Abstract

We apply a polarization Wigner formalism to the propagation of polarization in a Young interferometer within paraxial approximation. With a very simple ray picture, we obtain complete and rigorous information about polarization evolution via the superposition of the spatial–angular Stokes parameters associated with three light rays. We compare the degree of polarization in the interference region with several measures of the degree of coherence for vectorial fields.

© 2006 Optical Society of America

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References

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  1. D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Prog. Opt. 37, 1-56 (1997).
    [CrossRef]
  2. M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986).
    [CrossRef]
  3. M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space," J. Opt. Soc. Am. A 21, 2233-2243 (2004).
    [CrossRef]
  4. E. C. G. Sudarshan, "Quantum theory of radiative transfer," Phys. Rev. A 23, 2802-2809 (1981).
    [CrossRef]
  5. E. C. G. Sudarshan, "Quantum electrodynamics and light rays," Physica A 96, 315-320 (1979).
    [CrossRef]
  6. E. C. G. Sudarshan, "Pencils of rays in wave optics," Phys. Lett. 73A, 269-272 (1979).
  7. A Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005).
    [CrossRef]
  8. A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
    [CrossRef]
  9. A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141-146 (2006).
    [CrossRef]
  10. H. M. Pedersen, "Exact geometrical theory of free-space radiative energy transfer," J. Opt. Soc. Am. A 8, 176-185 (1991).
    [CrossRef]
  11. M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128-3135 (2003).
    [CrossRef] [PubMed]
  12. A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, "Statistical wave-theoretical derivation of the free-space transport equation of radiometry," J. Opt. Soc. Am. A 9, 1386-1393 (1992).
    [CrossRef]
  13. H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and of polarization," Opt. Commun. 252, 268-274 (2005).
    [CrossRef]
  14. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, "Effects of coherence on the degree of polarization in a Young interference pattern," Opt. Lett. 31, 688-690 (2006).
    [CrossRef] [PubMed]
  15. B. Karczewski, "Degree of coherence of the electromagnetic field," Phys. Lett. 5, 191-192 (1963).
    [CrossRef]
  16. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  17. S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
    [CrossRef]
  18. J. Tervo, T. Setälä, and A. T. Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003).
    [CrossRef] [PubMed]
  19. T. Setälä, J. Tervo, and A. T. Friberg, "Complete electromagnetic coherence in the space-frequency domain," Opt. Lett. 29, 328-330 (2004).
    [CrossRef] [PubMed]
  20. A. Luis, "Degree of coherence for vectorial electromagnetic fields as a distance between correlation matrices" (submitted to J. Soc. Am. Opt. A).
  21. O. Korotkova and E. Wolf, "Generalized Stokes parameters of random electromagnetic beams," Opt. Lett. 30, 198-200 (2005).
    [CrossRef] [PubMed]
  22. A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
    [CrossRef]
  23. A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
    [CrossRef]
  24. A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
    [CrossRef]
  25. A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
    [CrossRef]
  26. A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
    [CrossRef]
  27. A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
    [CrossRef]
  28. A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
    [CrossRef]

2006 (2)

2005 (6)

O. Korotkova and E. Wolf, "Generalized Stokes parameters of random electromagnetic beams," Opt. Lett. 30, 198-200 (2005).
[CrossRef] [PubMed]

H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and of polarization," Opt. Commun. 252, 268-274 (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, "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]

2004 (3)

2003 (6)

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

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

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]

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]

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]

1997 (1)

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

1992 (1)

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, "Statistical wave-theoretical derivation of the free-space transport equation of radiometry," J. Opt. Soc. Am. A 9, 1386-1393 (1992).
[CrossRef]

1991 (1)

1986 (1)

1981 (1)

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

1979 (2)

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

E. C. G. Sudarshan, "Pencils of rays in wave optics," Phys. Lett. 73A, 269-272 (1979).

1963 (1)

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

Agarwal, G. S.

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, "Statistical wave-theoretical derivation of the free-space transport equation of radiometry," J. Opt. Soc. Am. A 9, 1386-1393 (1992).
[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]

Foley, J. T.

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, "Statistical wave-theoretical derivation of the free-space transport equation of radiometry," J. Opt. Soc. Am. A 9, 1386-1393 (1992).
[CrossRef]

Friberg, A. T.

Gori, F.

Karczewski, B.

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

Korotkova, O.

Luis, A

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

Luis, A.

A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141-146 (2006).
[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, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (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, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
[CrossRef]

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

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

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

A. Luis, "Degree of coherence for vectorial electromagnetic fields as a distance between correlation matrices" (submitted to J. Soc. Am. Opt. A).

Pedersen, H. M.

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]

Roychowdhury, H.

H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and of polarization," Opt. Commun. 252, 268-274 (2005).
[CrossRef]

Santarsiero, M.

Setälä, T.

Sudarshan, E. C. G.

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

E. C. G. Sudarshan, "Pencils of rays in wave optics," Phys. Lett. 73A, 269-272 (1979).

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

Tervo, J.

