Abstract

We propose to analyze Shannon entropy properties of partially coherent and partially polarized light with Gaussian probability distributions. It is shown that the Shannon entropy is a sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degrees of coherence that have been recently introduced. This analysis clearly demonstrates the contribution of partial polarization and of partial coherence to the characterization of disorder of the light provided by the Shannon entropy, which is a standard measure of randomness. We illustrate these results on two simple examples.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. P. Vamihaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," Pure Appl. Opt. 6, 41-44 (2004).
    [CrossRef]
  15. C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423, C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 623-656 (1948).
  16. T. M. Cover and J. A. Thomas, "The asymptotic equipartition theory," in Elements of Information Theory (Wiley, 1991), pp. 50-59.
    [CrossRef]
  17. R. Barakat and C. Brosseau, "Von Neumann entropy of n interacting pencils of radiation," J. Opt. Soc. Am. A 10, 529-532 (1993).
    [CrossRef]
  18. R. Barakat, "Polarization entropy transfer and relative polarization entropy," Opt. Commun. 123, 443-448 (1996).
    [CrossRef]
  19. R. Barakat, "Some entropic aspects of optical imagery," Opt. Commun. 156, 235-239 (1998).
    [CrossRef]
  20. C. Brosseau, "Entropy of the radiation field," in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 165-176.
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    [CrossRef]
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  23. T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]

2006 (1)

2005 (3)

2004 (5)

2003 (3)

2002 (1)

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

1998 (1)

R. Barakat, "Some entropic aspects of optical imagery," Opt. Commun. 156, 235-239 (1998).
[CrossRef]

1996 (1)

R. Barakat, "Polarization entropy transfer and relative polarization entropy," Opt. Commun. 123, 443-448 (1996).
[CrossRef]

1993 (1)

1963 (1)

R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Barakat, R.

R. Barakat, "Some entropic aspects of optical imagery," Opt. Commun. 156, 235-239 (1998).
[CrossRef]

R. Barakat, "Polarization entropy transfer and relative polarization entropy," Opt. Commun. 123, 443-448 (1996).
[CrossRef]

R. Barakat and C. Brosseau, "Von Neumann entropy of n interacting pencils of radiation," J. Opt. Soc. Am. A 10, 529-532 (1993).
[CrossRef]

Borghi, R.

Brosseau, C.

R. Barakat and C. Brosseau, "Von Neumann entropy of n interacting pencils of radiation," J. Opt. Soc. Am. A 10, 529-532 (1993).
[CrossRef]

C. Brosseau, "Entropy of the radiation field," in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 165-176.

Chavel, P.

Cover, T. M.

T. M. Cover and J. A. Thomas, "The asymptotic equipartition theory," in Elements of Information Theory (Wiley, 1991), pp. 50-59.
[CrossRef]

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Friberg, A.

Friberg, A. T.

Glauber, R. J.

R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Goodman, J. W.

J. W. Goodman, "Some first-order properties of light waves," in Statistical Optics (Wiley, 1985), pp.116-156.

J. W. Goodman, "Coherence of optical waves," in Statistical Optics (Wiley, 1985) pp. 157-236.

Gori, F.

Goudail, F.

Guattari, G.

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Piquero, G.

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Réfrégier, Ph.

Roueff, A.

Santarsiero, M.

Setälä, T.

Shannon, C. E.

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423, C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 623-656 (1948).

Shirai, T.

Simon, R.

Tervo, J.

Thomas, J. A.

T. M. Cover and J. A. Thomas, "The asymptotic equipartition theory," in Elements of Information Theory (Wiley, 1991), pp. 50-59.
[CrossRef]

Vamihaa, P.

P. Vamihaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," Pure Appl. Opt. 6, 41-44 (2004).
[CrossRef]

Wolf, E.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Bell Syst. Tech. J. (1)

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423, C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 623-656 (1948).

