Abstract

A representation of partially spatially coherent and partially polarized stationary electromagnetic fields is given in terms of mutually uncorrelated, transversely shifted, fully coherent, and polarized elementary electric-field modes. This representation allows one to propagate non-paraxial partially coherent vector fields using techniques for spatially fully coherent fields, which are numerically far more efficient than methods for propagating correlation functions. A procedure is given to determine the elementary modes from the radiant intensity and the far-zone polarization properties of the entire field. The method is applied to quasi-homogeneous fields with rotationally symmetric cosnθ radiant intensity distributions (θ being the diffraction angle with respect to the optical axis and n being an integer). This is an adequate model for fields emitted by, e.g., many light-emitting diodes.

© 2010 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  3. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent, partially polarized sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
    [CrossRef]
  4. 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]
  5. F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  6. F. Gori, “Mode propagation of the light field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  7. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
    [CrossRef]
  8. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 73, 1538–1544 (1983).
  9. J. Huttunen, A. T. Friberg, and J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
    [CrossRef]
  10. P. Vahimaa and J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
    [CrossRef]
  11. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
    [CrossRef]
  12. F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
    [CrossRef]
  13. P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376–1381 (2006).
    [CrossRef] [PubMed]
  14. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
    [CrossRef] [PubMed]
  15. J. Turunen and P. Vahimaa, “Independent-elementary-field model for three-dimensional spatially partially coherent sources,” Opt. Express 16, 6433–6442 (2008).
    [CrossRef] [PubMed]
  16. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11, 085706 (2009).
    [CrossRef]
  17. T. Shirai, F. Gori, V. Ramírez-Sánchez, and M. Santarsiero, “A recipe for synthesizing genuine cross-spectral density matrices,” in Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization (Joensuu U. Press, 2009), pp. 80–81.
  18. M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008).
    [CrossRef]
  19. J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
    [CrossRef]
  20. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef] [PubMed]
  21. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [CrossRef] [PubMed]
  22. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).
  23. I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Series, and Products, 7th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2007).
  24. W. N. Bailey, in Generalized Hypergeometric Series, Vol. 32 of Cambridge Tracts in Mathematics and Mathematical Physics, G.H.Hardy and E.Cunningham, eds. (Cambridge U. Press, 1935).

2009 (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11, 085706 (2009).
[CrossRef]

2008 (2)

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008).
[CrossRef]

J. Turunen and P. Vahimaa, “Independent-elementary-field model for three-dimensional spatially partially coherent sources,” Opt. Express 16, 6433–6442 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

2004 (2)

2003 (2)

2002 (1)

J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

1997 (1)

1995 (1)

J. Huttunen, A. T. Friberg, and J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

1983 (2)

F. Gori, “Mode propagation of the light field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 73, 1538–1544 (1983).

1982 (2)

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

1980 (2)

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

1978 (1)

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Alonso, M. A.

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Bailey, W. N.

W. N. Bailey, in Generalized Hypergeometric Series, Vol. 32 of Cambridge Tracts in Mathematics and Mathematical Physics, G.H.Hardy and E.Cunningham, eds. (Cambridge U. Press, 1935).

Borghi, R.

Friberg, A. T.

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11, 085706 (2009).
[CrossRef]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[CrossRef] [PubMed]

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

F. Gori, “Mode propagation of the light field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

T. Shirai, F. Gori, V. Ramírez-Sánchez, and M. Santarsiero, “A recipe for synthesizing genuine cross-spectral density matrices,” in Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization (Joensuu U. Press, 2009), pp. 80–81.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Series, and Products, 7th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2007).

Guattari, G.

Huttunen, J.

J. Huttunen, A. T. Friberg, and J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Mandel, L.

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

Palma, C.

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Piquero, G.

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11, 085706 (2009).
[CrossRef]

T. Shirai, F. Gori, V. Ramírez-Sánchez, and M. Santarsiero, “A recipe for synthesizing genuine cross-spectral density matrices,” in Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization (Joensuu U. Press, 2009), pp. 80–81.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Series, and Products, 7th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2007).

