Abstract

Polarimetric measurements are designed to obtain information pertaining to the system under study, however noise in the system limits the precision and hence information obtainable. Exploitation of a priori knowledge of the system allows for an improvement in the precision of experimental data. In this vein we present a framework for system design and optimisation based upon the Fisher information matrix, which allows easy incorporation of such a priori information. As such the proposed figure of merit is more complete than the commonly used condition number. Conditions of equivalence are considered, however a number of examples highlight the failings of the condition number under more general scenarios. Bounds on the achievable informational gains via multiple polarimeter arms are also given. Finally we present analytic results concerning error distribution in a Mueller matrix polar decomposition, allowing for a more accurate noise analysis in polarimetric experiments.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]

2008

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

2007

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

M. R. Foreman, S. S. Sherif, and P. Török, "Photon statistics in single molecule orientational imaging," Opt. Express 15, 13597-13606 (2007).
[CrossRef] [PubMed]

2006

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

2005

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

R. M. A. Azzam and F. F. Sudradjat, "Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter," Appl. Opt. 44, 190-196 (2005).
[CrossRef] [PubMed]

2004

2003

S. M. Nee, "Error analysis for Mueller matrix measurement," J. Opt. Soc. Am. A 20, 1651-1657 (2003).
[CrossRef]

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

2002

2001

J. M. Bueno, "Depolarization effects in the human eye," Vision Research 41, 2687-2696 (2001).
[CrossRef] [PubMed]

2000

1998

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

J. S. Tyo, "Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system," J. Opt. Soc. Am. A 15,359-366 (1998).
[CrossRef]

1997

D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).

1996

1995

1994

V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).

1992

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

1990

E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990).
[CrossRef]

1988

B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
[CrossRef]

1986

1982

R. M. A. Azzam, "Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light," J. Mod. Opt. 29, 685-689 (1982).

J. Tinbergen, "Interstellar polarization in the immediate solar neighbourhood," Astron. Astrophys. 105, 53-64 (1982).

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

1974

R. Mehra, "Optimal input signals for parameter estimation in dynamic systems-Survey and new results," IEEE T. Automat. Contr. 19, 753-768 (1974).
[CrossRef]

1949

E. W. Barankin, "Locally best unbiased estimates," Ann. Math. Stat. 20, 477-501 (1949).
[CrossRef]

1948

H. Mueller, "The foundations of optics" J. Opt. Soc. Am. 38, 661-661 (1948).

1945

C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37, 81-89 (1945)

1925

R. Fisher, "Theory of statistical estimation," Proc. Cam. Phil. Soc. 22, 700-725 (1925)
[CrossRef]

1922

R. Fisher, "On the mathematical foundations of theoretical statistics," Phil. Trans. R. Soc. Lond. 222, 309-368 (1922)
[CrossRef]

Abbey, C. K.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Aïinouz, S.

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

Alouini, M

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Alouini, M.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter," Opt. Eng. 34,1651-1658 (1995).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and F. F. Sudradjat, "Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter," Appl. Opt. 44, 190-196 (2005).
[CrossRef] [PubMed]

R. M. A. Azzam, "Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light," J. Mod. Opt. 29, 685-689 (1982).

Bachor, H. A.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Barankin, E. W.

E. W. Barankin, "Locally best unbiased estimates," Ann. Math. Stat. 20, 477-501 (1949).
[CrossRef]

Barrett, H. H.

Beniere, A

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Bénière, A.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

Bhapkar, V.

V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).

Boccara, C.

Bueno, J. M.

J. M. Bueno, "Depolarization effects in the human eye," Vision Research 41, 2687-2696 (2001).
[CrossRef] [PubMed]

Cariou, J

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

Chipman, R. A.

Cox, I. J.

De Martino, A.

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, "Mueller Polarimetric Imaging System with Liquid Crystals," Appl. Opt. 43, 2824-2832 (2004)
[CrossRef] [PubMed]

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

Delaubert, V.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Denny, J. L.

