Abstract

Analytical nonparaxial propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam in free space is derived based on the generalized Raleigh-Sommerfeld diffraction integrals. Statistical properties, such as average intensity, degree of polarization and degree of coherence, of a nonparaxial cylindrical vector partially coherent field are illustrated numerically, and compared with that of a paraxial cylindrical vector partially coherent beam. It is found that the statistical properties of a nonparaxial cylindrical vector partially coherent field are much different from that of a paraxial cylindrical vector partially coherent beam, and are closely determined by the initial beam width and correlation coefficients. Our results will be useful for modulating the properties of a nonparaxial cylindrical vector partially coherent field.

© 2012 OSA

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  1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009).
    [CrossRef]
  2. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [PubMed]
  3. Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5(3), 229–232 (2003).
    [CrossRef]
  4. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998).
    [CrossRef]
  5. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [CrossRef] [PubMed]
  6. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [CrossRef] [PubMed]
  7. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001).
    [CrossRef] [PubMed]
  8. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
    [CrossRef] [PubMed]
  9. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
    [CrossRef] [PubMed]
  10. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23(6), 1228–1234 (2006).
    [CrossRef]
  11. D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45(3), 470–479 (2006).
    [CrossRef] [PubMed]
  12. B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. 12(7), 073012 (2010).
    [CrossRef]
  13. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
    [CrossRef] [PubMed]
  14. X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
    [CrossRef]
  15. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
    [CrossRef]
  16. R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express 18(10), 10834–10838 (2010).
    [CrossRef] [PubMed]
  17. K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
    [CrossRef]
  18. R. S. Kurti, K. Halterman, R. K. Shori, and M. J. Wardlaw, “Discrete cylindrical vector beam generation from an array of optical fibers,” Opt. Express 17(16), 13982–13988 (2009).
    [CrossRef] [PubMed]
  19. G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237(1-3), 89–95 (2004).
    [CrossRef]
  20. C. F. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel-Gaussian vector beams,” Opt. Lett. 32(24), 3543–3545 (2007).
    [CrossRef] [PubMed]
  21. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
  22. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [CrossRef]
  23. 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]
  24. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
    [CrossRef] [PubMed]
  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. 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]
  27. 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]
  28. 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]
  29. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
    [CrossRef]
  30. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
    [CrossRef] [PubMed]
  31. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
    [CrossRef]
  32. S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
    [CrossRef]
  33. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
    [CrossRef] [PubMed]
  34. G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011).
    [CrossRef] [PubMed]
  35. S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010).
    [CrossRef] [PubMed]
  36. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
    [CrossRef]
  37. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
    [CrossRef] [PubMed]
  38. L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
    [CrossRef]
  39. F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
    [CrossRef] [PubMed]
  40. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
    [CrossRef]
  41. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
    [CrossRef] [PubMed]
  42. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
    [CrossRef]
  43. H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
    [CrossRef]
  44. H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
    [CrossRef]
  45. H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
    [CrossRef]
  46. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21(10), 2029–2037 (2004).
    [CrossRef] [PubMed]
  47. K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29(8), 800–802 (2004).
    [CrossRef] [PubMed]
  48. Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett. 29(23), 2710–2712 (2004).
    [CrossRef] [PubMed]
  49. K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21(10), 1924–1932 (2004).
    [CrossRef] [PubMed]
  50. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
    [CrossRef]
  51. L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B 103(4), 1001–1008 (2011).
    [CrossRef]
  52. L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (2011).
    [CrossRef] [PubMed]
  53. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
    [CrossRef] [PubMed]
  54. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
    [CrossRef] [PubMed]
  55. 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).
    [CrossRef] [PubMed]
  56. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
    [CrossRef]
  57. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).
  58. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  59. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
  60. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
    [CrossRef] [PubMed]
  61. O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21(12), 2382–2385 (2004).
    [CrossRef] [PubMed]
  62. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
    [CrossRef] [PubMed]
  63. Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
    [CrossRef]

2012 (1)

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[CrossRef]

