Abstract

Cylindrical vector partially coherent beam is introduced as a natural extension of cylindrical vector coherent beam based on the unified theory of coherence and polarization. Analytical propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam passing through paraxial ABCD optical system is derived based on the generalized Collins integral formula. As an application example, the statistics properties, such as the average intensity, spreading and the degree of polarization, of a cylindrical vector partially coherent beam propagating in free space are studied in detail. It is found that the statistics properties of a cylindrical vector partially coherent beam are much different from a cylindrical vector coherent beam. Our results may find applications in connection with laser beam shaping and optical trapping.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009).
    [CrossRef]
  2. 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]
  3. B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” N. J. Phys. 12(7), 073012 (2010).
    [CrossRef]
  4. 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]
  5. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
    [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
    [CrossRef]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007).
    [CrossRef] [PubMed]
  14. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998).
    [CrossRef]
  15. 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]
  16. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001).
    [CrossRef] [PubMed]
  17. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [PubMed]
  18. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
    [CrossRef]
  19. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [CrossRef] [PubMed]
  20. W. Cheng, J. W. Haus, and Q. W. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
    [CrossRef] [PubMed]
  21. A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
    [CrossRef]
  22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  23. C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).
  24. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [CrossRef]
  25. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  26. 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]
  27. 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]
  28. 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]
  29. 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]
  30. 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]
  31. 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]
  32. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
  33. 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]
  34. 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]
  35. W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270(2), 474–478 (2007).
    [CrossRef]
  36. 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]
  37. 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]
  38. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
    [CrossRef]
  39. Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
    [CrossRef]
  40. S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B , doi:.
  41. S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010).
    [CrossRef]
  42. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
    [CrossRef]
  43. 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]
  44. 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]
  45. P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
    [CrossRef]
  46. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
    [CrossRef]
  47. Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007).
    [CrossRef]
  48. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  49. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
  50. 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]
  51. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2003), Vol. 44, pp.119–204.

2011

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]

P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
[CrossRef]

2010

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

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

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

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]

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (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]

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]

2009

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[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]

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

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[CrossRef] [PubMed]

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

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]

2008

2007

2006

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]

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]

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

2005

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]

2004

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]

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]

2003

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]

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

2002

2001

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]

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

2000

1998

Baykal, Y.

Bernet, S.

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

Biss, D. P.

Borghi, R.

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]

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]

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.

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]

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

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

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]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[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]

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

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 and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007).
[CrossRef]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef]

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B , doi:.

Chen, C.

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]

Chen, W.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

Cheng, W.

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]

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007).
[CrossRef] [PubMed]

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]

Furhapter, S.

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

Gao, W.

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270(2), 474–478 (2007).
[CrossRef]

Gbur, G.

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]

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.

Gu, Y.

Guo, C. S.

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,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Korotkova, O.

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]

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]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (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]

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[CrossRef] [PubMed]

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

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]

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]

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

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270(2), 474–478 (2007).
[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]

Kozawa, Y.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[CrossRef]

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]

Kurti, R. S.

Leger, J. R.

Li, C. F.

Lin, Q.

Liu, P.

P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
[CrossRef]

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]

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. 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]

Ni, W. J.

Ohtsu, A.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[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]

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]

Ramírez-Sánchez, V.

Ritsch-Marte, M.

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

Rong, J.

P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
[CrossRef]

Roxworthy, B. J.

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

Salem, M.

Santarsiero, M.

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]

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]

Sato, S.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[CrossRef]

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]

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]

Tong, Z.

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

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[CrossRef]

Toussaint, K. C.

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” N. 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.

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

Wang, G.

P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
[CrossRef]

Wang, H.

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, H. T.

Wang, J.

P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
[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]

Wang, X. L.

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.

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]

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

Xu, L.

Yang, H.

P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
[CrossRef]

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.

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

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

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]

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

Zhan, Q. W.

Zhang, L.

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]

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B , doi:.

Zhao, C.

Zheng, R.

Zhou, F.

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B , doi:.

Zhu, S.

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

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]

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B , doi:.

Adv. Opt. Photon.

Appl. Opt.

Appl. Phys. B

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (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]

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]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[CrossRef]

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B , doi:.

Appl. Phys. Lett.

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. Opt. A, Pure Appl. Opt.

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]

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

F. Gori, 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]

J. Opt. Soc. Am. A

N. J. Phys.

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” N. 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,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Opt. Commun.