Dolfi, D

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Dolfi, D.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

Drevillon, B.

Drévillon, B.

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

Dubois, A.

Fabre, C.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Fisher, R.

R. Fisher, "Theory of statistical estimation," Proc. Cam. Phil. Soc. 22, 700-725 (1925)
[CrossRef]

R. Fisher, "On the mathematical foundations of theoretical statistics," Phil. Trans. R. Soc. Lond. 222, 309-368 (1922)
[CrossRef]

Foreman, M. R.

Garcia-Caurel, E.

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

Goudail, F

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Goudail, F.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

J. Morio and F. Goudail, "Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices," Opt. Lett. 29, 2234-2236 (2004).
[CrossRef] [PubMed]

Grieve, K.

Kijewski, M. F.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

Laude, B.

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

Laude-Boulesteix, B.

Le Brun, G

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

Lecaque, R.

Lohmann, A. W.

D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).

Look, D. C.

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter," Opt. Eng. 34,1651-1658 (1995).
[CrossRef]

Lotrian, J

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

Lu, S. Y.

Meers, B. J.

B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
[CrossRef]

Mehra, R.

R. Mehra, "Optimal input signals for parameter estimation in dynamic systems-Survey and new results," IEEE T. Automat. Contr. 19, 753-768 (1974).
[CrossRef]

Mendlovic, D.

D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).

Moneron, G.

Moore, S. C.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

Morio, J.

Mueller, H.

H. Mueller, "The foundations of optics" J. Opt. Soc. Am. 38, 661-661 (1948).

Muller, S. P.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

Munro, P. R. T.

P. R. T. Munro and P. Torok, "Properties of confocal Mueller-matrix polarimeters," (submitted toOpt. Lett.).

Myers, K. J.

Nee, S. M.

Pronzatom, L.

E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990).
[CrossRef]

Rao, C.

C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37, 81-89 (1945)

Réfrégier, P.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Rybicki, F. J.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Savenkov, S. N.

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

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

Schwartz, L.

Sheppard, C. J. R.

Sherif, S. S.

Smith, M.

Srinivasan, C.

V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).

Stoll, M. Ph.

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

Sudradjat, F. F.

Tinbergen, J.

J. Tinbergen, "Interstellar polarization in the immediate solar neighbourhood," Astron. Astrophys. 105, 53-64 (1982).

Torok, P.

P. R. T. Munro and P. Torok, "Properties of confocal Mueller-matrix polarimeters," (submitted toOpt. Lett.).

Török, P.

Treps, N.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram of a polarimeter setup.

Fig. 2.
Fig. 2.

Fig. 2. Dependence of channel capacity on degree of polarisation for S 0/Id = 104.

Fig. 3.
Fig. 3.

Poincaré sphere showing possible polarimeter configurations (a) for a Stokes polarimeter matched to the template Stokes vector (1,0,0,1) i.e. left circularly polarised light and (b) a linear polarimeter, assuming the ratio S 0/Id = 105. Each arrow denotes the basis Stokes vector of a polarimeter arm.

Equations (63)

