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).
  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).
  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).
  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.
  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).
  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.

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).

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).

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).

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).

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).

Look, D. C.

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]

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).

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).

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).

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).

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]

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).

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).

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]

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).

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).

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).

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)

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]

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

Opt. Lett. (2)

Other (21)

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).

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).

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

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.

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).

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

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

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).

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).

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

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).

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