Abstract

We elaborate the overall degree of coherence for vectorial electromagnetic waves within paraxial approximation, expressing it in terms of a polarization Wigner function and the spatial–angular Stokes parameters.

© 2007 Optical Society of America

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References

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  1. B. Karczewski, "Degree of coherence of the electromagnetic field," Phys. Lett. 5, 191-192 (1963).
    [CrossRef]
  2. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  3. S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
    [CrossRef]
  4. E. Wolf, "Comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1712 (2004).
    [CrossRef] [PubMed]
  5. J. Tervo, T. Setälä, and A. T. Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003).
    [CrossRef] [PubMed]
  6. T. Setälä, J. Tervo, and A. T. Friberg, "Complete electromagnetic coherence in the space-frequency domain," Opt. Lett. 29, 328-330 (2004).
    [CrossRef] [PubMed]
  7. T. Setälä, J. Tervo, and A. T. Friberg, "Reply to comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1713-1714 (2004).
    [CrossRef]
  8. A. Luis, "Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices," J. Opt. Soc. Am. A 24, 1063-1068 (2007).
    [CrossRef]
  9. A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855-2860 (2006).
    [CrossRef]
  10. P. Réfrégier and F. Goudail, "Invariant degrees of coherence of partially polarized light," Opt. Express 13, 6051-6060 (2005).
    [CrossRef] [PubMed]
  11. P. Réfrégier and J. Morio, "Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations," J. Opt. Soc. Am. A 23, 3036-3044 (2006).
    [CrossRef]
  12. M. J. Bastiaans, "New class of uncertainty relations for partially coherent light," J. Opt. Soc. Am. A 1, 711-715 (1984).
    [CrossRef]
  13. M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986).
    [CrossRef]
  14. H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of spatially and spectrally partially coherent pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
    [CrossRef]
  15. M. A. Alonso, "Radiometry and wide-angle wave fields III: partial coherence," J. Opt. Soc. Am. A 18, 2502-2511 (2001).
    [CrossRef]
  16. J. Tervo, T. Setälä, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
    [CrossRef]
  17. A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005).
    [CrossRef]
  18. A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
    [CrossRef]
  19. A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141-146 (2006).
    [CrossRef]
  20. A. Luis, "Complementary Huygens principle for geometrical and nongeometrical optics," Eur. J. Phys. 28, 231-240 (2007).
    [CrossRef]
  21. E. C. G. Sudarshan, "Quantum theory of radiative transfer," Phys. Rev. A 23, 2802-2809 (1981).
    [CrossRef]
  22. E. C. G. Sudarshan, "Quantum electrodynamics and light rays," Physica A 96, 315-320 (1979).
    [CrossRef]
  23. A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
    [CrossRef]
  24. A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
    [CrossRef]
  25. A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
    [CrossRef]
  26. A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
    [CrossRef]
  27. A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
    [CrossRef]
  28. A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
    [CrossRef]
  29. A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
    [CrossRef]
  30. P. Vahimaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," J. Opt. A, Pure Appl. Opt. 6, S41-S44 (2004).
    [CrossRef]
  31. D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Prog. Opt. 37, 1-56 (1997).
    [CrossRef]
  32. A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).
  33. M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
    [CrossRef]
  34. M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
    [CrossRef]
  35. R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters," J. Opt. Soc. Am. A 17, 2440-2463 (2000).
    [CrossRef]
  36. H. M. Pedersen, "Exact geometrical theory of free-space radiative energy transfer," J. Opt. Soc. Am. A 8, 176-185 (1991).
    [CrossRef]
  37. K. B. Wolf, M. A. Alonso, and G. W. Forbes, "Wigner functions for Helmholtz wave fields," J. Opt. Soc. Am. A 16, 2476-2487 (1999).
    [CrossRef]
  38. M. A. Alonso, "Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions," J. Opt. Soc. Am. A 18, 902-909 (2001).
    [CrossRef]
  39. M. A. Alonso, "Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions," J. Opt. Soc. Am. A 18, 910-918 (2001).
    [CrossRef]
  40. C. J. R. Sheppard and K. G. Larkin, "Wigner function for highly convergent three-dimensional wave fields," Opt. Lett. 26, 968-970 (2001).
    [CrossRef]
  41. R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Cross polarization in laser beams," Appl. Opt. 26, 1589-1593 (1987).
    [CrossRef] [PubMed]
  42. M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128-3135 (2003).
    [CrossRef] [PubMed]
  43. M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic fields in free space," J. Opt. Soc. Am. A 21, 2233-2243 (2004).
    [CrossRef]
  44. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, "Coherent-mode decomposition of partially polarized, partially coherent sources," J. Opt. Soc. Am. A 20, 78-84 (2003).
    [CrossRef]

