Abstract

A class of electromagnetic sources with sinc Schell-model correlations is introduced. The conditions on source parameters guaranteeing that the source generates a physical beam are derived. The evolution behaviors of statistical properties for the electromagnetic stochastic beams generated by this new source on propagating in free space and in atmosphere turbulence are investigated with the help of the weighted superposition method and by numerical simulations. It is demonstrated that the intensity distributions of such beams exhibit unique features on propagating in free space and produce a double-layer flat-top profile of being shape-invariant in the far field. This feature makes this new beam particularly suitable for some special laser processing applications. The influences of the atmosphere turbulence with a non-Kolmogorov power spectrum on statistical properties of the new beams are analyzed in detail.

© 2014 Optical Society of America

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References

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  1. J. A. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antenn. Propag. 15(1), 187–188 (1967).
    [Crossref]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  4. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  5. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [Crossref] [PubMed]
  6. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
  7. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [Crossref] [PubMed]
  8. Z. Mei, “Light sources generating self-splitting beams and their propagation in non-Kolmogorov turbulence,” Opt. Express 22(11), 13029–13040 (2014).
    [Crossref] [PubMed]
  9. E. Wolf, Introduction to the Theories of Coherence and Polarization of Light (Cambridge University, Cambridge, 2007).
  10. O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24(9), 2728–2736 (2007).
    [Crossref] [PubMed]
  11. Y. Zhu and D. Zhao, “Propagation of a random electromagnetic beam through a misaligned optical system in turbulent atmosphere,” J. Opt. Soc. Am. A 25(10), 2408–2414 (2008).
    [Crossref] [PubMed]
  12. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
    [Crossref] [PubMed]
  13. G. Zhang and J. Pu, “Stochastic electromagnetic beams focused by a bifocal lens,” J. Opt. Soc. Am. A 25(7), 1710–1715 (2008).
    [Crossref] [PubMed]
  14. B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26(4), 862–869 (2009).
    [Crossref] [PubMed]
  15. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
    [Crossref] [PubMed]
  16. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
    [Crossref]
  17. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  18. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]
  19. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
    [Crossref]
  20. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
    [Crossref] [PubMed]
  21. Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012).
    [Crossref] [PubMed]
  22. Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013).
    [Crossref] [PubMed]
  23. Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
    [Crossref] [PubMed]
  24. X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atomosphere,” Opt. Commun. 275(2), 292–300 (2007).
    [Crossref]
  25. J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
    [Crossref]
  26. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
    [Crossref] [PubMed]
  27. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
    [Crossref]
  28. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
    [Crossref] [PubMed]
  29. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17(6), 4257–4262 (2009).
    [Crossref] [PubMed]
  30. M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29(9), 1838–1842 (2012).
    [Crossref] [PubMed]
  31. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
    [Crossref] [PubMed]
  32. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
    [Crossref]

2014 (5)

2013 (4)

2012 (4)

2010 (1)

2009 (4)

X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17(6), 4257–4262 (2009).
[Crossref] [PubMed]

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26(4), 862–869 (2009).
[Crossref] [PubMed]

2008 (6)

2007 (3)

2005 (2)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[Crossref]

1967 (1)

J. A. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antenn. Propag. 15(1), 187–188 (1967).
[Crossref]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Baykal, Y.

Borghi, R.

Cai, Y.

Charnotskii, M.

Chen, B.

Du, X.

Eyyuboglu, H. T.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Gbur, G.

Golbraikh, E.

Gori, F.

Ji, X.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atomosphere,” Opt. Commun. 275(2), 292–300 (2007).
[Crossref]

Kopeika, N. S.

Korotkova, O.

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[Crossref] [PubMed]

O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24(9), 2728–2736 (2007).
[Crossref] [PubMed]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Liang, C.

Liu, X.

Lü, B.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atomosphere,” Opt. Commun. 275(2), 292–300 (2007).
[Crossref]

Mei, Z.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Pu, J.

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(8), 085706 (2009).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref] [PubMed]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Sahin, S.

Santarsiero, M.

Schell, J. A.

J. A. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antenn. Propag. 15(1), 187–188 (1967).
[Crossref]

Shchepakina, E.

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(8), 085706 (2009).
[Crossref]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[Crossref]

Tong, Z.

Toselli, I.

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Wang, F.

Wolf, E.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[Crossref]

Yao, M.

Zhang, E.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atomosphere,” Opt. Commun. 275(2), 292–300 (2007).
[Crossref]

Zhang, G.