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]

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]

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[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, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270(2), 474–478 (2007).
[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]

P. Liu, H. Yang, J. Rong, G. Wang, and J. Wang, “Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective,” Opt. Commun. 284(4), 909–914 (2011).
[CrossRef]

Opt. Express

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]

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

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]

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]

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]

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

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

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. 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]

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]

Opt. Lett.

Phys. Lett. A

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

Other

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

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

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2003), Vol. 44, pp.119–204.

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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Normalized intensity distribution (cross line v = 0 ) of a cylindrical vector partially coherent LG beam for different values of the correlation coefficients σ x x ,   σ y y at several propagation distances in free space (a) z = 0 , (b) z = 3.2 m , (c) z = 6.5 m , (d) z = 20 m .

Fig. 2
Fig. 2

Effective beam size of a cylindrical vector partially coherent LG beam versus the propagation distance z for different values of the correlation coefficients σ x x ,   σ y y and the mode orders p ,   n ± 1 in free space.

Fig. 3
Fig. 3

Degree of polarization (cross line, v = 0 ) of a cylindrical vector partially coherent LG beam for different values of the initial correlation coefficients σ x x σ x y σ y y at several propagation distances in free space. In Fig. 3(a)3(d), σ x x σ x y σ y y = 1 m m , in Fig. 3(e)3(h), σ x x σ x y σ y y = 0.5 m m .

Fig. 4
Fig. 4

Degree of polarization (cross line, v = 0) of a partially coherent cylindrically polarized LG beams at z = 6.5 m in free space for different mode orders of p and n ± 1 with λ = 632.8 n m , w 0 = 2 m m , B x y 1 , σ x x σ x y σ y y = 1 m m .

Equations (29)

