Abstract

We present a method for calibrating a polarization state analyzer that uses a set of well- characterized reference polarization states and makes no assumptions about the optics contained in the polarimeter other than their linearity. The method requires that a matrix be constructed that contains the data acquired for each of the reference polarization states and that this matrix be pseudoinverted. Since this matrix is usually singular, we improve the method by performing the pseudoinversion by singular value decomposition, keeping only the four largest singular values. We demonstrate the calibration technique using an imaging polarimeter based upon liquid crystal variable retarders and with light emitting diode (LED) illumination centered at 472  nm, 525  nm, and 630   nm. We generate the reference polarization states by using an unpolarized source, a single polarizer, and a Fresnel rhomb. This method is particularly useful when calibrations are performed on field-grade instruments at a centrally maintained facility and when a traceability chain needs to be maintained.

© 2007 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).
  2. R. A. Chipman, "Polarimetry," in Handbook of Optics (McGraw-Hill, 1995), Vol. 2, Chap. 22.
  3. A. En Naciri, L. Johann, R. Kleim, M. Sieskind, and M. Arnann, "Spectroscopic ellipsometry of anisotropic materials: application to the optical constants of Hgl2," Appl. Opt. 38, 647-654 (1999).
    [CrossRef]
  4. T. W. Cronin, E. J. Warrant, and B. Greiner, "Celestial polarization patterns during twilight," Appl. Opt. 45, 5582-5589 (2006).
    [CrossRef] [PubMed]
  5. J. Cariou, B. Le Jeune, J. Lotrian, and Y. Guern, "Polarization effects of seawater and underwater targets," Appl. Opt. 29, 1689-1695 (1990).
    [CrossRef] [PubMed]
  6. G. Le Brun, B. Le Jeune, J. Cariou, and J. Lotrian, "Laser imaging procedure for evaluating the polarization signature of immersed targets," Pure Appl. Opt. 2, 445-470 (1993).
  7. P. Olivard, P. Y. Gerligand, B. Le Jeune, J. Cariou, and J. Lotrian, "Measurement of optical fibre parameters using an optical polarimeter and Stokes-Mueller formalism," J. Phys. D 32, 1618-1625 (1999).
    [CrossRef]
  8. F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
    [CrossRef]
  9. V. Sankaran, J. T. Walsh, Jr., and D. J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
    [CrossRef] [PubMed]
  10. F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
    [CrossRef]
  11. P. Y. Gerligand, M. Smith, and R. Chipman, "Polarimetric images of a cone," Opt. Express 4, 420-430 (1999).
    [CrossRef] [PubMed]
  12. O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging," Appl. Opt. 45, 4062-4068 (2006).
    [CrossRef] [PubMed]
  13. P. J. Wu and J. T. Walsh, Jr., "Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light," J. Biomed. Opt. 11, 014031 (2006).
    [CrossRef] [PubMed]
  14. J. S. Tyo and T. S. Turner, Jr., "Variable-retardance, Fourier-transform imaging spectropolarimeters for visible spectrum remote sensing," Appl. Opt. 40, 1450-1458 (2001).
    [CrossRef]
  15. 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]
  16. J. S. Tyo, "Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters," Opt. Lett. 25, 1198-1200 (2000).
    [CrossRef]
  17. 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]
  18. D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
    [CrossRef]
  19. B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
    [CrossRef]
  20. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11-29.
  21. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, 1992), Chap. 2, pp. 59-71.
  22. B. Boulbry, T. A. Germer, and J. C. Ramella-Roman, "A novel hemispherical spectro-polarimetric scattering instrument for skin lesion imaging," Proc. SPIE 6078, 128-134 (2006).
  23. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystals," Appl. Opt. 43, 2824-2832 (2004).
    [CrossRef] [PubMed]
  24. S. Huard, Polarization of Light (Wiley, 1997).
  25. R. J. King, "Quarter-wave retardation systems based on the Fresnel rhomb principle," J. Sci. Instrum. 43, 617-622 (1966).
    [CrossRef]

2006 (5)

P. J. Wu and J. T. Walsh, Jr., "Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light," J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

B. Boulbry, T. A. Germer, and J. C. Ramella-Roman, "A novel hemispherical spectro-polarimetric scattering instrument for skin lesion imaging," Proc. SPIE 6078, 128-134 (2006).