Zhang, Z.

Zhao, D.

Zhu, Y.

Zilberman, A.

Appl. Opt. (1)

IEEE Trans. Antenn. Propag. (1)

J. A. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antenn. Propag. 15(1), 187–188 (1967).
[Crossref]

J. Opt. (1)

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

J. Opt. A, Pure Appl. Opt. (2)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[Crossref]

J. Opt. Soc. Am. A (9)

O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24(9), 2728–2736 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref] [PubMed]

G. Zhang and J. Pu, “Stochastic electromagnetic beams focused by a bifocal lens,” J. Opt. Soc. Am. A 25(7), 1710–1715 (2008).
[Crossref] [PubMed]

Y. Zhu and D. Zhao, “Propagation of a random electromagnetic beam through a misaligned optical system in turbulent atmosphere,” J. Opt. Soc. Am. A 25(10), 2408–2414 (2008).
[Crossref] [PubMed]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[Crossref] [PubMed]

M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29(9), 1838–1842 (2012).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
[Crossref] [PubMed]

B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26(4), 862–869 (2009).
[Crossref] [PubMed]

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

Opt. Commun. (3)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atomosphere,” Opt. Commun. 275(2), 292–300 (2007).
[Crossref]

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Opt. Express (5)

Opt. Lett. (7)

Other (2)

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

E. Wolf, Introduction to the Theories of Coherence and Polarization of Light (Cambridge University, Cambridge, 2007).

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Figures (8)

Fig. 1
Fig. 1 Transverse distribution of the spectral density S of an EM SSM beam with δ y y = 2.5 m m at several different propagation distances in free space. (a) z = 0m; (b) z = 250m; (c) z = 350m; (d) z = 500m.
Fig. 2
Fig. 2 Transverse distribution of the spectral density S of EM SSM beams with different δ y y propagating in free space at the plane z = 1km. (a) δ y y = 1 m m ; (b) δ y y = 2.5 m m ; (c) δ y y = 5 m m .
Fig. 3
Fig. 3 Transverse distribution of the spectral density S of an EM SSM beam with δ y y = 2.5 m m propagating in the atmosphere at the distance z = 5km for different values of atmosphere parameters.
Fig. 4
Fig. 4 Transverse distribution of the degree of polarization of the beams with different δ y y and δ x y propagating in free space at the plane z = 1km. (a) δ y y = δ x y = 1 m m ; (b) δ y y = δ x y = 2.5 m m ; (c) δ y y = δ x y = 5 m m .
Fig. 5
Fig. 5 Transverse distribution of the degree of polarization P of an EM SSM beam with δ y y = δ x y = 2.5 m m propagating in the atmosphere at the distance z = 5km for different values of turbulent parameters.
Fig. 6
Fig. 6 Changes in the degree of polarization P along the z-axis of the same beam as in Fig. 5 propagating in the atmosphere turbulence for different values of turbulent parameters. (a) α = 3.1 ; (b) C ˜ n 2 = 10 12 m 3 α .
Fig. 7
Fig. 7 Evolutions of the degree of coherence μ as a function of ρ d , for an EM SSM beam with δ y y = 2.5 m m propagating in the atmosphere turbulent with different values of atmosphere parameters.
Fig. 8
Fig. 8 Changes in the degree of coherence μ along the z-axis of the same beam as in Fig. 7 propagating in the atmosphere turbulent for different values of atmosphere parameters when ρ d = 0.5 m m . (a) α = 3.1 ; (b) C ˜ n 2 = 10 12 m 3 α .

Equations (27)