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

E ( r , ϕ , z ) = 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 ( n ϕ ) e ϕ sin ( n ϕ ) e r ± sin ( n ϕ ) e ϕ + cos ( n ϕ ) e r } ,
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 ] { sin [ ( n ± 1 ) ϕ ] e x + cos [ ( n ± 1 ) ϕ ] e y cos [ ( n ± 1 ) ϕ ] e x + sin [ ( n ± 1 ) ϕ ] e y } = 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 ] { exp [ i ( n ± 1 ) ϕ ] exp [ i ( n ± 1 ) ϕ ] 2 i e x + exp [ i ( n ± 1 ) ϕ ] + exp [ i ( n ± 1 ) ϕ ] 2 e y exp [ i ( n ± 1 ) ϕ ] + exp [ i ( n ± 1 ) ϕ ] 2 e x + exp [ i ( n ± 1 ) ϕ ] exp [ i ( n ± 1 ) ϕ ] 2 i e y } ,
[ 2 r 2 w 0 2 ] ( n ± 1 ) / 2 L p n ± 1 [ 2 r 2 w 0 2 ] exp [ i ( n ± 1 ) ϕ ] = ( 1 ) p 2 2 p + n ± 1 p ! m = 0 p s = 0 n ± 1 i s ( p m ) ( n ± 1 s ) H 2 m + n ± 1 s ( 2 x w 0 ) H 2 p 2 m + s ( 2 y w 0 ) ,
E ( x , y , 0 ) = { E x ( x , y , 0 ) e x + E y ( x , y , 0 ) e y E y ( x , y , 0 ) e x + E x ( x , y , 0 ) e y }                 = exp ( x 2 + y 2 w 0 2 ) ( 1 ) p 2 2 p + n ± 1 p ! { 1 2 i m = 0 p s = 0 n ± 1 i s [ 1 ( 1 ) s ] ( p m ) ( n ± 1 s ) H 2 m + n ± 1 s ( 2 x w 0 ) H 2 p 2 m + s ( 2 y w 0 ) e x 1 2 m = 0 p s = 0 n ± 1 i s [ 1 + ( 1 ) s ] ( p m ) ( n ± 1 s ) H 2 m + n ± 1 s ( 2 x w 0 ) H 2 p 2 m + s ( 2 y w 0 ) e x                      + 1 2 m = 0 p s = 0 n ± 1 i s [ 1 + ( 1 ) s ] ( p m ) ( n ± 1 s ) H 2 m + n ± 1 s ( 2 x w 0 ) H 2 p 2 m + s ( 2 y w 0 ) e y + 1 2 i m = 0 p s = 0 n ± 1 i s [ 1 ( 1 ) s ] ( p m ) ( n ± 1 s ) H 2 m + n ± 1 s ( 2 x w 0 ) H 2 p 2 m + s ( 2 y w 0 ) e y } .
W ^ ( x 1 , y 1 , x 2 , y 2 , 0 ) = ( W x x ( x 1 , y 1 , x 2 , y 2 , 0 )             W x y ( x 1 , y 1 , x 2 , y 2 , 0 ) W y x ( x 1 , y 1 , x 2 , y 2 , 0 )             W y y ( x 1 , y 1 , x 2 , y 2 , 0 ) ) ,
W α β ( x 1 , y 1 , x 2 , y 2 , 0 ) = E α ( x 1 , y 1 , 0 ) E β * ( x 2 , y 2 , 0 ) ,    ( α = x , y ; β = x , y ) ,
W α β ( x 1 , y 1 , x 2 , y 2 , 0 ) = E α ( x 1 , y 1 , 0 ) E β ( x 2 , y 2 , 0 ) μ α β ( x 1 x 2 , y 1 y 2 , 0 ) ( α = x , y ; β = x , y ) ,
μ α β ( x 1 x 2 , y 1 y 2 , 0 ) = B α β exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ α β 2 ] , ( α = x , y ; β = x , y ) ,
W x x ( x 1 , y 1 , x 2 , y 2 , 0 ) = 1 4 exp [ x 1 2 + x 2 2 + y 1 2 + y 2 2 w 0 2 ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ x x 2 ] × 1 2 4 p + 2 n ± 2 ( p ! ) 2 m = 0 p s = 0 n ± 1 l = 0 p h = 0 n ± 1 i s ( i ) h [ 1 ( 1 ) s ] [ 1 ( 1 ) h ] ( p m ) ( p l ) ( n ± 1 s ) ( n ± 1 h ) H 2 m + n ± 1 s ( 2 x 1 w 0 ) × H 2 l + n ± 1 h ( 2 x 2 w 0 ) H 2 p 2 m + s ( 2 y 1 w 0 ) H 2 p 2 l + h ( 2 y 2 w 0 ) ,
