Abstract

We analyze estimation error as a function of spectral bandwidth for division-of-amplitude (DoAm) Stokes polarimeters. Our approach allows quantitative assessment of the competing effects of noise and deterministic error, or bias, as bandwidth is varied. We use the signal-to-rms error (SRR) as a metric. Rather than calculating the SRR of the estimated Stokes parameters themselves, we use the singular-value decomposition to calculate the SRRs of the coefficients of the measured data vector projected onto the measurement matrix left singular vectors. We argue that calculating the SRRs for left singular vector coefficients will allow development of reconstruction filters to minimize Stokes estimation error. For the example case of a source with constant polarization over a relatively wide band, we show that as the spectral filter bandwidth is increased to include wavelengths significantly different than the design wavelength, the SRRs of the estimated left singular vector coefficients will a.) increase monotonically if relatively few photo-detection events (PDEs) are recorded, b.) after a sharp peak close to the design wavelength, decrease monotonically if relatively many PDEs are recorded, and c.) have well-defined maxima for nominal PDE counts. Given some idea of the source brightness relative to detector noise, one can specify a spectral filter bandwidth minimizing the variance and bias effects and optimizing Stokes parameter estimation. Our approach also allows one to specify the bandwidth over which the response of “achromatic” optics must be reasonably invariant with wavelength for rms Stokes estimation error to remain below some desired maximum. Finally, we point out that our method can be generalized not only to other types of polarimeters, but also to any sensing scheme that can be represented by a linear system for limiting values of a certain parameter.

© 2010 Optical Society of America

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References

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  1. R. M. A. Azzam, “Division-of-amplitude photo-polarimeter for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689 (1982).
  2. R. A. Chipman, “Data reduction for light-measuring polarimeters,” in Handbook of Optics, 3 ed., Vol. 1, Ch. 15, Sec. 20, McGraw-Hill (1995)
  3. A. Ambirajan, and D. C. Look, “Optimum angles for a polarimeter: Part I,” Opt. Eng. 34, 1651–1655 (1995).
    [CrossRef]
  4. A. Ambirajan, and D. C. Look, “Optimum angles for a polarimeter: Part II,” Opt. Eng. 34, 1656–1658 (1995).
    [CrossRef]
  5. D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. L. Dereniak, S. A. Kemma, and G. S. Phipps, “Figures of merit for complete Stokes polarimeter optimization,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan and D. H. Goldstein, eds., Proc. SPIE 4133 75–81 (2000).
    [CrossRef]
  6. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
    [CrossRef]
  7. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25, 1198–1200 (2000).
    [CrossRef]
  8. J. S. Tyo, “Design of optimal polarimeters: Maximization of signal-to-noise and minimization of deterministic error,” Appl. Opt. 41, 619–630 (2002).
    [CrossRef] [PubMed]
  9. V. L. Gamiz, and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
    [CrossRef]
  10. M. R. Foreman, C. M. Romero, and P. Török, “A priori information and optimization in polarimetry,” Opt. Express 16, 15212–15226 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-19-15212.
    [CrossRef] [PubMed]
  11. E. Wolf, “Unified theory of polarization and coherence,” in Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press 2007)pp. 174–201.
  12. 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(22), 5453–5469 (2006).
    [CrossRef] [PubMed]
  13. J. Boger, D. Bowers, M. P. Fetrow, and K. Bishop, “Issues in a broadband 4-channel reduced Stokes polarimeter,” in Polarization Analysis, Measurement, and Remote Sensing IV, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE 4481, pp. 311–321 (2002).
    [CrossRef]
  14. G. H. Golub, and C. F. Van Loan, “Orthogonality and the SVD,” in Matrix Computations, 3rd ed., Johns Hopkins University Press, p. 70 (1996).
  15. D. J. Kadrmas, E. C. Frey, and B. M. W. Tsui, “An SVD investigation of modeling scatter in multiple energy windows for improved SPECT images,” IEEE Trans. Nucl. Sci. 43, 2275–2284 (1996).
    [CrossRef]
  16. A. M. Phenis, M. Virgen, and E. de Leon, “Achromatic instantaneous Stokes imaging polarimeter,” in Novel Optical Systems Design and Optimization VIII,” J. Sasian, R. Koshen, & R. Juergen, eds., Proc. SPIE 5875, pp. 587502–1–587502–8 (2005).
  17. E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous Stokes imaging polarimeter,” in Polarization Science and Remote Sensing III, J. Shaw, J. S. Tyo, eds., Proc. SPIE 6682, 668215–668215–9 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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  20. J. D. Mudge, M. A. Virgen, and P. Dean, “Near-infrared simultaneous Stokes imaging polarimeter,” in Polarization Science and Remote Sensing IV, J.A. Shaw & J.S. Tyo, eds., Proc. SPIE 7461,74610L (2009).
    [CrossRef]
  21. F. Goudail, and A. Bénière, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett. 34(9), 1471–1473 (2009).
    [CrossRef] [PubMed]
  22. J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial Mueller polarimeters,” Appl. Opt. 49, 2326–2333 (2010).
    [CrossRef] [PubMed]
  23. J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, “Iterative image restoration using approximate inverse preconditioning,” IEEE Trans. Image Process. 5, 1151–1162 (1996).
    [CrossRef]