Wolf, E.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, "Effects of coherence on the degree of polarization in a Young interference pattern," Opt. Lett. 31, 688-690 (2006).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, "Generalized Stokes parameters of random electromagnetic beams," Opt. Lett. 30, 198-200 (2005).
[CrossRef] [PubMed]

H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and of polarization," Opt. Commun. 252, 268-274 (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]

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

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, "Statistical wave-theoretical derivation of the free-space transport equation of radiometry," J. Opt. Soc. Am. A 9, 1386-1393 (1992).
[CrossRef]

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

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]

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

H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and of polarization," Opt. Commun. 252, 268-274 (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, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141-146 (2006).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. (2)

E. C. G. Sudarshan, "Pencils of rays in wave optics," Phys. Lett. 73A, 269-272 (1979).

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]

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

E. C. G. Sudarshan, "Quantum theory of radiative transfer," Phys. Rev. A 23, 2802-2809 (1981).
[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. Luis, "Degree of coherence for vectorial electromagnetic fields as a distance between correlation matrices" (submitted to J. Soc. Am. Opt. A).

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Figures (3)

Fig. 1
Fig. 1

Young interferometer showing the three rays reaching the observation point.

Fig. 2
Fig. 2

Plot of the degree of polarization [Eq. (28)] for B = 0 and δ y = 3 δ x as a function of μ ( 0 ) (solid curve) and D ( 0 ) (dashed curve).

Fig. 3
Fig. 3

Plots of D ( 0 ) (dashed curve) and μ ( 0 ) (solid curve) as functions of a δ x for δ y = 3 δ x and B = 0 .

Equations (57)

