## Statistics properties of a cylindrical vector partially coherent beam |

Optics Express, Vol. 19, Issue 7, pp. 5979-5992 (2011)

http://dx.doi.org/10.1364/OE.19.005979

Acrobat PDF (1133 KB)

### 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

## 1. Introduction

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

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]

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

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]

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

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]

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]

## 2. Theory

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]

*r*and

*ϕ*are the radial and azimuthal (angle) coordinates,

*p*and

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

*x*and

*y*directions perpendicular to the

*z*-axis. Here the asterisk denotes the complex conjugate and the angular brackets denote ensemble average.

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

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]

*ϕ*is the phase difference between the

*x-*and

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

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]

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]

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]

*λ*being the wavelength.

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

## 3. Statistics properties of a cylindrical vector partially coherent LG beam in free space

**1**(1), 1–57 (2009). [CrossRef]

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]

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]

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]

## 4. Summary

## Acknowledgments

## References and links

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

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

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

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

5. | Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. |

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

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. |

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

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

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. |

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 |

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

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. |

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

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

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

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

18. | W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. |

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

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

21. | A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B |

22. | L. Mandel and E. Wolf, |

23. | C. Brosseau, |

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

25. | E. Wolf, |

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. |

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

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

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

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

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

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

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

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

35. | W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. |

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

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. |

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 |

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 |

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, “M |

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

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

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

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. |

46. | Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. |

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 |

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

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

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

51. | J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(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: February 11, 2011

Revised Manuscript: March 9, 2011

Manuscript Accepted: March 10, 2011

Published: March 16, 2011

**Citation**

Yiming Dong, Yangjian Cai, Chengliang Zhao, and Min Yao, "Statistics properties of a cylindrical vector partially coherent beam," Opt. Express **19**, 5979-5992 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-5979

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### References

- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [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]
- 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]
- Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009). [CrossRef] [PubMed]
- 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]
- 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,” N. 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. Express 18(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. Express 17(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]
- 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]
- A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998). [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]
- 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]
- W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006). [CrossRef]
- 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]
- 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]
- 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).
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
- 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. A 25(5), 1016–1021 (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]
- 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, “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]
- 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. 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]
- 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]
- 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]
- 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:.
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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.

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