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

R. Barakat, "Polarization entropy transfer and relative polarization entropy," Opt. Commun. 123, 443-448 (1996).
[CrossRef]

R. Barakat, "Some entropic aspects of optical imagery," Opt. Commun. 156, 235-239 (1998).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Phys. Rev. (1)

R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Phys. Rev. Lett. (1)

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

P. Vamihaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," Pure Appl. Opt. 6, 41-44 (2004).
[CrossRef]

Other (6)

C. Brosseau, "Entropy of the radiation field," in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 165-176.

J. W. Goodman, "Some first-order properties of light waves," in Statistical Optics (Wiley, 1985), pp.116-156.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004).

J. W. Goodman, "Coherence of optical waves," in Statistical Optics (Wiley, 1985) pp. 157-236.

T. M. Cover and J. A. Thomas, "The asymptotic equipartition theory," in Elements of Information Theory (Wiley, 1991), pp. 50-59.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Evolution of the part of the entropy due to partial temporal coherence (i.e. of the entropy difference S [ P x 1 , x 2 ] S I , P ) as a function of ξ = τ τ X and of α = τ Y τ X = 1 (solid curve) and for α = τ Y τ X = 100 (dashed curve).

Fig. 2
Fig. 2

Evolution of the part of the entropy due to partial spatial coherence (i.e., of the entropy difference S [ P x 1 , x 2 ] S I , P ) as a function of r 1 r 2 σ 1 for different values of σ 1 2 σ 2 2 and of Φ 1 2 and Φ 2 2 . (a) σ 1 2 σ 2 2 = 1 ; from top curve (solid curve) to bottom curve (dashed curve), Φ 1 2 = Φ 2 2 = 10 , 1, 0.1. (b) Φ 1 2 = Φ 2 2 = 1 ; from top curve (solid curve) to bottom curve (dashed curve); σ 1 2 σ 2 2 = 100 , 1, 0.01.

Fig. 3
Fig. 3

Analogous curves to Fig. 2 but (a) σ 1 2 σ 1 2 = 1 , Φ 1 2 = 1 ; and from top curve (solid curve) to bottom curve (dashed curve), Φ 2 2 = 10 , 1, 0.1. (b) Φ 1 2 = 1 ; and from top curve (solid curve) to bottom curve (dashed curve), Φ 2 2 = 10 , 1, 0.1 and σ 1 2 σ 2 2 = 10 , 1, 0.1.

Equations (76)

