## Elliptical Laguerre-Gaussian correlated Schell-model beam |

Optics Express, Vol. 22, Issue 11, pp. 13975-13987 (2014)

http://dx.doi.org/10.1364/OE.22.013975

Acrobat PDF (3689 KB)

### Abstract

A new kind of partially coherent beam with non-conventional correlation function named elliptical Laguerre-Gaussian correlated Schell-model (LGCSM) beam is introduced. Analytical propagation formula for an elliptical LGCSM beam passing through a stigmatic ABCD optical system is derived. The elliptical LGCSM beam exhibits unique features on propagation, e.g., its intensity in the far field (or in the focal plane) displays an elliptical ring-shaped beam profile, being qualitatively different from the circular ring-shaped beam profile of the circular LGCSM beam. Furthermore, we carry out experimental generation of an elliptical LGCSM beam with controllable ellipticity, and measure its focusing properties. Our experimental results are consistent with the theoretical predictions. The elliptical LGCSM beam will be useful in atomic optics.

© 2014 Optical Society of America

## 1. Introduction

32. Y. Cai, Z. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express **16**(19), 15254–15267 (2008). [CrossRef] [PubMed]

26. T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. **78**(25), 4713–4716 (1997). [CrossRef]

32. Y. Cai, Z. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express **16**(19), 15254–15267 (2008). [CrossRef] [PubMed]

33. Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A **21**(6), 1058–1065 (2004). [CrossRef] [PubMed]

36. J. C. Gutiérrez-Vega, “Characterization of elliptical dark hollow beams,” Proc. SPIE **7062**, 706207 (2008). [CrossRef]

37. C. Zhao, X. Lu, L. Wang, and H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. **40**(3), 575–580 (2008). [CrossRef]

39. R. Chakraborty and A. Ghosh, “Generation of an elliptical hollow beam using Mathieu and Bessel functions,” J. Opt. Soc. Am. A **23**(9), 2278–2282 (2006). [CrossRef]

40. Z. Wang, Q. Lin, and Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. **240**(4-6), 357–362 (2004). [CrossRef]

41. Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A **25**, 1497–1503 (2008). [CrossRef]

42. X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A **369**(1-2), 157–166 (2007). [CrossRef]

43. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. **33**(12), 1389–1391 (2008). [CrossRef] [PubMed]

33. Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A **21**(6), 1058–1065 (2004). [CrossRef] [PubMed]

35. Z. Mei and D. Zhao, “Controllable elliptical dark-hollow beams,” J. Opt. Soc. Am. A **23**(4), 919–925 (2006). [CrossRef] [PubMed]

## 2. Ellipitcal Laguerre-Gaussian correlated Schell-model beam: theory

1. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. **32**(24), 3531–3533 (2007). [CrossRef] [PubMed]

*H*is an arbitrary kernel, and

*I*is a nonnegative function,

16. 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]

*H*and

*I*. Here

*H*and

*I*denote the response function of the optical path and the intensity of the incoherent source, respectively.

*H*and

*I*as follows withwhere

*H*denotes the response function of the optical path which consists of free space with length

*f*, a thin lens with focal length

*f*and a Gaussian amplitude filter with transmission function

*T*(see Fig. 1 of Ref [16

16. 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]

*I*denotes the intensity of an incoherent elliptical DH beam [33

33. Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A **21**(6), 1058–1065 (2004). [CrossRef] [PubMed]

*x*and

*y*directions, respectively.

*x*and

*y*directions, respectively,

*n*and 0. We call the partially coherent beam whose mutual coherence function and degree of coherence are given by Eqs. (7) and (8) as elliptical LGCSM beam. Under the condition of

*n*= 0, elliptical LGCSM beam reduces to elliptical GSM beam [16

16. 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]

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

**38**(11), 1814–1816 (2013). [CrossRef] [PubMed]

*n*= 0, elliptical LGCSM beam reduces to circular GSM beam [44]. Figure 1 shows the density plot of the square of the modulus of the degree of coherence of the elliptical LGCSM beam for different values of

*n*= 5. One finds from Fig. 1 that the density plot is of elliptical symmetry and the ellipticity is controlled by

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

*A*,

*B*,

*C*and

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

*n*, Eq. (15) becomes

*f*located in the source plane. The output plane is located in the geometrical plane. Then the transfer matrix between the source plane and the output plane reads as

*n*= 5,

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

**38**(11), 1814–1816 (2013). [CrossRef] [PubMed]

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

*n*= 5) of different values of

*L*, the generated elliptical DH beam illuminates the RGGD, producing an incoherent elliptical DH beam. Here

*L*is used to control the beam spot size on the RGGD through varying the distance between

*L*and RGGD. The transmitted beam from the RGGD can be regarded as an incoherent elliptical DH beam if the diameter of the beam spot on the RGGD is larger than the inhomogeneity scale of the RGGD [48

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

*L*

_{1}and the GAF, the incoherent elliptical DH beam becomes an elliptical LGCSM beam.

