Abstract

While several analyses of polarimeter noise-reduction have been published, little data has been presented to support the analytical results, particularly for a laser polarimeter based on measurements taken at discrete, independent rotation angles of two birefringent waveplates. This paper derives and experimentally demonstrates the reduction of both system and speckle noise in this type of laser polarimeter, achieved by optimizing the rotation angles of the waveplates by minimizing the condition numbers of the appropriate matrix equation. Results are demonstrated experimentally in signal-to-noise ratio (SNR) variations for a range of materials and spatial bandwidths. Use of optimal waveplate angles is found to improve the average SNR of the normalized Mueller matrix over speckle by a factor of up to 8 for a non-depolarizing material, but to provide little improvement for a depolarizing material. In the limit of zero spatial bandwidth, the average SNR of the normalized Mueller matrix over speckle is found to be greater than one for a non-depolarizing material and less than one for a depolarizing material.

© 2008 Optical Society of America

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References

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  1. D. B. Chenault, J. L. Pezzaniti, and R. A. Chipman, "Mueller matrix algorithms," in Polarization Analysis and Measurement, D. Goldstein and R. Chipman, eds.,Proc. SPIE 1746, 231-246 (1992).
    [CrossRef]
  2. R. A. Chipman, "Polarimetry," in Handbook of Optics, (McGraw-Hill, New York, 1994), Chap. 22.
  3. S. Breugnot and P. Clemenceau, "Modeling and performances of a polarization active imager at λ = 806nm," Opt. Eng. 39, 2681-2688 (2000).
    [CrossRef]
  4. A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part 1," Opt. Eng. 34, 1651-1655 (1995).
    [CrossRef]
  5. J. Zallat, S. Aïnouz, and M. Ph. Stoll, "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A 8, 807-814 (2006).
    [CrossRef]
  6. Y. Takakura and J. E. Ahmad, "Noise distribution of Mueller matrices retrieved with active rotating polarimeters," Appl. Opt. 46, 7354-7364 (2007)
    [CrossRef] [PubMed]
  7. M. H. Smith, "Optimization of a dual-rotating-retarder Mueller matrix polarimeter" Appl. Opt. 41, 2488-2493 (2002)
    [CrossRef] [PubMed]
  8. R. M. A. Azzam, "Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal," Opt. Lett. 2, 148-150 (1977)
    [CrossRef]
  9. J.W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J. C. Dainty, ed., (Springer-Verlag, New York, 1984), Chap. 2.
  10. E. Isaacson and H. B. Keller, Analysis of Numerical Methods (John Wiley and Sons, 1966).
  11. G. W. Stewart, Introduction to Matrix Computations (Academic Press, New York, 1973).
  12. R. Kress, Numerical Analysis (Springer-Verlag, Berlin, 1998).
    [CrossRef]
  13. P. D. Lax, Functional Analysis (Wiley-Interscience, 2002).
  14. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, 1990).
  15. D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
    [CrossRef]
  16. B. Boulbry, B. Le Jeune, F. Pellen, J. Cariou, and J. Lotrian, "Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source," J. Phys. D 35, 2508-2515 (2002).
    [CrossRef]
  17. 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]
  18. 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]
  19. J. S. Tyo, "Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters," Opt. Lett. 25, 1198-1200 (2000).
    [CrossRef]
  20. B. G. Hoover, R. A. Peredo, L. F. DeSandre, and L. J. Ulibarri, "Active polarimetric assessment of surface weathering," in Laser Radar Techniques for Atmospheric Sensing, U. N. Singh, ed., Proc. SPIE 5575, 38-43 (2004).
    [CrossRef]
  21. B. G. Hoover and J. S. Tyo, "Polarization components analysis for invariant discrimination," Appl. Opt. 46, 8364-8373 (2007).
    [CrossRef] [PubMed]
  22. J. J. Gil, "Characteristic properties of Mueller matrices," J. Opt. Soc. Am. A 17, 328-334 (2000).
    [CrossRef]
  23. J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
    [CrossRef]
  24. J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," Opt. Acta 33, 185-189 (1986).
    [CrossRef]
  25. M , Arioli, M. Baboulin, and S. Gratton, "A partial condition number for linear least squares problems," SIAM J. Matrix Anal. Appl. 29, 413-433 (2007).
    [CrossRef]

