Abstract

It is possible to introduce tolerancing with the realizability test of Givens and Kostinski [J. Mod. Opt. 40, 471 (1993)] with a minor modification: it suffices to add a special matrix to [G][M]T[G][M]. Consequently, it is possible to check experimental Mueller matrices with a more flexible tool. Application to the emblematic case of the Mueller images of the free space shows that more than 99% of the pixels are in fact physical, while only 2% initially passed the test.

© 2009 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  2. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, 1995).
  3. H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661 (1948).
  4. D. H. Goldstein, “Mueller matrix dual-rotating polarimeter,” Appl. Opt. 31, 6676-6683 (1992).
    [CrossRef] [PubMed]
  5. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A 2, 216-222 (2000).
    [CrossRef]
  6. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  7. M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470(1992).
    [CrossRef]
  8. S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26-36 (1986).
  9. S. R. Cloude, “Conditions for the physical realizability of matrix operator in polarimetry,” Proc. SPIE 1166, 177-185(1989).
  10. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305-2319 (1994).
    [CrossRef]
  11. Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461-484 (1992).
    [CrossRef]
  12. B. J. Howell, “Measurement of the polarisation effects of an instrument using partially polarized light,” Appl. Opt. 18, 809-812 (1979).
    [CrossRef] [PubMed]
  13. C. R. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
    [CrossRef]
  14. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
    [CrossRef]
  15. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).
  16. A. B. Kostinski, B. D. James, and W.-M. Boerner, “Optimal reception of partially polarized waves,” J. Opt. Soc. Am. A 5, 58-64 (1988).
    [CrossRef]
  17. A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506-3508 (1992).
    [CrossRef] [PubMed]
  18. K. E. Yushtin and S. N. Savenkov, “Analysis of Mueller matrix elements measurement error influence on its physical realisability,” in Proceedings of Mathematical Methods in Electromagnetic Theory (MMET'98), Kharkov, Ukraine (1998), pp. 435-437.
  19. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
    [CrossRef]
  20. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likehood estimation of Mueller matrices,” Opt. Lett. 31, 817-819 (2006).
    [CrossRef] [PubMed]
  21. Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. 46, 7354-7364 (2007).
    [CrossRef] [PubMed]
  22. L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231-287 (1965).
    [CrossRef]
  23. J. M. Steele, The Cauchy-Schwartz Master Class (Cambridge University Press, 2004).
    [CrossRef]
  24. A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus,” http://www.arxiv.org/abs/math/math-ph/0412061.
  25. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488-493 (1941).
    [CrossRef]
  26. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
  27. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
    [CrossRef]
  28. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
  29. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press), 2002).
  30. S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599-1610 (1995).
    [CrossRef]
  31. J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Trans. Circuits Syst. CAS-25, 772-781(1978).
    [CrossRef]
  32. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1985).
  33. P. S. Hauge, “Automated Mueller matrix ellipsometry,” Opt. Commun 17, 74-76 (1976).
    [CrossRef]
  34. E. Landi Degl'Innocenti and J. C. del Toro Iniesta, “Physical significance of experimental Mueller matrices,” J. Opt. Soc. Am. A 15, 533-537 (1998).
    [CrossRef]
  35. C. N. Delzell, “A continuous, constructive solution to Hilbert's 17th problem,” Invent. Math. 76, 365-384 (1984).
    [CrossRef]
  36. J. E. Ahmad and Y. Takakura, “Estimation of physically realizable Mueller matrices from experiments using global constrained optimization,” Opt. Express 16, 14274-14287(2008).
    [CrossRef] [PubMed]

2008 (1)

2007 (1)

2006 (1)

2000 (1)

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A 2, 216-222 (2000).
[CrossRef]

1998 (2)

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

E. Landi Degl'Innocenti and J. C. del Toro Iniesta, “Physical significance of experimental Mueller matrices,” J. Opt. Soc. Am. A 15, 533-537 (1998).
[CrossRef]

1997 (1)

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

1995 (1)

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599-1610 (1995).
[CrossRef]

1994 (1)

1993 (2)

C. R. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
[CrossRef]

1992 (4)

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461-484 (1992).
[CrossRef]

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470(1992).
[CrossRef]

D. H. Goldstein, “Mueller matrix dual-rotating polarimeter,” Appl. Opt. 31, 6676-6683 (1992).
[CrossRef] [PubMed]

A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506-3508 (1992).
[CrossRef] [PubMed]

1989 (1)

S. R. Cloude, “Conditions for the physical realizability of matrix operator in polarimetry,” Proc. SPIE 1166, 177-185(1989).