2007 (2)

A. Luis, "Complementary Huygens principle for geometrical and nongeometrical optics," Eur. J. Phys. 28, 231-240 (2007).
[CrossRef]

A. Luis, "Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices," J. Opt. Soc. Am. A 24, 1063-1068 (2007).
[CrossRef]

2006 (3)

2005 (7)

A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
[CrossRef]

A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
[CrossRef]

A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005).
[CrossRef]

A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
[CrossRef]

H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of spatially and spectrally partially coherent pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

P. Réfrégier and F. Goudail, "Invariant degrees of coherence of partially polarized light," Opt. Express 13, 6051-6060 (2005).
[CrossRef] [PubMed]

2004 (7)

2003 (7)

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, "Coherent-mode decomposition of partially polarized, partially coherent sources," J. Opt. Soc. Am. A 20, 78-84 (2003).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003).
[CrossRef] [PubMed]

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

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
[CrossRef]

A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
[CrossRef]

M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128-3135 (2003).
[CrossRef] [PubMed]

2002 (2)

A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
[CrossRef]

A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
[CrossRef]

2001 (4)

2000 (1)

1999 (1)

1997 (1)

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Prog. Opt. 37, 1-56 (1997).
[CrossRef]

1991 (1)

1987 (1)

1986 (1)

1984 (1)

1981 (1)

E. C. G. Sudarshan, "Quantum theory of radiative transfer," Phys. Rev. A 23, 2802-2809 (1981).
[CrossRef]

1979 (2)

1978 (1)

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

1963 (1)

B. Karczewski, "Degree of coherence of the electromagnetic field," Phys. Lett. 5, 191-192 (1963).
[CrossRef]

Alonso, M. A.

Bastiaans, M. J.

Borghi, R.

Dragoman, D.

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Prog. Opt. 37, 1-56 (1997).
[CrossRef]

Forbes, G. W.

Friberg, A. T.

Gori, F.

Goudail, F.

Guattari, G.

Karczewski, B.

B. Karczewski, "Degree of coherence of the electromagnetic field," Phys. Lett. 5, 191-192 (1963).
[CrossRef]

Lajunen, H.

Larkin, K. G.

Luis, A.

A. Luis, "Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices," J. Opt. Soc. Am. A 24, 1063-1068 (2007).
[CrossRef]

A. Luis, "Complementary Huygens principle for geometrical and nongeometrical optics," Eur. J. Phys. 28, 231-240 (2007).
[CrossRef]

A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141-146 (2006).
[CrossRef]

A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855-2860 (2006).
[CrossRef]

A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
[CrossRef]

A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
[CrossRef]

A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005).
[CrossRef]

A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
[CrossRef]

A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
[CrossRef]

A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
[CrossRef]

A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
[CrossRef]

A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
[CrossRef]

A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
[CrossRef]

Morio, J.

Mukunda, N.

Pedersen, H. M.

Piquero, G.

Ponomarenko, S. A.

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

Réfrégier, P.

Santarsiero, M.

Setälä, T.

Sheppard, C. J. R.

Simon, R.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Cross polarization in laser beams," Appl. Opt. 26, 1589-1593 (1987).
[CrossRef] [PubMed]

E. C. G. Sudarshan, "Quantum theory of radiative transfer," Phys. Rev. A 23, 2802-2809 (1981).
[CrossRef]

E. C. G. Sudarshan, "Quantum electrodynamics and light rays," Physica A 96, 315-320 (1979).
[CrossRef]

Tervo, J.

Torre, A.

A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

Vahimaa, P.

H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of spatially and spectrally partially coherent pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

P. Vahimaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," J. Opt. A, Pure Appl. Opt. 6, S41-S44 (2004).
[CrossRef]

Wolf, E.

E. Wolf, "Comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1712 (2004).
[CrossRef] [PubMed]

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

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

Wolf, K. B.