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Ω ( x 1 , x 2 ) = E ( x 2 ) E ( x 1 ) ,
Ω ( x 1 , x 2 ) = [ E X ( x 2 ) E X * ( x 1 ) E X ( x 2 ) E Y * ( x 1 ) E Y ( x 2 ) E X * ( x 1 ) E Y ( x 2 ) E Y * ( x 1 ) ] ,
Γ ( x i ) = E ( x i ) E ( x i ) = [ E X ( x i ) E X * ( x i ) E X ( x i ) E Y * ( x i ) E Y ( x i ) E X * ( x i ) E Y ( x i ) E Y * ( x i ) ] .
M ( x 1 , x 2 ) = Γ 1 2 ( x 2 ) Ω ( x 1 , x 2 ) Γ 1 2 ( x 1 ) .
S [ P ] = n P ( n ) log [ P ( n ) ] .
S D [ P x ] = P x ( E ) log [ P x ( E ) ] d E ,
P x , δ ( E ) = δ 4 P x ( E ) .
S δ ( P ) [ P x ] = E i P x , δ ( E i ) log [ P x , δ ( E i ) ] ,
S δ ( P ) [ P x ] = P x ( E ) log [ P x , δ ( E ) ] d E .
P x ( E ) = 1 π 2 det [ Γ ( x ) ] exp [ E Γ 1 ( x ) E ] ,
P x , δ ( E ) = δ 4 π 2 det [ Γ ( x ) ] exp [ E Γ 1 ( x ) E ] .
S δ ( P ) [ P x ] = P x ( E ) log [ P x , δ ( E ) ] d E .
S δ ( P ) [ P x ] = log { π 2 e 2 det [ Γ ( x ) ] δ 4 } .
P ( x ) = 1 4 det [ Γ ( x ) ] tr [ Γ ( x ) ] 2 ,
S δ ( P ) [ P x ] = 2 log ( e π 2 ) + 2 log [ I ( x ) δ 2 ] + log [ 1 P 2 ( x ) ]
E ( x ) = E ( x ) u ( x ) ,
η ( x 1 , x 2 ) = E ( x 2 ) E * ( x 1 ) E ( x 1 ) 2 E ( x 2 ) 2 .
P x 1 , x 2 ( E ) = 1 π 2 det [ Φ ( x 1 , x 2 ) ] exp [ E Φ 1 ( x 1 , x 2 ) E ] ,
Φ ( x 1 , x 2 ) = E ( x 1 , x 2 ) E ( x 1 , x 2 ) = [ E ( x 1 ) E * ( x 1 ) E ( x 1 ) E * ( x 1 ) E ( x 2 ) E * ( x 1 ) E ( x 2 ) E * ( x 2 ) ] .
S δ ( C ) [ P x 1 , x 2 ] = log { π 2 e 2 det [ Φ ( x 1 , x 2 ) ] δ 4 } ,
det [ Φ ( x 1 , x 2 ) ] = E ( x 1 ) 2 E ( x 2 ) 2 [ 1 η ( x 1 , x 2 ) 2 ] ,
S δ ( C ) [ P x 1 , x 2 ] = 2 log ( e π ) + log [ I ( x 1 ) δ 2 ] + log [ I ( x 2 ) δ 2 ] + log [ 1 η ( x 1 , x 2 ) 2 ] ,
S δ ( CP ) [ P x 1 , x 2 ] = P x 1 , x 2 ( E 1 , E 2 ) log [ δ 8 P x 1 , x 2 ( E 1 , E 2 ) ] d E 1 d E 2 .
E ( x 1 , x 2 ) = [ E X ( x 1 ) , E Y ( x 1 ) , E X ( x 2 ) , E Y ( x 2 ) ] T .
Υ ( x 1 , x 2 ) = E ( x 1 , x 2 ) E ( x 1 , x 2 ) ,
Υ ( x 1 , x 2 ) = [ γ X X ( x 1 , x 1 ) γ X Y ( x 1 , x 1 ) γ X X ( x 1 , x 2 ) γ X Y ( x 1 , x 2 ) γ Y X ( x 1 , x 1 ) γ Y Y ( x 1 , x 1 ) γ Y X ( x 1 , x 2 ) γ Y Y ( x 1 , x 2 ) γ X X ( x 2 , x 1 ) γ X Y ( x 2 , x 1 ) γ X X ( x 2 , x 2 ) γ X Y ( x 2 , x 2 ) γ Y X ( x 2 , x 1 ) γ Y Y ( x 2 , x 1 ) γ Y X ( x 2 , x 2 ) γ Y Y ( x 2 , x 2 ) ] ,