1988 (1)

1987 (1)

1986 (1)

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26-36 (1986).

1984 (1)

C. N. Delzell, “A continuous, constructive solution to Hilbert's 17th problem,” Invent. Math. 76, 365-384 (1984).
[CrossRef]

1979 (1)

1978 (1)

J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Trans. Circuits Syst. CAS-25, 772-781(1978).
[CrossRef]

1976 (1)

P. S. Hauge, “Automated Mueller matrix ellipsometry,” Opt. Commun 17, 74-76 (1976).
[CrossRef]

1965 (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231-287 (1965).
[CrossRef]

1948 (1)

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

1941 (1)

Ahmad, J. E.

Aiello, A.

A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likehood estimation of Mueller matrices,” Opt. Lett. 31, 817-819 (2006).
[CrossRef] [PubMed]

A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus,” http://www.arxiv.org/abs/math/math-ph/0412061.

Anderson, D. G. M.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Barakat, R.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Boerner, W.-M.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press), 2002).

Brewer, J. W.

J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Trans. Circuits Syst. CAS-25, 772-781(1978).
[CrossRef]

Bueno, J. M.

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A 2, 216-222 (2000).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, 1995).

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Cloude, S. R.

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599-1610 (1995).
[CrossRef]

S. R. Cloude, “Conditions for the physical realizability of matrix operator in polarimetry,” Proc. SPIE 1166, 177-185(1989).

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26-36 (1986).

Degl'Innocenti, E. Landi

del Toro Iniesta, J. C.

Delzell, C. N.

C. N. Delzell, “A continuous, constructive solution to Hilbert's 17th problem,” Invent. Math. 76, 365-384 (1984).
[CrossRef]

Givens, C. R.

C. R. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Goldstein, D. H.

Gopala Rao, A. V.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

Hauge, P. S.

P. S. Hauge, “Automated Mueller matrix ellipsometry,” Opt. Commun 17, 74-76 (1976).
[CrossRef]

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1985).

Howell, B. J.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

James, B. D.

Jeune, B. Le

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1985).

Jones, R. C.

Kim, K.

Kostinski, A.

C. R. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Kostinski, A. B.

Kumar, M. Sanjay

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470(1992).
[CrossRef]

Mallesh, K. S.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

Mandel, L.

Mueller, H.

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

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Pottier, E.

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599-1610 (1995).
[CrossRef]

Puentes, G.

Roy-Brehonnet, F. Le

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Savenkov, S. N.

K. E. Yushtin and S. N. Savenkov, “Analysis of Mueller matrix elements measurement error influence on its physical realisability,” in Proceedings of Mathematical Methods in Electromagnetic Theory (MMET'98), Kharkov, Ukraine (1998), pp. 435-437.

Simon, R.

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470(1992).
[CrossRef]

Steele, J. M.

J. M. Steele, The Cauchy-Schwartz Master Class (Cambridge University Press, 2004).
[CrossRef]

Sudha,

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

Takakura, Y.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

van der Mee, C. V. M.

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
[CrossRef]

Voigt, D.

Woerdman, J. P.

A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likehood estimation of Mueller matrices,” Opt. Lett. 31, 817-819 (2006).
[CrossRef] [PubMed]

A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus,” http://www.arxiv.org/abs/math/math-ph/0412061.

Wolf, E.

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
[CrossRef]

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231-287 (1965).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press), 2002).

Xing, Z.-F.

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461-484 (1992).
[CrossRef]

Yushtin, K. E.

K. E. Yushtin and S. N. Savenkov, “Analysis of Mueller matrix elements measurement error influence on its physical realisability,” in Proceedings of Mathematical Methods in Electromagnetic Theory (MMET'98), Kharkov, Ukraine (1998), pp. 435-437.

Appl. Opt. (4)

IEEE Trans. Circuits Syst. (1)

J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Trans. Circuits Syst. CAS-25, 772-781(1978).
[CrossRef]

Invent. Math. (1)

C. N. Delzell, “A continuous, constructive solution to Hilbert's 17th problem,” Invent. Math. 76, 365-384 (1984).
[CrossRef]

J. Math. Phys. (1)

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
[CrossRef]

J. Mod. Opt. (3)

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

C. R. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461-484 (1992).
[CrossRef]

J. Opt. A (1)

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A 2, 216-222 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun (1)

P. S. Hauge, “Automated Mueller matrix ellipsometry,” Opt. Commun 17, 74-76 (1976).
[CrossRef]

Opt. Commun. (1)

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470(1992).
[CrossRef]

Opt. Eng. (1)

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599-1610 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

S. R. Cloude, “Conditions for the physical realizability of matrix operator in polarimetry,” Proc. SPIE 1166, 177-185(1989).