2010 (1)

2009 (1)

2008 (1)

2006 (1)

2002 (2)

J. S. Tyo, “Design of optimal polarimeters: Maximization of signal-to-noise and minimization of deterministic error,” Appl. Opt. 41, 619–630 (2002).
[CrossRef] [PubMed]

V. L. Gamiz, and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

2000 (2)

1996 (2)

D. J. Kadrmas, E. C. Frey, and B. M. W. Tsui, “An SVD investigation of modeling scatter in multiple energy windows for improved SPECT images,” IEEE Trans. Nucl. Sci. 43, 2275–2284 (1996).
[CrossRef]

J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, “Iterative image restoration using approximate inverse preconditioning,” IEEE Trans. Image Process. 5, 1151–1162 (1996).
[CrossRef]

1995 (2)

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, 1656–1658 (1995).
[CrossRef]

1992 (1)

1982 (1)

R. M. A. Azzam, “Division-of-amplitude photo-polarimeter for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689 (1982).

Ambirajan, A.

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, 1656–1658 (1995).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Division-of-amplitude photo-polarimeter for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689 (1982).

Belsher, J. F.

V. L. Gamiz, and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

Bénière, A.

Bilmont, M. F.

Chenault, D. B.

Dereniak, E. L.

Descour, M. R.

Foreman, M. R.

Frey, E. C.

D. J. Kadrmas, E. C. Frey, and B. M. W. Tsui, “An SVD investigation of modeling scatter in multiple energy windows for improved SPECT images,” IEEE Trans. Nucl. Sci. 43, 2275–2284 (1996).
[CrossRef]

Gamiz, V. L.

V. L. Gamiz, and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

Goldstein, D. L.

Goudail, F.

Hoover, B. G.

Johnson, S. J.

Kadrmas, D. J.

D. J. Kadrmas, E. C. Frey, and B. M. W. Tsui, “An SVD investigation of modeling scatter in multiple energy windows for improved SPECT images,” IEEE Trans. Nucl. Sci. 43, 2275–2284 (1996).
[CrossRef]

Kemme, S. A.

Look, D. C.

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

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

Nagy, J. G.

J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, “Iterative image restoration using approximate inverse preconditioning,” IEEE Trans. Image Process. 5, 1151–1162 (1996).
[CrossRef]

Phipps, G. S.

Plemmons, R. J.

J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, “Iterative image restoration using approximate inverse preconditioning,” IEEE Trans. Image Process. 5, 1151–1162 (1996).
[CrossRef]

Roggemann, M. C.

Romero, C. M.

Sabatke, D. S.

Shaw, J. A.

Sweatt, W. C.

Torgersen, T. C.

J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, “Iterative image restoration using approximate inverse preconditioning,” IEEE Trans. Image Process. 5, 1151–1162 (1996).
[CrossRef]

Török, P.

Tsui, B. M. W.

D. J. Kadrmas, E. C. Frey, and B. M. W. Tsui, “An SVD investigation of modeling scatter in multiple energy windows for improved SPECT images,” IEEE Trans. Nucl. Sci. 43, 2275–2284 (1996).
[CrossRef]

Tyler, D. W.

Tyo, J. S.

Wang, Z.

Appl. Opt. (4)

IEEE Trans. Image Process. (1)

J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, “Iterative image restoration using approximate inverse preconditioning,” IEEE Trans. Image Process. 5, 1151–1162 (1996).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. J. Kadrmas, E. C. Frey, and B. M. W. Tsui, “An SVD investigation of modeling scatter in multiple energy windows for improved SPECT images,” IEEE Trans. Nucl. Sci. 43, 2275–2284 (1996).
[CrossRef]

J. Mod. Opt. (1)

R. M. A. Azzam, “Division-of-amplitude photo-polarimeter for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689 (1982).