2011 (8)

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B 103(4), 1001–1008 (2011).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011).
[CrossRef] [PubMed]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (2011).
[CrossRef] [PubMed]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[CrossRef] [PubMed]

2010 (9)

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
[CrossRef] [PubMed]

R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express 18(10), 10834–10838 (2010).
[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]

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010).
[CrossRef] [PubMed]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
[CrossRef]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[CrossRef]

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
[CrossRef]

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. 12(7), 073012 (2010).
[CrossRef]

2009 (8)

2008 (6)

2007 (2)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

C. F. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel-Gaussian vector beams,” Opt. Lett. 32(24), 3543–3545 (2007).
[CrossRef] [PubMed]

2006 (3)

D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45(3), 470–479 (2006).
[CrossRef] [PubMed]

D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23(6), 1228–1234 (2006).
[CrossRef]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
[CrossRef]

2005 (3)

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]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

2004 (7)

2003 (4)

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5(3), 229–232 (2003).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

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

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[CrossRef] [PubMed]

2002 (2)

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

2001 (3)

2000 (1)

1998 (2)

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

Baykal, Y.

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Biss, D. P.

Bogatyryova, G. V.

Borghi, R.

Brown, T. G.

Bu, J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
[CrossRef]

Burge, R. E.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
[CrossRef]

Cai, Y.

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B 103(4), 1001–1008 (2011).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011).
[CrossRef] [PubMed]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (2011).
[CrossRef] [PubMed]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[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]

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010).
[CrossRef] [PubMed]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[CrossRef]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[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).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

Chen, J.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
[CrossRef]

Cheng, W.

Deng, D.

Ding, J.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
[CrossRef]

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

Dong, Y.

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Duan, K.

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

Eyyuboglu, H. T.

Fan, Y.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
[CrossRef]

Fel’de, C. V.

Friberg, A. T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Gori, F.

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]

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]

Gu, C.

Halterman, K.

Haus, J. W.

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Korotkova, O.

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[CrossRef]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
[CrossRef]

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, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[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).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

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

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]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21(12), 2382–2385 (2004).
[CrossRef] [PubMed]

Kozawa, Y.

Kurti, R. S.

Laabs, H.

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

Leger, J. R.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Li, C. F.

Li, X.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

Lin, H.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[CrossRef]

Lin, Q.

Liu, D.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

Liu, X.

Low, D. K. Y.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
[CrossRef]

Lü, B.

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Ming, H.

Moh, K. J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
[CrossRef]

Mondello, A.

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]

Petrov, D.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237(1-3), 89–95 (2004).
[CrossRef]

Piquero, G.

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]

Polyanskii, P. V.

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

Ponomarenko, S. A.

Pu, J.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
[CrossRef] [PubMed]

Qin, J.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Ramírez-Sánchez, V.

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Roxworthy, B. J.

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. 12(7), 073012 (2010).
[CrossRef]

Salem, M.

Santarsiero, M.

Sato, S.

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Shirai, T.

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]

Shori, R. K.

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]

Soskin, M. S.

Tong, Z.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
[CrossRef]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[CrossRef]

Toussaint, K. C.

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. 12(7), 073012 (2010).
[CrossRef]

Tovar, A. A.

Volpe, G.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237(1-3), 89–95 (2004).
[CrossRef]

Wang, A.

Wang, F.

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[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).
[CrossRef] [PubMed]

Wang, H.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
[CrossRef]

Wang, X.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
[CrossRef] [PubMed]

Wardlaw, M. J.

Watson, E.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

Wolf, E.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (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]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21(12), 2382–2385 (2004).
[CrossRef] [PubMed]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[CrossRef] [PubMed]

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

Wu, G.

Xu, L.

Yao, M.

Youngworth, K. S.

Yuan, X. C.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
[CrossRef]

Zhan, Q.

Zhang, L.

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B 103(4), 1001–1008 (2011).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (2011).
[CrossRef] [PubMed]

Zhang, Y.

Zhang, Z.

Zhao, C.