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11, 085706 (2009).
[CrossRef]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[CrossRef] [PubMed]

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

T. Shirai, F. Gori, V. Ramírez-Sánchez, and M. Santarsiero, “A recipe for synthesizing genuine cross-spectral density matrices,” in Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization (Joensuu U. Press, 2009), pp. 80–81.

Setälä, T.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11, 085706 (2009).
[CrossRef]

T. Shirai, F. Gori, V. Ramírez-Sánchez, and M. Santarsiero, “A recipe for synthesizing genuine cross-spectral density matrices,” in Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization (Joensuu U. Press, 2009), pp. 80–81.

Simon, R.

Starikov, A.

A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 73, 1538–1544 (1983).

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

Tervo, J.

Turunen, J.

Vahimaa, P.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Wolf, E.

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008).
[CrossRef]

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

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

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

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11, 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. (2)

A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 73, 1538–1544 (1983).

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

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

Opt. Acta (1)

F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

Opt. Commun. (5)

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Mode propagation of the light field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008).
[CrossRef]

J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. E (1)

J. Huttunen, A. T. Friberg, and J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Other (5)

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

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Series, and Products, 7th ed., A.Jeffrey and D.Zwillinger, eds. (Academic, 2007).

W. N. Bailey, in Generalized Hypergeometric Series, Vol. 32 of Cambridge Tracts in Mathematics and Mathematical Physics, G.H.Hardy and E.Cunningham, eds. (Cambridge U. Press, 1935).

T. Shirai, F. Gori, V. Ramírez-Sánchez, and M. Santarsiero, “A recipe for synthesizing genuine cross-spectral density matrices,” in Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization (Joensuu U. Press, 2009), pp. 80–81.

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Figures (6)

Fig. 1
Fig. 1

Notations used for (a) position and (b) wave vector coordinates.

Fig. 2
Fig. 2

Relative amplitudes of the radial (solid line), azimuthal (dashed line), and longitudinal (dotted line) components of the elementary fields as a function of the normalized radial coordinate k ρ , as well as the function w ( k ρ ) (thick solid line) for n = 1 , which corresponds to a Lambertian source.

Fig. 3
Fig. 3

Same as Fig. 2, but with n = 2 , which corresponds to an incoherent source in scalar theory.

Fig. 4
Fig. 4

Same as Fig. 2, but with n = 5 . Thus the source has a somewhat directional radiation pattern.

Fig. 5
Fig. 5

Same as Fig. 4, but with n = 20 . The radiation pattern is increasingly directional and could be produced approximately by a LED with an integrated collimating lens.

Fig. 6
Fig. 6

(a) A generic geometry of a broad-area surface-emitting LED: p n is the active emitting area and n is the refractive index of the semiconductor material. The solid and dashed curves illustrate the azimuthally and radially polarized contributions to the radiant intensity distribution J ( θ ) . (b) Geometrical-optics predictions of the azimuthally (dashed curve) and radially (dotted curve) polarized contributions to the radiant intensity, their average (solid curve), and the degree of polarization P ( θ ) in the far zone (thick solid curve).

Equations (98)

