Abstract

Methods are presented for optimizing the design of Mueller matrix polarimeters and and in particular selecting the retardances and orientation angles of polarization components to ensure accurate reconstruction of a sample’s Mueller matrix in the presence of error sources. Metrics related to the condition number and to the singular value decomposition are used to guide the design process for Mueller matrix polarimeters with the goal of specifying polarization elements, comparing polarimeter configurations, estimating polarimeter errors, and compensating for known error sources. The use of these metrics is illustrated with analyses of two example polarimeters: a dual rotating retarder polarimeter, and a dual variable retarder polarimeter.

© 2008 Optical Society of America

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References

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  1. 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. A 5, 681-689 (1988).
    [CrossRef]
  2. A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part I," Opt. Eng. 34, 1651-1655 (1995).
    [CrossRef]
  3. A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part II," Opt. Eng. 34, 1656-1659 (1995).
    [CrossRef]
  4. D. S. Sabatke, M. R. Descour, E. 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]
  5. D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
    [CrossRef]
  6. M. H. Smith, "Optimization of a dual-rotating-retarder Mueller matrix polarimeter," Appl. Opt. 41, 2488-2493 (2002).
    [CrossRef] [PubMed]
  7. J. S. Tyo, "Noise equialization in Stokes parameter images obtained by use of variable-retardance polarimeters," Opt. Lett. 25, 1198-2000 (2000).
    [CrossRef]
  8. J. S. Tyo, "Considerations in polarimeter design," Proc. SPIE 4133, 65-74 (2000).
    [CrossRef]
  9. 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]
  10. S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002).
    [CrossRef]
  11. A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
    [CrossRef] [PubMed]
  12. A. De Martino, B. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).
  13. E. Garcia-Caurel, A. De Martina, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
  14. 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]
  15. J. Zallat, S. Ainouz, 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]
  16. L. Broch and L. Johann, "Optimizing precision of rotating compensator ellipsometry," Phys. Status Solidi C 5, 1036-1040 (2008).
    [CrossRef]
  17. G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Phys. Status Solidi C 5, 1027-1030 (2008).
    [CrossRef]
  18. Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
    [CrossRef]
  19. R. A. Chipman, "Polarimetry" in Handbook of Optics 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), pp. 22.1 - 22.37.
  20. G. H. Golub and C. F. Van Loan, Matrix Computations, (Johns Hopkins University Press, Baltimore, Maryland, 1983).
  21. R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).
  22. K. M. Twietmeyer, GDx-MM: An Imaging Mueller Matrix Retinal Polarimeter, Ph.D. Dissertation (University of Arizona, 2007).

2008

L. Broch and L. Johann, "Optimizing precision of rotating compensator ellipsometry," Phys. Status Solidi C 5, 1036-1040 (2008).
[CrossRef]

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Phys. Status Solidi C 5, 1027-1030 (2008).
[CrossRef]

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

2006

J. Zallat, S. Ainouz, 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

A. De Martino, B. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).

E. Garcia-Caurel, A. De Martina, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).

2003

2002

2000

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

J. S. Tyo, "Considerations in polarimeter design," Proc. SPIE 4133, 65-74 (2000).
[CrossRef]

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

D. S. Sabatke, M. R. Descour, E. 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]

1998

1995

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

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

1988

Ainouz, S.

J. Zallat, S. Ainouz, 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]

Alvarez-Herrero, A.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Ambirajan, A.

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

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

Azzam, R. M. A.

Broch, L.

L. Broch and L. Johann, "Optimizing precision of rotating compensator ellipsometry," Phys. Status Solidi C 5, 1036-1040 (2008).
[CrossRef]

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Phys. Status Solidi C 5, 1027-1030 (2008).
[CrossRef]

De Martina, A.

E. Garcia-Caurel, A. De Martina, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).

De Martino, A.

A. De Martino, B. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

del Toro Iniesta, J. C.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Dereniak, E.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

D. S. Sabatke, M. R. Descour, E. 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]

Descour, M. R.

D. S. Sabatke, M. R. Descour, E. 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]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

Domingo, V.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Drevillon, B.

A. De Martino, B. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).

E. Garcia-Caurel, A. De Martina, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

Elminyawi, I. M.

El-Saba, A. M.

Garcia, J. P.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

Garcia-Caurel, B.

A. De Martino, B. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).

Garcia-Caurel, E.

E. Garcia-Caurel, A. De Martina, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

Gasent, J. L.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Heredero, R. L.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Jochum, L.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Johann, L.

