Abstract

The relationship between system condition and signal-to-noise ratio (SNR) in reconstructed Stokes parameter images is investigated for rotating compensator, variable retardance, and rotating analyzer Stokes vector (SV) polarimeters. A variety of optimal configurations are presented for each class of systems. The operation of polarimeters is discussed in terms of a four-dimensional conical vector space; and the concept of nonorthogonal bases, frames, and tight frames is introduced to describe the operation of SV polarimeters. Although SNR is an important consideration, performance of a polarimeter in the presence of errors in the calibration and alignment of the optical components is also important. The relationship between system condition and error performance is investigated, and it is shown that an optimum system from the point of view of SNR is not always an optimum system with respect to error performance. A detailed theory of error performance is presented, and the error of a SV polarimeter is shown to be related to the stability and condition number of the polarization processing matrices. The rms error is found to fall off as the inverse of the number of measurements taken. Finally, the concepts used to optimize SV polarimeters are extended to be useful for full Mueller matrix polarimeters.

© 2002 Optical Society of America

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  1. D. H. Goldstein, R. A. Chipman, eds., Polarization: Measurement, Analysis, and Remote Sensing, Proc. SPIE3121 (1997).
  2. D. H. Goldstein, D. B. Chenault, eds., Polarization: Measurement, Analysis, and Remote Sensing II, Proc. SPIE3754 (1999).
  3. D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Polarization Analysis, Measurement, and Remote Sensing III, Proc. SPIE4133 (2000).
  4. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  5. A. Ambirajan, D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651–1655 (1995).
    [CrossRef]
  6. A. Ambirajan, D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1659 (1995).
    [CrossRef]
  7. D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
    [CrossRef]
  8. D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
    [CrossRef]
  9. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25, 1198–2000 (2000).
    [CrossRef]
  10. J. S. Tyo, “Considerations in polarimeter design,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 65–74 (2000).
    [CrossRef]
  11. J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization-sensitive vision or imaging system,” J. Opt. Soc. Am. A 15, 359–366 (1998).
    [CrossRef]
  12. J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
    [CrossRef]
  13. J. S. Tyo, E. N. Pugh, N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15, 367–374 (1998).
    [CrossRef]
  14. J. S. Tyo, “Polarization difference imaging,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1997).
  15. M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
    [CrossRef]
  16. E. A. West, J. G. Porter, J. M. Davis, A. Gary, M. Adams, “Development of a polarimeter for magnetic field measurements in the ultraviolet,” in Polarization Analysis, Measurement, and Remote Sensing, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE4481 (to be published).
  17. In fact, there is only one orthogonal polarization direction. Two states are orthogonally polarized regardless of their relative intensities; only the direction within the Stokes cone matters.
  18. M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).
  19. J. S. Tyo, T. S. Turner, “Variable-retardance, Fourier-transform imaging spectropolarimeters for visible spectrum remote sensing,” Appl. Opt. 40, 1450–1458 (2001).
    [CrossRef]
  20. R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
    [CrossRef]
  21. L. B. Wolff, “Polarization camera for computer vision with a beam splitter,” J. Opt. Soc. Am. A 11, 2935–2945 (1994).
    [CrossRef]
  22. R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
    [CrossRef]
  23. G. P. Nordin, J. T. Meier, P. C. Deguzman, M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 169–177 (1999).
  24. Note that the rows and columns of the Mueller matrix are numbered 0, … , 3, in agreement with conventions.4
  25. S.-Y. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on the polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  26. R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980), pp. 184–196.
  27. I. Daubecheis, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992), pp. 53–106.
    [CrossRef]
  28. Sabatke et al.7,8 introduced the concept of equal-weighted variance for polarimeter optimization. The equal-weighted variance is equivalent to the Frobenius norm of the synthesis matrix  B̲ , defined in Eq. (25). Because the Frobenius norm of all analysis matrices is N2/2, the Frobenius norm of the synthesis matrix is equivalent to the Frobenius condition number  A̲ .
  29. G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, pp. 11–29.
  30. Azzam et al.22 first proposed the tetrahedron on the Poincaré sphere for choosing the optimal set of calibration states for a SV polarimeter.
  31. T. S. Turner, K. W. Peters, J. S. Tyo, “Portable, visible, imaging spectropolarimeters for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
    [CrossRef]
  32. D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
    [CrossRef]
  33. If the input states are not uniformly distributed, the expectation can simply be evaluated with the appropriate statistical distribution of polarization state.
  34. J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
    [CrossRef]
  35. Note that the full set of N vectors forms a frame in ℝM . The last three columns form a tight frame in ℝM-1 .

