Abstract

The full Mueller matrix for a Spectralon white reflectance standard was measured in the incidence plane, to obtain the polarization state of the scattered light for different angles of illumination. The experimental setup was a Mueller matrix ellipsometer, by which measurements were performed for scattering angles measured relative to the normal of the Spectralon surface from −90° to 90° sampled at every 2.5° for an illumination wavelength of 532 nm. Previously, the polarization of light scattered from Spectralon white reflectance standards was measured only for four of the elements of the Muller matrix. As in previous investigations, the reflection properties of the Spectralon white reflectance standard was found to be close to those of a Lambertian surface for small scattering and illumination angles. At large scattering and illumination angles, all elements of the Mueller matrix were found to deviate from those of a Lambertian surface. A simple empirical model with only two parameters, was developed, and used to simulate the measured results with fairly good accuracy.

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References

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  1. “A guide to diffuse reflectance coatings & materials;” http://www.prolite.co.uk/File/coatings materials documentation.php .
  2. “Reflectance standards product sheet 8.pdf” http://www.labsphere.com/data/userFiles/ .
  3. E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).
  4. D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of Spectralon illuminated by coherent light,” Appl. Opt. 38(30), 6350–6356 (1999).
    [CrossRef] [PubMed]
  5. B. Gordon, “Integrating sphere diffuse reflectance technology for use with UV-Visible spectroscopy;” Tech. Note: 51450, Thermo Fisher Scientific, WI, USA.
  6. D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
    [CrossRef] [PubMed]
  7. K. J. Voss and H. Zhang, “Bidirectional reflectance of dry and submerged Labsphere Spectralon plaque,” Appl. Opt. 45(30), 7924–7927 (2006).
    [CrossRef] [PubMed]
  8. G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–502 (2004).
    [CrossRef]
  9. AA. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. 50(16), 2431–2442 (2011).
    [CrossRef] [PubMed]
  10. G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 588, 588203 (2005).
  11. B. T. McGuckin, D. A. Haner, R. T. Menzies, C. Esproles, and A. M. Brothers, “Directional reflectance characterization facility and measurement methodology,” Appl. Opt. 35(24), 4827–4834 (1996).
    [CrossRef] [PubMed]
  12. M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res. 112(C5), 5026–5039 (2007).
    [CrossRef]
  13. G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. 6(4), 741–746 (1967).
    [CrossRef] [PubMed]
  14. G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. 241(4-6), 255–261 (2004).
    [CrossRef]
  15. G. W. Kattawar and D. J. Gray, “Mueller matrix imaging of targets in turbid media: effect of the volume scattering function,” Appl. Opt. 42(36), 7225–7230 (2003).
    [CrossRef] [PubMed]
  16. P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt. 48(2), 250–260 (2009).
    [CrossRef] [PubMed]
  17. 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]
  18. F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. 47(7), 073604 (2008).
    [CrossRef]
  19. P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. 96(1-3), 81–107 (1980).
    [CrossRef]
  20. R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
    [CrossRef]
  21. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25(2), 473–482 (2008).
    [CrossRef] [PubMed]
  22. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
    [CrossRef]
  23. F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

2011 (1)

2009 (1)

2008 (3)

R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25(2), 473–482 (2008).
[CrossRef] [PubMed]

F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. 47(7), 073604 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
[CrossRef]

2007 (1)

M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res. 112(C5), 5026–5039 (2007).
[CrossRef]

2006 (2)

2005 (1)

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 588, 588203 (2005).

2004 (2)

G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. 241(4-6), 255–261 (2004).
[CrossRef]

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–502 (2004).
[CrossRef]

2003 (1)

2000 (1)

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

1999 (1)

1998 (1)

1996 (2)

B. T. McGuckin, D. A. Haner, R. T. Menzies, C. Esproles, and A. M. Brothers, “Directional reflectance characterization facility and measurement methodology,” Appl. Opt. 35(24), 4827–4834 (1996).
[CrossRef] [PubMed]

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

1986 (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
[CrossRef]

1980 (1)

P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. 96(1-3), 81–107 (1980).
[CrossRef]

1967 (1)

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
[CrossRef]

Barnes, P. Y.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Ben Hatit, S.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
[CrossRef]

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
[CrossRef]

Bhandari, AA.

Biggar, S. F.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Brothers, A. M.

Bruegge, C. J.

Butler, J. J.

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 588, 588203 (2005).

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–502 (2004).
[CrossRef]

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Cariou, J.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

Chami, M.

M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res. 112(C5), 5026–5039 (2007).
[CrossRef]

Chenault, D. B.

De Martino, A.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
[CrossRef]

Duval, V.

Early, E. A.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Eliés, P.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

Esproles, C.

Frette, Ø.

Garcia-Caurel, E.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
[CrossRef]

Georgiev, G. T.

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 588, 588203 (2005).

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–502 (2004).
[CrossRef]

Gil, J. J.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
[CrossRef]

Gilbert, G. D.

Goldstein, D. L.

Gray, D. J.

Hamre, B.

Haner, D. A.

Hauge, P. S.

P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. 96(1-3), 81–107 (1980).
[CrossRef]

Johnson, B. C.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Kattawar, G. W.

Kildemo, M.

Le Jeune, B.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

Le Roy-Bréhonnet, F.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

Lindgren, M.

F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. 47(7), 073604 (2008).
[CrossRef]

Lotrain, J.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

McGuckin, B. T.

Menzies, R. T.

Muller, R. H.

P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. 96(1-3), 81–107 (1980).
[CrossRef]

Nerbo?, I. S.