γ U V ( x i , x j ) = E U ( x i ) [ E V ( x j ) ] * ,
Υ ( x 1 , x 2 ) = [ Γ ( x 1 ) Ω ( x 1 , x 2 ) Ω ( x 1 , x 2 ) Γ ( x 2 ) ] .
P x 1 , x 2 ( E ) = 1 π 4 det [ Υ ( x 1 , x 2 ) ] exp [ E Υ 1 ( x 1 , x 2 ) E ] .
S δ ( CP ) [ P x 1 , x 2 ] = log { π 4 e 4 det [ Υ ( x 1 , x 2 ) ] δ 8 } .
Γ T ( x 1 , x 2 ) = [ Γ ( x 1 ) 0 0 Γ ( x 2 ) ] .
Ξ ( x 1 , x 2 ) = Γ T 1 2 ( x 1 , x 2 ) Υ ( x 1 , x 2 ) Γ T 1 2 ( x 1 , x 2 ) .
Ξ ( x 1 , x 2 ) = [ I d ( 2 ) M ( x 1 , x 2 ) M ( x 1 , x 2 ) I d ( 2 ) ] ,
S δ ( CP ) [ P x 1 , x 2 ] = log { π 2 e 2 det [ Γ ( x 1 ) ] δ 4 } + log { π 2 e 2 det [ Γ ( x 2 ) ] δ 4 } + log { det [ Ξ ( x 1 , x 2 ) ] } .
S δ ( CP ) [ P x 1 , x 2 ] = 2 log [ e π 2 I ( x 1 ) δ 2 ] + log [ 1 P 2 ( x 1 ) ] + 2 log [ e π 2 I ( x 2 ) δ 2 ] + log [ 1 P 2 ( x 2 ) ] + log { det [ Ξ ( x 1 , x 2 ) ] } .
M ( x 1 , x 2 ) = [ η X X ( x 2 , x 1 ) η X Y ( x 2 , x 1 ) η Y X ( x 2 , x 1 ) η Y Y ( x 2 , x 1 ) ] .
det [ Ξ ( x 1 , x 2 ) ] = 1 η X X ( x 1 , x 2 ) 2 η X Y ( x 1 , x 2 ) 2 η Y X ( x 1 , x 2 ) 2 η Y Y ( x 1 , x 2 ) 2 + η X X ( x 1 , x 2 ) 2 + η Y Y ( x 1 , x 2 ) 2 + η X Y ( x 1 , x 2 ) 2 η Y X ( x 1 , x 2 ) 2 η X X ( x 1 , x 2 ) η X Y ( x 1 , x 2 ) * η Y X ( x 1 , x 2 ) * η Y Y ( x 1 , x 2 ) η X X ( x 1 , x 2 ) * η X Y ( x 1 , x 2 ) η Y X ( x 1 , x 2 ) η Y Y ( x 1 , x 2 ) * .
det [ Ξ ( x 1 , x 2 ) ] = det [ I d ( 2 ) M ( x 1 , x 2 ) M ( x 1 , x 2 ) ] .
M ( x 1 , x 2 ) = N 2 ( x 1 , x 2 ) D ( x 1 , x 2 ) N 1 ( x 1 , x 2 ) ,
D ( x 1 , x 2 ) = [ μ S ( x 1 , x 2 ) 0 0 μ I ( x 1 , x 2 ) ] ,
S δ ( CP ) [ P x 1 , x 2 ] = 2 log [ π e 2 I ( x 1 ) δ 2 ] + 2 log [ π e 2 I ( x 2 ) δ 2 ] + log [ 1 P 2 ( x 2 ) ] + log [ 1 P 2 ( x 1 ) ] + log [ 1 μ S 2 ( x 1 , x 2 ) ] + log [ 1 μ I 2 ( x 1 , x 2 ) ] .
S δ ( CP ) [ P x 1 , x 2 ] = S δ ( P ) [ P x 1 ] + S δ ( P ) [ P x 2 ] + log [ 1 μ S 2 ( x 1 , x 2 ) ] + log [ 1 μ I 2 ( x 1 , x 2 ) ] .
E X ( x 2 ) E X * ( x 1 ) = I X f X ( τ ) , E Y ( x 2 ) E Y * ( x 1 ) = I Y f Y ( τ ) ,
Ω ( x 1 , x 2 ) = E ( x 2 ) E ( x 1 ) = [ I X f X ( τ ) 0 0 I Y f Y ( τ ) ] .
Γ ( x i ) = [ I X 0 0 I Y ]
M ( x 1 , x 2 ) = [ f X ( τ ) 0 0 f Y ( τ ) ] ,
D ( x 1 , x 2 ) = [ f X ( τ ) 0 0 f Y ( τ ) ] .
