ( Media 9), and (c) plots the RMS Mueller error against the norm of (��in)+ for each simulated set ( Media 10).

Equations (29)

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

S out = M S in ,
κ ( A ) A A 1 ,
A p = max S A S p S p .
[ S 1 out S n out ] M [ S 1 in S n in ] .
ε f ( M + ε η ) 2 | ε = 0 = 1 4 η , 𝕊 in ( 𝕊 in ) M 𝕊 in ( 𝕊 out ) = 0
M = 𝕊 out ( 𝕊 in ) [ 𝕊 in ( 𝕊 in ) ] 1 = 𝕊 out ( 𝕊 in ) + ,
[ S 1 out + δ S 1 out S n out + δ S n out ] ( M + δ M ) [ S 1 in + δ S 1 in S n in + δ S n in ] ,
𝕊 out + δ 𝕊 out ( M + δ M ) ( 𝕊 in + δ 𝕊 in ) .
( M + δ M ) = ( 𝕊 out + δ 𝕊 out ) ( 𝕊 in + δ 𝕊 in ) + .
( A + B ) + A + + ( A + ) | A = B ,
( A + ) = A + ( A ) A + + ( I n A + A ) ( A ) ( A + ) A + ,
δ M [ δ 𝕊 out M ( δ 𝕊 in ) ] ( 𝕊 in ) + .
δ M n 1 / 2 2 ( δ 𝕊 out + 4 M δ 𝕊 in ) ( 𝕊 in ) + .
( 𝕊 in ) + 2 = Tr { [ 𝕊 in ( 𝕊 in ) ] 1 } .
( 𝕊 in ) + 2 = Tr ( D ) 1 = i = 1 4 1 λ i .
[ 𝕊 in ( 𝕊 in ) ] μ ν = r μ r ν
det ( X ) X = det ( X ) ( X 1 ) .
[ 1 1 1 q 1 q 2 q n u 1 u 2 u n v 1 v 2 v n ] ,
r 2 2 + r 3 2 + r 4 2 = i = 1 n q i 2 + i = 1 n u i 2 + i = 1 n v i 2 = i = 1 n ( q i 2 + u i 2 + v i 2 ) = n ,
Γ ( λ 1 , , λ 4 , γ ) = i = 1 4 1 λ i + γ [ ( i = 2 4 λ i ) λ 1 ] .
i = 1 n q i = i = 1 n u i = i = 1 n v i = 0 ,
i = 1 n q i u i = i = 1 n u i v i = i = 1 n v i q i = 0 ,
i = 1 n q i 2 = i = 1 n u i 2 = i = 1 n v i 2 = n 3 ,
( 𝕊 in ) + min ( n ) = ( 10 n ) 1 / 2 .
δ M ( 5 2 ) 1 / 2 ( δ 𝕊 out + 4 M δ 𝕊 in ) .
R = [ 1 0 0 R ] ,
( R 𝕊 in ) + = ( 𝕊 in ) + ( R ) .
{ [ 1 0.68 0.70 0.21 ] . [ 1 0.43 0.75 0.50 ] , [ 1 0.73 0.40 0.56 ] , [ 1 0.39 0.35 0.85 ] } .
{ [ 1 0.58 0.58 0.58 ] , [ 1 0.58 0.58 0.58 ] , [ 1 0.58 0.58 0.58 ] , [ 1 0.58 0.58 0.58 ] , [ 1 0.58 0.58 0.58 ] , [ 1 0.58 0.58 0.58 ] , [ 1 0.58 0.58 0.58 ] , [ 1 0.58 0.58 0.58 ] } ,

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