W x y ( x 1 , y 1 , x 2 , y 2 , 0 ) = B x y 4 i exp [ x 1 2 + x 2 2 + y 1 2 + y 2 2 w 0 2 ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ x y 2 ] × 1 2 4 p + 2 n ± 2 ( p ! ) 2 m = 0 p s = 0 n ± 1 l = 0 p h = 0 n ± 1 i s ( i ) h [ 1 ( 1 ) s ] [ 1 + ( 1 ) h ] ( p m ) ( p l ) ( n ± 1 s ) ( n ± 1 h ) H 2 m + n ± 1 s ( 2 x 1 w 0 ) × H 2 l + n ± 1 h ( 2 x 2 w 0 ) H 2 p 2 m + s ( 2 y 1 w 0 ) H 2 p 2 l + h ( 2 y 2 w 0 ) ,
W y x ( x 1 , y 1 , x 2 , y 2 , 0 ) = [ W x y ( x 1 , y 1 , x 2 , y 2 , 0 ) ] * ,
W y y ( x 1 , y 1 , x 2 , y 2 , 0 ) = 1 4 exp [ x 1 2 + x 2 2 + y 1 2 + y 2 2 w 0 2 ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ y y 2 ] × 1 2 4 p + 2 n ± 2 ( p ! ) 2 m = 0 p s = 0 n ± 1 l = 0 p h = 0 n ± 1 i s ( i ) h [ 1 + ( 1 ) s ] [ 1 + ( 1 ) h ] ( p m ) ( p l ) ( n ± 1 s ) ( n ± 1 h ) H 2 m + n ± 1 s ( 2 x 1 w 0 ) × H 2 l + n ± 1 h ( 2 x 2 w 0 ) H 2 p 2 m + s ( 2 y 1 w 0 ) H 2 p 2 l + h ( 2 y 2 w 0 ) .
W 1 x x ( x 1 , y 1 , x 2 , y 2 , 0 ) = W y y ( x 1 , y 1 , x 2 , y 2 , 0 ) ,   W 1 y y ( x 1 , y 1 , x 2 , y 2 , 0 ) = W x x ( x 1 , y 1 , x 2 , y 2 , 0 ) , W 1 x y ( x 1 , y 1 , x 2 , y 2 , 0 ) = W y x ( x 1 , y 1 , x 2 , y 2 , 0 ) ,   W 1 x y ( x 1 , y 1 , x 2 , y 2 , 0 ) = W x y ( x 1 , y 1 , x 2 , y 2 , 0 ) .
W α β ( u 1 , v 1 , u 2 , v 2 , z ) = ( 1 λ | B | ) 2 W α β ( x 1 , y 1 , x 2 , y 2 , 0 )                                  × exp [ i k 2 B ( A x 1 2 + A y 1 2 2 x 1 u 1 2 y 1 v 1 + D u 1 2 + D v 1 2 ) ]                                  × exp [ i k 2 B * ( A * x 2 2 + A * y 2 2 2 x 2 u 2 2 y 2 v 2 + D * u 2 2 + D * v 2 2 ) ] d x 1 d x 2 d y 1 d y 2 ,
W x x ( u 1 , v 1 , u 2 , v 2 , z ) = 1 4 ( 1 λ | B | ) 2 π 2 M 1 x x 1 2 5 ( n ± 1 + 2 p ) / 2 ( p ! ) 2 ( 1 2 M 1 x x w 0 2 ) ( n ± 1 + 2 p ) / 2 × exp [ i k D 2 B ( u 1 2 + v 1 2 ) ] exp [ i k D * 2 B * ( u 2 2 + v 2 2 ) ] exp [ k 2 4 M 1 x x B 2 ( u 1 2 + v 1 2 ) ] × exp { k 2 4 M 2 x x ( u 2 B * u 1 2 M 1 x x B σ x x 2 ) 2 } exp { k 2 4 M 2 x x ( v 2 B * v 1 2 M 1 x x B σ x x 2 ) 2 } × m = 0 p s = 0 n ± 1 l = 0 p h = 0 n ± 1 c 1 = 0 2 m + n ± 1 s d 1 = 0 c 1 / 2 e 1 = 0 ( 2 l + n ± 1 h ) / 2 c 2 = 0 2 p 2 m + s d 2 = 0 c 2 / 2 e 2 = 0 ( 2 p 2 l + h ) / 2 ( 2 p 2 m + s c 2 ) ( 2 m + n ± 1 s c 1 ) × ( p m ) ( p l ) ( n ± 1 s ) ( n ± 1 h ) [ 1 ( 1 ) s ] [ 1 ( 1 ) h ] ( 1 ) d 1 + d 2 + e 1 + e 2 i s ( i ) h ( 2 i ) ( c 1 + c 2 