2007

2006

J. Zallat, S. Aïnouz, and M. Ph. Stoll, "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A 8, 807-814 (2006).
[CrossRef]

2002

B. Boulbry, B. Le Jeune, F. Pellen, J. Cariou, and J. Lotrian, "Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source," J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

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]

M. H. Smith, "Optimization of a dual-rotating-retarder Mueller matrix polarimeter" Appl. Opt. 41, 2488-2493 (2002)
[CrossRef] [PubMed]

2000

1995

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part 1," Opt. Eng. 34, 1651-1655 (1995).
[CrossRef]

1990

1986

J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," Opt. Acta 33, 185-189 (1986).
[CrossRef]

1985

J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
[CrossRef]

1977

Appl. Opt.

J. Opt. A

J. Zallat, S. Aïnouz, and M. Ph. Stoll, "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A 8, 807-814 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

B. Boulbry, B. Le Jeune, F. Pellen, J. Cariou, and J. Lotrian, "Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source," J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

Opt. Acta

J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
[CrossRef]

J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," Opt. Acta 33, 185-189 (1986).
[CrossRef]

Opt. Eng.

S. Breugnot and P. Clemenceau, "Modeling and performances of a polarization active imager at λ = 806nm," Opt. Eng. 39, 2681-2688 (2000).
[CrossRef]

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part 1," Opt. Eng. 34, 1651-1655 (1995).
[CrossRef]

Opt. Lett.

SIAM J. Matrix Anal. Appl.

M , Arioli, M. Baboulin, and S. Gratton, "A partial condition number for linear least squares problems," SIAM J. Matrix Anal. Appl. 29, 413-433 (2007).
[CrossRef]

Other

B. G. Hoover, R. A. Peredo, L. F. DeSandre, and L. J. Ulibarri, "Active polarimetric assessment of surface weathering," in Laser Radar Techniques for Atmospheric Sensing, U. N. Singh, ed., Proc. SPIE 5575, 38-43 (2004).
[CrossRef]

D. B. Chenault, J. L. Pezzaniti, and R. A. Chipman, "Mueller matrix algorithms," in Polarization Analysis and Measurement, D. Goldstein and R. Chipman, eds.,Proc. SPIE 1746, 231-246 (1992).
[CrossRef]

R. A. Chipman, "Polarimetry," in Handbook of Optics, (McGraw-Hill, New York, 1994), Chap. 22.

J.W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J. C. Dainty, ed., (Springer-Verlag, New York, 1984), Chap. 2.

E. Isaacson and H. B. Keller, Analysis of Numerical Methods (John Wiley and Sons, 1966).

G. W. Stewart, Introduction to Matrix Computations (Academic Press, New York, 1973).

R. Kress, Numerical Analysis (Springer-Verlag, Berlin, 1998).
[CrossRef]

P. D. Lax, Functional Analysis (Wiley-Interscience, 2002).

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, 1990).

Supplementary Material (2)

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of the DRR polarimeter. (a) View from behind the object, with “FA” denoting the fast axes of the waveplates. (b) Top view, with PSG = polarization-state generator, PSA = polarization-state analyzer, zp = 290mm, and F1 = 1m, F2 = 350mm the focal lengths of lenses L1 and L2. The aperture stop is a variable iris.

Fig. 2.
Fig. 2.

Mueller-averaged SNR of the due to black flat paint measured in the static configuration using different waveplate-angle sets.

Fig. 3.
Fig. 3.

Mueller-averaged SNR due to materials with different depolarization indices, measured in the static configuration using different waveplate-angle sets.

Fig. 4.
Fig. 4.

Mueller-averaged SNR of the Mueller matrices due to materials with different depolarization indices, measured in the static configuration using the 1-norm angle set, with equalized power on the detector.