Zheng, R.

Zhou, Z.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

Zhu, S.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. B (6)

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B 103(4), 1001–1008 (2011).
[CrossRef]

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[CrossRef]

Appl. Phys. Lett. (2)

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[CrossRef]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89(25), 251114 (2006).
[CrossRef]

J. Mod. Opt. (1)

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5(3), 229–232 (2003).
[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]

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 (6)

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

New J. Phys. (2)

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. 12(7), 073012 (2010).
[CrossRef]

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Opt. Commun. (8)

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282(17), 3421–3425 (2009).
[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]

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237(1-3), 89–95 (2004).
[CrossRef]

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

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

Opt. Express (16)

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001).
[CrossRef] [PubMed]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[PubMed]

R. S. Kurti, K. Halterman, R. K. Shori, and M. J. Wardlaw, “Discrete cylindrical vector beam generation from an array of optical fibers,” Opt. Express 17(16), 13982–13988 (2009).
[CrossRef] [PubMed]

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[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).
[CrossRef] [PubMed]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
[CrossRef] [PubMed]

R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express 18(10), 10834–10838 (2010).
[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]

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010).
[CrossRef] [PubMed]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011).
[CrossRef] [PubMed]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (2011).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

Opt. Lett. (8)

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[CrossRef] [PubMed]

C. F. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel-Gaussian vector beams,” Opt. Lett. 32(24), 3543–3545 (2007).
[CrossRef] [PubMed]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
[CrossRef] [PubMed]

Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett. 29(23), 2710–2712 (2004).
[CrossRef] [PubMed]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[CrossRef] [PubMed]

K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29(8), 800–802 (2004).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

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]

Phys. Rev. A (1)

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Other (4)

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

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Figures (11)

Fig. 1
Fig. 1

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =10λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

Fig. 2
Fig. 2

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

Fig. 3
Fig. 3

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =0.5λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

Fig. 4
Fig. 4

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at z=10 z r with ρ= x 2 + y 2 and w 0 =λ for different values of the correlation coefficients σ xx , σ xy , σ yy .

Fig. 5
Fig. 5

Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at the source plane (z = 0).

Fig. 6
Fig. 6

Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w 0 =10λ and ρ= x 2 + y 2 .

Fig. 7
Fig. 7

Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with w 0 =λ and ρ= x 2 + y 2 .

Fig. 8
Fig. 8

Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w 0 =0.5λ and ρ= x 2 + y 2 .

Fig. 9
Fig. 9

Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at z= z r for different values of the correlation coefficients σ xx , σ xy , σ yy .

Fig. 10
Fig. 10

Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1,0) and (x2,0) at several propagation distances for different values of w 0 .

Fig. 11
Fig. 11

Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1, 0) and (x2, 0) at z= z r for different values of the correlation coefficients σ xx , σ xy , σ yy .

Equations (47)

