## The cross correlation function of partially coherent vortex beam |

Optics Express, Vol. 22, Issue 2, pp. 1350-1358 (2014)

http://dx.doi.org/10.1364/OE.22.001350

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### Abstract

This work presents theoretical analysis on the cross correlation function (CCF) of partially coherent vortex beam (PCVB), where the relation of the number of the rings of CCF dislocations and orbital angular momentum (OAM) of PCVB is analyzed in detail. It is shown that rings of CCF dislocations do not always exist, and depend on the coherence length, the order of PCVB and location of observation plane, although the CCF indicates topological charge to some degree. Comprehensive analysis of the CCF of PCVB and numerical simulations all validate such phenomenon.

© 2014 Optical Society of America

## 1. Introduction

## 2. The CCF of PCVB during propagation

*m*th-order vortex, of which the mutual correlation function (MCF) in the original plane can be described aswhere

18. Z. Y. Chen and J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A **372**(15), 2734–2740 (2008). [CrossRef]

## 3 Correlation between CCF and topological charge of PCVB

### 3.1 The case while m = 1

_{R}is the Rayleigh distance).

### 3.2 The case of high order

_{R}[Fig. 3(a)], the CCF of the 3rd order PCVB exhibits three zero points, however, the CCFs of the 4th and 5th order PCVB show no clear zero points. While the propagation length increases to 0.02Z

_{R}[Fig. 3(b)], it is interesting to find that the CCFs of the 3rd, 4th, 5th order PCVB show clearly 3,4,5 zero points, respectively. With the continuous increasing of propagation length [Figs. 3(c) and 3(d)], the CCFs (m = 3, 4, 5) only obviously display the anterior zero points and other zero points are blurry and almost missing. It can be observed that the CCF of the higher order PCVB with low coherence has high resolution of zero points in the near field and there exists an optimum observation plane [Fig. 3(b)].

_{R}, the CCF of the3rd order PCVB takes the lead to show all zero points [Fig. 4(b)]. The CCFs of the 4th and 5th order PCVB follow to show the zero points with the increasing of propagation length, and while the propagation length increases to 0.5Z

_{R}, the CCFs show all the zero points obviously [Fig. 4(c)]). With the continuous increasing of propagation length, the CCFs can still show all the zero points but the resolution of the posterior zero points tends to decrease [Fig. 4(d)]. It can be concluded from Figs. 4(a)-4(d) that the best distance should be between the near field and Rayleigh distance to display all the zero points of CCF of higher order PCVB with moderate coherence.

16. Y. J. Yang, M. Mazilu, and K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. **37**(23), 4949–4951 (2012). [PubMed]

_{R}, the CCFs show no clear zero points in Figs. 5(a) and 5(b). While the propagation length increases to 5Z

_{R}, the CCF of the 3rd PCVB takes the lead to show all zero points [Fig. 5(c)], which is a little similar to that in Fig. 4(b). However, with the continuous increasing of propagation length (z = 20Z

_{R}), the CCFs of the 4th and 5th order PCVB have no inclination to show the zero points at all. This characteristic does not change a little even at the propagation length beyond 100Z

_{R}(calculated, but not plotted in Fig. 5), which accords with that in [16

16. Y. J. Yang, M. Mazilu, and K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. **37**(23), 4949–4951 (2012). [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

2. | E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. |

3. | M. Šiler, P. Jákl, O. Brzobohatý, and P. Zemánek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express |

4. | K. Y. Bliokh and F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A |

5. | H. W. Yan, E. T. Zhang, B. Y. Zhao, and K. L. Duan, “Free-space propagation of guided optical vortices excited in an annular core fiber,” Opt. Express |

6. | Y. F. Jiang, K. K. Huang, and X. H. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express |

7. | E. Mari, F. Tamburini, G. A. Swartzlander Jr, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express |

8. | Y. J. Han and G. H. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett. |

9. | Y. X. Liu, J. X. Pu, and B. D. Lü, “Method for exploring the orbital angular momentum of an optical vortex beam with a triangular multipoint plate,” Appl. Opt. |

10. | S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. |

11. | L. Shi, L. H. Tian, and X. F. Chen, “Characterizing topological charge of optical vortex using non-uniformly distributed multi-pinhole plate,” Chin. Opt. Lett. |

12. | H. Tao and J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B |

13. | M. E. Anderson, H. Bigman, L. E. E. De Araujo, and J. L. Chaloupka, “Measuring the topological charge of ultra-broadband, optical-vortex beams with a triangular aperture,” J. Opt. Soc. Am. B |

14. | D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. |

15. | C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. |

16. | Y. J. Yang, M. Mazilu, and K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. |

17. | Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, and K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. |

18. | Z. Y. Chen and J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A |

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

20. | E. Wolf, |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(050.4865) Diffraction and gratings : Optical vortices

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: October 14, 2013

Revised Manuscript: December 21, 2013

Manuscript Accepted: December 21, 2013

Published: January 14, 2014

**Citation**

Pan Feng Ding and Jixiong Pu, "The cross correlation function of partially coherent vortex beam," Opt. Express **22**, 1350-1358 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1350

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### References

- J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). [CrossRef]
- E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef] [PubMed]
- M. Šiler, P. Jákl, O. Brzobohatý, P. Zemánek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20(22), 24304–24319 (2012). [CrossRef] [PubMed]
- K. Y. Bliokh, F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012). [CrossRef]
- H. W. Yan, E. T. Zhang, B. Y. Zhao, K. L. Duan, “Free-space propagation of guided optical vortices excited in an annular core fiber,” Opt. Express 20(16), 17904–17915 (2012). [CrossRef] [PubMed]
- Y. F. Jiang, K. K. Huang, X. H. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012). [CrossRef] [PubMed]
- E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012). [CrossRef] [PubMed]
- Y. J. Han, G. H. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett. 36(11), 2017–2019 (2011). [CrossRef] [PubMed]
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- S. Prabhakar, A. Kumar, J. Banerji, R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36(22), 4398–4400 (2011). [CrossRef] [PubMed]
- L. Shi, L. H. Tian, X. F. Chen, “Characterizing topological charge of optical vortex using non-uniformly distributed multi-pinhole plate,” Chin. Opt. Lett. 10(12), 120501 (2012). [CrossRef]
- H. Tao, J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012). [CrossRef]
- M. E. Anderson, H. Bigman, L. E. E. De Araujo, J. L. Chaloupka, “Measuring the topological charge of ultra-broadband, optical-vortex beams with a triangular aperture,” J. Opt. Soc. Am. B 29(8), 1968–1976 (2012).
- D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004). [CrossRef] [PubMed]
- C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012). [CrossRef]
- Y. J. Yang, M. Mazilu, K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012). [PubMed]
- Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013). [CrossRef]
- Z. Y. Chen, J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

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