Abstract

The recently introduced concept of spatial coherence wavelets is generalized to describe the propagation of electromagnetic fields in the free space. For this aim, the spatial coherence wavelet tensor is introduced as an elementary amount, in terms of which the formerly known quantities for this domain can be expressed. It allows for the analysis of the relationship between the spatial coherence properties and the polarization state of the electromagnetic wave. This approach is completely consistent with the recently introduced unified theory of coherence and polarization for random electromagnetic beams, but it provides further insight about the causal relationship between the polarization states at different planes along the propagation path.

© 2006 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995). A complete presentation of the subject is shown in Chap. 6, with all the corresponding references. Some applications are also analyzed in the subsequent chapters.
  3. R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).
  4. R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
    [CrossRef]
  5. R. Castañeda and J. Garcia-Sucerquia, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
    [CrossRef]
  6. R. Castañeda, "Partially coherent imaging and spatial coherence wavelets," Opt. Commun. 230, 7-18 (2004).
    [CrossRef]
  7. R. Castaneda and J. Garcia-Sucerquia, "Spatial coherence wavelets and radiometry," Opt. Commun. 248, 147-165 (2005).
    [CrossRef]
  8. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  9. F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 241-243 (1998).
    [CrossRef]
  10. G. P. Agrawal and E. Wolf, "Propagation-induced polarization changes in partially coherent optical beams," J. Opt. Soc. Am. A 17, 2019-2023 (2000).
    [CrossRef]
  11. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, "Use of the van Cittert-Zernike theorem for partially polarized sources," Opt. Lett. 25, 1291-1293 (2000).
    [CrossRef]
  12. G. Gbur and D. F. V. James, "Unpolarized sources that generate highly polarized fields outside the source," J. Mod. Opt. 47, 1171-1177 (2000).
  13. 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]
  14. J. Tervo, T. Setälä, and A. T. Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003).
    [CrossRef] [PubMed]
  15. M. Mujat, A. Dogariu, and E. Wolf, "A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws," J. Opt. Soc. Am. A 21, 2414-2417 (2004).
    [CrossRef]
  16. H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
    [CrossRef]
  17. S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic field," Opt. Commun. 227, 73-74 (2003).
    [CrossRef]
  18. O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
    [CrossRef]
  19. G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, "Generation of complete coherence in Young's interference experiment with random mutually uncorrelated electromagnetic beam," Opt. Lett. 30, 120-122 (2005).
    [CrossRef] [PubMed]
  20. O. Korotkova and E. Wolf, "Generalized Stokes parameters of random electromagnetic beams," Opt. Lett. 30, 198-200 (2005).
    [CrossRef] [PubMed]
  21. G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, 1970).
  22. R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters," J. Opt. Soc. Am. A 17, 2440-2463 (2000).
    [CrossRef]
  23. D. Dragoman, "The Wigner distribution function in optics and optoelectronics," in Progress in Optics, E.Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
    [CrossRef]
  24. A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005). Scalar quantum Wigner functions with spin are used for alternatively representing the polarization properties of electromagnetic waves.
    [CrossRef]
  25. J. Garcia-Sucerquia, R. Castañeda, and F. F. Medina, "Fresnel-Fraunhofer diffraction and spatial coherence," Opt. Commun. 205, 239-245 (2002).
    [CrossRef]

2005 (5)

R. Castaneda and J. Garcia-Sucerquia, "Spatial coherence wavelets and radiometry," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005). Scalar quantum Wigner functions with spin are used for alternatively representing the polarization properties of electromagnetic waves.
[CrossRef]

G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, "Generation of complete coherence in Young's interference experiment with random mutually uncorrelated electromagnetic beam," Opt. Lett. 30, 120-122 (2005).
[CrossRef] [PubMed]

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

2004 (2)

2003 (8)

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]

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic field," 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]

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

2002 (1)

J. Garcia-Sucerquia, R. Castañeda, and F. F. Medina, "Fresnel-Fraunhofer diffraction and spatial coherence," Opt. Commun. 205, 239-245 (2002).
[CrossRef]

2000 (4)

1998 (1)

Agarwal, G. S.