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W ( x 1 , y 1 , x 2 , y 2 ,z )=( W xx ( x 1 , y 1 , x 2 , y 2 ,z ) W xy ( x 1 , y 1 , x 2 , y 2 ,z ) W xz ( x 1 , y 1 , x 2 , y 2 ,z ) W yx ( x 1 , y 1 , x 2 , y 2 ,z ) W yy ( x 1 , y 1 , x 2 , y 2 ,z ) W yz ( x 1 , y 1 , x 2 , y 2 ,z ) W zx ( x 1 , y 1 , x 2 , y 2 ,z ) W zy ( x 1 , y 1 , x 2 , y 2 ,z ) W zz ( x 1 , y 1 , x 2 , y 2 ,z ) ),
W αβ ( x 1 , y 1 , x 2 , y 2 ,z )= E α * ( x 1 , y 1 ,z ) E β ( x 2 , y 2 ,z ) ,( α,β=x,y,z )
E α (x,y,z)= 1 2π E α ( x 0 , y 0 ,0) z [ exp( ikR ) R ]d x 0 d y 0 ,(α=x,y)
E z ( x,y,z )= 1 2π { E x ( x 0 , y 0 ,0 ) x [ exp( ikR ) R ] + E y ( x 0 , y 0 ,0 ) y [ exp( ikR ) R ] }d x 0 d y 0 ,
α [ exp( ikR ) R ]= ikexp( ikR ) R 2 ( α α 0 ),(α=x,y)
z [ exp( ikR ) R ]= ikzexp( ikR ) R 2 .
W αβ ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 z 2 4 π 2 W αβ ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ×d x 10 d y 10 d x 20 d y 20 ,(α,β=x,y)
W αz ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 z 4 π 2 [ W αx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( x 2 x 20 ) + W αy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( y 2 y 20 ) ]d x 10 d y 10 d x 20 d y 20 ,(α=x,y)
W zz ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 4 π 2 [ W xx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ×( x 1 x 10 )( x 2 x 20 )+ W xy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( x 1 x 10 )( y 2 y 20 ) + W yx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( y 1 y 10 )( x 2 x 20 ) + W yy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( y 1 y 10 )( y 2 y 20 ) ]d x 10 d y 10 d x 20 d y 20 ,
W αβ ( x 10 , y 10 , x 20 , y 20 ,0 )= E α * ( x 10 , y 10 ,0 ) E β ( x 20 , y 20 ,0 ) . ( α,β=x,y )
E ( r,ϕ,0 )=exp( r 2 w 0 2 ) ( 2 r 2 w 0 2 ) ( n±1 )/2 L p n±1 ( 2 r 2 w 0 2 ){ cos( nϕ ) e ϕ sin( nϕ ) e r ±sin( nϕ ) e ϕ +cos( nϕ ) e r },
E ( x 0 , y 0 ,0 )={ E x ( x 0 , y 0 ,0 ) e x + E y ( x 0 , y 0 ,0 ) e y E y ( x 0 , y 0 ,0 ) e x + E x ( x 0 , y 0 ,0 ) e y } =exp( x 0 2 + y 0 2 w 0 2 ) (1) p 2 2p+n±1 p! { 1 2i m=0 p s=0 n±1 i s [ 1 (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e x 1 2 m=0 p s=0 n±1 i s [ 1+ (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e x + 1 2 m=0 p s=0 n±1 i s [ 1+ (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e y + 1 2i m=0 p s=0 n±1 i s [ 1 (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e y },