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging," Appl. Opt. 45, 4062-4068 (2006).
[CrossRef] [PubMed]

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]

T. W. Cronin, E. J. Warrant, and B. Greiner, "Celestial polarization patterns during twilight," Appl. Opt. 45, 5582-5589 (2006).
[CrossRef] [PubMed]

2005 (1)

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

2004 (2)

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystals," Appl. Opt. 43, 2824-2832 (2004).
[CrossRef] [PubMed]

2002 (3)

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]

V. Sankaran, J. T. Walsh, Jr., and D. J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (3)

1997 (1)

S. Huard, Polarization of Light (Wiley, 1997).

1995 (1)

R. A. Chipman, "Polarimetry," in Handbook of Optics (McGraw-Hill, 1995), Vol. 2, Chap. 22.

1993 (1)

G. Le Brun, B. Le Jeune, J. Cariou, and J. Lotrian, "Laser imaging procedure for evaluating the polarization signature of immersed targets," Pure Appl. Opt. 2, 445-470 (1993).

1992 (1)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, 1992), Chap. 2, pp. 59-71.

1990 (2)

1987 (1)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).

1983 (1)

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11-29.

1966 (1)

R. J. King, "Quarter-wave retardation systems based on the Fresnel rhomb principle," J. Sci. Instrum. 43, 617-622 (1966).
[CrossRef]

Arnann, M.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).

Bentivegna, F.

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

Boucher, Y.

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

Boulbry, B.

B. Boulbry, T. A. Germer, and J. C. Ramella-Roman, "A novel hemispherical spectro-polarimetric scattering instrument for skin lesion imaging," Proc. SPIE 6078, 128-134 (2006).

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Boulvert, F.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

Bousquet, B.

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Cariou, J.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

P. Olivard, P. Y. Gerligand, B. Le Jeune, J. Cariou, and J. Lotrian, "Measurement of optical fibre parameters using an optical polarimeter and Stokes-Mueller formalism," J. Phys. D 32, 1618-1625 (1999).
[CrossRef]

G. Le Brun, B. Le Jeune, J. Cariou, and J. Lotrian, "Laser imaging procedure for evaluating the polarization signature of immersed targets," Pure Appl. Opt. 2, 445-470 (1993).

J. Cariou, B. Le Jeune, J. Lotrian, and Y. Guern, "Polarization effects of seawater and underwater targets," Appl. Opt. 29, 1689-1695 (1990).
[CrossRef] [PubMed]

Chenault, D. B.

Chipman, R.

Chipman, R. A.

R. A. Chipman, "Polarimetry," in Handbook of Optics (McGraw-Hill, 1995), Vol. 2, Chap. 22.

D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
[CrossRef]

Cronin, T. W.

De Martino, A.

Drévillon, B.

En Naciri, A.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, 1992), Chap. 2, pp. 59-71.

Gerligand, P. Y.

P. Y. Gerligand, M. Smith, and R. Chipman, "Polarimetric images of a cone," Opt. Express 4, 420-430 (1999).
[CrossRef] [PubMed]

P. Olivard, P. Y. Gerligand, B. Le Jeune, J. Cariou, and J. Lotrian, "Measurement of optical fibre parameters using an optical polarimeter and Stokes-Mueller formalism," J. Phys. D 32, 1618-1625 (1999).
[CrossRef]

Germer, T. A.

B. Boulbry, T. A. Germer, and J. C. Ramella-Roman, "A novel hemispherical spectro-polarimetric scattering instrument for skin lesion imaging," Proc. SPIE 6078, 128-134 (2006).

Goldstein, D. H.

Goldstein, D. L.

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11-29.

Gorria, P.

Greiner, B.

Guegan, M.

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

Guern, Y.

Huard, S.

S. Huard, Polarization of Light (Wiley, 1997).

Johann, L.

King, R. J.

R. J. King, "Quarter-wave retardation systems based on the Fresnel rhomb principle," J. Sci. Instrum. 43, 617-622 (1966).
[CrossRef]

Kleim, R.

Laude-Boulesteix, B.

Le Brun, G.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

G. Le Brun, B. Le Jeune, J. Cariou, and J. Lotrian, "Laser imaging procedure for evaluating the polarization signature of immersed targets," Pure Appl. Opt. 2, 445-470 (1993).

Le Jeune, B.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

P. Olivard, P. Y. Gerligand, B. Le Jeune, J. Cariou, and J. Lotrian, "Measurement of optical fibre parameters using an optical polarimeter and Stokes-Mueller formalism," J. Phys. D 32, 1618-1625 (1999).
[CrossRef]

G. Le Brun, B. Le Jeune, J. Cariou, and J. Lotrian, "Laser imaging procedure for evaluating the polarization signature of immersed targets," Pure Appl. Opt. 2, 445-470 (1993).