Agrawal, G. P.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, 1970).

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Castaneda, R.

R. Castaneda and J. Garcia-Sucerquia, "Spatial coherence wavelets and radiometry," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

Castañeda, R.

R. Castañeda, "Partially coherent imaging and spatial coherence wavelets," Opt. Commun. 230, 7-18 (2004).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

J. Garcia-Sucerquia, R. Castañeda, and F. F. Medina, "Fresnel-Fraunhofer diffraction and spatial coherence," Opt. Commun. 205, 239-245 (2002).
[CrossRef]

Dogariu, A.

Dragoman, D.

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," in Progress in Optics, E.Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

Friberg, A. T.

Garcia-Sucerquia, J.

R. Castaneda and J. Garcia-Sucerquia, "Spatial coherence wavelets and radiometry," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

J. Garcia-Sucerquia, R. Castañeda, and F. F. Medina, "Fresnel-Fraunhofer diffraction and spatial coherence," Opt. Commun. 205, 239-245 (2002).
[CrossRef]

Gbur, G.

G. Gbur and D. F. V. James, "Unpolarized sources that generate highly polarized fields outside the source," J. Mod. Opt. 47, 1171-1177 (2000).

Gori, F.

Guattari, G.

James, D. F. V.

G. Gbur and D. F. V. James, "Unpolarized sources that generate highly polarized fields outside the source," J. Mod. Opt. 47, 1171-1177 (2000).

Korotkova, O.

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

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

Luis, A.

A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005). Scalar quantum Wigner functions with spin are used for alternatively representing the polarization properties of electromagnetic waves.
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995). A complete presentation of the subject is shown in Chap. 6, with all the corresponding references. Some applications are also analyzed in the subsequent chapters.

Medina, F. F.

J. Garcia-Sucerquia, R. Castañeda, and F. F. Medina, "Fresnel-Fraunhofer diffraction and spatial coherence," Opt. Commun. 205, 239-245 (2002).
[CrossRef]

Mujat, M.

Mukunda, N.

Piquero, G.

Ponomarenko, S. A.

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

Roychowdhury, H.

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

Santarsiero, M.

Setälä, T.

Simon, R.

Tervo, J.

Visser, T. D.

Wolf, E.

G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, "Generation of complete coherence in Young's interference experiment with random mutually uncorrelated electromagnetic beam," Opt. Lett. 30, 120-122 (2005).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

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

M. Mujat, A. Dogariu, and E. Wolf, "A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws," J. Opt. Soc. Am. A 21, 2414-2417 (2004).
[CrossRef]

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic field," 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]

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

G. P. Agrawal and E. Wolf, "Propagation-induced polarization changes in partially coherent optical beams," J. Opt. Soc. Am. A 17, 2019-2023 (2000).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995). A complete presentation of the subject is shown in Chap. 6, with all the corresponding references. Some applications are also analyzed in the subsequent chapters.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

J. Mod. Opt. (3)

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

R. Castañeda and J. Garcia-Sucerquia, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

G. Gbur and D. F. V. James, "Unpolarized sources that generate highly polarized fields outside the source," J. Mod. Opt. 47, 1171-1177 (2000).

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

Opt. Commun. (8)

A. Luis, "Scalar Wigner function for vectorial fields and spatial-angular Stokes parameters," Opt. Commun. 246, 437-443 (2005). Scalar quantum Wigner functions with spin are used for alternatively representing the polarization properties of electromagnetic waves.
[CrossRef]

J. Garcia-Sucerquia, R. Castañeda, and F. F. Medina, "Fresnel-Fraunhofer diffraction and spatial coherence," Opt. Commun. 205, 239-245 (2002).
[CrossRef]

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

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

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

R. Castañeda, "Partially coherent imaging and spatial coherence wavelets," Opt. Commun. 230, 7-18 (2004).
[CrossRef]

R. Castaneda and J. Garcia-Sucerquia, "Spatial coherence wavelets and radiometry," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

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

Other (4)

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, 1970).