2001 (1)

2000 (2)

1998 (2)

1997 (1)

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

1996 (1)

1995 (3)

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

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

A. Ambirajan, D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1659 (1995).
[CrossRef]

1994 (1)

1990 (1)

1988 (1)

1981 (1)

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
[CrossRef]

Adams, M.

E. A. West, J. G. Porter, J. M. Davis, A. Gary, M. Adams, “Development of a polarimeter for magnetic field measurements in the ultraviolet,” in Polarization Analysis, Measurement, and Remote Sensing, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE4481 (to be published).

Ambirajan, A.

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

A. Ambirajan, D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1659 (1995).
[CrossRef]

Ax, G. R.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

Azzam, R. M. A.

Bashara, N. M.

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

Chipman, R. A.

Daubecheis, I.

I. Daubecheis, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992), pp. 53–106.
[CrossRef]

Davis, J. M.

E. A. West, J. G. Porter, J. M. Davis, A. Gary, M. Adams, “Development of a polarimeter for magnetic field measurements in the ultraviolet,” in Polarization Analysis, Measurement, and Remote Sensing, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE4481 (to be published).

Deguzman, P. C.

G. P. Nordin, J. T. Meier, P. C. Deguzman, M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 169–177 (1999).

Dereniak, E.

D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Descour, M. R.

D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Elminyawi, I. M.

El-Saba, A. M.

Engheta, N.

Garcia, J. P.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Gary, A.

E. A. West, J. G. Porter, J. M. Davis, A. Gary, M. Adams, “Development of a polarimeter for magnetic field measurements in the ultraviolet,” in Polarization Analysis, Measurement, and Remote Sensing, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE4481 (to be published).

Goldstein, D. H.

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, pp. 11–29.

Howe, J. D.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

Jones, M.

G. P. Nordin, J. T. Meier, P. C. Deguzman, M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 169–177 (1999).

Kemme, S. A.

D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Locke, A. M.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Look, D. C.

A. Ambirajan, D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1659 (1995).
[CrossRef]

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

Lu, S.-Y.

Meier, J. T.

G. P. Nordin, J. T. Meier, P. C. Deguzman, M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 169–177 (1999).

Miller, M. A.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

Nordin, G. P.

G. P. Nordin, J. T. Meier, P. C. Deguzman, M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 169–177 (1999).

Peters, K. W.

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable, visible, imaging spectropolarimeters for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

Petty, T. E.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Phipps, G. S.

D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Porter, J. G.

E. A. West, J. G. Porter, J. M. Davis, A. Gary, M. Adams, “Development of a polarimeter for magnetic field measurements in the ultraviolet,” in Polarization Analysis, Measurement, and Remote Sensing, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE4481 (to be published).

Pugh, E. N.

Sabatke, D. S.

D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Silverman, M. P.

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Smith, M. H.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

Sornsin, E. A.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

Strange, W.

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Sweatt, W. C.

D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

Turner, T. S.

J. S. Tyo, T. S. Turner, “Variable-retardance, Fourier-transform imaging spectropolarimeters for visible spectrum remote sensing,” Appl. Opt. 40, 1450–1458 (2001).
[CrossRef]

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable, visible, imaging spectropolarimeters for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
[CrossRef]

Tyo, J. S.

J. S. Tyo, T. S. Turner, “Variable-retardance, Fourier-transform imaging spectropolarimeters for visible spectrum remote sensing,” Appl. Opt. 40, 1450–1458 (2001).
[CrossRef]

J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25, 1198–2000 (2000).
[CrossRef]

J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization-sensitive vision or imaging system,” J. Opt. Soc. Am. A 15, 359–366 (1998).
[CrossRef]

J. S. Tyo, E. N. Pugh, N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15, 367–374 (1998).
[CrossRef]

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable, visible, imaging spectropolarimeters for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

J. S. Tyo, “Considerations in polarimeter design,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 65–74 (2000).
[CrossRef]

J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
[CrossRef]

J. S. Tyo, “Polarization difference imaging,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1997).

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, pp. 11–29.

Walraven, R.

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
[CrossRef]

West, E. A.

E. A. West, J. G. Porter, J. M. Davis, A. Gary, M. Adams, “Development of a polarimeter for magnetic field measurements in the ultraviolet,” in Polarization Analysis, Measurement, and Remote Sensing, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE4481 (to be published).

Wolff, L. B.

Woodruff, J. B.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

Young, R. M.

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980), pp. 184–196.