J. Cariou, B. Le Jeune, J. Lotrian, and Y. Guern, "Polarization effects of seawater and underwater targets," Appl. Opt. 29, 1689-1695 (1990).
[CrossRef] [PubMed]

Lotrian, J.

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

P. Olivard, P. Y. Gerligand, B. Le Jeune, J. Cariou, and J. Lotrian, "Measurement of optical fibre parameters using an optical polarimeter and Stokes-Mueller formalism," J. Phys. D 32, 1618-1625 (1999).
[CrossRef]

G. Le Brun, B. Le Jeune, J. Cariou, and J. Lotrian, "Laser imaging procedure for evaluating the polarization signature of immersed targets," Pure Appl. Opt. 2, 445-470 (1993).

J. Cariou, B. Le Jeune, J. Lotrian, and Y. Guern, "Polarization effects of seawater and underwater targets," Appl. Opt. 29, 1689-1695 (1990).
[CrossRef] [PubMed]

Maitland, D. J.

V. Sankaran, J. T. Walsh, Jr., and D. J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

Meriaudeau, F.

Morel, O.

Olivard, P.

P. Olivard, P. Y. Gerligand, B. Le Jeune, J. Cariou, and J. Lotrian, "Measurement of optical fibre parameters using an optical polarimeter and Stokes-Mueller formalism," J. Phys. D 32, 1618-1625 (1999).
[CrossRef]

Pellen, F.

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, 1992), Chap. 2, pp. 59-71.

Ramella-Roman, J. C.

B. Boulbry, T. A. Germer, and J. C. Ramella-Roman, "A novel hemispherical spectro-polarimetric scattering instrument for skin lesion imaging," Proc. SPIE 6078, 128-134 (2006).

Rivet, S.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

Sankaran, V.

V. Sankaran, J. T. Walsh, Jr., and D. J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

Schwartz, L.

Sharaiha, A.

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

Shaw, J. A.

Sieskind, M.

Smith, M.

Stolz, C.

Tariaki, M.

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, 1992), Chap. 2, pp. 59-71.

Turner, T. S.

Tyo, J. S.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11-29.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, 1992), Chap. 2, pp. 59-71.

Walsh, J. T.

P. J. Wu and J. T. Walsh, Jr., "Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light," J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

V. Sankaran, J. T. Walsh, Jr., and D. J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

Warrant, E. J.

Wu, P. J.

P. J. Wu and J. T. Walsh, Jr., "Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light," J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

Appl. Opt. (8)

J. Biomed. Opt. (2)

V. Sankaran, J. T. Walsh, Jr., and D. J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef] [PubMed]

P. J. Wu and J. T. Walsh, Jr., "Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light," J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

J. Opt. A: Pure Appl. Opt. (1)

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, "Analysis of the depolarizing properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21-28 (2005).
[CrossRef]

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

J. Phys. D (1)

P. Olivard, P. Y. Gerligand, B. Le Jeune, J. Cariou, and J. Lotrian, "Measurement of optical fibre parameters using an optical polarimeter and Stokes-Mueller formalism," J. Phys. D 32, 1618-1625 (1999).
[CrossRef]

J. Sci. Instrum. (1)

R. J. King, "Quarter-wave retardation systems based on the Fresnel rhomb principle," J. Sci. Instrum. 43, 617-622 (1966).
[CrossRef]

Meas. Sci. Technol. (1)

B. Boulbry, B. Le Jeune, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (2)

B. Boulbry, T. A. Germer, and J. C. Ramella-Roman, "A novel hemispherical spectro-polarimetric scattering instrument for skin lesion imaging," Proc. SPIE 6078, 128-134 (2006).

F. Bentivegna, F. Boulvert, M. Guegan, B. Boulbry, A. Sharaiha, M. Tariaki, F. Pellen, B. Le Jeune, and Y. Boucher, "Polarimetric analysis of a semiconductor optical amplifier based on the Mueller-Stokes formalism," Proc. SPIE 5452, 486-497 (2004).
[CrossRef]

Pure Appl. Opt. (1)

G. Le Brun, B. Le Jeune, J. Cariou, and J. Lotrian, "Laser imaging procedure for evaluating the polarization signature of immersed targets," Pure Appl. Opt. 2, 445-470 (1993).

Other (5)

S. Huard, Polarization of Light (Wiley, 1997).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).

R. A. Chipman, "Polarimetry," in Handbook of Optics (McGraw-Hill, 1995), Vol. 2, Chap. 22.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11-29.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, 1992), Chap. 2, pp. 59-71.