Prog. Quantum Electron. (1)

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231-287 (1965).
[CrossRef]

Other (11)

J. M. Steele, The Cauchy-Schwartz Master Class (Cambridge University Press, 2004).
[CrossRef]

A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus,” http://www.arxiv.org/abs/math/math-ph/0412061.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press), 2002).

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1985).

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26-36 (1986).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, 1995).

K. E. Yushtin and S. N. Savenkov, “Analysis of Mueller matrix elements measurement error influence on its physical realisability,” in Proceedings of Mathematical Methods in Electromagnetic Theory (MMET'98), Kharkov, Ukraine (1998), pp. 435-437.

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Figures (5)

Fig. 1
Fig. 1

Broadening the Poincaré sphere. Adding a special matrix to [ G ] [ M ] T [ G ] [ M ] within the frame of the G–K test is geometrically equivalent to inflating the Poincaré sphere. The resulting ellipsoid is shifted toward positive s 1 as the projection on the plane s 3 = 2 indicates.

Fig. 2
Fig. 2

Basis of an active rotating polarimeter from Ref. [21]. The PSG and PSA share the same architecture: a polarizer followed by a retarder. They are in the direct order for the PSG and in the reverse order for the PSA.

Fig. 3
Fig. 3

Experimental Mueller images of the free space. The color bar on the right applies to all images. Diagonal elements have been framed in order to distinguish them from the background. Fluctuations are clearly visible on the middle ones.

Fig. 4
Fig. 4

Spatial distribution of the intensity variance σ 2 . The maximal value is about 0.0008, which makes σ 0.0283 . The shape of the illumination beam can be identified.

Fig. 5
Fig. 5

Output of the G–K realizability tests. For better visibility, pixels have been replaced by 8 × 8 crosses. Black, nonphysical; white, physical. There are less than 2% of acceptable pixels on the top left and almost 100% on the bottom right (see text).

Equations (35)