Appl. Opt. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

M. P. Silverman, W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997).
[CrossRef]

Opt. Eng. (4)

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

A. Ambirajan, D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34, 1656–1659 (1995).
[CrossRef]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).
[CrossRef]

Opt. Lett. (2)

Other (21)

J. S. Tyo, “Considerations in polarimeter design,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 65–74 (2000).
[CrossRef]

J. S. Tyo, T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” in Imaging Spectrometry V, M. R. Descour, S. S. Shen, eds., Proc. SPIE3753, 214–225 (1999).
[CrossRef]

E. A. West, J. G. Porter, J. M. Davis, A. Gary, M. Adams, “Development of a polarimeter for magnetic field measurements in the ultraviolet,” in Polarization Analysis, Measurement, and Remote Sensing, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE4481 (to be published).

In fact, there is only one orthogonal polarization direction. Two states are orthogonally polarized regardless of their relative intensities; only the direction within the Stokes cone matters.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 137–143 (1999).

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 75–81 (2000).
[CrossRef]

D. H. Goldstein, R. A. Chipman, eds., Polarization: Measurement, Analysis, and Remote Sensing, Proc. SPIE3121 (1997).

D. H. Goldstein, D. B. Chenault, eds., Polarization: Measurement, Analysis, and Remote Sensing II, Proc. SPIE3754 (1999).

D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Polarization Analysis, Measurement, and Remote Sensing III, Proc. SPIE4133 (2000).

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

J. S. Tyo, “Polarization difference imaging,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1997).

G. P. Nordin, J. T. Meier, P. C. Deguzman, M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 169–177 (1999).

Note that the rows and columns of the Mueller matrix are numbered 0, … , 3, in agreement with conventions.4

Note that the full set of N vectors forms a frame in ℝM . The last three columns form a tight frame in ℝM-1 .

If the input states are not uniformly distributed, the expectation can simply be evaluated with the appropriate statistical distribution of polarization state.

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980), pp. 184–196.

I. Daubecheis, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992), pp. 53–106.
[CrossRef]

Sabatke et al.7,8 introduced the concept of equal-weighted variance for polarimeter optimization. The equal-weighted variance is equivalent to the Frobenius norm of the synthesis matrix  B̲ , defined in Eq. (25). Because the Frobenius norm of all analysis matrices is N2/2, the Frobenius norm of the synthesis matrix is equivalent to the Frobenius condition number  A̲ .

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, pp. 11–29.

Azzam et al.22 first proposed the tetrahedron on the Poincaré sphere for choosing the optimal set of calibration states for a SV polarimeter.

T. S. Turner, K. W. Peters, J. S. Tyo, “Portable, visible, imaging spectropolarimeters for remote sensing applications,” in Sensors, Systems, and Next-Generation Satellites II, H. Fujisada, ed., Proc. SPIE3498, 223–230 (1998).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic of the 3-D Stokes cone that represents all possible partially linearly polarized states. The three Cartesian axes are given by s0, s1, and s2, but only states that satisfy inequality (2) are physically realizable, resulting in the conical space. States on the surface of the cone are completely linearly polarized. This figure presents three cross sections. The first is parallel to the s1 and s2 axes and represents a 2-D Poincaré sphere (circle) of partially linearly polarized states. This plane is the projection of the traditional 3-D Poincaré sphere onto its equatorial plane. The second and third projections are in the s0s1 and s0s2 planes. These projections are 2-D Stokes cones that represent polarization differences between orthogonal linear polarization states.11,13,14 The 3-D conical space is itself a projection of the 4-D Stokes cone onto the s0s1s2 hyperplane.

Fig. 2
Fig. 2

Schematic of the 3-D Stokes cone with some important bases and frames. The basis formed by the vectors marked 0°, 60°, and 120° form the optimum three-measurement system.11 The vectors marked 0°, 45°, and 90° form a second common basis that was shown to be suboptimal.11 This basis, along with the fourth vector marked 135°, forms a four-element frame, the optimal system for N = 4, as proposed by Walraven.20

Fig. 3
Fig. 3

Optimal condition number for four-measurement rotating compensator systems. There is a clear optimum at δ = 0.3661λ. I obtained the optimization by minimizing the L2 condition number of the system matrices, but achieved the same result as Sabatke et al.,7 where the equal-weighted variance—equivalent to the Frobenius condition number of the system matrices—was minimized.

Fig. 4
Fig. 4

Optimal condition number for VR systems showing the lowest possible condition number for a VR polarimeter with the given fast-axis orientation values of ϕ1 and ϕ2. The jaggedness of the contours is due to coarse sampling in ϕ1 and ϕ2, but that should not detract from the overall message. All ϕ12 pairs inside the lowest contour have a continuum of optimal configurations.9 Although the condition number of all optimal configurations is the same, the best set of angles is ϕ1 = 45°, ϕ2 = 0°. This point is the geometric center of the optimal region and provides access to the entire Poincaré sphere. For a detailed description of this optimization, see Tyo.10

Fig. 5
Fig. 5

Numerical simulation and analytic prediction of rms error in the reconstructed SV for RR systems. I calculated the rms error by generating 106 random polarization states and simulating the polarimeter operation with 50 different realizations of error in the angular setting of the compensator. The error was generated by use of normal statistics with a standard deviation of σ. Analytic curves were generated with Eq. (23).