L. Broch and L. Johann, "Optimizing precision of rotating compensator ellipsometry," Phys. Status Solidi C 5, 1036-1040 (2008).
[CrossRef]

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Phys. Status Solidi C 5, 1027-1030 (2008).
[CrossRef]

Kemme, S. A.

D. S. Sabatke, M. R. Descour, E. 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]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

Laude, B.

A. De Martino, B. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

Locke, A. M.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

Look, D. C.

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

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

López Jiménez, A. C.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Martínez Pillet, V.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Phipps, G. S.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

D. S. Sabatke, M. R. Descour, E. 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]

Piller, G.

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Phys. Status Solidi C 5, 1027-1030 (2008).
[CrossRef]

Sabatke, D. S.

D. S. Sabatke, M. R. Descour, E. 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]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

Savenkov, S. N.

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

Smith, M. H.

Stoll, M. P.

J. Zallat, S. Ainouz, 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]

Sweatt, W. C.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

D. S. Sabatke, M. R. Descour, E. 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]

Tyo, J. S.

Uribe-Patarroyo, N.

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Zallat, J.

J. Zallat, S. Ainouz, 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. Opt. A: Pure Appl. Opt.

J. Zallat, S. Ainouz, 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

Opt. Eng.

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 I," Opt. Eng. 34, 1651-1655 (1995).
[CrossRef]

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

Opt. Lett.

Phys. Status Solidi C

L. Broch and L. Johann, "Optimizing precision of rotating compensator ellipsometry," Phys. Status Solidi C 5, 1036-1040 (2008).
[CrossRef]

G. Piller, L. Broch, and L. Johann, "Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer," Phys. Status Solidi C 5, 1027-1030 (2008).
[CrossRef]

Q4Q5. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, V. Martínez Pillet, and The IMaX Team, "IMaX: a polarimeter based on liquid crystal variable retarders for an aerospace mission," Phys. Status Solidi C 5, 1041-1045 (2008).
[CrossRef]

Proc. SPIE

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, "Figures of merit for complete Stokes polarimeters," Proc. SPIE 4133, 75-81 (2000).
[CrossRef]

J. S. Tyo, "Considerations in polarimeter design," Proc. SPIE 4133, 65-74 (2000).
[CrossRef]

Thin Solid Films

A. De Martino, B. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).

E. Garcia-Caurel, A. De Martina, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).

Other

R. A. Chipman, "Polarimetry" in Handbook of Optics 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), pp. 22.1 - 22.37.

G. H. Golub and C. F. Van Loan, Matrix Computations, (Johns Hopkins University Press, Baltimore, Maryland, 1983).

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).

K. M. Twietmeyer, GDx-MM: An Imaging Mueller Matrix Retinal Polarimeter, Ph.D. Dissertation (University of Arizona, 2007).

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Figures (8)

Fig. 1.
Fig. 1.

Polarimeter configurations used in illustrative examples. (a) Dual Rotating Retarder (DRR) polarimeter. Components include fixed polarizers at orientation θ, and rotating retarders with retardance γ and orientations φg and φa. (b) Dual Variable Retarder (DVR) polarimeter. Components include fixed polarizers with orientation θ, outer retarders with fixed orientation φ1 and retardances γg1 and γa1, and inner retarders with fixed orientation φ2 and retardances γg2 and γa2.

Fig. 2.
Fig. 2.

Trajectories on the Poincarè sphere. (a) DRR polarimeter with one variable traces a figure-eight trajectory; trajectory shown is for γ = 127°. (b) DVR polarimeter with two variables traces a surface; surface shown is for φ1 = 27.4°, φ2 = 72.4°.

Fig. 3.
Fig. 3.

Base 10 log of the condition number as a function of waveplate retardance γ for the DRR polarimeter. The optimum solution is a retardance of 127°; condition number increases significantly with distance from the optimum solution.

Fig. 4.
Fig. 4.

Condition number as a function of rotating retarder increment for the DRR polarimeter. Horizontal and vertical axes are the rotational increments of the two retarders (Δθg and Δθa) in degrees. Blue indicates a high (undesirable) condition number; red indicates a low condition number. Color scale is nonlinear to emphasize the low condition number solutions. (a) For N = 16, the best solutions are highly localized; one example is Δθg = 34° and Δθa = 26°. (b) For N = 30 there are many extended regions with good solutions.

Fig. 5.
Fig. 5.

