Abstract

An error minimization method is presented for Stokes polarimeters applicable when the detected signals are affected by a combination of shot and Gaussian noise. The expectation of the Stokes vector variance is used as a performance measure. This measure is compared with the condition number of a polarization state analyzer matrix that is commonly used as a figure of merit. We show that a polarimeter with the minimum condition number is not necessarily optimal. The approach is used to optimize existing prism based polarimeters giving improvements in the performance when shot-noise cannot be neglected.

© 2009 Optical Society of America

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References

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  1. J. S. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt. 41(4), 619-630 (2002).
    [CrossRef]
  2. A. D. Martino, E. 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).
    [CrossRef]
  3. 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(11), 802-804 (2000).
    [CrossRef]
  4. K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).
  5. A. E. Elsner, S. A. Burns, J. J. Weiter, and F. C. Delori, "Infrared imaging of sub-retinal structures in the human ocular fundus," Vision Research 36(1), 191-205 (1996).
    [CrossRef]
  6. F. C. Delori and K. P. Pflibsen, "Spectral reflectance of the human ocular fundus," Appl. Opt. 28(6), 1061-1077 (1989).
    [CrossRef]
  7. M. R. Foreman, C. M. Romero, and P. Torok, "A priori information and optimisation in polarimetry," Opt. Express 16(19), 15,212-15,227 (2008).
  8. R. M. A. Azzam, "Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all Stokes parameters of light," Opt. Acta 29(5), 685-689 (1982).
    [CrossRef]
  9. E. Compain and B. Drevillon, "Broadband division-of-amplitude polarimeter based on uncoated prisms," Appl. Opt. 37(25), 5938-5944 (1998).
    [CrossRef]
  10. E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters and Mueller-matrix ellipsometers," Appl. Opt. 38(16), 3490-3502 (1999).
    [CrossRef]
  11. B. Laude-Boulesteix, A. de Martino, B. Drevillon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystals," Appl. Opt. 43(14), 2824-2832 (2004).
    [CrossRef]
  12. D. Lara and C. Dainty, "Axially resolved complete Mueller matrix confocal microscopy," Appl. Opt. 45(9), 1917-1930 (2006).
    [CrossRef]
  13. Perkin Elmer, "Avalanche Photodiodes: A User's Guide," Technical information, PerkinElmer Optoelectronics (2006).
  14. Hamamatsu, "Characteristics and use of Si APD (Avalanche Photodiode)," Technical Information SD-28, Hamamatsu Photonics K. K., Solid Division (2004).
  15. R. J. McIntyre, "Multiplication noise in uniform avalanche diodes," IEEE Trans. Electron Dev. 13(1), 164-168 (1966).
    [CrossRef]
  16. S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations. (Pitman, London, 1979).
  17. T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy, 1st ed. (Academic press, London NW1, 1984).
  18. F. Goudail, "Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise," Opt. Lett. 34(5), 647-649 (2009).
    [CrossRef]
  19. F. Delplancke, "Automated high-speed Mueller matrix scatterometer," Appl. Opt. 36(22), 5388-5395 (1997).
    [CrossRef]
  20. J. Ellis and A. Dogariu, "Discrimination of globally unpolarized fields through Stokes vector element correlations," J. Opt. Soc. Am A 22(3), 491-496 (2005).
    [CrossRef]

2009

2008

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).

M. R. Foreman, C. M. Romero, and P. Torok, "A priori information and optimisation in polarimetry," Opt. Express 16(19), 15,212-15,227 (2008).

2006

2005

J. Ellis and A. Dogariu, "Discrimination of globally unpolarized fields through Stokes vector element correlations," J. Opt. Soc. Am A 22(3), 491-496 (2005).
[CrossRef]

2004

A. D. Martino, E. 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).
[CrossRef]

B. Laude-Boulesteix, A. de Martino, B. Drevillon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystals," Appl. Opt. 43(14), 2824-2832 (2004).
[CrossRef]

2002

2000

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(11), 802-804 (2000).
[CrossRef]

1999

1998

1997

1996

A. E. Elsner, S. A. Burns, J. J. Weiter, and F. C. Delori, "Infrared imaging of sub-retinal structures in the human ocular fundus," Vision Research 36(1), 191-205 (1996).
[CrossRef]

1989

1982

R. M. A. Azzam, "Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all Stokes parameters of light," Opt. Acta 29(5), 685-689 (1982).
[CrossRef]

1966

R. J. McIntyre, "Multiplication noise in uniform avalanche diodes," IEEE Trans. Electron Dev. 13(1), 164-168 (1966).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, "Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all Stokes parameters of light," Opt. Acta 29(5), 685-689 (1982).
[CrossRef]

Burns, S. A.

