Abstract

A procedure for modeling the general electromagnetic Schell-model vortex light source is proposed. Based on this method, we introduce a new class of stochastic electromagnetic vortex light sources with multi-Gaussian Schell-model coherence function. The far-field statistical properties of the beams generated by such sources are studied in detail by numerical examples. Our results can be used to determine the mode structure of a new class of stochastic electromagnetic vortex light sources and of the radiation fields generated by them.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  10. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2013).
  11. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
    [Crossref]
  12. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  18. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, 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).
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    [Crossref]
  25. 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]
  26. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  32. M. Luo and D. Zhao, “Determining the topological charge of stochastic electromagnetic vortex beams with the degree of cross-polarization,” Opt. Lett. 39(17), 5070–5073 (2014).
    [Crossref] [PubMed]
  33. X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
    [Crossref] [PubMed]
  34. L. Guo, Y. Chen, X. Liu, L. Liu, and Y. Cai, “Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam,” Opt. Express 24(13), 13714–13728 (2016).
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    [Crossref] [PubMed]
  36. C. S. D. Stahl and G. Gbur, “Partially coherent vortex beams of arbitrary order,” J. Opt. Soc. Am. A 34(10), 1793–1799 (2017).
    [Crossref] [PubMed]
  37. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]

2017 (1)

2016 (2)

2015 (2)

2014 (5)

2013 (5)

2012 (2)

2010 (1)

2009 (3)

2008 (4)

2007 (2)

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

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

2005 (3)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (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]

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

2002 (1)

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

1986 (1)

1967 (1)

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]

Baykal, Y.

Borghi, R.

M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009).
[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. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Cai, Y.

Chen, Y.

Chen, Z.

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[Crossref]

Du, X.

Eyyuboglu, H. T.

Forbes, A.

Gbur, G.

Gori, F.

F. Gori, V. R. Sanchez, 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]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

C. Palma, F. Gori, G. Guattari, and P. D. Santis, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3(8), 1258–1262 (1986).
[Crossref]

Guattari, G.

Guo, L.

Hernandez-Aranda, R. I.

Ji, X.

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

Korotkova, O.

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[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, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013).
[Crossref]

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

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[Crossref] [PubMed]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, 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]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

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

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (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]

Lahiri, M.

Li, J.

Liu, L.

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 atmosphere,” Opt. Commun. 275(2), 292–300 (2007).
[Crossref]

Luo, M.

Mei, Z.

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Palma, C.

Perez-Garcia, B.

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Pu, J.

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[Crossref]

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]

Ramírez-Sánchez, V.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

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.

Sanchez, V. R.

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

Santarsiero, M.

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

M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009).
[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. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Santis, P. D.

Schell, A.

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.

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

Shirai, T.

F. Gori, V. R. Sanchez, 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]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Stahl, C. S. D.

Swartzlander, G. A.

Tang, M.

Tong, Z.

Wang, F.

Wolf, E.

M. Lahiri and E. Wolf, “Propagation of electromagnetic beams of any state of spatial coherence and polarization through multilayered stratified media,” J. Opt. Soc. Am. A 30(12), 2547–2555 (2013).
[Crossref] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2013).

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]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

Yao, M.

Yepiz, A.

Zhang, E.

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

Zhang, G.

Zhang, Y.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Zhao, C.

Zhao, D.

Zhao, Q.

Zhou, M.

Zhu, S.

Appl. Opt. (1)

IEEE Trans. Antenn. Propag. (1)

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. (3)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[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]

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

J. Opt. Soc. Am. A (10)

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

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, 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. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009).
[Crossref] [PubMed]

J. Li, Y. Chen, Q. Zhao, and M. Zhou, “Effect of astigmatism on states of polarization of aberrant stochastic electromagnetic beams in turbulent atmosphere,” J. Opt. Soc. Am. A 26(10), 2121–2127 (2009).
[Crossref] [PubMed]

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[Crossref] [PubMed]

M. Lahiri and E. Wolf, “Propagation of electromagnetic beams of any state of spatial coherence and polarization through multilayered stratified media,” J. Opt. Soc. Am. A 30(12), 2547–2555 (2013).
[Crossref] [PubMed]

C. Palma, F. Gori, G. Guattari, and P. D. Santis, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3(8), 1258–1262 (1986).
[Crossref]

C. S. D. Stahl and G. Gbur, “Partially coherent vortex beams of arbitrary order,” J. Opt. Soc. Am. A 34(10), 1793–1799 (2017).
[Crossref] [PubMed]

Opt. Commun. (4)

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1), 9–16 (2002).
[Crossref]

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

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

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

Opt. Express (5)

Opt. Lett. (6)

Phys. Lett. A (3)

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2013).

