## Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space |

Optics Express, Vol. 20, Issue 14, pp. 15908-15927 (2012)

http://dx.doi.org/10.1364/OE.20.015908

Acrobat PDF (6719 KB)

### 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

## 1. Introduction

1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**(1), 1–57 (2009). [CrossRef]

1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**(1), 1–57 (2009). [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]

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]

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]

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

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

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

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

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]

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]

## 2. Nonparaxial propagation theory of a cylindrical vector partially coherent beam

49. K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A **21**(10), 1924–1932 (2004). [CrossRef] [PubMed]

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]

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]

*x*,

*y*and z directions, respectively .

49. K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A **21**(10), 1924–1932 (2004). [CrossRef] [PubMed]

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]

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]

*m*,

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]

*x-*and

*y-*components of the field and is removable in most case,

*x*component of the field, of the

*y*component of the field and of the mutual correlation function of

*x*and

*y*field components, respectively.

**19**(7), 5979–5992 (2011). [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]

**19**(7), 5979–5992 (2011). [CrossRef] [PubMed]

## 3. Statistical properties of a nonparaxial cylindrical vector partially coherent field

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]

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]

*y*=

*x*) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with

*y*=

*x*) calculated by the paraxial propagation formulae (Eqs. (15) and (18) of Ref [41

**19**(7), 5979–5992 (2011). [CrossRef] [PubMed]

*y*=

*x*) of a nonparaxial cylindrical vector partially coherent LG field at

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]

*y*= 0) of a nonparaxial cylindrical vector partially coherent LG field at

*x*

_{1}, 0) and (

*x*

_{2}, 0) at several propagation distances for different values of

*x*

_{1}, 0) and (

*x*

_{2}, 0) at

## 4. Summary

## Acknowledgments

## References and links

1. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

2. | Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express |

3. | Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. |

4. | A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A |

5. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

6. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

7. | D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express |

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 |

9. | W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express |

10. | D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B |

11. | D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. |

12. | B. J. Roxworthy and K. C. Toussaint Jr., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. |

13. | Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express |

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. |

15. | C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. |

16. | R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express |

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. |

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 |

19. | G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. |

20. | C. F. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel-Gaussian vector beams,” Opt. Lett. |

21. | E. Wolf, |

22. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

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. |

24. | O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. |

25. | F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A |

26. | T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. |

27. | O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. |

28. | S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express |

29. | O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. |

30. | Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express |

31. | O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B |

32. | S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B |

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. |

34. | G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express |

35. | S. Zhu and Y. Cai, “M |

36. | Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. |

37. | C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express |

38. | L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. |

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. |

40. | Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A |

41. | Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express |

42. | F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. |

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 |

44. | H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. |

45. | H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. |

46. | R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A |

47. | K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. |

48. | Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett. |

49. | K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A |

50. | X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B |

51. | L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B |

52. | L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express |

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. |

54. | S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A |

55. | F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express |

56. | J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. |

57. | R. K. Luneburg, |

58. | A. Erdelyi, W. Magnus, and F. Oberhettinger, |

59. | M. Abramowitz and I. Stegun, |

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. |

61. | O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A |

62. | Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. |

63. | Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(140.3300) Lasers and laser optics : Laser beam shaping

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: April 24, 2012

Revised Manuscript: June 12, 2012

Manuscript Accepted: June 14, 2012

Published: June 28, 2012

**Citation**

Yiming Dong, Fanlong Feng, Yahong Chen, Chengliang Zhao, and Yangjian Cai, "Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space," Opt. Express **20**, 15908-15927 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15908

Sort: Year | Journal | Reset

### References

- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
- Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002). [PubMed]
- 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]
- A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A15(10), 2705–2711 (1998). [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]
- K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
- D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001). [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. Express16(11), 7665–7673 (2008). [CrossRef] [PubMed]
- W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express17(20), 17829–17836 (2009). [CrossRef] [PubMed]
- D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B23(6), 1228–1234 (2006). [CrossRef]
- 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]
- B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys.12(7), 073012 (2010). [CrossRef]
- Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express18(10), 10828–10833 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express18(10), 10834–10838 (2010). [CrossRef] [PubMed]
- 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]
- R. S. Kurti, K. Halterman, R. K. Shori, and M. J. Wardlaw, “Discrete cylindrical vector beam generation from an array of optical fibers,” Opt. Express17(16), 13982–13988 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5-6), 263–267 (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]
- 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]
- F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A25(5), 1016–1021 (2008). [CrossRef] [PubMed]
- T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.7(5), 232–237 (2005). [CrossRef]
- 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]
- S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express18(12), 12587–12598 (2010). [CrossRef] [PubMed]
- O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun.281(9), 2342–2348 (2008). [CrossRef]
- Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express16(20), 15834–15846 (2008). [CrossRef] [PubMed]
- O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B94(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. B99(1-2), 317–323 (2010). [CrossRef]
- 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]
- G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express19(9), 8700–8714 (2011). [CrossRef] [PubMed]
- S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express18(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]
- C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express17(24), 21472–21487 (2009). [CrossRef] [PubMed]
- 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]
- Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010). [CrossRef]
- Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express19(7), 5979–5992 (2011). [CrossRef] [PubMed]
- 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]
- H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B101(1-2), 361–369 (2010). [CrossRef]
- 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]
- H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998). [CrossRef]
- R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A21(10), 2029–2037 (2004). [CrossRef] [PubMed]
- K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett.29(8), 800–802 (2004). [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]
- K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A21(10), 1924–1932 (2004). [CrossRef] [PubMed]
- X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(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. B103(4), 1001–1008 (2011). [CrossRef]
- L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express19(14), 13312–13325 (2011). [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]
- S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A18(1), 150–156 (2001). [CrossRef] [PubMed]
- F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009). [CrossRef] [PubMed]
- 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]
- 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).
- 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]
- O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A21(12), 2382–2385 (2004). [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. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B105(2), 405–414 (2011). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.