Appl. Opt. (1)

Eur. J. Phys. (1)

A. Luis, "Complementary Huygens principle for geometrical and nongeometrical optics," Eur. J. Phys. 28, 231-240 (2007).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

P. Vahimaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," J. Opt. A, Pure Appl. Opt. 6, S41-S44 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (15)

R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters," J. Opt. Soc. Am. A 17, 2440-2463 (2000).
[CrossRef]

M. A. Alonso, "Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions," J. Opt. Soc. Am. A 18, 902-909 (2001).
[CrossRef]

M. A. Alonso, "Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions," J. Opt. Soc. Am. A 18, 910-918 (2001).
[CrossRef]

H. M. Pedersen, "Exact geometrical theory of free-space radiative energy transfer," J. Opt. Soc. Am. A 8, 176-185 (1991).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
[CrossRef]

M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic fields in free space," J. Opt. Soc. Am. A 21, 2233-2243 (2004).
[CrossRef]

H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of spatially and spectrally partially coherent pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

M. A. Alonso, "Radiometry and wide-angle wave fields III: partial coherence," J. Opt. Soc. Am. A 18, 2502-2511 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, "Coherent-mode decomposition of partially polarized, partially coherent sources," J. Opt. Soc. Am. A 20, 78-84 (2003).
[CrossRef]

A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855-2860 (2006).
[CrossRef]

P. Réfrégier and J. Morio, "Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations," J. Opt. Soc. Am. A 23, 3036-3044 (2006).
[CrossRef]

A. Luis, "Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices," J. Opt. Soc. Am. A 24, 1063-1068 (2007).
[CrossRef]

M. J. Bastiaans, "New class of uncertainty relations for partially coherent light," J. Opt. Soc. Am. A 1, 711-715 (1984).
[CrossRef]

K. B. Wolf, M. A. Alonso, and G. W. Forbes, "Wigner functions for Helmholtz wave fields," J. Opt. Soc. Am. A 16, 2476-2487 (1999).
[CrossRef]

M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986).
[CrossRef]

J. Phys. A (1)

A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
[CrossRef]

Opt. Commun. (7)

A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
[CrossRef]

A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005).
[CrossRef]

A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
[CrossRef]

A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141-146 (2006).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
[CrossRef]

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Lett. (1)

B. Karczewski, "Degree of coherence of the electromagnetic field," Phys. Lett. 5, 191-192 (1963).
[CrossRef]

Phys. Lett. A (2)

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

A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
[CrossRef]

Phys. Rev. A (4)

A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
[CrossRef]

A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
[CrossRef]

E. C. G. Sudarshan, "Quantum theory of radiative transfer," Phys. Rev. A 23, 2802-2809 (1981).
[CrossRef]

A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Physica A (1)

E. C. G. Sudarshan, "Quantum electrodynamics and light rays," Physica A 96, 315-320 (1979).
[CrossRef]

Prog. Opt. (1)

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Prog. Opt. 37, 1-56 (1997).
[CrossRef]

Other (1)

A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

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Equations (42)