A. E. Elsner, S. A. Burns, J. J. Weiter, and F. C. Delori, "Infrared imaging of sub-retinal structures in the human ocular fundus," Vision Research 36(1), 191-205 (1996).
[CrossRef]

Chipman, R. A.

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).

Compain, E.

Dainty, C.

de Martino, A.

Delori, F. C.

A. E. Elsner, S. A. Burns, J. J. Weiter, and F. C. Delori, "Infrared imaging of sub-retinal structures in the human ocular fundus," Vision Research 36(1), 191-205 (1996).
[CrossRef]

F. C. Delori and K. P. Pflibsen, "Spectral reflectance of the human ocular fundus," Appl. Opt. 28(6), 1061-1077 (1989).
[CrossRef]

Delplancke, F.

Dereniak, E. L.

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(11), 802-804 (2000).
[CrossRef]

Descour, M. R.

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(11), 802-804 (2000).
[CrossRef]

Dogariu, A.

J. Ellis and A. Dogariu, "Discrimination of globally unpolarized fields through Stokes vector element correlations," J. Opt. Soc. Am A 22(3), 491-496 (2005).
[CrossRef]

Drevillon, B.

Ellis, J.

J. Ellis and A. Dogariu, "Discrimination of globally unpolarized fields through Stokes vector element correlations," J. Opt. Soc. Am A 22(3), 491-496 (2005).
[CrossRef]

Elsner, A. E.

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).

A. E. Elsner, S. A. Burns, J. J. Weiter, and F. C. Delori, "Infrared imaging of sub-retinal structures in the human ocular fundus," Vision Research 36(1), 191-205 (1996).
[CrossRef]

Foreman, M. R.

M. R. Foreman, C. M. Romero, and P. Torok, "A priori information and optimisation in polarimetry," Opt. Express 16(19), 15,212-15,227 (2008).

Garcia-Caurel, E.

A. D. Martino, E. 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).
[CrossRef]

Goudail, F.

Kemme, S. A.

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(11), 802-804 (2000).
[CrossRef]

Lara, D.

Laude, B.

A. D. Martino, E. 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).
[CrossRef]

Laude-Boulesteix, B.

Martino, A. D.

A. D. Martino, E. 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).
[CrossRef]

McIntyre, R. J.

R. J. McIntyre, "Multiplication noise in uniform avalanche diodes," IEEE Trans. Electron Dev. 13(1), 164-168 (1966).
[CrossRef]

Pflibsen, K. P.

Phipps, G. S.

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(11), 802-804 (2000).
[CrossRef]

Poirier, S.

Romero, C. M.

M. R. Foreman, C. M. Romero, and P. Torok, "A priori information and optimisation in polarimetry," Opt. Express 16(19), 15,212-15,227 (2008).

Sabatke, D. S.

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(11), 802-804 (2000).
[CrossRef]

Schwartz, L.

Sweatt, W. C.

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(11), 802-804 (2000).
[CrossRef]

Torok, P.

M. R. Foreman, C. M. Romero, and P. Torok, "A priori information and optimisation in polarimetry," Opt. Express 16(19), 15,212-15,227 (2008).

Twietmeyer, K. M.

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).

Tyo, J. S.

VanNasdale, D.

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).

Weiter, J. J.

A. E. Elsner, S. A. Burns, J. J. Weiter, and F. C. Delori, "Infrared imaging of sub-retinal structures in the human ocular fundus," Vision Research 36(1), 191-205 (1996).
[CrossRef]

Zhao, Y.

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).

Appl. Opt.

IEEE Trans. Electron Dev.

