## Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam |

Optics Express, Vol. 22, Issue 5, pp. 5826-5838 (2014)

http://dx.doi.org/10.1364/OE.22.005826

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### Abstract

Laguerre-Gaussian correlated Schell-model (LGCSM) vortex beam is introduced as an extension of LGCSM beam which was proposed [Opt. Lett. 38, 91 (2013) Opt. Lett. 38, 1814 (2013)] just recently. Explicit formula for a LGCSM vortex beam propagating through a stigmatic ABCD optical system is derived, and the propagation properties of such beam in free space and the focusing properties of such beam are studied numerically. Furthermore, we carry out experimental generation of a LGCSM vortex beam, and studied its focusing properties. It is found that the propagation and focusing properties of a LGCSM vortex beam are different from that of a LGCSM beam, and we can shape the beam profile of a LGCSM vortex at the focal plane (or in the far field) by varying its initial spatial coherence. Our experimental results are consistent with the theoretical predictions, and our results will be useful for particle trapping.

© 2014 Optical Society of America

## 1. Introduction

24. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. **104**(10), 103601 (2010). [CrossRef] [PubMed]

28. Y. Yang, Y. Dong, C. Zhao, Y. Liu, and Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express **22**(3), 2925–2932 (2014). [CrossRef]

*l*being the topological charge [24

24. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. **104**(10), 103601 (2010). [CrossRef] [PubMed]

29. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. **222**(1-6), 117–125 (2003). [CrossRef]

34. 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. **101**(26), 261104 (2012). [CrossRef]

33. F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. **36**(16), 3281–3283 (2011). [CrossRef] [PubMed]

34. 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. **101**(26), 261104 (2012). [CrossRef]

35. X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. **38**(24), 5323–5326 (2013). [CrossRef] [PubMed]

## 2. Laguerre-Gaussian correlated Schell-model vortex beam: Theory

20. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. **38**(2), 91–93 (2013). [CrossRef] [PubMed]

*z*= 0,

*n*and 0. The degree of coherence of the LGCSM beam at

*z*= 0 is given asFrom Eq. (2), one finds that the degree of coherence of the LGCSM beam doesn’t satisfy Gaussian distribution. Under the condition of

*n*= 0, Eq. (1) reduces to the expression for the mutual coherence function of a GSM beam [1,2].

*m*denotes the topological charge and

*n*= 0, Eq. (3) reduces to the expression for the mutual coherence function of a GSM vortex beam [33

33. F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. **36**(16), 3281–3283 (2011). [CrossRef] [PubMed]

36. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. **60**(9), 1168–1177 (1970). [CrossRef]

37. 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] [PubMed]

*A*,

*B*,

*C*and

*D*are the elements of a transfer matrix for the optical system,

33. F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. **36**(16), 3281–3283 (2011). [CrossRef] [PubMed]

*m*= 0, Eq. (15) reduces to the following expression for the average intensity of a LGCSM beam in the output plane

*z*reads as

20. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. **38**(2), 91–93 (2013). [CrossRef] [PubMed]

*f*is located at the source plane (

*z*= 0), and the output plane is located at the geometrical focal plane, then the transfer matrix between the source plane and output plane reads as

*f*= 40cm. One finds from Figs. 3 and 4 that the dependence of the focused intensity of a LGCSM vortex beam or LGCSM beam on the initial coherence width is similar to the dependence of the far-field intensity of such beam on initial coherence width. In fact the intensity profile of a beam in the in the focal plane of a converging lens is necessarily the same (with suitable scaling factors) as the intensity profile of the beam in the far field. When the initial coherence width is small, the intensity of a LGCSM beam or LGCSM vortex beam at the geometrical focal plane has a dark hollow beam profile. With the increase of the initial coherence width, the dark hollow beam profile of a LGCSM beam or LGCSM vortex beam at the geometrical focal plane disappears gradually. When the initial coherence is large, at the geometrical focal plane, the intensity of a LGCSM beam has a Gaussian beam profile, while the intensity of a LGCSM vortex beam has a dark hollow beam profile. For suitable values of the initial coherence width, the intensity of a LGCSM beam or a LGCSM vortex exhibits flat-topped beam profile. Thus, modulating the spatial coherence of a LGCSM vortex beam or a LGCSM beam provides one way for shaping its focused beam profile, which will be useful for particle trapping, where a focused Gaussian or flat-topped beam spot is used to trap a Rayleigh particle whose refractive index is larger than that of the ambient and a dark hollow beam spot is used to trap a Rayleigh particle whose refractive index is smaller than that of the ambient [38

38. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express **12**(15), 3377–3382 (2004). [CrossRef] [PubMed]

40. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. **36**(12), 2251–2253 (2011). [CrossRef] [PubMed]

