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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Improved method for calibrating a Stokes polarimeter

Bruno Boulbry, Jessica C. Ramella-Roman, and Thomas A. Germer
Appl. Opt. 46(35) 8533-8541 (2007)

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2008 (1)

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. Europ. Opt. Soc. Rap. Public. 308002 (2008).
[Crossref]

2007 (2)

M. R. Foreman, S. S. Sherif, and P. Török, “Photon statistics in single molecule orientational imaging,” Opt. Express 1513597–13606 (2007).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007).
[Crossref]

2006 (2)

P. Török, M. Salt, E.E. Kriezis, P.R.T. Munro, H.P. Herzig, and C. Rockstuhl, “Optical disk and reader therefor,” Worldwide Patent No. WO 2006/010882 (2006).

J. Zallat, S. Aïnouz, and M. Ph. Stoll “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt. 8807–814 (2006).
[Crossref]

2005 (2)

S. P. Müller, 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. 503697–3715 (2005).
[Crossref] [PubMed]

R. M. A. Azzam and F. F. Sudradjat, “Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter,” Appl. Opt. 44190–196 (2005).
[Crossref] [PubMed]

2004 (4)

2003 (2)

S. M. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 201651–1657 (2003).
[Crossref]

S. N. Savenkov and K. E. Yushtin, “Mueller matrix elements error distribution for polarimetric measurements,” Proc. SPIE 5158251–259 (2003).
[Crossref]

2002 (3)

2001 (1)

J. M. Bueno, “Depolarization effects in the human eye,” Vision Research 412687–2696 (2001).
[Crossref] [PubMed]

2000 (1)

1998 (2)

M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (1998).
[Crossref]

J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system,” J. Opt. Soc. Am. A 15359–366 (1998).
[Crossref]

1997 (1)

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

1996 (1)

1995 (2)

1994 (1)

V. Bhapkar and C. Srinivasan, “On Fisher information inequalities in the presence of nuisance parameters,” Ann. Inst. Statist. Math 46593–604 (1994).

1992 (1)

M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768153 (1992).
[Crossref]

1990 (1)

E. Walter and L. Pronzatom “Qualitative and quantitative experiment design for phenomenological models - a survey,” Automatica 26195–213 (1990).
[Crossref]

1988 (1)

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 382317–2326 (1988).
[Crossref]

1986 (1)

1982 (3)

J. Tinbergen, “Interstellar polarization in the immediate solar neighbourhood,” Astron. Astrophys. 105, 53–64 (1982).

R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29685–689 (1982).

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” J. Mod. Opt. 29685–689 (1982)

1979 (1)

E. Collett, “Automatic determination of the polarization state of nanosecond laser pulses,” U.S. Patent No. 4158506 (1979).

1974 (1)

R. Mehra, “Optimal input signals for parameter estimation in dynamic systems—Survey and new results,” IEEE T. Automat. Contr. 19753–768 (1974).
[Crossref]

1949 (1)

E. W. Barankin, “Locally best unbiased estimates,” Ann. Math. Stat. 20477–501 (1949).
[Crossref]

1948 (1)

H. Mueller, “The foundations of optics” J. Opt. Soc. Am. 38, 661–661 (1948).

1945 (1)

C. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 3781–89 (1945)

1925 (1)

R. Fisher, “Theory of statistical estimation,” Proc. Cam. Phil. Soc. 22700–725 (1925)
[Crossref]

1922 (1)

R. Fisher, “On the mathematical foundations of theoretical statistics,” Phil. Trans. R. Soc. Lond. 222309–368 (1922)
[Crossref]

Abbey, C. K.

S. P. Müller, 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. 503697–3715 (2005).
[Crossref] [PubMed]

Aïnouz, S.

J. Zallat, S. Aïnouz, and M. Ph. Stoll “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt. 8807–814 (2006).
[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. Europ. Opt. Soc. Rap. Public. 308002 (2008).
[Crossref]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007).
[Crossref]

Ambirajan, A.

A. Ambirajan and D.C. Look, “Optimum angles for a polarimeter,” Opt. Eng. 341651–1658 (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. 44190–196 (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. 29685–689 (1982).

Azzam, R.M.A.

R.M.A. Azzam and N.M. Bashara, Ellipsometry and polarised light, (Elsevier, North Holland, 1987).

Bachor, H. A.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” J. Mod. Opt. 29685–689 (1982)

Barankin, E. W.

E. W. Barankin, “Locally best unbiased estimates,” Ann. Math. Stat. 20477–501 (1949).
[Crossref]

Barrett, H. H.

Bashara, N.M.

R.M.A. Azzam and N.M. Bashara, Ellipsometry and polarised light, (Elsevier, North Holland, 1987).

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. Europ. Opt. Soc. Rap. Public. 308002 (2008).
[Crossref]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007).
[Crossref]

Bhapkar, V.

V. Bhapkar and C. Srinivasan, “On Fisher information inequalities in the presence of nuisance parameters,” Ann. Inst. Statist. Math 46593–604 (1994).

Boccara, C.

Bueno, J. M.

J. M. Bueno, “Depolarization effects in the human eye,” Vision Research 412687–2696 (2001).
[Crossref] [PubMed]

Cariou, J.