S [ P x 1 , x 2 ] = S I , P + log [ 1 f X ( τ ) 2 ] + log [ 1 f Y ( τ ) 2 ] ,
S I , P = 4 log [ π e 2 ( I X + I Y ) δ 2 ] + 2 log ( 1 P 2 ) .
f U ( τ ) = exp [ ( τ τ U ) 2 ] exp ( i 2 π ν 0 τ ) ,
S [ P x 1 , x 2 ] = S I , P + log { 1 exp [ 2 ( τ τ X ) 2 ] } + log { 1 exp [ 2 ( τ τ Y ) 2 ] } .
E ( T ) ( r , t ) = T ( r ) E ( I ) ( r , t ) ,
Ω ( T ) ( r 1 , r 2 ) = T ( r 2 ) Ω ( I ) ( r 1 , r 2 ) T ( r 1 ) ,
Γ ( T ) ( r ) = T ( r ) Γ ( I ) ( r ) T ( r ) ,
M ( l ) ( r 1 , r 2 ) = [ Γ ( l ) ( r 2 ) ] 1 2 Ω ( l ) ( r 1 , r 2 ) [ Γ ( l ) ( r 1 ) ] 1 2 ,
T ( r ) = [ exp i ϕ 1 ( r ) 0 0 exp i ϕ 2 ( r ) ] ,
δ ϕ n ( r 1 ) δ ϕ n ( r 2 ) = Φ n 2 exp ( r 1 r 2 2 2 σ n 2 ) ,
δ ϕ 1 ( r 1 ) δ ϕ 2 ( r 2 ) = 0 .
exp [ i δ ϕ n ( r ) ] = exp ( Φ n 2 2 ) ,
exp [ i δ ϕ n ( r 1 ) i δ ϕ n ( r 2 ) ] = F n ( r 1 r 2 ) ,
F n ( r 1 r 2 ) = exp { Φ n 2 [ exp ( r 1 r 2 2 2 σ n 2 ) 1 ] } .
Ω ( I ) ( r 1 , r 2 ) = I 0 2 exp ( r 1 2 + r 2 2 4 σ S 2 ) I 2 ,
Ω ( T ) ( r 1 , r 2 ) = I 0 2 exp ( r 1 2 + r 2 2 4 σ S 2 ) L T ( r 1 , r 2 ) ,
L T ( r 1 , r 2 ) = T 0 ( r 2 ) L ( r 1 , r 2 ) T 0 ( r 1 ) ,
L ( r 1 , r 2 ) = [ F 1 ( r 1 r 2 ) 0 0 F 2 ( r 1 r 2 ) ] ,
T 0 ( r ) = [ exp [ i φ 1 ( r ) ] 0 0 exp [ i φ 2 ( r ) ] ] .
S [ P x 1 , x 2 ] = S I , P + log [ 1 F 1 ( r 1 r 2 ) 2 ] + log [ 1 F 2 ( r 1 r 2 ) 2 ] ,
Ξ ( x 1 , x 2 ) = Γ T 1 2 ( x 1 , x 2 ) Υ ( x 1 , x 2 ) Γ T 1 2 ( x 1 , x 2 ) .
Υ ( x 1 , x 2 ) = [ Γ ( x 1 ) Ω ( x 1 , x 2 ) Ω ( x 1 , x 2 ) Γ ( x 2 ) ] ,
Γ T ( x 1 , x 2 ) = [ Γ ( x 1 ) 0 0 Γ ( x 2 ) ] .
Γ T 1 2 ( x 1 , x 2 ) = [ Γ 1 2 ( x 1 ) 0 0 Γ 1 2 ( x 2 ) ] ,
Ξ ( x 1 , x 2 ) = [ Γ 1 2 ( x 1 ) Γ ( x 1 ) Γ 1 2 ( x 1 ) Γ 1 2 ( x 1 ) Ω ( x 1 , x 2 ) Γ 1 2 ( x 2 ) Γ 1 2 ( x 2 ) Ω ( x 1 , x 2 ) Γ 1 2 ( x 1 ) Γ 1 2 ( x 2 ) Γ ( x 2 ) Γ 1 2 ( x 2 ) ] ,
M ( x 1 , x 2 ) M ( x 1 , x 2 ) = N 2 ( x 1 , x 2 ) D 2 ( x 1 , x 2 ) N 2 ( x 1 , x 2 ) .
det [ M ( x 1 , x 2 ) M ( x 1 , x 2 ) λ I d ( 2 ) ] = 0 .
det [ I d ( 2 ) M ( x 1 , x 2 ) M ( x 1 , x 2 ) ] = [ 1 μ S 2 ( x 1 , x 2 ) ] [ 1 μ I 2 ( x 1 , x 2 ) ] ,
log { det [ Ξ ( x 1 , x 2 ) ] } = log [ 1 μ S 2 ( x 1 , x 2 ) ] + log [ 1 μ I 2 ( x 1 , x 2 ) ] ,

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