## 3. Laguerre-Gaussian correlated Schell-model vortex beam: experiment

21. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. **38**(11), 1814–1816 (2013). [CrossRef] [PubMed]

*n*can be formed when an incoherent beam whose intensity distribution has a dark hollow beam profile and is expressed as

*f*, a thin lens with focal length

*f*and a Gaussian amplitude filter (GAF), and the spatial coherence width of the generated LGCSM beam can be approximated as

*n*= 1 is shown as inset in Fig. 5. The first order of the beam from the SLM is a dark hollow beam with

*n*= 1 and is selected out by a circular aperture. After passing through a thin lens L

_{1}, the generated dark hollow beam illuminates a rotating ground-glass disk (RGGD), producing an incoherent beam with dark hollow beam profile. The beam from the RGGD can be regarded as a spatially incoherent beam if the diameter of the beam spot on the RGGD is larger than the inhomogeneity scale of the ground glass [41

41. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. **29**(3), 256–260 (1979). [CrossRef]

*f*

_{2}, the thin lens L

_{2}, and the GAF, the generated incoherent dark hollow beam becomes a LGCSM beam with

*n*= 1 [21

21. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. **38**(11), 1814–1816 (2013). [CrossRef] [PubMed]

*m*= 3, the generated LGCSM beam becomes a LGCSM vortex beam. The SPP just adds a vortex phase to the LGCSM beam, and it doesn’t alter its spatial coherence and its intensity distribution in the source plane, thus the spatial coherence width and the intensity distribution of generated LGCSM vortex beam are the same with those of the generated LGCSM beam. The spatial coherence width of the generated LGCSM beam is modulated by varying the beam spot on the RGGD through varying the distance between the thin lens L

_{1}and the RGGD.

42. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A **24**(7), 1937–1944 (2007). [CrossRef] [PubMed]

21. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. **38**(11), 1814–1816 (2013). [CrossRef] [PubMed]

42. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A **24**(7), 1937–1944 (2007). [CrossRef] [PubMed]

_{3}with focal length

*n*= 1 at the geometrical focal plane for different values of the initial coherence width

*n*= 1 and

*m*= 3 at the geometrical focal plane for different values of the initial coherence width

## 4. Summary

## Acknowledgments

## References and links

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

2. | E. Wolf, |

3. | Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. |

4. | Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE |

5. | F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. |

6. | F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. |

7. | L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics |

8. | Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A |

9. | H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. |

10. | Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. |

11. | Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A |

12. | Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. |

13. | S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. |

14. | O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

15. | S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. |

16. | Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. |

17. | Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. |

18. | Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express |

19. | C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. |

20. | Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. |

21. | F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. |

22. | J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. |

23. | R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express |

24. | J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. |

25. | D. G. Grier, “A revolution in optical manipulation,” Nature |

26. | A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. |

27. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

28. | Y. Yang, Y. Dong, C. Zhao, Y. Liu, and Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express |

29. | G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. |

30. | W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. |

31. | 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. |

32. | T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A |

33. | F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. |

34. | 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. |

35. | X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. |

36. | S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. |

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

38. | Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express |

39. | Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A |

40. | C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. |

41. | P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. |

42. | F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A |

**OCIS Codes**

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

(030.1670) Coherence and statistical optics : Coherent optical effects

(140.3300) Lasers and laser optics : Laser beam shaping

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 21, 2014

Revised Manuscript: February 27, 2014

Manuscript Accepted: February 27, 2014

Published: March 5, 2014

**Virtual Issues**

Vol. 9, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Yahong Chen, Fei Wang, Chengliang Zhao, and Yangjian Cai, "Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam," Opt. Express **22**, 5826-5838 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-5826

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007)
- Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010). [CrossRef]
- Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011). [CrossRef]
- F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]
- F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]
- L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012). [CrossRef]
- Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]
- H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]
- Z. Tong, O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]
- Z. Tong, O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]
- Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef] [PubMed]
- S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]
- O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]
- S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013). [CrossRef]
- Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). [CrossRef]
- Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]
- Z. Mei, E. Shchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]
- C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]
- Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]
- F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef] [PubMed]
- J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]
- R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014). [CrossRef] [PubMed]
- J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010). [CrossRef] [PubMed]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]
- A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003). [CrossRef] [PubMed]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
- Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014). [CrossRef]
- G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003). [CrossRef]
- W. Wang, M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006). [CrossRef] [PubMed]
- G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003). [CrossRef] [PubMed]
- T. van Dijk, T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009). [CrossRef] [PubMed]
- F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011). [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]
- X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013). [CrossRef] [PubMed]
- S. A. Collins., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]
- Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef] [PubMed]
- Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]
- Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). [CrossRef]
- C. Zhao, Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]
- P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979). [CrossRef]
- F. Wang, Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef] [PubMed]

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