W ( x 10 , y 10 , x 20 , y 20 ,0 )=( W xx ( x 10 , y 10 , x 20 , y 20 ,0 ) W xy ( x 10 , y 10 , x 20 , y 20 ,0 )0 W yx ( x 10 , y 10 , x 20 , y 20 ,0 ) W yy ( x 10 , y 10 , x 20 , y 20 ,0 )0 000 ),
W αβ ( x 10 , y 10 , x 20 , y 20 ,0 )= A αβ 1 4 exp[ x 10 2 + x 20 2 + y 10 2 + y 20 2 w 0 2 ( x 10 x 20 ) 2 + ( y 10 y 20 ) 2 2 σ αβ 2 ] × 1 2 4p+2n±2 (p!) 2 m=0 p s=0 n±1 l=0 p h=0 n±1 i s (i) h [ 1+ C α (1) s ][ 1+ C β (1) h ]( p m )( p l )( n±1 s )( n±1 h ) × H 2m+n±1s ( 2 x 10 w 0 ) H 2l+n±1h ( 2 x 20 w 0 ) H 2p2m+s ( 2 y 10 w 0 ) H 2p2l+h ( 2 y 20 w 0 ),(α,β=x,y)
W yx ( x 10 , y 10 , x 20 , y 20 ,0 )= [ W xy ( x 20 , y 20 , x 10 , y 10 ,0 ) ] * ,
A αβ ={ 1α=β=x, i B xy α=x,β=y 1α=β=y, , C α ={ 1α=x, 1α=y,
W 1xx ( x 10 , y 10 , x 20 , y 20 ,0 )= W yy ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1yy ( x 10 , y 10 , x 20 , y 20 ,0 )= W xx ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1xy ( x 10 , y 10 , x 20 , y 20 ,0 )= W yx ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1xy ( x 10 , y 10 , x 20 , y 20 ,0 )= W xy ( x 10 , y 10 , x 20 , y 20 ,0 ),
R i r i + x i0 2 + y i0 2 2 x i x i0 2 y i y i0 2 r i ,(i=1,2)
W αβ ( x 1 , y 1 , x 2 , y 2 ,z )= A αβ k 2 z 2 exp[ ik( r 1 r 2 ) ] 16 r 1 2 r 2 2 M 1αβ 1 2 5( 2p+n±1 )/2 (p!) 2 ( 1 2 M 1αβ w 0 2 ) ( n±1+2p )/2 ×exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1αβ r 1 2 k 2 4 M 2αβ [ ( x 2 r 2 x 1 2 M 1αβ r 1 σ αβ 2 ) 2 + ( y 2 r 2 y 1 2 M 1αβ r 1 σ αβ 2 ) 2 ] } × m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 2m+n±1s d 1 =0 c 1 /2 e 1 =0 (2l+n±1h)/2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 ( 2p2m+s c 2 )( 2m+n±1s c 1 ) ×( p m )( p l )( n±1 s )( n±1 h )[ 1+ C α (1) s ][ 1+ C β (1) h ] (1) d 1 + d 2 + e 1 + e 2 i s (i) h (2i) ( c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 ) × ( 2 2 w 0 ) 2p+n±12 e 2 2 e 1 c 1 ! d 1 !( c 1 2 d 1 )! c 2 ! d 2 !( c 2 2 d 2 )! (2l+n±1h)! e 1 !(2l+n±1h2 e 1 )! (2p2l+h)! e 2 !(2p2l+h2 e 2 )! × 1 ( M 2αβ ) c 1 + c 2 +n±1+2p2 d 1 2 d 2 2 e 1 2 e 2 +2 [ 2 σ αβ 2 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ] c 1 + c 2 2 d 1 2 d 2 × H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2αβ ( x 1 2 M 1αβ r 1 σ αβ 2 x 2 r 2 ) ] H 2m+n±1s c 1 [ ik x 1 r 1 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ] × H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2αβ ( y 1 2 M 1αβ r 1 σ αβ 2 y 2 r 2 ) ] H 2p2m+s c 2 [ ik y 1 r 1 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ],(α,β=x,y)
W yx ( x 1 , y 1 , x 2 , y 2 ,z)= W xy * ( x 2 , y 2 , x 1 , y 1 ,z),
