Abstract

A class of random source for circular optical frame is generalized to electromagnetic domain. Analytical formulas for the propagation of the electromagnetic source for circular frames combinations through atmospheric turbulence are derived. As two examples, the statistic characteristics of a single circular frame and two nested frames are comparatively studied in free space and in non-Kolmogorov’s atmospheric turbulence. The evolutions of the degree of polarization and the degree of coherence of such circular frames exhibit unique features. The impacts, arising from the refractive-index structure constant, the fractal constant of the atmospheric spectrum and the upper index in the source degree of coherence, on the statistical characteristics are analyzed in detail.

© 2015 Optical Society of America

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References

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  1. E. Wolf, Introduction to the Theories of Coherences and Polarization of Light (Cambridge University Press, Cambridge, 2007).
  2. C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
    [Crossref]
  3. F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
    [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).
    [PubMed]
  6. Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
    [Crossref] [PubMed]
  7. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  8. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  9. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
  10. O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
    [Crossref] [PubMed]
  11. F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
    [PubMed]
  12. 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]
  13. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
    [Crossref] [PubMed]
  14. Z. Mei and Y. Mao, “Electromagnetic sinc Schell-model beams and their statistical properties,” Opt. Express 22(19), 22534–22546 (2014).
    [Crossref] [PubMed]
  15. X. Liu and D. Zhao, “Fractional Fourier transforms of electromagnetic rectangular Gaussian Schell model beam,” Opt. Commun. 344, 181–187 (2015).
    [Crossref]
  16. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [Crossref]
  17. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28(13), 1078–1080 (2003).
    [Crossref] [PubMed]
  18. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
    [Crossref]
  19. 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]
  20. 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]
  21. 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]
  22. 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]
  23. M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
    [Crossref] [PubMed]
  24. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
    [Crossref]
  25. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
    [PubMed]
  26. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
    [Crossref] [PubMed]
  27. H. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
    [Crossref]
  28. G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
    [Crossref]
  29. 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]
  30. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  31. M. Yao and O. Korotkova, “Random optical frames in atmospheric turbulence,” J. Opt. 16(10), 105713 (2014).
    [Crossref]
  32. O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
    [Crossref]

2015 (1)

X. Liu and D. Zhao, “Fractional Fourier transforms of electromagnetic rectangular Gaussian Schell model beam,” Opt. Commun. 344, 181–187 (2015).
[Crossref]

2014 (8)

2013 (3)

2012 (3)

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]

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (2)

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[PubMed]

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]

2008 (3)

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]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[PubMed]

2007 (3)

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]

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. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

2006 (1)

2004 (1)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

2003 (2)

1996 (1)

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Alavynejad, M.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
[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.

H. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

Borghi, R.

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[PubMed]

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Cai, Y.

Cincotti, G.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Du, X.

Eyyuboglu, H.

H. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

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]

Ghafary, B.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
[Crossref]

Gori, F.

He, S.

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]

Kashani, F. D.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
[Crossref]

Korotkova, O.

M. Yao and O. Korotkova, “Random optical frames in atmospheric turbulence,” J. Opt. 16(10), 105713 (2014).
[Crossref]

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref] [PubMed]

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

O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (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 and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

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

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

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

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[PubMed]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]

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]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

Lajunen, H.

Liu, X.

X. Liu and D. Zhao, “Fractional Fourier transforms of electromagnetic rectangular Gaussian Schell model beam,” Opt. Commun. 344, 181–187 (2015).
[Crossref]

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]

Mao, Y.

Mei, Z.

Palma, C.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

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]

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]

Saastamoinen, T.

Sahin, S.

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

Santarsiero, M.

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]

Taherabadi, G.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
[Crossref]

Tong, Z.

Toselli, I.

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (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.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28(13), 1078–1080 (2003).
[Crossref] [PubMed]

Yao, M.

Yousefi, M.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
[Crossref]

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]

Zhao, D.

Appl. Phys. B (1)

H. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

J. Opt. (2)

M. Yao and O. Korotkova, “Random optical frames in atmospheric turbulence,” J. Opt. 16(10), 105713 (2014).
[Crossref]

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

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

J. Opt. Soc. Am. A (1)

Opt. Commun. (6)

X. Liu and D. Zhao, “Fractional Fourier transforms of electromagnetic rectangular Gaussian Schell model beam,” Opt. Commun. 344, 181–187 (2015).
[Crossref]

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285(8), 2017–2021 (2012).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[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]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[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 (8)

Opt. Lett. (9)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

Other (2)

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

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

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Figures (6)