Fig. 5.
Fig. 5.

Mueller-averaged SNR due to materials with different depolarization indices, measured using different waveplate-angle sets in the speckle configuration with a spatial bandwidth of (a) 3.0mm−1 and (b) 65.5mm−1.

Fig. 6.
Fig. 6.

Mueller-averaged SNR over speckle due to materials with different depolarization indices, measured using the 2-norm waveplate-angle set and variable spatial bandwidth. The green line at the top is the Mueller-averaged SNR over system noise of the titanium data. The theoretical result for polarized-intensity SNR over speckle in the limit of zero spatial bandwidth is annotated (the classical intensity limit).

Fig. 7.
Fig. 7.

Speckle patterns as seen through the modulated polarimeter due to (a) Spectralon (b) Titanium. Click on the pictures above to view the videos. [Media 1][Media 2]

Fig. 8.
Fig. 8.

Mueller-averaged SNR over speckle compared with polarized-intensity SNR over speckle at low bandwidth. (a) Spectralon/highly depolarizing and (b) Titanium/non-depolarizing.

Fig. 9.
Fig. 9.

SNR over speckle of the normalized Mueller matrix elements due to titanium and Spectralon at low bandwidth.

Tables (5)

Tables Icon

Table 1. Optimum angle sets for a receiver quarter-waveplate (in degrees) under different optimizations.

Tables Icon

Table 2. Optimum angle sets for a transmitter quarter-waveplate (in degrees) under different optimizations.

Tables Icon

Table 3. Optimum angle sets for a receiver waveplate with arbitrary retardance (in degrees) under different optimizations.

Tables Icon

Table 4. Optimum angle sets for a transmitter waveplate with arbitrary retardance (in degrees) under different optimizations.

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Table 5. Test materials: measured depolarization indices and relative polarized intensities.

Equations (29)