R. J. McIntyre, "Multiplication noise in uniform avalanche diodes," IEEE Trans. Electron Dev. 13(1), 164-168 (1966).
[CrossRef]

J. Opt. Soc. Am A

J. Ellis and A. Dogariu, "Discrimination of globally unpolarized fields through Stokes vector element correlations," J. Opt. Soc. Am A 22(3), 491-496 (2005).
[CrossRef]

Opt. Acta

R. M. A. Azzam, "Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all Stokes parameters of light," Opt. Acta 29(5), 685-689 (1982).
[CrossRef]

Opt. Express

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, "Mueller matrix retinal imager with optimized polarization conditions," Opt. Express 16(26), 21,339-21,354 (2008).

M. R. Foreman, C. M. Romero, and P. Torok, "A priori information and optimisation in polarimetry," Opt. Express 16(19), 15,212-15,227 (2008).

Opt. Lett

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(11), 802-804 (2000).
[CrossRef]

Opt. Lett.

Thin Solid Films

A. D. Martino, E. 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).
[CrossRef]

Vision Research

A. E. Elsner, S. A. Burns, J. J. Weiter, and F. C. Delori, "Infrared imaging of sub-retinal structures in the human ocular fundus," Vision Research 36(1), 191-205 (1996).
[CrossRef]

Other

Perkin Elmer, "Avalanche Photodiodes: A User's Guide," Technical information, PerkinElmer Optoelectronics (2006).

Hamamatsu, "Characteristics and use of Si APD (Avalanche Photodiode)," Technical Information SD-28, Hamamatsu Photonics K. K., Solid Division (2004).

S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations. (Pitman, London, 1979).

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy, 1st ed. (Academic press, London NW1, 1984).

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Figures (7)

Fig. 1.
Fig. 1.

Signal to noise ratio of an APD and transimpedance amplifier combination operating at 8MHz sampling rate and with 5 nW of optical power falling on the detector. Realistic model of APD Perkin Elmer C30902E and a transimpedance amplifier with noise current of 3pA/Hz1/2.

Fig. 2.
Fig. 2.

Sampling of states of polarization represented on the surface of the Poincaré sphere. Uniformly distributed samples are taken along a right-handed spiral locus that starts at the south pole and finishes at the north pole, with 128 samples per revolution (For clarity the figure shows only 24 revolutions, though 128 were used in the numerical evaluation). Parameters u and θ are uniformly distributed within the intervals shown.

Fig. 3.
Fig. 3.

(Color online) BS polarimeter. (a) Eigenstates of the polarization analyzers of the DOAP on the Poincaré sphere, and (b) the noise on the Stokes parameters for states of polarization along the spiral locus of Figure 2 (30 nW input power at 8MHz).

Fig. 4.
Fig. 4.

(Color online) Compain polarimeter. (a) and (b) as per Figure 3.

Fig. 5.
Fig. 5.

(Color online) Six-detector polarimeter. (a) and (b) as per Fig. 3.

Fig. 6.
Fig. 6.

Shape parameters for the optimized polarimeter.

Fig. 7.
Fig. 7.

(Color online) Noise on the Stokes parameters for optimized PSA (Optim3) with 30 nW optical power at 8MHz.

Tables (1)

Tables Icon

Table 1. Cost function and condition number (δ) of the polarimeters evaluated for 30 nW and 10 nW input optical power sampled at 8MHz.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

i k = D kj s j ,
s ̂ j = F jk i k ,
n s j 2 = F jk 2 n k 2 .
ε S 2 = j = 0 3 ( s j s ̂ j ) 2 + ( n s j ) 2 ;
σ PSA 2 = ε S 2 p ( S ) d n S ,
D BS = [ 0.1978 0.1978 0 0 0.1978 0.1978 0 0 0.1514 0.0102 0.1497 0.0206 0.1508 0.0014 0.0205 0.1494 ] .
D Compain = [ 0.1875 0.1079 0.1534 0 0.1875 0.1079 0.1534 0 0.1950 0.1203 0 0.1535 0.1950 0.1203 0 0.1535 ] .
D 6 det = 1 6 [ 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 ] .
D Optim 3 = [ 0.2337 0.1259 0.1969 0 0.2337 0.1259 0.1969 0 0.2419 0.1427 0 0.1653 0.2419 0.1427 0 0.1953 ] .

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