Opt. Eng. (3)

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, 1656–1658 (1995).
[CrossRef]

V. L. Gamiz, and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Other (9)

J. Boger, D. Bowers, M. P. Fetrow, and K. Bishop, “Issues in a broadband 4-channel reduced Stokes polarimeter,” in Polarization Analysis, Measurement, and Remote Sensing IV, D. H. Goldstein, D. B. Chenault, W. G. Egan, M. J. Duggin, eds., Proc. SPIE 4481, pp. 311–321 (2002).
[CrossRef]

G. H. Golub, and C. F. Van Loan, “Orthogonality and the SVD,” in Matrix Computations, 3rd ed., Johns Hopkins University Press, p. 70 (1996).

A. M. Phenis, M. Virgen, and E. de Leon, “Achromatic instantaneous Stokes imaging polarimeter,” in Novel Optical Systems Design and Optimization VIII,” J. Sasian, R. Koshen, & R. Juergen, eds., Proc. SPIE 5875, pp. 587502–1–587502–8 (2005).

E. de Leon, R. Brandt, A. Phenis, and M. Virgen, “Initial results of a simultaneous Stokes imaging polarimeter,” in Polarization Science and Remote Sensing III, J. Shaw, J. S. Tyo, eds., Proc. SPIE 6682, 668215–668215–9 (2007).
[CrossRef]

P. C. Hansen, “The smoothing property of the kernel,” in Rank-Deficient and Discrete Ill-Posed Problems,” SIAM Press, Philadelphia, p. 8 (1998).

J. D. Mudge, M. A. Virgen, and P. Dean, “Near-infrared simultaneous Stokes imaging polarimeter,” in Polarization Science and Remote Sensing IV, J.A. Shaw & J.S. Tyo, eds., Proc. SPIE 7461,74610L (2009).
[CrossRef]

E. Wolf, “Unified theory of polarization and coherence,” in Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press 2007)pp. 174–201.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. L. Dereniak, S. A. Kemma, and G. S. Phipps, “Figures of merit for complete Stokes polarimeter optimization,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan and D. H. Goldstein, eds., Proc. SPIE 4133 75–81 (2000).
[CrossRef]

R. A. Chipman, “Data reduction for light-measuring polarimeters,” in Handbook of Optics, 3 ed., Vol. 1, Ch. 15, Sec. 20, McGraw-Hill (1995)

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Figures (8)

Fig. 1.
Fig. 1.

Condition number of the W matrix as a function of wavelength for the polarimeter described in this paper. Note the clear minimum at the design wavelength of 633 nm.

Fig. 2.
Fig. 2.

Photo-detection events (PDEs) among 4 detectors of a DoAm polarimeter for unpolarized light measured over nine 10-nm spectral bands. The bands are each centered at the diamond symbols on each plot.

Fig. 3.
Fig. 3.

Absolute difference between detector PDE counts for unpolarized light as a function of optical bandwidth and relative to the rms shot noise averaged over all four detectors. The bandwidth is increased from 10 nm centered at 633 nm by successively adding 10 nm subbands to alternating sides of the previous band.

Fig. 4.
Fig. 4.

Signal to rms-error (SRR) for the coefficients of U T d as spectral filter width is increased in 5-nm increments for a constant, DoP=1 source and 1 × 105 photo-detection events (PDEs) in the 5-nm band centered at the reference wavelength and all added subbands.

Fig. 5.
Fig. 5.

Signal to rms-error (SRR) for the coefficients of U T d as spectral filter width is increased in 5-nm increments for a constant, DoP=1 source and 5 × 104 PDEs in the 5-nm band centered at the reference wavelength and all added sub-bands.

Fig. 6.
Fig. 6.

Signal to rms-error (SRR) for the coefficients of U T d as spectral filter width is increased in 5-nm increments for a constant, DoP=1 source and 1 × 103 PDEs in the 5-nm band centered at the reference wavelength and all added sub-bands.

Fig. 7.
Fig. 7.

Signal to rms-error (SRR) for the coefficients of U T d as spectral filter width is increased in 5-nm increments for a source with DoP=1 inside and DoP=0 outside the 5-nm band about the reference wavelength. The plots are for 1 × 103 PDEs in the reference band and all added sub-bands.

Fig. 8.
Fig. 8.