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W ( r 1 , r 2 ) = 1 ( 2 π ) 4 A ( κ 1 , κ 2 ) × exp [ i ( k 2 r 2 k 1 r 1 ) ] d 2 κ 1 d 2 κ 2 .
A ( κ 1 , κ 2 ) = g ( Δ κ ) f ( κ 1 ) f ( κ 2 ) .
J ( r s ̂ ) = 2 n π 2 k 2 cos 2 θ A ( k σ , k σ ) ,
J ( r s ̂ ) = 2 n π 2 k 2 cos 2 θ | f ( k σ ) | 2 .
e ( r ) = 1 ( 2 π ) 2 f ( κ ) exp ( i k r ) d 2 κ ,
p ( ρ ) = 1 ( 2 π ) 2 g ( Δ κ ) exp ( i Δ κ ρ ) d 2 Δ κ ,
W ( r 1 , r 2 ) = p ( ρ ) e ( r 1 ρ ) e ( r 2 ρ ) d 2 ρ
W ( r 1 , r 2 ) = E ( r 1 ) E T ( r 2 ) .
W ( r 1 , r 2 ) = 1 ( 2 π ) 4 A ( κ 1 , κ 2 ) × exp [ i ( k 2 r 2 k 1 r 1 ) ] d 2 κ 1 d 2 κ 2 ,
A ( κ 1 , κ 2 ) = W ( ρ 1 , ρ 2 , 0 ) × exp [ i ( κ 1 ρ 1 κ 2 ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 ,
W ( r 1 s ̂ 1 , r 2 s ̂ 2 ) = ( 2 π k ) 2 cos θ 1   cos θ 2 A ( k σ 1 , k σ 2 ) exp [ i k ( r 2 r 1 ) ] r 1 r 2 ,
P ( r s ̂ ) = n s ̂ 2 ϵ 0 μ 0   tr W ( r s ̂ , r s ̂ ) = s ̂ cos 2 θ 2 n π 2 k 2 r 2 ϵ 0 μ 0   tr A ( k σ , k σ ) ,
J ( r s ̂ ) = lim r [ r 2 P ( r s ̂ ) ] = 2 n π 2 k 2 cos 2 θ ϵ 0 μ 0   tr A ( k σ , k σ ) .
A ( κ 1 , κ 2 ) = A ( κ 1 ) A T ( κ 2 ) ,
A ( κ 1 , κ 2 ) = A ( κ 2 , κ 1 ) ,
a ( κ 1 ) A ( κ 1 , κ 2 ) a ( κ 2 ) d 2 κ 1 d 2 κ 2 0 ,
A ( κ , κ ) = j = 1 2 I j ( κ ) F j ( κ ) F j T ( κ ) ,
F p ( κ ) F q ( κ ) = δ p q .
J ( r s ̂ ) = J 0 cos 2 θ [ I 1 ( k σ ) + I 2 ( k σ ) ] ,
P ( r ) = { 1 4   det W ( r , r ) [ tr W ( r , r ) ] 2 } 1 / 2 .
P ( r s ̂ ) = { 1 4   det A ( k σ , k σ ) [ tr A ( k σ , k σ ) ] 2 } 1 / 2 .
P ( r s ̂ ) = | I 1 ( k σ ) I 2 ( k σ ) I 1 ( k σ ) + I 2 ( k σ ) | .
A ( κ 1 , κ 2 ) = A 1 ( κ 1 , κ 2 ) + A 2 ( κ 1 , κ 2 ) ,
A ( κ , κ ) = U ( κ ) D ( κ , κ ) U T ( κ ) ,
U T ( κ ) A ( κ , κ ) U ( κ ) = D ( κ , κ ) .
A ( κ 1 , κ 2 ) = j = 1 2 G j ( κ 1 , κ 2 ) F j ( κ 1 ) F j T ( κ 2 ) ,
U T ( κ 1 ) A ( κ 1 , κ 2 ) U ( κ 2 ) = D ( κ 1 , κ 2 ) ,
G j 2 ( κ 1 , κ 2 ) I j ( κ 1 ) I j ( κ 2 ) .
g j ( κ 1 , κ 2 ) = G j ( κ 1 , κ 2 ) [ I j ( κ 1 ) I j ( κ 2 ) ] 1 / 2 ,