Condition number as a function of polarimeter variables for the DVR polarimeter, for N = 16. Blue indicates a high (undesirable) condition number, and red indicates a low condition number. Color scale is nonlinear to emphasize the low condition number solutions. (a) Optimization of retarder angular orientations φ1 and φ2 assuming retardances of γ1 = 225°, γ2 = 45°. Horizontal and vertical axes are the angular orientations of the two retarders in degrees. (b) Optimization of retarder retardances γ1 and γ2 for the optimum orientations of φ1 = 27.4°, φ2 = 72.4°. Horizontal and vertical axes are the retardances of the two retarders in degrees.

Fig. 6.
Fig. 6.

Singular values for the example polarimeter configurations. Lower slope and absolute singular value range indicates a more stable configuration.

Fig. 7.
Fig. 7.

Calculated mean and SD error for DRR (a,b) and DVR (c,d) polarimeters. (a), (c) Sample is a linear polarizer rotated from 0° to 90° in 15° increments; (b), (d) Sample is a quarter wave plate rotated from 0° to 90° in 15° increments. Solid red and blue lines indicate worst case error from systematic sources; error bars indicate worst case error from random sources.

Fig. 8.
Fig. 8.

EM metric to optimize retardance for the DRR polarimeter when information is known about the error source and/or sample to be measured. (a) The dependence of optimal retardance on M; and (b) The dependence of optimal retardance on N for a horizontal linear polarizer.

Tables (2)

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Table 1. Summary of polarimeter polarization components

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Table 2. Summary of polarimeter error sources

Equations (29)

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W i = [ a i , 0 g i , 0 a i , 0 g i , 1 a i , 0 g i , 2 a i , 0 g i , 3 a i , 1 g i , 0 a i , 3 g i , 3 ] .
WM = P .
M = W 1 P ,
W p 1 = ( W T W ) 1 W T ,
κ p ( A ) = A p A 1 p ,
A p = sup x D ( A ) Ax p x p ( matrix p norm ) , ( x p ) p = i x i p ( vector p norm )
A = UDV T = U [ μ 0 μ 1 μ K 1 μ K 0 0 0 0 ] V T ,
P = WM = U μ 1 0 0 0 0 μ 2 0 0 0 0 0 μ 16 V T M
κ ( W ) = κ ( G T A ) = κ ( G T ) κ ( A ) = κ ( G ) κ ( A ) .
V 1 = { 1 , 1 , 0 , 0 } , V 2 = { 1 , 1 3 , 2 2 3 , 0 } , V 3 = { 1 , 1 3 , 2 3 , 2 3 } , V 4 = { 1 , 1 3 , 2 3 , 2 3 } .
{ 4 , 4 3 3 , 4 3 3 , 4 3 3 , 4 3 3 , 4 3 3 , 4 3 3 , 4 3 , 4 3 , 4 3 , 4 3 , 4 3 , 4 3 , 4 3 , 4 3 , 4 3 , 4 3 } ,
{ 1 . , 0.333 , 0.943 ,0,1 . , 0.333 , 0.943,0,0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , } ,
{ 1 , 1 . , 0.0005 , 0 , 1 . , 1 . , 0.0005 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0,0 } .
{ 1 , 1 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 4.0557 , 2.7273 , 2.3094 , 2.3094 , 2.3094 , 2.3094 , 2.0221 , 1.4988 , 1.3333 , 1.3333 , 1.3333 , 1.3333 , 1.3333 , 1.3333 , 1.3333 , 0.0004 } ,
( W + δ W ) M + δ P = P M ,
M R = W p 1 P M
= W p 1 . [ ( W + δ W ) M + δ P ]
= M + δ M = M + W p 1 . [ δ WM + δ P ] ,
δ W ij r = 1 R δ rij δ W ij δx r x r = ϕ r .
δ P i = ε i ,
δ M k = i = 1 N W ki 1 [ j = 1 16 r = 1 R δ rij δ W ij δx r | x r = ϕ r M j ] + i = 1 N W ki 1 ε i k : 1 16 .
C M , jk = < δM j δM k > < δM j > < δM k > j , k : 1 16 .
δ E i = j = 1 16 δ W ij M j + δ P i .
δ E i = j = 1 16 r = 1 R a rji δ r m j + b i ε ,
C E , jk = δ E j δ E k δ E j δ E k j , k : 16 .
C E , ii = k = 1 16 j = 1 16 m j m k [ r = 1 R c rjki σ r 2 + d i σ ε 2 ] ,
C M = W 1 C E ( W 1 ) T .
EM = i C M , ii .

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