Signal to rms-error (SRR) for the coefficients of U T d as spectral filter width is increased in 5-nm increments for a constant, DoP=1 source and 1 × 102 PDEs in the 5-nm band centered at the reference wavelength and all added sub-bands. The detected photons are corrupted by 25 electrons rms read noise.

Equations (40)

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d = Ws ,
s ˜ = W 1 d ,
SNR θ ˜ = θ ˜ [ E { ( θ ˜ θ ˜ ) 2 } ] 1 2 = θ ˜ σ θ ˜ 2 .
SRR θ ˜ = θ ˜ [ E { ( θ ˜ θ ) 2 } ] 1 2 = θ ˜ ε θ ˜ 2 ,
ε θ ˜ 2 = σ θ ˜ 2 ( θ ˜ θ ) 2 ,
W T W x i = u i x i
W x i 2 = x i W T W x i = u i ,
W = U S V T ,
W 1 = V S 1 U T ,
s ˜ = V S 1 U T d
g ̂ = U T d ,
s ˜ = V S 1 F U T d ,
V 633 = ( 1.0 0.0 0.0 0.0 0.0 8.53 × 10 7 1.0 5.56 × 10 9 0.0 2.78 × 10 9 5.56 × 10 9 1.0 0.0 1.0 8.53 × 10 7 2.78 × 10 9 )
p ̂ = S 1 U T d .
s ˜ = V p ̂ .
s ˜ = P S 1 U T d = P p ̂
V 638 = ( 0.99 1.22 × 10 3 0.003 0.0029 6.87 × 10 4 0.992 0.1054 0.0724 5.46 × 10 5 0.019 0.68 0.73 4.32 × 10 3 0.126 0.7242 0.678 )
V ¯ 85 = ( 0.999870 0.00325933 0.0155427 0.00282588 0.0153775 0.0375014 0.903992 0.425623 0.00226081 0.994201 0.0767387 0.0753072 0.00431352 0.100738 0.420318 0.901757 )
g b = SV T n = 1 N f n s n ,
g = n = 1 N S n V n T f n s n ,
g ̂ = U r T n = 1 N [ W n f n s n + n n ] ,
W n = U n S n V n T = U n S n R n ( 4 ) R n ( 4 ) T V n T = U n G n ( V n R n ( 4 ) ) T
g b = S r n = 1 N p nr = S r n = 1 N ( V n R n ( 4 ) ) T f n s n
b j = k = 1 K U rjk T [ n = 1 N f n W n s n ] k [ S r n = 1 N ( V n R n ( 4 ) ) T f n s n ] j ,
v = diag { C m } = diag { U T n [ U T n ] T U T n [ U T n ] T }
v = diag { ( k = 1 K U 1 k T n k k = 1 K U 2 k T n k k = 1 K U 3 k T n k k = 1 K U 4 k T n k ) T ( k = 1 K U 1 k T n k k = 1 K U 2 k T n k k = 1 K U 3 k T n k k = 1 K U 4 k T n k ) } .
v j = k = 1 K U jk T n k q = 1 K U jq T n q
= [ U j 1 T n 1 + U j 2 T n 2 + + U j K T n K ] [ U j 1 T n 1 + U j 2 T n 2 + + U j K T n K ] .
v j = k = 1 K ( U jk T ) 2 σ k 2 ,
v j = k = 1 K U jk T m = 1 K U jq T n k n q k = 1 K U jk T n k q = 1 K U jq T n q .
v j = k = 1 K ( U jk T ) 2 [ n k 2 n k 2 ] = k = 1 K ( U jk T ) 2 d k ,
v j = k = 1 K ( U jk T ) 2 n = 1 N f n ( W n s n ) k .
SRR j = k = 1 K U jk T [ n = 1 N ( W n f n s n ) ] k k = 1 K ( U jk T ) 2 [ n = 1 N ( f n W n s n ) k + σ k 2 ] k + b j 2 .
V r T = R n ( 4 ) T V n T .
R n ( 4 ) T = V r T V n .
C mrij = k = 1 K U rik T U rjk T σ k ( total ) 2 ,
σ k ( total ) 2 = σ k 2 + n = 1 N ( f n W n s n ) k
C S = ( V r S r 1 U r T n ) ( V r S r 1 U r T n ) T = ( V r S r 1 ) C mr ( V r S r 1 ) T .
C S = ( V r S 1 F ) C mr ( V r S 1 F ) T .
s = ( 1 1 3 1 3 1 3 ) .

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