A ( κ 1 , κ 2 ) = j = 1 2 g j ( Δ κ ) f j ( κ 1 ) f j T ( κ 2 ) ,
f j ( κ ) = [ I j ( κ ) ] 1 / 2 F j ( κ ) .
W ( r 1 , r 2 ) = W 1 ( r 1 , r 2 ) + W 2 ( r 1 , r 2 ) .
W j ( r 1 , r 2 ) = p j ( ρ ) e j ( r 1 ρ ) e j T ( r 2 ρ ) d 2 ρ
f j ( κ ) = e j ( r ) exp ( i k r ) d 2 ρ
e 1 ( ρ ρ , 0 ) e 2 ( ρ , 0 ) exp ( i κ ρ ) d 2 ρ d 2 ρ = 0.
2 e j ( r ) + k 2 e j ( r ) = 0.
1 T W ( r 1 , r 2 ) = 0 ,
e j ( r ) = 0.
W j e ( r 1 , r 2 ) = e j ( r 1 ) e j T ( r 2 ) ,
W ( r 1 , r 2 ) = j = 1 2 p j ( ρ ) W j e ( r 1 ρ , r 2 ρ ) d 2 ρ ,
S ( r ) = j = 1 2 p j ( ρ ) S j e ( r ρ ) d 2 ρ ,
μ ( r 1 , r 2 ) = { tr [ μ ( r 1 , r 2 ) μ ( r 2 , r 1 ) ] } 1 / 2 .
μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) [ S ( r 1 ) S ( r 2 ) ] 1 / 2
α ( κ 1 , κ 2 ) = { tr [ α ( κ 1 , κ 2 ) α ( κ 2 , κ 1 ) ] } 1 / 2 ,
α ( κ 1 , κ 2 ) = A ( κ 1 , κ 2 ) [ tr A ( κ 1 , κ 1 ) tr A ( κ 2 , κ 2 ) ] 1 / 2
f j ( κ ) = f j , θ ( κ ) θ ̂ + f j , ψ ( κ ) ψ ̂ .
e j ( r ) = k 2 π m = i m   exp ( i m ϕ ) 0 π / 2 exp ( i k z   cos θ ) × { ϱ ̂ [ m ρ f j , ψ , m ( θ ) i   cos θ f j , θ , m ( θ ) d d ρ ] + ϕ ̂ [ m ρ cos θ f j , θ , m ( θ ) i f j , ψ , m ( θ ) d d ρ ] z ̂ k sin 2 θ f j , θ , m ( θ ) } J m ( k ρ   sin θ ) d θ ,
f j , ξ , m ( θ ) = 1 2 π 0 2 π f j , ξ ( θ , ψ ) exp ( i m ψ ) d ψ
e j ( r ) = k 2 2 π 0 π / 2 sin θ   exp ( i k z   cos θ ) { i J 1 ( k ρ   sin θ ) × [ cos θ f j , θ , 0 ( θ ) ρ ̂ + f j , ψ , 0 ( θ ) ϕ ̂ ] sin θ J 0 ( k ρ   sin θ ) f j , θ , 0 ( θ ) z ̂ } d θ .
e 1 ( r ) = k 2 2 π 0 π / 2 f j , θ , 0 ( θ ) sin θ [ i ρ ̂ J 1 ( k ρ   sin θ ) cos θ z ̂   sin θ J 0 ( k ρ   sin θ ) ] exp ( i k z   cos θ ) d θ ,
e 2 ( r ) = ϕ ̂ i k 2 2 π 0 π / 2 f j , ψ , 0 ( θ ) sin θ J 1 ( k ρ   sin θ ) exp ( i k z   cos θ ) d θ .
W ( ρ 1 , ρ 2 , 0 ) S ( ρ ¯ , 0 ) μ ( Δ ρ , 0 ) .
A ( κ 1 , κ 2 ) = S ̃ ( Δ κ ) μ ̃ ( κ ¯ ) ,
S ̃ ( Δ κ ) = S ( ρ ¯ , 0 ) exp ( i Δ κ ρ ¯ ) d 2 ρ ¯ ,