W xz ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 2m+n±1s d 1 =0 c 1 /2 e 1 =0 ( 2l+n±1h ) /2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 Q 1 ×[ 1 (1) s ]{ Q 3xx Q 4xx [ 1 (1) h ] H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2xx ( y 1 2 M 1xx r 1 σ xx 2 y 2 r 2 ) ] Q 2 Q 5xy [ 1+ (1) h ] H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 x 2 r 2 ) ] },
W yz ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 2m+n±1s d 1 =0 c 1 /2 e 1 =0 ( 2l+n±1h ) /2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 Q 1 [ 1+ (1) s ]{ Q 2 Q 4xy [ 1 (1) h ] H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2xy ( y 1 2 M 1xy r 1 σ xy 2 y 2 r 2 ) ] + Q 3yy Q 5yy [ 1+ (1) h ] H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2yy ( x 1 2 M 1yy r 1 σ yy 2 x 2 r 2 ) ] },
W zx ( x 1 , y 1 , x 2 , y 2 ,z)= W xz * ( x 2 , y 2 , x 1 , y 1 ,z),
W zy ( x 1 , y 1 , x 2 , y 2 ,z)= W yz * ( x 2 , y 2 , x 1 , y 1 ,z),
W zz ( x 1 , y 1 , x 2 , y 2 ,z)= W zzxx ( x 1 , y 1 , x 2 , y 2 ,z)+ W zzxy ( x 1 , y 1 , x 2 , y 2 ,z) + W zzyx ( x 1 , y 1 , x 2 , y 2 ,z)+ W zzyy ( x 1 , y 1 , x 2 , y 2 ,z),
W zzxx ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 ( 2m+n±1s )/2 d 1 =0 ( 2l+n±1h )/2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 Q 6xx Q 7xx × H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2xx ( y 1 2 M 1xx r 1 σ xx 2 y 2 r 2 ) ]{ x 1 f 1 =0 2m+n±1s2 c 1 g 1 =0 f 1 /2 Q 8xx ( 2m+n±1s2 c 1 f 1 ) × H 2m+n±1s2 c 1 f 1 ( k x 1 2 M 1xx r 1 ) [ 2i M 2xx x 2 H f 1 2 g 1 +2l+n±1h2 d 1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) H f 1 2 g 1 +2l+n±1h2 d 1 +1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) ] f 2 =0 2m+n±1s2 c 1 +1 g 2 =0 f 2 /2 Q 9xx ( 2m+n±1s2 c 1 +1 f 2 ) × H 2m+n±1s2 c 1 +1 f 2 ( k x 1 2 M 1xx r 1 ) [ 2i M 2xx x 2 H f 2 2 g 2 +2l+n±1h2 d 1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) H f 2 2 g 2 +2l+n±1h2 d 1 +1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) ] },
W zzyy ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 2 =0 2m+n±1s d 2 =0 c 2 /2 e 2 =0 ( 2l+n±1h ) /2 c 1 =0 ( 2p2m+s )/2 d 1 =0 ( 2p2l+h )/2 Q 6yy ( 2m+n±1s c 2 ) × ( 2i ) ( c 2 2 d 2 +n±12 e 2 +4p2m+s2 c 1 2 d 1 +2) ( 1 2 w 0 2 M 1yy ) ( 2m+n±1s ) /2 ( 2 2 w 0 ) n±12 e 2 +4p2m+s2 c 1 2 d 1 × ( 2l+n±1h )! e 2 !(2l+n±1h2 e 2 )! ( 2p2m+s )! c 1 !( 2p2m+s2 c 1 )! ( 2p2l+h )! d 1 !( 2p2l+h2 d 1 )! 