Fig. 1
Fig. 1 Evolution of the spectral density of the single circular optical frame (the top row) and the two overlapped circular frames (the bottom row) propagating in free space and turbulent atmosphere. (a) and (d) the longitudinal distribution, C ˜ n 2 = 10 13 m 3 α , α = 3.667 and M = 10 . (b) and (e) the 2-Dimensional distribution at the plane z = 1 km ; (c) and (f) the transverse distribution for different values of parameter C ˜ n 2 .
Fig. 2
Fig. 2 The transverse distribution of the circular optical frame’s spectral density at the plane z = 5 km for different parameters C ˜ n 2 , α and M . (a) and (d) for different C ˜ n 2 with α = 3.667 , M = 20 ; (b) and (e) for different α with C ˜ n 2 = 10 13 m 3 α , M = 20 ; (c) and (f) for different M with C ˜ n 2 = 10 13 m 3 α , α = 3.667 . The top row corresponding to the single circular frame and the bottom row corresponding to the two combined circular frame.
Fig. 3
Fig. 3 The change in the spectral degree of polarization of the single circular frame (the top row) and the two superimposed ones (the bottom row) along z-axis for different parameters C ˜ n 2 , α and M . (a) α = 3.667 , M = 10 ; (b) C ˜ n 2 = 10 13 m 3 α , M = 10 ; (c) C ˜ n 2 = 10 13 m 3 α , α = 3.667 .
Fig. 4
Fig. 4 The transverse distribution of the spectral degree of polarization of the single circular frame (the top row) and the two overlapped frame (the bottom row) at the plane z = 5 km . (b) and (c) C ˜ n 2 = 10 14 m 3 α , the other parameters set as Fig. 2.
Fig. 5
Fig. 5 Change in the spectral degree of coherence of a single circular frame along the z-axis for different parameters as in Fig. 3.
Fig. 6
Fig. 6 Evolution of the transverse degree of coherence as a function of r 2 at the plane z = 1 km for different parameters as in Fig. 5.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