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Figures (5)

Fig. 1
Fig. 1

Reference polarization state generator (bottom) and the imaging polarimeter (top). The polarizer (POL) in the reference polarization state generator is placed in positions 1 and 2 for the two configurations.

Fig. 2
Fig. 2

Calibration data (symbols) obtained for λ = 630   nm and the best fit (curves) to the first calibration method. The top frame is for the rhomb–polarizer configuration, and the bottom frame is for the polarizer–rhomb configuration. Each of the curves corresponds to one of the six LC retardance combinations ( δ 1 , δ 2 ) : (squares and solid curves) ( 0 ° , 0 ° ) , (circles and long dashed curves) ( 0 ° , 180 ° ) , (up-triangles and dotted curve) ( 90 ° , 90 ° ) , (down-triangles and dash-dot curves) ( 90 ° , 90 ° ) , (diamonds and dash-dot-dot curves) ( 0 ° , 90 ° ) , and (left-triangles and short-dotted curves) ( 0 ° , 90 ° ) .

Fig. 3
Fig. 3

Residual normalized Stokes parameters for λ = 630   nm as a function of polarizer angle. The open symbols are (squares) S 0 , (circles) S 1 , (up triangles) S 2 , and (down triangles) S 3 , for the first calibration method, and the solid symbols are respective values for the second calibration method. The top frame is for the rhomb–polarizer configuration, and the bottom frame is for the polarizer–rhomb configuration.

Fig. 4
Fig. 4

Illustration of how errors in the reference polarizations affect subsequent measurements. The Poincaré space (top) shows nominal paths traced out by the reference polarizations (12 and 13). Due to an error in the retardance, nominal path 13 is actually path 13′, lying along a tilted plane. Errors in subsequent measurements for polarizations along path 23 are shown (bottom).

Fig. 5
Fig. 5

Root-mean-square error in normalized Stokes parameters, (squares) S 1 / S 0 , (down triangle) S 2 / S 0 , and (up triangle) S 3 / S 0 , calculated as functions of the width of the reflectance edge. The insets show the spectrum I ( λ ) and the reflectance R ( λ ) used in the calculation for illustration.

Tables (2)

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Table 1 Best-Fit Parameters Obtained from a Nonlinear Least Squares Fit to the Calibration Data

Tables Icon

Table 2 Root-Mean-Square Error of the Stokes Vector Elements based upon the Two Calibration Methods Described in the Text

Equations (29)

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S = ( S 0 S 1 S 2 S 3 ) = ( I x + I y I x I y I 45 ° I 45 ° I rcp I lcp ) ,
S = W · I ,
I = ( I x I y I 45 ° I 45 ° I rcp I lcp ) T
W = ( 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 )
I x + I y = I 45 ° + I 45 ° = I rcp + I lcp ,
W = ( 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 5 / 3 1 / 3 1 / 3 1 / 3 1 / 3 2 / 3 2 / 3 4 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 1 / 3 1 / 3 4 / 3 2 / 3 )
u s = ( 1.414 s , 1.414 s , 1.414 s , 1.414 s ) T
u s = ( 0.816 s , 1.826 s , 1.826 s , 1.826 s ) T
W = ( 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 ) .
S i = M i · S ,
I i = S sense, i · S ,
S sense, i = ( M i , 00 M i , 01 M i , 02 M i , 03 ) .
I = A · S ,
A = ( M 1 , 00 M 1 , 01 M 1 , 02 M 1 , 03 M N , 00 M N , 01 M N , 02 M N , 03 ) .
W = ( A T · A ) 1 · A T .
[ S 1 S 1 S M ] = W · [ I 1 I 2 I M ]
S = W · I ,
W = S · I 1 .
I + = I T · ( I · I T ) 1 .
I = U · D · V T .
I + = V · diag ( 1 / σ 1 ,   …   ,   1 / σ N ) · U T .
I ^ + = V · diag ( 1 / σ 1 ,   …   ,   1 / σ 4 , 0 ,   …   ,   0 ) · U T
W = S · I ^ + .
M = M pol ( θ 3 ) · M ret ( θ 2 , δ 2 ) · M ret ( θ 1 , δ 1 ) ,
A = ( 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 ) .
S before ( θ ) = [ 1   cos   2 θ   0   sin   2 θ ] T ,
S after ( θ ) = [ 1   cos   2 θ   sin   2 θ 0 ] T ,
I ( λ ) = A   exp [ 2 .773 ( λ λ 0 ) 2 / λ FWHM 2 ] ,
R ( λ ) = { erf [ ( λ λ 0 ) / Δ λ ) + 1 } / 2 ,

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