1 ( M 1yy ) 2p2m+s2 c 1 +3 × H c 2 2 d 2 +2l+n±1h2 e 2 [ k 2 M 2yy ( x 1 2 M 1yy r 1 σ yy 2 x 2 r 2 ) ] H 2m+n±1s c 2 [ ik x 1 ( w 0 2 M 1yy 2 r 1 2 2 M 1yy r 1 2 ) 1/2 ] ×{ y 1 f 1 =0 2p2m+s2 c 1 g 1 =0 f 1 /2 Q 8yy ( 2p2m+s2 c 1 f 1 ) H 2p2m+s2 c 1 f 1 ( k y 1 2 M 1yy r 1 ) [ 2i M 2yy y 2 H 2p2l+h2 d 2 + f 1 2 g 1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) H 2p2l+h2 d 1 + f 1 2 g 1 +1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) ] f 2 =0 2p2m+s2 c 1 +1 g 2 =0 f 2 /2 Q 9yy ( 2p2m+s2 c 1 +1 f 2 ) H 2p2m+s2 c 1 +1 f 2 ( k y 1 2 M 1yy r 1 ) [ 2i M 2yy y 2 H 2p2l+h2 d 1 + f 2 2 g 2 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) H 2p2l+h2 d 1 + f 2 2 g 2 +1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) ] }.
W zzyx ( x 1 , y 1 , x 2 , y 2 ,z)= [ W zzxy ( x 2 , y 2 , x 1 , y 1 ,z) ] * ,
W zzxy ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 ( 2m+n±1s )/2 d 1 =0 ( 2l+n±1h )/2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 ( 2p2l+h ) /2 Q 5xy Q 6xy Q 7xy ×{ x 1 f 1 =0 2m+n±1s2 c 1 g 1 =0 f 1 /2 Q 8xy ( 2m+n±1s2 c 1 f 1 ) H 2m+n±1s2 c 1 f 1 ( k x 1 2 M 1xy r 1 ) H f 1 2 g 1 +2l+n±1h2 d 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 x 2 r 2 ) ] f 2 =0 2m+n±1s2 c 1 +1 g 2 =0 f 2 /2 Q 9xy ( 2m+n±1s2 c 1 +1 f 2 ) × H 2m+n±1s2 c 1 +1 f 2 ( k x 1 2 M 1xy r 1 ) H f 2 2 g 2 +2l+n±1h2 d 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 x 2 r 2 ) ] },
M 1αβ =1/ w 0 2 +1/(2 σ αβ 2 )ik/(2 r 1 ), (α,β=x,y)
M 2αβ =1/ w 0 2 +1/(2 σ αβ 2 )+ik/(2 r 2 )1/(4 M 1αβ σ αβ 4 ), (α,β=x,y)
Q 1 = k 2 zexp[ ik( r 1 r 2 ) ] 16 r 1 2 r 2 2 2 5( 2p+n±1 ) /2 (p!) 2 (1) d 1 + d 2 + e 1 + e 2 i s (i) h ( 2i ) ( c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 +1) × ( 2 2 w 0 ) n±1+2p2 e 1 2 e 2 ( 2m+n±1s c 1 )( 2p2m+s c 2 )( p m )( p l )( n±1 s )( n±1 h ) × c 1 ! d 1 !( c 1 2 d 1 )! c 2 ! d 2 !( c 2 2 d 2 )! ( 2l+n±1h )! e 1 !(2l+n±1h2 e 1 )! (2p2l+h)! e 2 !(2p2l+h2 e 2 )! ,
Q 2 = i B xy M 1xy 1 ( M 2xy ) c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 +3 [ 2 σ xy 2 ( w 0 2 M 1xy 2 2 M 1xy ) 1/2 ] c 1 + c 2 2 d 1 2 d 2 × ( 1 2 M 1xy w 0 2 ) ( n±1+2p )/2 exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1xy r 1 2 k 2 4 M 2xy [ ( x 2 r 2 x 1 2 M 1xy r 1 σ xy 2 ) 2 + ( y 2 r 2 y 1 2 M 1xy r 1 σ xy 2 ) 2 ] } × H 2m+n±1s c 1 [ ik x 1 ( w 0 2 M 1xy 2 r 1 2 2 M 1xy r 1 2 ) 1/2 ] H 2p2m+s c 2 ( ik y 1 r 1 ( w 0 2 M 1xy 2 2 M 1xy ) 1/2 ),