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," in Progress in Optics, E.Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995). A complete presentation of the subject is shown in Chap. 6, with all the corresponding references. Some applications are also analyzed in the subsequent chapters.

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Figures (3)

Fig. 1
Fig. 1

Cartesian coordinate axes and center and difference coordinates.

Fig. 2
Fig. 2

Illustration of S ( r A , r A ; ν ) . The straight line from r A to r A makes an angle θ.

Fig. 3
Fig. 3

Illustration of the nonlocal character of the polarization parameter.

Equations (51)

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W l m A B ( r A + r D 2 , r A r D 2 , r A ; ν ) = S l m A B ( r A , r A , ν ) exp ( j k z r D r A ) ,
S l m A B ( r A , r A ; ν ) = A W l m A B ( r A + r D 2 , r A r D 2 ; ν ) exp [ i k z ( r A r A ) r D ] d 2 r D
S ( r A , r A ; ν ) = A W ( r A + r D 2 , r A r D 2 ; ν ) exp [ i k z ( r A r A ) r D ] d 2 r D ,
W ( r A + r D 2 , r A r D 2 ; ν ) = ( 1 λ z ) 2 exp ( i k z r A r D ) × A A W ( r A + r D 2 , r A r D 2 ; ν ) × exp ( i k z r A r D ) × exp [ i k z ( r A r D + r D r A ) ] d 2 r A d 2 r D .
W ( r A + r D 2 , r A r D 2 ; ν ) = ( 1 λ z ) 2 exp ( i k z r A r D ) × A W ( r A + r D 2 , r A r D 2 , r A ; ν ) d 2 r A = ( 1 λ r ) 2 exp ( i k z r A r D ) × A S ( r A , r A ; ν ) exp ( i k z r D r A ) d 2 r A .
S ( r A ; ν ) = ( 1 λ z ) 2 A S ( r A , r A ; ν ) d 2 r A ,
S ( r A ; ν ) = tr [ S ( r A ; ν ) ] = ( 1 λ z ) 2 A tr [ S ( r A , r A ; ν ) ] d 2 r A .
W l m ( r A + r D 2 , r A r D 2 ; ν ) δ ( ν ν ) = E l ( r A + r D 2 , ν ) E m * ( r A r D 2 , ν ) ,
W l m ( r A + r D 2 , r A r D 2 ; ν ) = E l ( r A + r D 2 , ν ) 2 1 2 η l m ( r A + r D 2 , r A r D 2 , ν ) E m ( r A r D 2 , ν ) 2 1 2 ,
η ( r A + r D 2 , r A r D 2 , ν ) = [ η x x ( r A + r D 2 , r A r D 2 , ν ) η x y ( r A + r D 2 , r A r D 2 , ν ) η y x ( r A + r D 2 , r A r D 2 , ν ) η y y ( r A + r D 2 , r A r D 2 , ν ) ] ,
W ( r A + r D 2 , r A r D 2 , ν ) = E ( r A + r D 2 , ν ) η ( r A + r D 2 , r A r D 2 , ν ) E ( r A r D 2 , ν ) ,
E ( r A ± r D 2 , ν ) = [ E x ( r A ± r D 2 , ν ) 2 1 2 0 0 E y ( r A ± r D 2 , ν ) 2 1 2 ] .