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Γ ( r 1 , r 2 , ω ) = d τ E ( r 1 , t + τ ) E * ( r 2 , t ) exp ( i ω τ ) ,
μ 2 ( r 1 , r 2 ) = Γ ( r 1 , r 2 ) 2 Γ ( r 1 , r 1 ) Γ ( r 2 , r 2 ) .
μ g 2 = d 2 r 1 d 2 r 2 Γ ( r 1 , r 2 ) 2 [ d 2 r Γ ( r , r ) ] 2 .
μ g 2 = d 2 r 1 d 2 r 2 Γ ( r 1 , r 1 ) Γ ( r 2 , r 2 ) μ 2 ( r 1 , r 2 ) [ d 2 r Γ ( r , r ) ] 2 .
Γ ( r 1 , r 2 ) = m λ m ψ m ( r 1 ) ψ m * ( r 2 ) ,
d 2 r ψ l ( r ) ψ m * ( r ) = δ l , m ,
μ g 2 = m λ m 2 ( m λ m ) 2 .
μ g 2 = ( 2 π k ) 2 d 2 r d 2 p W 2 ( r , p ) [ d 2 r d 2 p W ( r , p ) ] 2 ,
W ( r , p ) = ( k 2 π ) 2 d 2 r Γ ( r r 2 , r + r 2 ) exp ( i k p r ) ,
W N ( r , p ) = W ( r , p ) d 2 r d 2 p W ( r , p ) ,
Γ ( r 1 , r 2 ) δ 2 ( r 1 r 2 ) ,
W Δ ( r , p ) = 1 π 2 Δ 4 exp ( r 2 + p 2 Δ 2 ) ,
d 2 r d 2 p W Δ ( r , p ) = 1 , d 2 r d 2 p W Δ 2 ( r , p ) 0 .
μ g 2 = lim Δ ( 2 π k ) 2 d 2 r d 2 p [ W N ( r , p ) W Δ ( r , p ) ] 2 .
W z ( r , p ) = W 0 ( r , p ) ,
( r p ) = [ A B C D ] ( r p ) ,
d 2 r d 2 p W z m ( r , p ) = d 2 r d 2 p W 0 m ( r , p ) ,
Γ l , m ( r 1 , r 2 ) = d τ E l ( r 1 , t + τ ) E m * ( r 2 , t ) exp ( i ω τ ) ,
μ G 2 = d 2 r 1 d 2 r 2 tr [ Γ ( r 1 , r 2 ) Γ ( r 1 , r 2 ) ] [ d 2 r tr Γ ( r , r ) ] 2 ,
μ ̃ 2 ( r 1 , r 2 ) = tr [ Γ ( r 1 , r 2 ) Γ ( r 1 , r 2 ) ] tr Γ ( r 1 , r 1 ) tr Γ ( r 2 , r 2 ) ,
μ G 2 = d 2 r 1 d 2 r 2 tr Γ ( r 1 , r 1 ) tr Γ ( r 2 , r 2 ) μ ̃ 2 ( r 1 , r 2 ) [ d 2 r tr Γ ( r , r ) ] 2 .
Γ ( r 1 , r 2 ) = m Λ m Ψ m ( r 1 ) Ψ m ( r 2 ) ,
d 2 r Ψ l ( r ) Ψ m ( r ) = δ l , m ,
μ G 2 = m Λ m 2 ( m Λ m ) 2 .
W ( r , p ) = ( k 2 π ) 2 d 2 r Γ ( r r 2 , r + r 2 ) exp ( i k p r ) ,
W ( r , p , Ω ) = Ω S ( r , p ) ,
S j ( r , p ) = tr [ σ j W ( r , p ) ] ,
Ω = 1 2 ( 1 3 sin θ cos ϕ 3 sin θ sin ϕ 3 cos θ ) ,
μ G 2 = 8 π 3 k 2 d 2 r d 2 p d 2 Ω W 2 ( r , p , Ω ) [ d 2 r d 2 p d 2 Ω W ( r , p , Ω ) ] 2
μ G 2 = 2 π 2 k 2 d 2 r d 2 p S 2 ( r , p ) [ d 2 r d 2 p S 0 ( r , p ) ] 2 ,
d 2 r d 2 p tr [ W 2 ( r , p ) ] = d 2 r d 2 p 1 2 S 2 ( r , p ) = k 2 4 π 2 d 2 r 1 d 2 r 2 tr [ Γ ( r 1 , r 2 ) Γ ( r 2 , r 1 ) ] .
μ G 2 = 2 π 2 k 2 d 2 r d 2 p S 0 2 ( r , p ) [ 1 + P 2 ( r , p ) ] [ d 2 r d 2 p S 0 ( r , p ) ] 2 ,
P 2 ( r , p ) = S 2 ( r , p ) S 0 2 ( r , p ) ,
μ G 2 = lim Δ 8 π 3 k 2 d 2 r d 2 p d 2 Ω [ W N ( r , p , Ω ) W Δ ( r , p , Ω ) ] 2 ,
W N ( r , p , Ω ) = W ( r , p , Ω ) d 2 r d 2 p d 2 Ω W ( r , p , Ω ) ,
W Δ ( r , p , Ω ) = 1 4 π 3 Δ 4 exp ( r 2 + p 2 Δ 2 ) ,
Γ ( r 1 , r 2 ) I δ 2 ( r 1 r 2 ) ,
W z ( r , p , Ω ) = W 0 ( r , p , M t Ω ) ,
S ( r , p ) = M S ( r , p ) ,
S 0 ( r , p ) = ν S 0 ( r , p ) , S 2 ( r , p ) = ν 2 S 2 ( r , p ) ,
W ( r , p , Ω ) = S Ω W ( r , p ) ,
μ G 2 = 1 2 S 2 S 0 2 μ g 2 = 1 2 ( 1 + P 2 ) μ g 2 ,

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