2 d 1 2 d 2 + n ± 1 + 2 p 2 e 1 2 e 2 )                                                       × ( 2 2 w 0 ) 2 p + n ± 1 2 e 1 2 e 2 c 1 ! d 1 ! ( c 1 2 d 1 ) ! c 2 ! d 2 ! ( c 2 2 d 2 ) ! ( 2 l + n ± 1 h ) ! e 1 ! ( 2 l + n ± 1 h 2 e 1 ) ! ( 2 p 2 l + h ) ! e 2 ! ( 2 p 2 l + h 2 e 2 ) ! × 1 ( M 2 x x ) c 1 + c 2 + n ± 1 + 2 p 2 d 1 2 d 2 2 e 1 2 e 2 + 2 ( 2 ( w 0 2 M 1 x x 2 σ x x 4 2 M 1 x x σ x x 4 ) 1 / 2 ) c 1 + c 2 2 d 1 2 d 2 × H c 1 2 d 1 + 2 l + n ± 1 h 2 e 1 ( k 2 M 2 x x ( u 1 2 M 1 x x B σ x x 2 u 2 B * ) ) H 2 m + n ± 1 s c 1 ( i k u 1 ( w 0 2 M 1 x x 2 B 2 2 M 1 x x B 2 ) 1 / 2 ) × H c 2 2 d 2 + 2 p 2 l + h 2 e 2 ( k 2 M 2 x x ( v 1 2 M 1 x x B σ x x 2 v 2 B * ) ) H 2 p 2 m + s c 2 ( i k v 1 ( w 0 2 M 1 x x 2 B 2 2 M 1 x x B 2 ) 1 / 2 ) ,
M 1 x x = 1 / w 0 2 + 1 / ( 2 σ x x 2 ) i k A / ( 2 B ) ,   M 2 x x = 1 / w 0 2 + 1 / ( 2 σ x x 2 ) + i k A * / ( 2 B * ) 1 / ( 4 M 1 x x σ x x 4 ) ,
W x y ( u 1 , v 1 , u 2 , v 2 , z ) = B x y 4 i ( 1 λ | B | ) 2 π 2 M 1 x y 1 2 5 ( n ± 1 + 2 p ) / 2 ( p ! ) 2 ( 1 2 M 1 x y w 0 2 ) ( n ± 1 + 2 p ) / 2 × exp [ i k D 2 B ( u 1 2 + v 1 2 ) ] exp [ i k D * 2 B * ( u 2 2 + v 2 2 ) ] exp [ k 2 4 M 1 x y B 2 ( u 1 2 + v 1 2 ) ] × exp { k 2 4 M 2 x y ( u 2 B * u 1 2 M 1 x y B σ x y 2 ) 2 } exp { k 2 4 M 2 x y ( v 2 B * v 1 2 M 1 x y B σ x y 2 ) 2 } × m = 0 p s = 0 n ± 1 l = 0 p h = 0 n ± 1 c 1 = 0 2 m + n ± 1 s d 1 = 0 c 1 / 2 e 1 = 0 ( 2 l + n ± 1 h ) / 2 c 2 = 0 2 p 2 m + s d 2 = 0 c 2 / 2 e 2 = 0 ( 2 p 2 l + h ) / 2 ( 2 p 2 m + s c 2 ) ( 2 m + n ± 1 s c 1 ) × ( p m ) ( p l ) ( n ± 1 s ) ( n ± 1 h ) [ 1 ( 1 ) s ] [ 1 + ( 1 ) h ] ( 1 ) d 1 + d 2 + e 1 + e 2 i s ( i ) h ( 2 i ) ( c 1 + c 2 2 d 1 2 d 2 + n ± 1 + 2 p 2 e 1 2 e 2 )                                                       × ( 2 2 w 0 ) 2 p + n ± 1 2 e 1 2 e 2 c 1 ! d 1 ! ( c 1 2 d 1 ) ! c 2 ! d 2 ! ( c 2 2 d 2 ) ! ( 2 l + n ± 1 h ) ! e 1 ! ( 2 l + n ± 1 h 2 e 1 ) ! ( 2 p 2 l + h ) ! e 2 ! ( 2 p 2 l + h 2 e 2 ) ! × 1 ( M 2 x y ) c 1 + c 2 2 d 1 2 d 2 + n ± 1 + 2 p 2 e 1 2 e 2 + 2 ( 2 ( w 0 2 M 1 x y 2 σ x y 4 2 M 1 x y σ x y 4 ) 1 / 2 ) c 1 + c 2 2 d 1 2 d 2 × H c 1 2 d 1 + 2 l + n ± 1 h 2 e 1 ( k 2 M 2 x y ( u 1 2 M 1 x y B σ x y 2 u 2 B * ) ) H 2 m + n ± 1 s c 1 ( i k u 1 ( w 0 2 M 1 x y 2 B 2 2 M 1 x y B 2 ) 1 / 2 ) × H c 2 2 d 2 + 2 p 2 l + h 2 e 2 ( k 2 M 2 x y ( v 1 2 M 1 x y B σ x y 2 v 2 B * ) ) H 2 p 2 m + s c 2 ( i k v 1 ( w 0 2 M 1 x y 2 B 2 2 M 1 x y B 2 ) 1 / 2 ) ,