*f*-imaging system). Thus, the degree of coherence of the beam in the plane of the CCD is the same as that just behind the GAF. The output signal from the CCD is sent to a personal computer to measure the normalized fourth-order correlation function (FOCF) of the beam which is defined aswhere

*x*and

*y*being pixel spatial coordinates. Here

*m*denotes each realization and ranges from 1 to 2000. Then the square of the modulus of the degree of coherence of the generated elliptical LGCSM beam is obtained aswhere Here

*n*= 5) just behind the GAF for different values of coherence widths

*n*= 5) in the geometrical focal plane for different values of coherence widths

## 4. Summary

## Acknowledgments

## References and links

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

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

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

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

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

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

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

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

9. | O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. |

10. | 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. |

11. | 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. |

12. | Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A |

13. | 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. |

14. | 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 |

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

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

17. | 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 |

18. | Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. |

19. | Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express |

20. | 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. |

21. | A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science |

22. | L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science |

23. | M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. |

24. | X. Xu, V. G. Minogin, K. Lee, Y. Wang, and W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A |

25. | J. Yin, Y. Zhu, W. Jhe, and Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A |

26. | T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. |

27. | F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. |

28. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

29. | Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. |

30. | Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A |

31. | Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. |

32. | Y. Cai, Z. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express |

33. | Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A |

34. | Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express |

35. | Z. Mei and D. Zhao, “Controllable elliptical dark-hollow beams,” J. Opt. Soc. Am. A |

36. | J. C. Gutiérrez-Vega, “Characterization of elliptical dark hollow beams,” Proc. SPIE |

37. | C. Zhao, X. Lu, L. Wang, and H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. |

38. | H. Li and J. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. |

39. | R. Chakraborty and A. Ghosh, “Generation of an elliptical hollow beam using Mathieu and Bessel functions,” J. Opt. Soc. Am. A |

40. | Z. Wang, Q. Lin, and Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. |

41. | Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A |

42. | X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A |

43. | C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. |

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

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

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

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

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

**OCIS Codes**

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

(140.3300) Lasers and laser optics : Laser beam shaping

(140.7010) Lasers and laser optics : Laser trapping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 14, 2014

Manuscript Accepted: May 23, 2014

Published: May 30, 2014

**Citation**

Yahong Chen, Lin Liu, Fei Wang, Chengliang Zhao, and Yangjian Cai, "Elliptical Laguerre-Gaussian correlated Schell-model beam," Opt. Express **22**, 13975-13987 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13975

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### References

- 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]
- 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 beams in uniformly correlated random 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] [PubMed]
- O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef] [PubMed]
- 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]
- Y. Zhang, L. Liu, C. Zhao, Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- Y. Chen, Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014). [CrossRef] [PubMed]
- Y. Chen, F. Wang, C. Zhao, Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014). [CrossRef] [PubMed]
- 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. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999). [CrossRef] [PubMed]
- L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001). [CrossRef] [PubMed]
- M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995). [CrossRef] [PubMed]
- X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999). [CrossRef]
- J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998). [CrossRef]
- T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997). [CrossRef]
- F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987). [CrossRef]
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). [CrossRef] [PubMed]
- Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28(13), 1084–1086 (2003). [CrossRef] [PubMed]
- Z. Mei, D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22(9), 1898–1902 (2005). [CrossRef] [PubMed]
- Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007). [CrossRef] [PubMed]
- Y. Cai, Z. Wang, Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16(19), 15254–15267 (2008). [CrossRef] [PubMed]
- Y. Cai, Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21(6), 1058–1065 (2004). [CrossRef] [PubMed]
- Y. Cai, S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]
- Z. Mei, D. Zhao, “Controllable elliptical dark-hollow beams,” J. Opt. Soc. Am. A 23(4), 919–925 (2006). [CrossRef] [PubMed]
- J. C. Gutiérrez-Vega, “Characterization of elliptical dark hollow beams,” Proc. SPIE 7062, 706207 (2008). [CrossRef]
- C. Zhao, X. Lu, L. Wang, H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. 40(3), 575–580 (2008). [CrossRef]
- H. Li, J. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. 36(4), 457–459 (2011). [CrossRef] [PubMed]
- R. Chakraborty, A. Ghosh, “Generation of an elliptical hollow beam using Mathieu and Bessel functions,” J. Opt. Soc. Am. A 23(9), 2278–2282 (2006). [CrossRef]
- Z. Wang, Q. Lin, Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. 240(4-6), 357–362 (2004). [CrossRef]
- Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A 25, 1497–1503 (2008). [CrossRef]
- X. Lü, Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007). [CrossRef]
- C. Zhao, Y. Cai, F. Wang, X. Lu, Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]
- L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University, 1995).
- S. A. Collins., “Lens-system diffraction integral written in terms ofmatrix 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]
- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
- P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979). [CrossRef]

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