S ( r A , r A ; ν ) = A E ( r A + r D 2 , ν ) η ( r A + r D 2 , r A r D 2 , ν ) E ( r A r D 2 , ν ) exp [ i k z ( r A r A ) r D ] d 2 r D .
S ( r A , r A ; ν ) = A Ψ ( r A + r D 2 , ν ) η ( r A + r D 2 , r A r D 2 , ν ) Ψ * ( r A r D 2 , ν ) exp ( i k z r A r D ) d 2 r D ,
S ̃ ( r A , r A ; ν ) = S ( r A , r A ; ν ) = S 0 [ 1 0 0 1 ] .
η ( r A + r D 2 , r A r D 2 , ν ) = η 0 ( r A + r D 2 , r A r D 2 , ν ) [ 1 0 0 1 ] .
W 0 ( r A + r D 2 , r A r D 2 ; ν ) = ( 1 λ z ) 2 exp ( i k z r A r D ) A A E 0 ( r A + r D 2 , ν ) 2 E 0 ( r A r D 2 , ν ) 2 η 0 ( r A + r D 2 , r A r D 2 , ν ) exp ( i k z r A r D ) exp [ i k z ( r A r D + r A r D ) ] d 2 r A d 2 r D
W x x ( r A + r D 2 , r A r D 2 , ν ) = W y y ( r A + r D 2 , r A r D 2 , ν ) = W 0 ( r A + r D 2 , r A r D 2 , ν ) ,
E l ( r A ± r D 2 , ν ) 2 = E 0 ( r A ± r D 2 , ν ) 2 .
W ( r A + r D 2 , r A r D 2 , r A ; ν ) = λ z E 0 ( r A , ν ) 2 exp ( i k z r D r A ) [ 1 0 0 1 ] .
W ( r A + r D 2 , r A r D 2 ; ν ) = 1 λ z exp ( i k z r A r D ) A E 0 ( r A , ν ) 2 exp ( i k z r D r A ) d 2 r A [ 1 0 0 1 ] .
η 0 ( r A + r D 2 , r A r D 2 , ν ) = η 0 ( r A + r D 2 , r A r D 2 , ν ) exp [ i β 0 ( r A + r D 2 , r A r D 2 , ν ) ]
S ( r A ; ν ) = 2 λ z A E 0 ( r A , ν ) 2 d 2 r A + 4 ( 1 λ z ) 2 A A r D 0 E 0 ( r A + r D 2 , ν ) 2 1 2 E 0 ( r A r D 2 , ν ) 2 1 2 × η 0 ( r A + r D 2 , r A r D 2 , ν ) cos [ k z ( r A r A ) r D + β 0 ] d 2 r D d 2 r A
η ( r A + r D 2 , r A r D 2 , ν ) = [ 1 exp ( i β ) exp ( i β ) 1 ]
S l l ( r A , r A ; ν ) = A E l ( r A + r D 2 , ν ) 2 1 2 E l ( r A r D 2 , ν ) 2 1 2 exp [ i k z ( r A r A ) r D ] d 2 r D .
W ( r A + r D 2 , r A r D 2 , r A ; ν ) = [ S x x S x x 1 2 S y y 1 2 exp ( i β ) S x x 1 2 S y y 1 2 exp ( i β ) S y y ] × exp ( i k z r D r A )
η ( r A + r D 2 , r A r D 2 , ν ) = [ η x x 0 0 η y y ] + [ 0 η x y η y x 0 ] ,
S ( r A , r A ; ν ) = [ S x x S x y S y x S y y ] = S 0 [ 1 0 0 1 ] + [ S x x ( pol ) S x y S y x S y y ( pol ) ] ,
S 0 2 S 0 ( S x x + S y y ) + S x x S y y = S 0 2 S 0 tr [ S ( r A , r A ; ν ) ] + det [ S ( r A , r A ; ν ) ] = 0 ,
S 0 ( ± ) = 1 2 { tr [ S ( r A , r A ; ν ) ] ± ( tr 2 [ S ( r A , r A ; ν ) ] 4 det [ S ( r A , r A ; ν ) ] ) 1 2 } .
S x x ( ± ) ( pol ) = 1 2 { S x x S y y ± ( tr 2 [ S ( r A , r A ; ν ) ] 4 det [ S ( r A , r A ; ν ) ] ) 1 2 } ,