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Γ m , m ( x 1 , x 2 ) = E m ( x 1 ) E m * ( x 2 ) ,
s 0 ( x ) = Γ x , x ( x , x ) + Γ y , y ( x , x ) ,
s 1 ( x ) = Γ x , x ( x , x ) Γ y , y ( x , x ) ,
s 2 ( x ) = Γ x , y ( x , x ) + Γ y , x ( x , x ) ,
s 3 ( x ) = i [ Γ x , y ( x , x ) Γ y , x ( x , x ) ] ,
s  2 ( x ) s 0 2 ( x ) , s 0 ( x ) 0 .
S ̃ 0 ( x 1 , x 2 ) = Γ x , x ( x 1 , x 2 ) + Γ y , y ( x 1 , x 2 ) ,
S ̃ 1 ( x 1 , x 2 ) = Γ x , x ( x 1 , x 2 ) Γ y , y ( x 1 , x 2 ) ,
S ̃ 2 ( x 1 , x 2 ) = Γ x , y ( x 1 , x 2 ) + Γ y , x ( x 1 , x 2 ) ,
S ̃ 3 ( x 1 , x 2 ) = i [ Γ x , y ( x 1 , x 2 ) Γ y , x ( x 1 , x 2 ) ] ,
W m , m ( x , p ) = k 2 π d x Γ m , m ( x x 2 , x + x 2 ) exp ( i k x p ) ,
S 0 ( x , p ) = W x , x ( x , p ) + W y , y ( x , p ) ,
S 1 ( x , p ) = W x , x ( x , p ) W y , y ( x , p ) ,
S 2 ( x , p ) = W x , y ( x , p ) + W y , x ( x , p ) ,
S 3 ( x , p ) = i [ W x , y ( x , p ) W y , x ( x , p ) ] .
s ( x ) = d p S ( x , p ) ,
S ( x , p ) = k 2 π d x S ̃ ( x x 2 , x + x 2 ) exp ( i k x p ) ,
S ̃ ( x 1 , x 2 ) = d p S ( x 1 2 + x 2 2 , p ) exp [ i k ( x 1 x 2 ) p ] .
t ( x ) = exp ( x 2 2 σ 2 ) ,
E x , y ( 0 ) ( x ) t ( x a ) E x , y ( a ) + t ( x + a ) E x , y ( a ) ,
W m , m ( 0 ) ( x , p ) = k σ π exp ( σ 2 k 2 p 2 ) { Γ m , m ( 0 ) ( a , a ) exp [ ( x a ) 2 σ 2 ] + Γ m , m ( 0 ) ( a , a ) exp [ ( x + a ) 2 σ 2 ] + exp ( x 2 σ 2 ) [ Γ m , m ( 0 ) ( a , a ) exp ( i 2 k p a ) + Γ m , m ( 0 ) ( a , a ) exp ( i 2 k p a ) ] } ,
S ( 0 ) ( x , p ) = k σ π exp ( σ 2 k 2 p 2 ) { s ( 0 ) ( a ) exp [ ( x a ) 2 σ 2 ] + s ( 0 ) ( a ) exp [ ( x + a ) 2 σ 2 ] + exp ( x 2 σ 2 ) [ S ̃ ( 0 ) ( a , a ) exp ( i 2 k p a ) + S ̃ ( 0 ) ( a , a ) exp ( i 2 k p a ) ] } ,
S ( 0 ) ( x x j , p ) = 0 , j = ± , 0 .
S ( z ) ( x , p ) = S ( 0 ) ( x z p , p ) .
x z p j = x j , p j = x x j z ,
S ( z ) ( x , p ) = { S ( 0 ) ( x j , p j ) if p = p j 0 otherwise ,
S ( z ) ( x , p ± ) k σ π s ( 0 ) ( ± a ) ,
S ( z ) ( x , p 0 ) k σ π [ S ̃ ( 0 ) ( a , a ) exp ( i 2 k x a z ) + S ̃ ( 0 ) ( a , a ) exp ( i 2 k x a z ) ] .
s ( z ) ( x ) π σ z j = ± , 0 S ( z ) ( x , p j ) = k σ 2 z [ s ( 0 ) ( a ) + s ( 0 ) ( a ) + S ̃ ( 0 ) ( a , a ) exp ( i 2 k x a z ) + S ̃ ( 0 ) ( a , a ) exp ( i 2 k x a z ) ] .
S ( 0 ) ( ± a , p ) k σ π s ( 0 ) ( ± a ) ,
S ( 0 ) ( 0 , p ) k σ π [ S ̃ ( 0 ) ( a , a ) exp ( i 2 k p a ) + S ̃ ( 0 ) ( a , a ) exp ( i 2 k p a ) ] .
Γ x , x ( 0 ) ( x 1 , x 2 ) = A ( 1 + B ) exp [ ( x 1 x 2 ) 2 2 δ x 2 ] ,
Γ x , y ( 0 ) ( x 1 , x 2 ) = Γ y , x ( 0 ) ( x 1 , x 2 ) = 0 ,
Γ y , y ( 0 ) ( x 1 , x 2 ) = A ( 1 B ) exp [ ( x 1 x 2 ) 2 2 δ y 2 ] ,
s ( 0 ) ( ± a ) = 2 A ( 1 , B , 0 , 0 ) ,
P ( 0 ) ( ± a ) = s ( 0 ) ( ± a ) s 0 ( 0 ) ( ± a ) = B .
S ̃ 0 ( 0 ) ( a , a ) = A [ ( 1 + B ) γ x + ( 1 B ) γ y ] ,
S ̃ 1 ( 0 ) ( a , a ) = A [ ( 1 + B ) γ x ( 1 B ) γ y ] ,
γ x = exp ( 2 a 2 δ x 2 ) , γ y = exp ( 2 a 2 δ y 2 ) ,
s 0 ( z ) ( 0 ) = 2 A k σ 2 z [ 2 + ( 1 + B ) γ x + ( 1 B ) γ y ] ,
s 1 ( z ) ( 0 ) = 2 A k σ 2 z [ 2 B + ( 1 + B ) γ x ( 1 B ) γ y ] ,
P ( z ) ( 0 ) = 2 B + ( 1 + B ) γ x ( 1 B ) γ y 2 + ( 1 + B ) γ x + ( 1 B ) γ y
P ( z ) ( 0 ) = γ x γ y 2 + γ x + γ y ,
S ( 0 ) ( ± a , p ) k σ π s ( 0 ) ( ± a ) = 0 .
S 0 ( 0 ) ( 0 , p ) = 2 k σ π A ( γ x + γ y ) cos ( 2 k a p ) ,
S 1 ( 0 ) ( 0 , p ) = 2 k σ π A ( γ x γ y ) cos ( 2 k a p ) .
Γ ( 0 ) ( a , a ) = Γ ( 0 ) ( a , a ) = A [ 1 + B 0 0 1 B ] ,
Γ ( 0 ) ( a , a ) = Γ ( 0 ) ( a , a ) = A [ ( 1 + B ) γ x 0 0 ( 1 B ) γ y ] .
μ ( x 1 , x 2 ) = tr Γ ( x 1 , x 2 ) tr Γ ( x 1 , x 1 ) tr Γ ( x 2 , x 2 ) .
μ ( 0 ) ( a , a ) = 1 2 [ ( 1 + B ) γ x + ( 1 B ) γ y ] .
μ ̃ ( x 1 , x 2 ) = tr [ Γ ( x 1 , x 2 ) Γ ( x 1 , x 2 ) ] tr Γ ( x 1 , x 1 ) tr Γ ( x 2 , x 2 ) .
μ ̃ ( 0 ) ( a , a ) = 1 2 ( 1 + B ) 2 γ x 2 + ( 1 B ) 2 γ y 2 .
M ( x 1 , x 2 ) = [ Γ ( x 1 , x 1 ) Γ ( x 1 , x 2 ) Γ ( x 1 , x 2 ) Γ ( x 2 , x 2 ) ]
D ( x 1 , x 2 ) = 2 tr [ ( 1 4 I 1 tr M M ) 2 ] = 2 [ tr ( M ) 2 ( tr M ) 2 1 4 ] ,
D ( 0 ) ( a , a ) = 1 4 [ ( 1 + B ) 2 ( 1 + γ x 2 ) + ( 1 B ) 2 ( 1 + γ y 2 ) 2 ] .
μ ̃ ( 0 ) ( a , a ) = D ( 0 ) ( a , a ) = 1 2 γ x 2 + γ y 2 ,
μ ( 0 ) ( a , a ) = 1 2 ( γ x + γ y ) .

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