F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. 47(7), 073604 (2008).
[CrossRef]

Ossikovski, R.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
[CrossRef]

R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25(2), 473–482 (2008).
[CrossRef] [PubMed]

Pavlov, M. M.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Pernicka, J. C.

Shaw, J. A.

Smith, C. G.

P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. 96(1-3), 81–107 (1980).
[CrossRef]

Spyak, P. R.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Stabo-Eeg, F.

F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. 47(7), 073604 (2008).
[CrossRef]

Stamnes, J. J.

Tyo, J. S.

Voss, K. J.

Yang, P.

Yao, G.

G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. 241(4-6), 255–261 (2004).
[CrossRef]

Zhai, P. W.

Zhang, H.

Zhao, L.

Am. Met. Soc. (1)

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth observing system program,” Am. Met. Soc. 17, 1078–1091 (2000).

Appl. Opt. (9)

G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. 6(4), 741–746 (1967).
[CrossRef] [PubMed]

B. T. McGuckin, D. A. Haner, R. T. Menzies, C. Esproles, and A. M. Brothers, “Directional reflectance characterization facility and measurement methodology,” Appl. Opt. 35(24), 4827–4834 (1996).
[CrossRef] [PubMed]

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef] [PubMed]

D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of Spectralon illuminated by coherent light,” Appl. Opt. 38(30), 6350–6356 (1999).
[CrossRef] [PubMed]

G. W. Kattawar and D. J. Gray, “Mueller matrix imaging of targets in turbid media: effect of the volume scattering function,” Appl. Opt. 42(36), 7225–7230 (2003).
[CrossRef] [PubMed]

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]

K. J. Voss and H. Zhang, “Bidirectional reflectance of dry and submerged Labsphere Spectralon plaque,” Appl. Opt. 45(30), 7924–7927 (2006).
[CrossRef] [PubMed]

P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt. 48(2), 250–260 (2009).
[CrossRef] [PubMed]

AA. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. 50(16), 2431–2442 (2011).
[CrossRef] [PubMed]

Appl. Phys. (Berl.) (1)

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliés, J. Cariou, and J. Lotrain, “Optical media and target characterization by Mueller matrix decomposition,” Appl. Phys. (Berl.) 29, 34–38 (1996).

J. Geophys. Res. (1)

M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res. 112(C5), 5026–5039 (2007).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (Lond.) (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986).
[CrossRef]

Opt. Commun. (1)

G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. 241(4-6), 255–261 (2004).
[CrossRef]

Opt. Eng. (1)

F. Stabo-Eeg, M. Kildemo, I. S. Nerbo̸, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. 47(7), 073604 (2008).
[CrossRef]

Phys. Status Solidi (1)

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarising Mueller matrices: how to decompose them?” Phys. Status Solidi 205(4), 720–727 (2008).
[CrossRef]

Proc. SPIE (2)

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–502 (2004).
[CrossRef]

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 588, 588203 (2005).

Surf. Sci. (1)

P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci. 96(1-3), 81–107 (1980).
[CrossRef]

Other (3)

B. Gordon, “Integrating sphere diffuse reflectance technology for use with UV-Visible spectroscopy;” Tech. Note: 51450, Thermo Fisher Scientific, WI, USA.

“A guide to diffuse reflectance coatings & materials;” http://www.prolite.co.uk/File/coatings materials documentation.php .

“Reflectance standards product sheet 8.pdf” http://www.labsphere.com/data/userFiles/ .

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Figures (3)

Fig. 1
Fig. 1

A: Depolarization index p for simulated and measured results. B: M11 element for a Lambertian surface (with depolarization index p = 0) and measured for a Spectralon surface with illumination at normal incidence (θ0 = 0°), as well as for θ0 = 30°, θ0 = 45 o, and θ0 = 60°. The y-axis is normalized to the Lambertian surface.

Fig. 2
Fig. 2

Measured (black) and simulated (red) normalized Mueller matrices for four different illumination angles θ0. A: Normal incidence (θ0 = 0). B: θ0 = 30°. The element M11 is the so-called scattering phase function (from particle scattering terminology), normalized to its maximum value. The y- axis is normalized to the m22 element.

Fig. 3
Fig. 3

Measured (black) and simulated (red) normalized Mueller matrices for four different illumination angles θ0. C: θ0 = 45°. D: θ0 = 60°. The element M11 is the so-called scattering phase function (from particle scattering terminology), normalized to its maximum value. The y- axis is normalized to the m22 element.

Equations (7)

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

M Δ ideal = M 11 [ 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ].
M DR = M 11 [ 1 m 21 0 0 m 12 m 22 0 0 0 0 m 33 m 34 0 0 m 43 m 44 ].
M Δ ideal =[ M 11 M 12 ? ? M 21 M 22 ? ? ? ? ? ? ? ? ? ? ],
p= ( ( tr( M t M )- M 11 2 ) 3 M 11 2 ) 1 2 = ( ( ( i,j=1 4 M i,j 2 )- M 11 2 ) 3 M 11 2 ) 1 2 .
M Spectralon = M 11 [ 1 m 21 0 0 m 12 m 22 m 23 m 24 0 m 32 m 33 m 34 0 m 42 m 43 m 44 ],
M Spectralon = M 11 ( (1p)[ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]+[ p (p κ 2 + κ 4 ) 0 κ 4 (p κ 2 ( κ 4 2 )) p 2 + κ 4 p κ 2 2 p κ 2 κ 3 κ 5 p κ 2 2 p 2 κ 4 p κ 2 0 p κ 2 κ 3 p κ 2 p 2 κ 1 ] ),
p=| (θ+ θ 0 ) 3 22 |.

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