S y y ( ± ) ( pol ) = 1 2 { S y y S x x ± ( tr 2 [ S ( r A , r A ; ν ) ] 4 det [ S ( r A , r A ; ν ) ] ) 1 2 } .
( tr 2 [ S ( r A , r A ; ν ) ] 4 det [ S ( r A , r A ; ν ) ] ) 1 2 = [ ( S x x S y y ) 2 + 4 S x y 2 ] 1 2 S x x S y y
tr [ S pol ( r A , r A ; ν ) ] = ( tr 2 [ S ( r A , r A ; ν ) ] 4 det [ S ( r A , r A ; ν ) ] ) 1 2 .
P ( r A , r A ; ν ) = tr [ S pol ( r A , r A ; ν ) ] tr [ S ( r A , r A ; ν ) ] = 1 4 det [ S ( r A , r A ; ν ) ] tr 2 [ S ( r A , r A ; ν ) ] ,
η ( r A + r D 2 , r A r D 2 , ν ) = λ z δ ( r D ) [ 1 η x y ( r A , ν ) η y x ( r A , ν ) 1 ] ,
W ( r A + r D 2 , r A r D 2 , r A ; ν ) = λ z exp ( i k z r D r A ) × [ E x ( r A , ν ) 2 E x ( r A , ν ) 2 1 2 η x y ( r A , ν ) E y ( r A , ν ) 2 1 2 E x ( r A , ν ) 2 1 2 η y x ( r A , ν ) E y ( r A , ν ) 2 1 2 E y ( r A , ν ) 2 ] ,
P ( r A ; ν ) = { 1 4 E x ( r A , ν ) 2 E y ( r A , ν ) 2 [ 1 η x y ( r A , ν ) 2 ] [ E x ( r A , ν ) 2 + E y ( r A , ν ) 2 ] 2 } 1 2 .
W ( r A + r D 2 , r A r D 2 ; ν ) = [ W x x 0 0 W x x ] + [ 0 W x y W y x 0 ] ,
W l l ( r A + r D 2 , r A r D 2 ; ν ) = 1 λ z exp ( i k z r A r D ) A E l ( r A , ν ) 2 × exp ( i k z r D r A ) d 2 r A ,
W l m ( r A + r D 2 , r A r D 2 ; ν ) = 1 λ z exp ( i k z r A r D ) A E l ( r A , ν ) 2 1 2 η l m ( r A , ν ) × E m ( r A , ν ) 2 1 2 exp ( i k z r D r A ) d 2 r A
E ( r A ± r D 2 , ν ) = E 0 ( ν ) 2 1 2 [ 1 0 0 1 ] ,
W ( r A + r D 2 , r A r D 2 , r A ; ν ) = exp ( i k z r D r A ) E 0 ( ν ) 2 [ η ̃ 0 ( r A ; ν ) η ̃ x y ( r A ; ν ) η ̃ x y * ( r A ; ν ) η ̃ 0 ( r A ; ν ) ] ,
η ̃ 0 ( r A ; ν ) = A η 0 ( r D , ν ) exp ( i k z r A r D ) d 2 r D ,
η ̃ x y ( r A ; ν ) = A η x y ( r D , ν ) exp ( i k z r A r D ) d 2 r D .
E ( r A ± r D 2 , ν ) = E 0 ( r A ± r D 2 , ν ) 2 1 2 [ 1 0 0 1 ] ,
η ( r A + r D 2 , r A r D 2 , ν ) = λ z δ ( r D ) [ 1 0 0 1 ] ,
[ T LP ( θ ) ] = [ cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ ] ,
W ( r A + r D 2 , r A r D 2 , r A ; ν ) = λ z E 0 ( r A , ν ) 2 [ cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ ] exp ( i k z r D r A ) ,
[ T LP ( φ ) ] = [ cos 2 φ cos φ sin φ cos φ sin φ sin 2 φ ]
S ( r A , r A ; ν ) = λ z E 0 ( r A , ν ) 2 cos ( θ φ ) [ cos θ cos φ cos θ sin φ sin θ cos φ sin θ sin φ ] .

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