Fig. 6
Fig. 6

Frobenius norm of the perturbation matrix Δ̲ as a function of retardance of the rotating compensator. Δ̲ was computed for N = 4, 6, and 8 at the optimum angles given in Table 1 and divided by the number of measurements. Note that at δ = 0.3661λ where the condition number of A̲ and B̲ is minimized, ‖Δ̲F is relatively large.

Fig. 7
Fig. 7

Trajectories traced out on the surface of the Poincaré sphere for RR polarimeters as the retarder angle is altered. After Sabatke et al.,7 Fig. 1. For δ = 0.125λ, the condition number is high, but the total trajectory length is small as the retarder rotates. For δ = 0.37λ, the condition number is minimized, but the trajectory on the Poincaré sphere is longer, hence ‖Δ̲F is large.

Fig. 8
Fig. 8

Predicted and simulated rms error for the optimum RR system with N = 4, 6, and 8. Data were generated with the same procedures as in Fig. 5 with system parameters as given in Table 1.

Fig. 9
Fig. 9

Predicted rms error for optimum RR system (δ = 0.3661λ) as a function of the number of measurements made. At large N, the optimization is slow, and it is not always possible to find an optimum where the condition number of A̲ is 3. However, in all cases the condition number used to generate this figure was within 0.1% of 3.

Tables (1)

Tables Icon

Table 1 Optimum Retarder Positioning Angles for Four-, Six-, and Eight-Measurement RR Systemsa

Equations (37)

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S=s0s1s2s3=|Ext|2+|Eyt|2|Ext|2-|Eyt|2|E45t|2-|E-45t|2|Elcpt|2-|Ercpt|2,
s12+s22+s32s02,
S=Su+Sp=s0,u 0 0 0T+s0-s0,u s1 s2 s3T,
i=1N ξi2r2, N1,
i=1N ξi2αξ02, N1.
x·y=ξ0,xξ0,y+ξ1,xξ1,y=0.
ξ1,y2=-ξ0,x/ξ1,x2ξ0,y>ξ0,y2.
ξ1,y=-ξ0,x/ξ1,xξ0,y=-ξ0,y.
s0,outi=M00is0+M01is1+M02is2+M03is3=[M̲0i]T·S,
I=A̲·S, S=B̲·I,
y=i=1Nx˜i·yxi,
Ay·yi=1N |x˜i·y|2By·y,
A=0.5625, B=0.75.
A=0.25, B=0.8536.
κA̲=A̲ A̲-1.
A̲2=supxA̲·x2x2,
S=BA̲·S,
=S-S=BA̲-IS=B̲Δ̲·S
Δ̲ij=A̲-A̲ij=δiM̲0jϕi+δi-M̲0jϕiδiδiM̲0jϕϕϕi,
22=BΔ̲·S22=i=03BΔ̲·Si2,
EδBΔ̲Si2=jB̲ijδjΔ̲j1s1+Δ̲j2s2+Δ̲j3s3×kB̲ikδkΔ̲k1s1+Δ̲k2s2+Δ̲k3s3 =jkB̲ijB̲ikΔ̲j1s1+Δ̲j2s2+Δ̲j3s3Δ̲k1s1+Δ̲k2s2+Δ̲k3s3Eδjδk =σ2j=03B̲ij2Δ̲j12s12+Δ̲j22s22+Δ̲j32s32+2Δ̲j1Δ̲j2s1s2+2Δ̲j1Δ̲j3s1s3+Δ̲j2Δ̲j3s2s3.
EBΔ̲·Si2=σ23j=03B̲ij2Δ̲j12+Δ̲j22+Δ̲j32;.
E22=EBΔ̲·S22=σ23i=03j=03B̲ij2k=03Δ̲jk2.
σ23i=03j=03B̲ij2k=03Δ̲jk2=EBΔ̲F2,
A̲F2=ij |A̲ij|2
2=b0N,
M̲·S̲i=S̲o, with
S̲i=Si1 Si2 Si3 Si4,
S̲o=So1 So2 So3 So4,
M̲=S̲o·S̲i-1,
A̲=USV̲T,
A̲·Su=1/2  1/2T=u1.
σ1=A̲·Su2Su2=N2,
σj=N2(M-1)1/2j1,
κ2A̲=σ1/σM=M-11/2.
A̲·Su22Su22=σ12=N/4.
A̲·Sp22Sp22=12σ12+σM2=18N+NM-1.

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