μ ̃ ( κ ¯ ) = μ ( Δ ρ , 0 ) exp ( i κ ¯ Δ ρ ) d 2 Δ ρ .
tr   A ( κ , κ ) = S ̃ ( 0 ) tr   μ ̃ ( κ ) = S ̃ ( 0 ) tr   μ ( Δ ρ , 0 ) × exp ( i κ Δ ρ ) d 2 Δ ρ ,
α ( κ 1 , κ 2 ) = S ̃ ( Δ κ ) S ̃ ( 0 ) μ ̃ ( κ ¯ ) μ ̃ ( 0 ) .
α ( κ 1 , κ 2 ) = S ̃ ( Δ κ ) S ̃ ( 0 ) { tr [ μ ̃ ( κ ¯ ) ] 2 } 1 / 2 tr   μ ̃ ( κ ¯ ) .
P ( κ ) = { 2 tr [ μ ̃ ( κ ) ] 2 [ tr μ ̃ ( κ ) ] 2 1 } 1 / 2 .
α ( κ 1 , κ 2 ) = S ̃ ( Δ κ ) S ̃ ( 0 ) [ D 2 ( κ ¯ ) + 1 2 ] 1 / 2 .
j = 1 2 e j ( ρ 1 ρ , 0 ) e j T ( ρ 2 ρ , 0 ) d 2 ρ = 1 ( 2 π ) 2 A ( κ , κ ) exp ( i κ Δ ρ ) d 2 κ = S ̃ ( 0 ) μ ( Δ ρ , 0 ) ,
W ( ρ 1 , ρ 2 , 0 ) S ( ρ ¯ , 0 ) S ̃ ( 0 ) j = 1 2 e j ( ρ 1 ρ , 0 ) × e j T ( ρ 2 ρ , 0 ) d 2 ρ .
J ( θ , ψ ) = J 0 [ A 1 2 cos a θ + A 2 2 cos b θ ] .
f 1 ( θ , ψ ) = θ ̂ A 1 cos a / 2 1 θ ,
f 2 ( θ , ψ ) = ψ ̂ A 2 cos b / 2 1 θ .
e 1 ( ρ , 0 ) = A 1 i k 2 8 π [ ρ ̂ k ρ 2 Γ ( 1 2 + a 4 ) F ̃ 1 2 ( 3 2 ; 2 , 2 + a 4 ; k 2 ρ 2 4 ) z ̂ Γ ( a 4 ) F ̃ 1 2 ( 3 2 ; 1 , 3 2 + a 4 ; k 2 ρ 2 4 ) ] ,
e 2 ( ρ , 0 ) = ϕ ̂ A 2 i k 3 ρ 16 π Γ ( b 4 ) F ̃ 1 2 ( 3 2 ; 2 , 3 2 + b 4 ; k 2 ρ 2 4 ) ,
w ( ρ , 0 ) = [ e 1 ( ρ , 0 ) 2 + e 2 ( ρ , 0 ) 2 ] 1 / 2
J j ( i ) ( θ ) = J 0 , j ( i ) cos 2 θ I j ( i ) ( θ ) = J 0 , j ( i ) cos 2 θ | A j ( i ) ( θ ) | 2 ,
J j ( θ ) = J 0 , j cos 2 θ I j ( θ ) = J 0 , j cos 2 θ | A j ( θ ) | 2 .
t 1 ( θ , θ ) = 2 n s   cos θ cos θ + n s   cos θ ,
t 2 ( θ , θ ) = 2   cos θ n s   cos θ + cos θ ,
P ( θ ) = | t 1 ( θ , θ ) | 2 | t 2 ( θ , θ ) | 2 | t 1 ( θ , θ ) | 2 + | t 2 ( θ , θ ) | 2 ,
J ( θ ) J ( i ) ( θ ) = 1 2 n s cos 2 θ cos 2 θ [ | t 1 ( θ , θ ) | 2 + | t 2 ( θ , θ ) | 2 ] .
J j ( i ) ( θ ) = 1 2 J 0 ( i )   cos θ ,
J j ( θ ) = 1 2 J 0 cos 2 θ cos θ | t j ( θ , θ ) | 2 .
[ ρ ̂ ϕ ̂ ] = R ( ϕ ) [ x ̂ y ̂ ] ,
[ κ ̂ ψ ̂ ] = R ( ψ ) [ x ̂ y ̂ ] ,
[ k ̂ θ ̂ ] = R ( θ ) [ z ̂ κ ̂ ] ,
R ( ξ ) = [ cos   ξ sin   ξ sin   ξ cos   ξ ]
f j ( κ ) = cos θ f j , θ ( κ ) κ ̂ + f j , ψ ( κ ) ψ ̂ sin θ f j , θ ( κ ) z ̂ .