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

A 1 max 1 j 4 i = 1 4 a ij
A 2 λ max ( A * A )
A max 1 i 4 j = 1 4 a ij
A Frob i = 1 4 j = 1 4 a ij 2 ,
κ ( A ) A A 1
S out = M p 0 M qwp ( θ r ) MM qwp ( θ t ) M p 90 S in
= 1 4 [ 1 cos 2 ( 2 θ r ) sin ( 4 θ r ) 2 sin ( 2 θ r ) 1 cos 2 ( 2 θ r ) sin ( 4 θ r ) 2 sin ( 2 θ r ) 0 0 0 0 0 0 0 0 ] M
× [ 1 1 0 0 cos 2 ( 2 θ t ) cos 2 ( 2 θ t ) 0 0 sin ( 4 θ t ) 2 sin ( 4 θ t ) 2 0 0 sin ( 2 θ t ) sin ( 2 θ t ) 0 0 ] S in ,
I = [ 1 cos 2 ( 2 θ 1 r ) sin ( 4 θ 1 r ) 2 sin ( 2 θ 1 r ) 1 cos 2 ( 2 θ 2 r ) sin ( 4 θ 2 r ) 2 sin ( 2 θ 2 r ) 1 cos 2 ( 2 θ 3 r ) sin ( 4 θ 3 r ) 2 sin ( 2 θ 3 r ) 1 cos 2 ( 2 θ 4 r ) sin ( 4 θ 4 r ) 2 sin ( 2 θ 4 r ) ] M × [ 1 1 1 1 cos 2 ( 2 θ 1 t ) cos 2 ( 2 θ 2 t ) cos 2 ( 2 θ 3 t ) cos 2 ( 2 θ 4 t ) sin ( 4 θ 1 t ) 2 sin ( 4 θ 2 t ) 2 sin ( 4 θ 3 t ) 2 sin ( 4 θ 4 t ) 2 sin ( 2 θ 1 t ) sin ( 2 θ 2 t ) sin ( 2 θ 3 t ) sin ( 2 θ 4 t ) ] = RMT ,
T = [ 1 cos 2 ( 2 θ 1 t ) sin 2 ( 2 θ 1 t ) cos δ t sin ( 4 θ 1 t ) ( 1 cos δ t ) 2 sin ( 2 θ 1 t ) sin δ t 1 cos 2 ( 2 θ 2 t ) sin 2 ( 2 θ 2 t ) cos δ t sin ( 4 θ 2 t ) ( 1 cos δ t ) 2 sin ( 2 θ 2 t ) sin δ t 1 cos 2 ( 2 θ 3 t ) sin 2 ( 2 θ 3 t ) cos δ t sin ( 4 θ 3 t ) ( 1 cos δ t ) 2 sin ( 2 θ 3 t ) sin δ t 1 cos 2 ( 2 θ 4 t ) sin t ( 2 θ 4 t ) cos δ t sin ( 4 θ 4 t ) ( 1 cos δ t ) 2 sin ( 2 θ 4 t ) sin δ t ] T
R = [ 1 cos 2 ( 2 θ 1 r ) + sin 2 ( 2 θ 1 r ) cos δ r sin ( 4 θ 1 r ) ( 1 cos δ r ) 2 sin ( 2 θ 1 r ) sin δ r 1 cos 2 ( 2 θ 2 r ) + sin 2 ( 2 θ 2 r ) cos δ r sin ( 4 θ 2 r ) ( 1 cos δ r ) 2 sin ( 2 θ 2 r ) sin δ r 1 cos 2 ( 2 θ 3 r ) + sin 2 ( 2 θ 3 r ) cos δ r sin ( 4 θ 3 r ) ( 1 cos δ r ) 2 sin ( 2 θ 3 r ) sin δ r 1 cos 2 ( 2 θ 4 r ) + sin 2 ( 2 θ 4 r ) cos δ r sin ( 4 θ 4 r ) ( 1 cos δ r ) 2 sin ( 2 θ 4 r ) sin δ r ] ,
I + Δ I = ( R + Δ R ) ( M + Δ M ) ( T + Δ T ) .
R 1 ( I + Δ I ) T 1 = ( 1 + R 1 Δ R ) Δ M ( 1 + Δ T T 1 ) + ( 1 + R 1 Δ R ) M ( 1 + Δ T T 1 ) ,
R 1 Δ R < 1 and T 1 Δ T < 1 .
Δ M = ( 1 + R 1 Δ R ) 1 [ R 1 Δ I T 1 R 1 Δ R M M Δ T T 1 R 1 Δ RM Δ TT 1 ] ( 1 + Δ TT 1 ) 1 ,
Δ M ( 1 + R 1 Δ R ) 1 [ R 1 Δ I T 1 + R 1 Δ R M + M Δ T T 1
+ R 1 Δ R M Δ T T 1 ] ( 1 + Δ TT 1 ) 1 .
( 1 + R 1 Δ R ) 1 1 1 R 1 Δ R and ( 1 + Δ TT 1 ) 1 1 1 T 1 Δ T ,
Δ M 1 h [ R 1 Δ I T 1 + R 1 Δ R M
                        + M Δ T T 1 + R 1 Δ R M Δ T T 1 ] ,
h ( 1 R 1 Δ R ) ( 1 T 1 Δ T ) = ( 1 κ ( R ) Δ R R ) ( 1 κ ( T ) Δ T T ) .
I = RMT R M T
1 M R T I ,
Δ M M 1 h [ κ ( R ) κ ( T ) Δ I I + R 1 Δ R + Δ T T 1 + R 1 Δ R Δ T T 1 ]
              = κ ( R ) κ ( T ) h ( Δ I I + Δ R T R T ) + κ ( T ) h Δ T T + κ ( R ) h Δ R R ,
Δ M M κ ( R ) κ ( T ) h ( Δ I I + Δ R Δ T R T ) + κ ( T ) h Δ T T + κ ( R ) h Δ R R
Di ( M ) = 1 3 M m 00 ID Frob ,
[ m 0 m 1 · · · m 14 m 15 ] [ 1 m 01 m 00 · · · m 32 m 00 m 33 m 00 ] ,
M ( M ) 1 15 i = 1 15 m - i σ i ,

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