Other (3)

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, 2007).

O. Korotkova, Random Light Beams: Theory and Applications (CRC Press, 2013).

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Figures (4)

Fig. 1
Fig. 1 Far-field spectral densities radiated by a random source defined by Eq. (9) for various coherence lengths and topological charge.
Fig. 2
Fig. 2 Far-field distributions of the degree of polarization corresponding to Fig. 1.
Fig. 3
Fig. 3 The modulus of the degree of coherence corresponding to Fig. 1.
Fig. 4
Fig. 4 The phase of the degree of coherence corresponding to Fig. 1.

Equations (17)

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W ^ (0) ( ρ 1 , ρ 2 ;ω)[ W xx (0) ( ρ 1 , ρ 2 ;ω) W xy (0) ( ρ 1 , ρ 2 ;ω) W yx (0) ( ρ 1 , ρ 2 ;ω) W yy (0) ( ρ 1 , ρ 2 ;ω) ].
W αβ (0) ( ρ 1 , ρ 2 ;ω)= E α ( ρ 1 ;ω) E β ( ρ 2 ;ω), (α=x,y; β=x,y),
W αβ (0) ( ρ 1 , ρ 2 )= p αβ (v) H α ( ρ 1 ,v ) H β ( ρ 2 ,v) d 2 v,
H α ( ρ ,v)= F α ( ρ )exp(2πiv ρ ),
F α ( ρ )= A α ( ρ / σ α ) l exp( ρ 2 / σ α 2 )exp(il ϕ ),
W αβ (0) ( ρ 1 , ρ 2 )= A α A β ( ρ 1 ρ 2 σ α σ β ) l exp[ ρ 1 2 σ α 2 ρ 2 2 σ β 2 2 ]F{ p αβ (v)}(| ρ 1 ρ 2 |)exp[ il( ϕ 1 ϕ 2 1 ) ],
p αβ (v)= π B αβ δ αβ 2 C 2 m=1 M ( M m ) (1) m1 exp(m π 2 δ αβ 2 v x 2 ) × m=1 M ( M m ) (1) m1 exp(m π 2 δ αβ 2 v y 2 ),
C= m=1 M (1) m1 m ( M m ) ,
W αβ (0) ( ρ 1 , ρ 2 )= A α A β B αβ C 2 exp( x 1 2 + y 1 2 σ α 2 )exp( x 2 2 + y 2 2 σ β 2 ) ( x 1 +i y 1 σ α 2 ) l ( x 2 i y 2 σ β 2 ) l × m=1 M (1) m1 m ( M m ) exp[ ( x 1 x 2 ) 2 m δ αβ 2 ] m=1 M (1) m1 m ( M m ) exp[ ( y 1 y 2 ) 2 m δ αβ 2 ].
W αβ () ( ρ 1 , ρ 2 )= (λz) 2 W αβ (0) ( ρ 1 , ρ 2 )exp[ ik( ρ 1 ρ 1 ρ 2 ρ 2 )/z ] d 2 ρ 1 d 2 ρ 2 ,
W αα () ( ρ 1 , ρ 2 )= A α 2 (λz σ α l ) 2 T x T y ( x 1 i y 1 ) l ( x 2 +i y 2 ) l d 2 ρ 1 d 2 ρ 2 2 ,
T t = 1 C m=1 M (1) m1 m ( M m ) exp[ t 1 2 + t 2 2 g 2 + 2 t 1 t 2 m δ αβ 2 ik( t 1 t 1 t 2 t 2 ) z ], t=x,y,
1 g 2 = 1 σ α 2 + 1 m δ αβ 2 .
W αα () ( ρ 1 , ρ 2 )= A α 2 (λz σ α l ) 2 p=0 l q=0 l ( p l )( q l ) (i) 2lpq T x x 1 p x 2 q d x 1 d x 2 T y y 1 lp (- y 2 ) lq d y 1 d y 2 1 .
S(ρ)= W xx (ρ,ρ)+ W yy (ρ,ρ),
P(ρ)=| W xx (ρ,ρ) W yy (ρ,ρ) W xx (ρ,ρ)+ W yy (ρ,ρ) |,
μ( ρ 1 , ρ 2 )= W xx ( ρ 1 , ρ 2 )+ W yy ( ρ 1 , ρ 2 ) [ W xx ( ρ 1 , ρ 1 )+ W yy ( ρ 1 , ρ 1 )][ W xx ( ρ 2 , ρ 2 )+ W yy ( ρ 2 , ρ 2 )] .