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D = T S
S = T + D
S = M R
D = T M R
M = T + D R +
f ( D | D ¯ ) = D ¯ D D ! exp ( D )
f ( D | D ¯ ) = n = 1 N D N R ( D ¯ n + I n d ) D n D n exp ( D ¯ n I n d )
J w = E D [ ln f ( D | w ) w ln f ( D | w ) T w ]
J w = G T J D ¯ G
( J D ¯ ) ij = δ ij D i + I d
J w = E w [ J w nr ] + J w r
C D ¯ = n = 1 N D 1 D n
C D ¯ N D 2 S 0 + N D I d .
C D ¯ N R N D 2 R 0 + N D I d
Δ w = w w e
v min = v n c J w 1 = v n c J w
J S = T T J D ¯ T
J M = ( R T T ) J D ¯ ( R T T )
J S = T 2 J D ¯
J M = R 8 T 8 J D ¯
J S = T 2 Π i = 1 4 E S [ 1 D i + I d ]
J M = R 8 T 8 i = 1 16 E M [ 1 D i + I d ]
J u = J 11 J 12 J 22 −1 J 21
J w = ( J 11 J 12 J 21 J 22 )
J u = J w J 22
E s [ 1 D i + I d ] = 1 S 0 P log [ ( 1 + P ) S 0 + 2 I d ( 1 P ) S 0 + 2 I d ]
= 2 S 0 P arctanh [ S 0 P S 0 + 2 I d ]
J S = 1 8 T 2 ( arc tanh [ S 0 P S 0 + 2 I d ] S 0 P ) 4
SNR = ( S 0 2 P arc tanh [ S 0 P S 0 + 2 I d ] ) 1 2
T 0 pt = ( 1 1 0 0 1 0.333 0.816 0.471 1 0.333 0 0.943 1 0.333 0.816 0.471 )
J S = T 2 i = 1 4 1 D i + I d
E [ 1 D i + I d ] = 1 ( S 0 + 2 I d ) 2 + S 0 2 P 2 cos 2 2 α i
J S ' = T 2 j = 1 4 [ ( S 0 + 2 I d ) 2 + S 0 2 P 2 cos 2 2 α j ] 1 2 j = 1 4 sin 2 ( 2 α j ) [ ( S 0 + 2 I d ) 2 + S 0 2 P 2 cos 2 2 α j ] 1 2
J z = s T z T T J D ¯ T S z
J z = M T z ( R T T ) J D ¯ ( R T T ) M z
M A = T u ( 1 A T A m A )
m A = 1 A 2 I + ( 1 1 A 2 ) A A T A 2
M A A k = T u ( 0 k T k m A A k ) , M A T u = M A T u
m A i j A k = [ A i δ j k A 2 + A j δ i k A 2 ] [ 1 1 A 2 ] + A k δ ij 1 A 2 A i A j A k A 4 [ 2 + A 2 2 1 A 2 ]
M R = ( 1 0 T 0 m R )
m R i j = δ ij cos R + R i R j R 2 ( 1 cos R ) + q = 1 3 ε i j q R q R sin R
m Rij R k = [ R i δ j k R 2 + R j δ i k R 2 ] ( 1 cos R ) R k δ ij R sin R R i R j R k R 2 [ 1 cos R R + sin R ]
+ q = 1 3 ε ijq R 2 [ R q R k cos R + ( R δ qk R k R ) sin R ]
M Δ = ( 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ) , a , b , c 1
M Δ ij a = δ i 2 δ j 2 , M Δ ij b = δ i 3 δ j 3 , = M Δ ij c = δ i 4 δ j 4 ,
J z = M A T z ( R M R T M Δ T T T ) J D ¯ ( R T T M Δ M R ) M A z
+ M R T z ( M A R M Δ T T T ) J D ¯ ( R T M A T T M Δ ) M R z ¯
+ M Δ T z ( M R M A R T T ) J D ¯ ( R T M A T M R T T ) M Δ z ¯
J z = ( J R 0 T 0 T 0 J A 0 T 0 0 J Δ )
L ( D , w ) = In f ( D w ) + In f ( w )
J w = i = 1 2 j = 1 2 E D , w [ L i w L j T w ]
= J 1 + J 2 + J 3 + J 4
J 1 = L 1 w L 1 T w f ( D , w ) d D d w
= [ L 1 w L 1 T w f ( D w ) d D ] f ( w ) d w
= E w [ J w nr ]
J 2 = L 1 w L 2 T w f ( D , w ) d D d w
= f ( D , w ) f ( D w ) f ( w ) f ( D w ) w f ( w ) w d D d w
= [ w f ( D | w ) d D ] f ( w ) T w d w
= 0
J 4 = L 2 w L 2 T w f ( D , w ) d D d w
= L 2 w L 2 T w f ( w ) d w = J w r
J w = E w [ J w n r ] + J w r
In f ( w ) w = 0

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