M 1 x y = 1 / w 0 2 + 1 / ( 2 σ x y 2 ) i k A / ( 2 B ) ,   M 2 x y = 1 / w 0 2 + 1 / ( 2 σ x y 2 ) + i k A * / ( 2 B * ) 1 / ( 4 M 1 x y σ x y 4 ) ,
W y x ( u 1 , v 1 , u 2 , v 2 , z ) = [ W x y ( u 1 , v 1 , u 2 , v 2 , z ) ] * ,
W y y ( u 1 , v 1 , u 2 , v 2 , z ) = 1 4 ( 1 λ | B | ) 2 π 2 M 1 y y 1 2 5 ( n ± 1 + 2 p ) / 2 ( p ! ) 2 ( 1 2 M 1 y y w 0 2 ) ( n ± 1 + 2 p ) / 2 × exp [ i k D 2 B ( u 1 2 + v 1 2 ) ] exp [ i k D * 2 B * ( u 2 2 + v 2 2 ) ] exp [ k 2 4 M 1 y y B 2 ( u 1 2 + v 1 2 ) ] × exp { k 2 4 M 2 y y ( u 2 B * u 1 2 M 1 y y B σ y y 2 ) 2 } exp { k 2 4 M 2 y y ( v 2 B * v 1 2 M 1 y y B σ y y 2 ) 2 } × m = 0 p s = 0 n ± 1 l = 0 p h = 0 n ± 1 c 1 = 0 2 m + n ± 1 s d 1 = 0 c 1 / 2 e 1 = 0 ( 2 l + n ± 1 h ) / 2 c 2 = 0 2 p 2 m + s d 2 = 0 c 2 / 2 e 2 = 0 ( 2 p 2 l + h ) / 2 ( 2 p 2 m + s c 2 ) ( 2 m + n ± 1 s c 1 ) × ( p m ) ( p l ) ( n ± 1 s ) ( n ± 1 h ) [ 1 + ( 1 ) s ] [ 1 + ( 1 ) h ] ( 1 ) d 1 + d 2 + e 1 + e 2 i s ( i ) h ( 2 i ) ( c 1 + c 2 2 d 1 2 d 2 + n ± 1 + 2 p 2 e 1 2 e 2 )                                                       × ( 2 2 w 0 ) 2 p + n ± 1 2 e 1 2 e 2 c 1 ! d 1 ! ( c 1 2 d 1 ) ! c 2 ! d 2 ! ( c 2 2 d 2 ) ! ( 2 l + n ± 1 h ) ! e 1 ! ( 2 l + n ± 1 h 2 e 1 ) ! ( 2 p 2 l + h ) ! e 2 ! ( 2 p 2 l + h 2 e 2 ) ! × 1 ( M 2 y y ) c 1 + c 2 2 d 1 2 d 2 + n ± 1 + 2 p 2 e 1 2 e 2 + 2 ( 2 ( w 0 2 M 1 y y 2 σ y y 4 2 M 1 y y σ y y 4 ) 1 / 2 ) c 1 + c 2 2 d 1 2 d 2 × H c 1 2 d 1 + 2 l + n ± 1 h 2 e 1 ( k 2 M 2 y y ( u 1 2 M 1 y y B σ y y 2 u 2 B * ) ) H 2 m + n ± 1 s c 1 ( i k u 1 ( w 0 2 M 1 y y 2 B 2 2 M 1 y y B 2 ) 1 / 2 ) × H c 2 2 d 2 + 2 p 2 l + h 2 e 2 ( k 2 M 2 y y ( v 1 2 M 1 y y B σ y y 2 v 2 B * ) ) H 2 p 2 m + s c 2 ( i k v 1 ( w 0 2 M 1 y y 2 B 2 2 M 1 y y B 2 ) 1 / 2 ) ,
M 1 y y = 1 / w 0 2 + 1 / ( 2 σ y y 2 ) i k A / ( 2 B ) ,   M 2 y y = 1 / w 0 2 + 1 / ( 2 σ y y 2 ) + i k A * / ( 2 B * ) 1 / ( 4 M 1 y y σ y y 4 ) .
exp [ ( x y ) 2 ] H n ( a x ) d x = π ( 1 a 2 ) n / 2 H n ( a y ( 1 a 2 ) 1 / 2 ) ,
x n exp [ ( x β ) 2 ] d x = ( 2 i ) n π H n ( i β ) ,
H n ( x + y ) = 1 2 n / 2 k = 0 n ( n k ) H k ( 2 x ) H n k ( 2 y ) ,
H n ( x ) = k = 0 n / 2 ( 1 ) k n ! k ! ( n 2 k ) ! ( 2 x ) n 2 k .
I ( u , v , z ) = W x x ( u , v , u , v , z ) + W y y ( u , v , u , v , z ) .
W s z ( z ) = 2 s 2 I ( u , v , z ) d u d v I ( u , v , z ) d u d v ,           ( s = u , v ) .
P ( u , v , z ) = 1 4 det [ W ^ ( u , v , u , v , z ) ] { T r [ W ^ ( u , v , u , v , z ) ] } 2 .
( A B C D ) = ( 1 z 0 1 ) .

Metrics