W i j ( 0 ) ( r 10 , r 20 ; ω ) = E i ( r 10 ; ω ) E j ( r 20 ; ω ) ; ( i , j = x , y ) ,
W i j ( 0 ) ( r 10 , r 20 ) = p i j ( v ) H i ( r 10 , v ) H j ( r 20 , v ) d v ,
H i ( r 10 , v ) = A i exp ( r 10 2 4 σ i 2 ) exp ( 2 π i v r 10 ) ,
H j ( r 20 , v ) = A j exp ( r 20 2 4 σ j 2 ) exp ( 2 π i v r 20 ) ,
W i j ( 0 ) ( r 10 , r 20 ) = A i A j exp [ r 10 2 + r 20 2 4 σ 2 ] μ i j ( 0 ) ( r 10 , r 20 ) ,
p ( v ) i j = B i j A 0 i j C 0 m = 1 M ( 1 ) m 1 ( M m ) [ exp ( m δ i j e 2 v 2 2 ) exp ( m δ i j o 2 v 2 2 ) ] .
μ i j ( 0 ) ( r 10 , r 20 ) = B i j A 0 i j C 0 m = 1 M ( 1 ) m 1 m ( M m ) { 1 δ i j e exp [ ( r 20 r 10 ) 2 2 m δ i j e 2 ] 1 δ i j o exp [ ( r 20 r 10 ) 2 2 m δ i j o 2 ] } ,
B x x = B y y = 1 , | B x y | = | B y x | , δ x y = δ y x .
p i j ( v ) 0 ,
p x x ( v ) p y y ( v ) p x y ( v ) p y x ( v ) 0 ,
δ i j e < δ i j o ,
B x x A 0 x x { [ 1 exp ( δ x x e 2 v 2 / 2 ) ] M [ 1 exp ( δ x x o 2 v 2 / 2 ) ] M } × B y y A 0 y y { [ 1 exp ( δ y y e 2 v 2 / 2 ) ] M [ 1 exp ( δ y y o 2 v 2 / 2 ) ] M } , | B x y A 0 x y | 2 { [ 1 exp ( δ x y e 2 v 2 / 2 ) ] M [ 1 exp ( δ x y o 2 v 2 / 2 ) ] M } 2
δ x y o δ x y e min { δ x x o δ x x e , δ y y o δ y y e } ,
B x y δ x y o δ x y e min { δ x x o δ x x e , δ y y o δ y y e } ,
[ 1 exp ( δ x y e 2 v 2 / 2 ) ] M [ 1 exp ( δ x y o 2 v 2 / 2 ) ] M min { [ 1 exp ( δ x x e 2 v 2 / 2 ) ] M [ 1 exp ( δ x x o 2 v 2 / 2 ) ] M , [ 1 exp ( δ y y e 2 v 2 / 2 ) ] M [ 1 exp ( δ y y o 2 v 2 / 2 ) ] M } .
W i j ( r 1 , r 2 , z ) = ( k 2 π z ) 2 d 2 r 10 d 2 r 20 W i j ( 0 ) ( r 10 , r 20 ) K ( r 10 , r 20 , r 1 , r 2 ) ,
K ( r 10 , r 20 , r 1 , r 2 ) = exp [ i k ( r 1 r 10 ) 2 ( r 2 r 20 ) 2 2 z ] × exp { π 2 k 2 z 3 [ ( r 1 r 2 ) 2 + ( r 1 r 2 ) ( r 10 r 20 ) + ( r 10 r 20 ) 2 ] 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 ,
I = 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 ] ,
W i j ( r 1 , r 2 , z ) = W i j e ( r 1 , r 2 , z ) W i j o ( r 1 , r 2 , z ) ,
W i j a ( r 1 , r 2 , z ) = A i A j B i j A 0 i j C 0 δ i j a Δ i j a ( z ) m = 1 M ( 1 ) m 1 m ( M m ) exp [ ( r 1 + r 2 ) 2 8 σ 2 Δ i j a 2 ( z ) ] exp [ i k ( r 2 2 r 1 2 ) 2 R i j a ( z ) ] × exp { [ 1 2 Δ i j a 2 ( z ) Ω i j a 2 + 1 3 π 2 k 2 z I ( 1 + 2 Δ i j a 2 ( z ) ) π 4 k 2 z 4 I 2 18 σ 2 Δ i j a 2 ( z ) ] ( r 1 r 2 ) 2 } , ( a = e , o ) ,
1 Ω i j a 2 = 1 4 σ 2 + 1 m δ i j a 2 , Δ i j a 2 ( z ) = 1 + z 2 k 2 σ 2 Ω i j a 2 + 2 π 2 z 3 I 3 σ 2 , R i j a ( z ) = σ 2 Δ i j a 2 ( z ) z σ 2 Δ i j a 2 ( z ) + 1 3 π 2 z 3 I σ 2 .
μ i j = B n = 1 N a n μ n i j , B = ( n = 1 N a n ) 1 ,
W i j ( 0 ) = B n = 1 N a n W n i j ( 0 )
W n i j ( 0 ) ( r 10 , r 20 ) = A i A j exp [ r 10 2 + r 20 2 4 σ 2 ] B n i j A n 0 i j C 0 × m = 1 M ( 1 ) m 1 m ( M m ) { 1 δ n i j e exp [ ( r 20 r 10 ) 2 2 m δ n i j e 2 ] 1 δ n i j o exp [ ( r 20 r 10 ) 2 2 m δ n i j o 2 ] }
W i j = B n = 1 N a n W n i j ,
W n i j ( r 1 , r 2 , z ) = W n i j e ( r 1 , r 2 , z ) W n i j o ( r 1 , r 2 , z ) ,
W n i j a ( r 1 , r 2 , z ) = A i A j B n i j A n 0 i j C 0 δ n i j a Δ n i j a ( z ) m = 1 M ( 1 ) m 1 m ( M m ) exp [ ( r 1 + r 2 ) 2 8 σ 2 Δ n i j a 2 ( z ) ] exp [ i k ( r 2 2 r 1 2 ) 2 R n i j a ( z ) ] × exp { [ 1 2 Δ n i j a 2 ( z ) Ω n i j a 2 + 1 3 π 2 k 2 z I ( 1 + 2 Δ n i j a 2 ( z ) ) π 4 k 2 z 4 I 2 18 σ 2 Δ n i j a 2 ( z ) ] ( r 1 r 2 ) 2 } ,
A n 0 i j = ( 1 δ n i j e 1 δ n i j o ) 1 , 1 Ω n i j a 2 = 1 4 σ 2 + 1 m δ n i j a 2 , Δ n i j a 2 ( z ) = 1 + z 2 k 2 σ 2 Ω n i j a 2 + 2 π 2 z 3 I 3 σ 2 , R n i j a ( z ) = σ 2 Δ n i j a 2 ( z ) z σ 2 Δ n i j a 2 ( z ) + 1 3 π 2 z 3 I σ 2 .
S ( r , z ) = Tr W ( r , r , z ) ,
P ( r , z ) = 1 4 Det W ( r , r , z ) [ Tr W ( r , r , z ) ] 2 ,
μ ( r 1 , r 2 , z ) = Tr W ( r 1 , r 2 , z ) Tr W ( r 1 , r 1 , z ) Tr W ( r 2 , r 2 , z ) ,

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