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W α β ( 0 ) ( ρ 1 , ρ 2 ; ω ) = E α * ( ρ 1 ; ω ) E β ( ρ 2 ; ω ) ; ( α = x , y ; β = x , y ) ,
W α β ( 0 ) ( ρ 1 , ρ 2 ) = p α β ( v ) H α ( ρ 1 , v ) H β ( ρ 2 , v ) d v ,
H α ( ρ 1 , v ) = A α τ ( ρ 1 ) exp ( 2 π i v ρ 1 ) ,
H β ( ρ 2 , v ) = A β τ ( ρ 2 ) exp ( 2 π i v ρ 2 ) ,
W α β ( 0 ) ( ρ 1 , ρ 2 ) = A α A β τ ( ρ 1 ) τ ( ρ 2 ) p ˜ α β ( ρ 1 ρ 2 ) .
p α β ( v ) = B α β δ α β rect ( δ α β v ) ,
W α β ( 0 ) ( ρ 1 , ρ 2 ) = A α A β B α β exp ( ρ 1 2 + ρ 2 2 2 σ 0 2 ) sin c ( ρ 1 ρ 2 δ α β ) .
B x x = B y y = 1 , | B x y | = | B y x | , δ x y = δ y x .
p α α ( v ) 0 ,
p x x ( v ) p y y ( v ) p x y ( v ) p y x ( v ) 0 ,
δ x x δ y y rect ( δ x x v ) rect ( δ y y v ) | B x y | 2 δ x y 2 [ rect ( δ x y v ) ] 2 .
max { δ x x , δ y y } δ x y δ x x δ y y | B x y | .
W α β ( ρ 1 , ρ 2 , z ) = W α β ( 0 ) ( ρ 1 , ρ 2 ) K ( ρ 1 , ρ 2 , ρ 1 , ρ 2 , z ) d 2 ρ 1 d 2 ρ 2 ,
K ( ρ 1 , ρ 2 , ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 exp [ i k ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 2 z ] × exp { π 2 k 2 z 3 [ ( ρ 1 ρ 2 ) 2 + ( ρ 1 ρ 2 ) ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ] 0 κ 3 Φ n ( κ ) d κ } .
W α β ( ρ 1 , ρ 2 , z ) = p α β ( v ) H α * ( ρ 1 , v , z ) H β ( ρ 2 , v , z ) d v ,
H α * ( ρ 1 , v , z ) H β ( ρ 2 , v , z ) = H α * ( ρ 1 , v ) H β ( ρ 2 , v ) K ( ρ 1 , ρ 2 , ρ 1 , ρ 2 , z ) d 2 ρ 1 d 2 ρ 2 .
H α * ( ρ 1 , v , z ) H β ( ρ 2 , v , z ) = A α A β σ 0 2 w 2 ( z ) × exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] exp [ ( k 2 σ 0 2 4 z 2 + k 2 π 2 z 3 0 κ 3 Φ n ( κ ) d κ ) ( ρ 1 ρ 2 ) 2 ] × exp { [ ρ 1 + ρ 2 2 + 2 π z v k i ( k σ 0 2 2 z k π 2 z 2 3 0 κ 3 Φ n ( κ ) d κ ) ( ρ 1 ρ 2 ) ] 2 / w 2 ( z ) } ,
w 2 ( z ) = σ 0 2 + z 2 k 2 σ 0 2 + 4 π 2 z 3 3 0 κ 3 Φ n ( κ ) d κ .
Φ n ( κ ) = A ( α ) C ˜ n 2 exp [ ( κ 2 / κ m 2 ) ] / ( κ 2 + κ 0 2 ) α / 2 , 0 κ < , 3 < α < 4 ,
c ( α ) = [ Γ ( 5 α 2 ) A ( α ) 2 π 3 ] 1 / ( α 5 ) ,
A ( α ) = Γ ( α 1 ) cos ( α π / 2 ) 4 π 2 ,
0 κ 3 Φ n ( κ ) d κ = A ( α ) 2 ( α 2 ) C ˜ n 2 [ κ m 2 α β exp ( κ 0 2 κ m 2 ) Γ ( 2 α 2 , κ 0 2 κ m 2 ) 2 κ 0 4 α ] ,
H α * ( ρ 1 , v , z ) H β ( ρ 2 , v , z ) = A α A β σ 0 2 w 2 ( z ) exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] exp [ k 2 σ 0 2 4 z 2 ( ρ 1 ρ 2 ) 2 ] × exp { [ ρ 1 + ρ 2 2 + 2 π z v k i k σ 0 2 2 z ( ρ 1 ρ 2 ) ] 2 / w 2 ( z ) } ,
w 2 ( z ) = σ 0 2 + z 2 / ( k 2 σ 0 2 ) .
S ( ρ , z ) = Tr W ^ ( ρ , ρ , z ) ,
P ( ρ , z ) = 1 4 Det W ^ ( ρ , ρ , z ) [ Tr W ^ ( ρ , ρ , z ) ] 2 ,
μ ( ρ 1 , ρ 2 , z ) = Tr W ^ ( ρ 1 , ρ 2 , z ) Tr W ^ ( ρ 1 , ρ 1 , z ) Tr W ^ ( ρ 2 , ρ 2 , z ) ,

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