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

Φ = ϵ ϵ = [ ϵ 1 ϵ 1 * ϵ 1 ϵ 2 * ϵ 2 ϵ 1 * ϵ 2 ϵ 2 * ] .
s 0 = 1 2 ( ϵ 1 ϵ 1 * + ϵ 2 ϵ 2 * ) , s 1 = 1 2 ( ϵ 1 ϵ 1 * ϵ 2 ϵ 2 * ) , s 2 = 1 2 ( ϵ 1 ϵ 2 * + ϵ 2 ϵ 1 * ) s 3 = 1 2 i ( ϵ 1 ϵ 2 * + ϵ 2 ϵ 1 * ) .
s = [ s 0 s 1 s 2 s 3 ] = 1 2 [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] [ A ] [ ϵ 1 ϵ 1 * ϵ 2 ϵ 1 * ϵ 1 ϵ 2 * ϵ 2 ϵ 2 * ] vec { Φ } ,
s 0 2 ( s 1 2 + s 2 2 + s 3 2 ) 0 ,
ϵ = [ J ] ϵ ,
Φ = [ J ] Φ [ J ] ,
[ σ 1 ] = [ 1 0 0 1 ] , [ σ 2 ] = [ 0 1 1 0 ] , [ σ 3 ] = [ 0 i i 0 ] ; [ σ 0 ] = [ 1 0 0 1 ] .
[ J ] = 𝒥 0 [ σ 0 ] + 𝒥 1 [ σ 1 ] + 𝒥 2 [ σ 2 ] + 𝒥 3 [ σ 3 ] ,
𝒥 0 = 1 2 ( J 11 + J 22 ) , 𝒥 1 = 1 2 ( J 11 J 22 ) , 𝒥 2 = 1 2 ( J 12 + J 21 ) , 𝒥 3 = 1 2 i ( J 12 J 21 ) ,
ϵ ( r , t ) = ( Σ n = 0 N 1 π n [ J n ] ) ϵ ( r , t ) ,
[ T ] = k k e = [ J 11 J 11 * e J 11 J 21 * e J 11 J 12 * e J 11 J 22 * e J 21 J 11 * e J 21 J 21 * e J 21 J 12 * e J 21 J 22 * e J 12 J 11 * e J 12 J 21 * e J 12 J 12 * e J 12 J 22 * e J 22 J 11 * e J 22 J 21 * e J 22 J 12 * e J 22 J 22 * e ] .
Per { [ T p q , k l ] } = [ T p k , q l ] ,
[ J * ] [ J ] = [ J p q * [ J ] ] .
vec { [ U ] [ X ] [ V ] } = [ V T U ] vec { X } ,
vec { [ Φ ] } = [ J * J ] vec { [ Φ ] } ,
s = [ A ] [ J * J ] [ A 1 ] s = [ M ] s .
[ M ] = Σ p , q = 0 3 𝒥 p * 𝒥 q [ A ] [ σ p * σ q ] [ A 1 ] .
[ T ] Cloude = 1 2 Σ p , q = 0 3 m p q [ A ] [ σ p * σ q ] [ A 1 ] ,
[ M ] = [ 1 0 0 0 0 9 / 20 1 / 2 1 / 2 0 1 / 2 9 / 20 1 / 2 0 1 / 2 1 / 2 9 / 20 ] .
[ T Cloude ] = 1 40 [ 47 20 i 20 i 20 i 20 i 11 0 0 20 i 0 11 0 20 i 0 0 11 ] .
s = [ M ] s = [ M ] ( s 0 [ 1 u ] ) = s 0 [ 1 [ m ] u ] ,
u 2 = u T [ m ] T [ m ] u = u T [ 281 / 400 1 / 4 1 / 4 1 / 4 281 / 400 1 / 4 1 / 4 1 / 4 281 / 400 ] u = 281 400 1 4 u T [ 0 1 1 1 0 1 1 1 0 [ Q ] ] u .
= s 0 2 ( s 1 2 + s 2 2 + s 3 2 ) ,
= s T [ M ] T [ G ] [ M ] s + s T [ Δ M ] T [ G ] [ Δ M ] s ,
[ ] = [ Δ M ] T [ G ] [ Δ M ] = σ 2 N 2 [ 45 60 0 0 60 120 0 0 0 0 120 0 0 0 0 30 ] ,
[ 1 u T ] [ ] [ 1 u ] = σ 2 N 2 ( 15 + 120 ( s 1 + 1 2 ) 2 + 120 s 2 2 + 30 s 3 2 ) ,
( s 1 u 1 ) 2 a 2 + s 2 2 a 2 + s 3 2 b 2 = 1 ,
u 1 = 60 σ N 2 1 + 120 σ N 2 , a 2 = 1 + 75 σ N 2 1800 σ N 2 ( 1 + 120 σ N 2 ) 2 , b 2 = 1 + 75 σ N 2 1800 σ N 4 ( 1 + 120 σ N 2 ) ( 1 + 30 σ N 2 ) ,
L ^ = s T [ M ] T [ G ] [ M ] s s T [ ] s .
Δ = s T [ Δ M ] T [ G ] [ M ] s + s T [ M ] T [ G ] [ Δ M ] s .
σ 2 = 4 σ N 2 ( s T [ M ] T [ 3 4 0 0 4 8 0 0 0 0 8 0 0 0 0 2 ] [ M ] s [ 𝒮 ] ) ( s T [ 3 4 0 0 4 8 0 0 0 0 8 0 0 0 0 2 ] s ) .
max = + σ N · s T ( [ M ] T [ 𝒮 ] [ M ] + [ 𝒮 ] ) s = s T ( [ M ] T [ G ] [ M ] + σ N ( [ M ] T [ 𝒮 ] [ M ] + [ 𝒮 ] ) ) s .
max = s T ( [ G ] + 2 σ N [ 𝒮 ] ) s .
u 1 = 8 σ N 1 16 σ N , a 2 = 1 10 σ N 32 σ N 2 ( 1 16 σ N ) 2 , b 2 = 1 10 σ N 32 σ N 2 ( 1 16 σ N ) ( 1 4 σ N ) .
I i , j ( θ q , θ p ) = A i , j B i , j cos ( 2 θ q ) 2 cos ( 2 θ p ) 2 ( C i , j / 4 ) sin ( 4 θ q ) sin ( 4 θ p ) + D i , j sin ( 2 θ q ) sin ( 2 θ p ) .

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