M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (1998).
[Crossref]

Chipman, R. A.

Collett, E.

E. Collett, “Automatic determination of the polarization state of nanosecond laser pulses,” U.S. Patent No. 4158506 (1979).

Cox, I. J.

Cramér, H.

H. Cramér, Mathematical Methods of Statistics, (Princeton Univ. Press. 1946), ISBN 0-691-08004-6.

De Martino, A.

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films,  455–456112–119 (2004).
[Crossref]

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller Polarimetric Imaging System with Liquid Crystals,” Appl. Opt. 43, 2824–2832 (2004)
[Crossref] [PubMed]

Delaubert, V.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” J. Mod. Opt. 29685–689 (1982)

Denny, J.L.

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. Europ. Opt. Soc. Rap. Public. 308002 (2008).
[Crossref]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007).
[Crossref]

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,  455–456112–119 (2004).
[Crossref]

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller Polarimetric Imaging System with Liquid Crystals,” Appl. Opt. 43, 2824–2832 (2004)
[Crossref] [PubMed]

Dubois, A.

Fabre, C.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” J. Mod. Opt. 29685–689 (1982)

Fisher, R.

R. Fisher, “Theory of statistical estimation,” Proc. Cam. Phil. Soc. 22700–725 (1925)
[Crossref]

R. Fisher, “On the mathematical foundations of theoretical statistics,” Phil. Trans. R. Soc. Lond. 222309–368 (1922)
[Crossref]

Floc’h, M.

M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (1998).
[Crossref]

Foreman, M. R.

Frieden, B. R.

B. R. FriedenPhysics from Fisher Information: A Unification, (Cambridge University Press1998).
[Crossref]

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,  455–456112–119 (2004).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 2000).

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. Europ. Opt. Soc. Rap. Public. 308002 (2008).
[Crossref]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007).
[Crossref]

J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 292234–2236 (2004).
[Crossref] [PubMed]

Grieve, K.

Herzig, H.P.

P. Török, M. Salt, E.E. Kriezis, P.R.T. Munro, H.P. Herzig, and C. Rockstuhl, “Optical disk and reader therefor,” Worldwide Patent No. WO 2006/010882 (2006).

Kijewski, M. F.

S. P. Müller, 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. 503697–3715 (2005).
[Crossref] [PubMed]

M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768153 (1992).
[Crossref]

Kriezis, E.E.

P. Török, M. Salt, E.E. Kriezis, P.R.T. Munro, H.P. Herzig, and C. Rockstuhl, “Optical disk and reader therefor,” Worldwide Patent No. WO 2006/010882 (2006).

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,  455–456112–119 (2004).
[Crossref]

Laude-Boulesteix, B.

Le Brun, G.

M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (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 (1997).

Look, D.C.

A. Ambirajan and D.C. Look, “Optimum angles for a polarimeter,” Opt. Eng. 341651–1658 (1995).
[Crossref]

Lotrian, J.

M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (1998).
[Crossref]

Lu, S. Y.

Meers, B. J.

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 382317–2326 (1988).
[Crossref]

Mehra, R.

R. Mehra, “Optimal input signals for parameter estimation in dynamic systems—Survey and new results,” IEEE T. Automat. Contr. 19753–768 (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 (1997).

Moneron, G.

Moore, S. C.

S. P. Müller, 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. 503697–3715 (2005).
[Crossref] [PubMed]

M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768153 (1992).
[Crossref]

Morio, J.

Mueller, H.

H. Mueller, “The foundations of optics” J. Opt. Soc. Am. 38, 661–661 (1948).

Müller, S. P.

S. P. Müller, 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. 503697–3715 (2005).
[Crossref] [PubMed]

M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768153 (1992).
[Crossref]

Munro, P. R. T.

P. R. T. Munro and P. Török, “Properties of confocal Mueller-matrix polarimeters,” (submitted to Opt. Lett.).

Munro, P.R.T.

P. Török, M. Salt, E.E. Kriezis, P.R.T. Munro, H.P. Herzig, and C. Rockstuhl, “Optical disk and reader therefor,” Worldwide Patent No. WO 2006/010882 (2006).

Myers, K.J.

Nee, S. M.

Pronzatom, L.

E. Walter and L. Pronzatom “Qualitative and quantitative experiment design for phenomenological models - a survey,” Automatica 26195–213 (1990).
[Crossref]

Rao, C.

C. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 3781–89 (1945)

Réfrégier, P.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” J. Mod. Opt. 29685–689 (1982)

Rockstuhl, C.

P. Török, M. Salt, E.E. Kriezis, P.R.T. Munro, H.P. Herzig, and C. Rockstuhl, “Optical disk and reader therefor,” Worldwide Patent No. WO 2006/010882 (2006).

Rothenberg, T. J.

T. J. Rothenberg, Efficient estimation with a priori information (New Haven, Yale University Press, 1973).

Rybicki, F. J.

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P. R. T. Munro and P. Török, “Properties of confocal Mueller-matrix polarimeters,” (submitted to Opt. Lett.).

A. D. Whalen, Detection of signals in noise, (Academic Press Inc., New York, 1971).

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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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