Q 3αα = 1 M 1αα 1 ( M 2αα ) c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 +3 [ 2 σ αα 2 ( w 0 2 M 1αα 2 2 M 1αα ) 1/2 ] c 1 + c 2 2 d 1 2 d 2 × ( w 0 2 M 1αα 2 w 0 2 M 1αα ) ( 2p+n±1 ) /2 exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1αα r 1 2 k 2 4 M 2αα [ ( x 2 r 2 x 1 2 M 1αα r 1 σ αα 2 ) 2 + ( y 2 r 2 y 1 2 M 1αα r 1 σ αα 2 ) 2 ] } × H 2m+n±1s c 1 [ ik x 1 r 1 ( w 0 2 M 1αα 2 2 M 1αα ) 1/2 ] H 2p2m+s c 2 [ ik y 1 r 1 ( w 0 2 M 1αα 2 2 M 1αα ) 1/2 ],(α=x,y)
Q 4xα =2i M 2xα x 2 H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2xα ( x 1 2 M 1xα r 1 σ xα 2 x 2 r 2 ) ] H c 1 2 d 1 +2l+n±1h2 e 1 +1 [ k 2 M 2xα ( x 1 2 M 1xα r 1 σ xα 2 x 2 r 2 ) ],(α=x,y)
Q 5αy =2i M 2αy y 2 H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2αy ( y 1 2 M 1αy r 1 σ αy 2 y 2 r 2 ) ] H c 2 2 d 2 +2p2l+h2 e 2 +1 [ k 2 M 2αy ( y 1 2 M 1αy r 1 σ αy 2 y 2 r 2 ) ],(α=x,y)
Q 6αβ = A αβ exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1αβ r 1 2 k 2 4 M 2αβ [ ( x 2 r 2 x 1 2 M 1αβ r 1 σ αβ 2 ) 2 + ( y 2 r 2 y 1 2 M 1αβ r 1 σ αβ 2 ) 2 ] } × k 2 exp[ ik( r 1 r 2 ) ] 16 r 1 2 r 2 2 2 5( 2p+n±1 )/2 (p!) 2 ( p m )( p l )( n±1 s )( n±1 h ) i s (i) h (1) c 1 + d 1 + d 2 + e 2 [ 1+ C α (1) s ] 1+ C β (1) h 2 ( 2 c 1 +1 )/2 × 1 ( M 2αβ ) n±12 d 1 + c 2 2 d 2 +2p2 e 2 +3 [ 2 σ αβ 2 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ] c 2 2 d 2 c 2 ! d 2 !( c 2 2 d 2 )! ,(α,β=x,y)
Q 7xα =( 2p2m+s c 2 ) ( 2i ) ( 2m+2( n±1 )s2 c 1 2 d 1 + c 2 2 d 2 +2p2 e 2 +2 ) ( 2 2 w 0 ) 2m+2( n±1 )s2 c 1 2 d 1 +2p2 e 2 × ( 2m+n±1s )! c 1 !( 2m+n±1s2 c 1 )! ( 2l+n±1h )! d 1 !( 2l+n±1h2 d 1 )! (2p2l+h)! e 2 !(2p2l+h2 e 2 )! ( 1 2 M 1xα w 0 2 ) ( 2p2m+s )/2 × 1 ( M 1xα ) 2m+n±1s2 c 1 +3 H 2p2m+s c 2 [ ik y 1 ( w 0 2 M 1xα 2 r 1 2 2 M 1xα r 1 2 ) 1/2 ],(α=x,y)
Q 8αβ = 2 M 1αβ (1) g 1 (2i) ( f 1 2 g 1 1) f 1 ! g 1 !( f 1 2 g 1 )! 1 ( M 2αβ ) f 1 2 g 1 ( 2 i M 1αβ σ αβ 2 ) f 1 2 g 1 ,(α,β=x,y)
Q 9αβ = (1) g 2 (2i) ( f 2 2 g 2 ) f 2 ! g 2 !( f 2 2 g 2 )! 1 ( M 2αβ ) f 2 2 g 2 ( 2 i M 1αβ σ αβ 2 ) f 2 2 g 2 .(α,β=x,y)
exp[ ( xy ) 2 ] H n ( ax )dx= π ( 1 a 2 ) n/2 H n ( ay ( 1 a 2 ) 1/2 ),
x n exp[ ( xβ ) 2 ] dx= ( 2i ) n π H n ( iβ ),
H n ( x+y )= 1 2 n/2 k=0 n ( n k ) H k ( 2 x ) H nk ( 2 y ),
H n ( x )= k=0 n/2 ( 1 ) k n! k!( n2k )! ( 2x ) n2k .
I(x,y,z)= I x (x,y,z)+ I y (x,y,z)+ I z (x,y,z) = W xx (x,y,x,y,z)+ W yy (x,y,x,y,z)+ W zz (x,y,x,y,z),
P(x,y,z)= p 1 (x,y,z) p 2 (x,y,z) p 1 (x,y,z)+ p 2 (x,y,z)+ p 3 (x,y,z) ,
μ( x 1 x 2 , y 1 y 2 ,z)= Tr W ( x 1 , y 1 , x 2 , y 2 ,z) Tr W ( x 1 , y 1 , x 1 , y 1 ,z) Tr W ( x 2 , y 2 , x 2 , y 2 ,z) ,

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