[ κ ̂ ψ ̂ ] = 1 2 [ 1 1 i i ] [ e i β ( ρ ̂ + i ψ ̂ ) e i β ( ρ ̂ i ψ ̂ ) ] ,
f j ( θ , ψ ) = 1 2 [ cos θ f j , θ ( θ , ψ ) i f j , ψ ( θ , ψ ) ] e i β ( ρ ̂ + i ψ ̂ ) + 1 2 [ cos θ f j , θ ( θ , ψ ) + i f j , ψ ( θ , ψ ) ] e i β ( ρ ̂ i ψ ̂ ) sin θ f j , θ ( θ , ψ ) z ̂ .
exp ( i k r ) = exp ( i k z   cos θ ) m = i m J m ( k ρ   sin θ ) e i m β ,
e j ( r ) = k 2 π m = i m 0 π / 2 g j , m ( r ) exp ( i k z   cos θ ) × J m ( k ρ   sin θ ) k   sin θ d θ ,
g j , m ( r ) = 1 2 π 0 2 π f j ( θ , ψ ) e i m β d ψ .
g j , m ( r ) = 1 2 ( ρ ̂ + i ϕ ̂ ) [ cos θ f j , θ , m + 1 ( θ ) i f j , ψ , m + 1 ( θ ) ] × exp [ i ( m + 1 ) ϕ ] J m ( k ρ   sin θ ) k   sin θ + 1 2 ( ρ ̂ i ϕ ̂ ) × [ cos θ f j , θ , m 1 ( θ ) + i f j , ψ , m 1 ( θ ) ] × exp [ i ( m 1 ) ϕ ] J m ( k ρ   sin θ ) k   sin θ z ̂   exp [ i m ϕ ] J m ( k ρ   sin θ ) k sin 2 θ .
e i ( m ± 1 ) ϕ J m ( k ρ   sin   θ ) i 1 e i m ϕ J m 1 ( k ρ   sin   θ )
J m 1 ( x ) + J m + 1 ( x ) = 2 m x J m ( x ) ,
J m 1 ( x ) J m + 1 ( x ) = 2 d d x J m ( x ) ,
J m ( γ ) = j = 0 ( 1 ) j j ! ( m + j ) ! ( γ 2 ) m + 2 j .
0 π / 2 sin p θ cos q θ d θ = 1 2 B ( p + 1 2 , q + 1 2 ) ,
B ( γ , ξ ) = Γ ( γ ) Γ ( ξ ) Γ ( γ + ξ )
0 π / 2 sin p θ cos q θ J m ( k ρ   sin θ ) d θ = ( k ρ ) m 2 m + 1 Γ ( q + 1 2 ) × j = 0 Γ [ j + 1 2 ( p + m + 1 ) ] j ! Γ ( m + j + 1 ) Γ [ j + 1 + 1 2 ( p + m + q ) ] ( k 2 ρ 2 4 ) j ,
0 π / 2 sin p θ cos q θ J m ( k ρ   sin θ ) d θ = ( k ρ ) m 2 m + 1 Γ ( q + 1 2 ) Γ ( p + m + 1 2 ) F ̃ 1 2 ( p + m + 1 2 ; m + 1 , 1 + p + q + m 2 ; k 2 ρ 2 4 ) ,
F ̃ p q ( a 1 , a 2 , , a p ; b 1 , b 2 , , b q ; γ ) = F p q ( a 1 , a 2 , , a p ; b 1 , b 2 , , b q ; γ ) Γ ( b 1 ) Γ ( b 2 ) Γ ( b q ) ,
F p q ( a 1 , a 2 , , a p ; b 1 , b 2 , , b q ; γ ) = j = 0 ( a 1 ) j ( a 2 ) j ( a p ) j γ j j ! ( b 1 ) j ( b 2 ) j ( b q ) j ,
( c ) j = Γ ( j + c ) Γ ( c ) .

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