## Partially coherent standard and elegant Laguerre-Gaussian beams of all orders

Optics Express, Vol. 17, Issue 25, pp. 22366-22379 (2009)

http://dx.doi.org/10.1364/OE.17.022366

Acrobat PDF (247 KB)

### Abstract

Partially coherent standard and elegant Laguerre-Gaussian (LG) beams of all orders are introduced as a natural extension of coherent standard and elegant LG beams to the stochastic domain. By expanding the LG modes into a finite sum of Hermite-Gaussian modes, the analytical formulae are obtained for the cross-spectral densities of partially coherent standard and elegant LG beams in the source plane and after passing through paraxial ABCD optical system, based on the generalized Collins integral formula. A comparative study of the propagation properties of the partially coherent standard and elegant LG beams in free space is carried out via a set of numerical examples. Our results indicate that the intensity and spreading properties of partially coherent standard and elegant LG beams are closely related to their initial coherence states, and are very different from the corresponding results for the coherent standard and elegant LG beams. In particular, an elegant LG beam spreads slower than a standard LG beam, while this advantage disappears when their initial coherences are very small. Our results may find applications in connection with laser beam shaping, singular optics and astrophysical measurements of angular momentum of radiation.

© 2009 OSA

## 1. Introduction

2. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A **25**(7), 1642–1651 (
2008). [CrossRef]

23. Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B **84**(3), 493–500 (
2006). [CrossRef]

2. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A **25**(7), 1642–1651 (
2008). [CrossRef]

3. A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express **13**(13), 4952–4962 (
2005). [CrossRef] [PubMed]

4. C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A **38**(11), 5960–5963 (
1988). [CrossRef] [PubMed]

5. C. Tamm and C. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B **7**(6), 1034–1038 (
1990). [CrossRef]

6. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna V, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A **43**(9), 5090–5113 (
1991). [CrossRef] [PubMed]

8. T. Hasegawa and T. Shimizu, “Frequency-doubled Hermite-Gaussian beam and the mode conversion to the Laguerre–Gaussian beam,” Opt. Commun. **160**(1-3), 103–108 (
1999). [CrossRef]

9. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. **42**(1), 217–223 (
1995). [CrossRef]

10. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. **41**, 1097–1103 (
2002). [CrossRef]

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

12. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. **78**(25), 4713–4716 (
1997). [CrossRef]

13. J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B **71**(4), 549–554 (
2000). [CrossRef]

14. Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D **53**(2), 127–131 (
2009). [CrossRef]

15. X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. **281**(15-16), 4103–4108 (
2008). [CrossRef]

16. D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Opt. Lett. **30**(22), 3039–3041 (
2005). [CrossRef] [PubMed]

25. C. J. R. Sheppard, “Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams,” Opt. Express **17**(5), 3690–3697 (
2009). [CrossRef] [PubMed]

26. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. **63**(9), 1093–1094 (
1973). [CrossRef]

27. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A **2**(6), 826–829 (
1985). [CrossRef]

29. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. **45**, 1999–2009 (
1998). [CrossRef]

30. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A **18**(1), 177–184 (
2001). [CrossRef]

31. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A **18**(7), 1627–1633 (
2001). [CrossRef]

32. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. **29**(19), 2213–2215 (
2004). [CrossRef] [PubMed]

37. Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express **17**(17), 14865–14871 (
2009). [CrossRef] [PubMed]

38. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implication for free-space laser communication,” J. Opt. Soc. Am. A **19**(9), 1794–1802 (
2002). [CrossRef]

39. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. **53**(11), 1057–1060 (
1984). [CrossRef]

40. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A **36**(1), 202–206 (
1987). [CrossRef] [PubMed]

41. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express **15**(23), 15480–15492 (
2007). [CrossRef] [PubMed]

42. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express **17**(3), 1753–1765 (
2009). [CrossRef] [PubMed]

44. C. Cheng, W. Liu, and W. Gui, “Diffraction halo function of partially coherent speckle photography,” Appl. Opt. **38**(32), 6687–6691 (
1999). [CrossRef] [PubMed]

45. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. **93**(6), 068103 (
2004). [CrossRef] [PubMed]

47. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express **17**(4), 2453–2464 (
2009). [CrossRef] [PubMed]

42. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express **17**(3), 1753–1765 (
2009). [CrossRef] [PubMed]

48. Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. **245**(1-6), 21–26 (
2005). [CrossRef]

63. Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. **282**(1), 69–73 (
2009). [CrossRef]

58. S. A. Ponamorenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. **18**(1), 150–156 (
2001). [CrossRef]

60. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. **32**(17), 2508–2510 (
2007). [CrossRef] [PubMed]

61. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A **26**(4), 741–744 (
2009). [CrossRef]

63. Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. **282**(1), 69–73 (
2009). [CrossRef]

61. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A **26**(4), 741–744 (
2009). [CrossRef]

62. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. **47**(3), 036002 (
2008). [CrossRef]

63. Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. **282**(1), 69–73 (
2009). [CrossRef]

50. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **25**(1), 21–28 (
2008). [CrossRef]

52. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. **278**(1), 17–22 (
2007). [CrossRef]

55. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. **40**(1), 156–166 (
2008). [CrossRef]

57. G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express **17**(13), 10529–10534 (
2009). [CrossRef] [PubMed]

62. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. **47**(3), 036002 (
2008). [CrossRef]

42. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express **17**(3), 1753–1765 (
2009). [CrossRef] [PubMed]

## 2. Theory

35. Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. **6**(11), 1005–1011 (
2004). [CrossRef]

*p*and

*l*,

*standard*LG beam; for

*elegant*LG beam; also for

65. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. **25**(3), 293–296 (
1978). [CrossRef]

70. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A **9**(5), 796–803 (
1992). [CrossRef]

49. 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]

71. 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 transfer matrix elements of optical system,

*λ*being the wavelength.

74. W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. **19**(7), 1027–1029 (
1980). [CrossRef] [PubMed]

*x*or

*y*, the effective beam size of a partially coherent standard or elegant LG beam at plane z is defined as

75. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. **31**(6), 685–687 (
2006). [CrossRef] [PubMed]

## 3. Propagation properties of partially coherent standard and elegant LG beams in free space

27. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A **2**(6), 826–829 (
1985). [CrossRef]

37. Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express **17**(17), 14865–14871 (
2009). [CrossRef] [PubMed]

*p*and

*l*. For the convenience of comparison, in Fig. 4(a)-(c), we have chosen

*p*and

*l*of partially coherent standard and elegant LG beams affect their spreading properties strongly when the initial degree of coherence is high. Both partially coherent standard and elegant LG beams spread more rapidly as their mode orders

*p*and

*l*increase when the initial degree of coherence is high (see Fig. 4(a), (b), (d) and (e))). When the initial coherence is small, the partially coherent LG beams, both standard and elegant, with different mode orders exhibit almost the same spreading features (see Fig. 4 (c) and (f)).

## 4. Summary

## Acknowledgments

## References and links

1. | A. E. Siegman, |

2. | N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A |

3. | A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express |

4. | C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A |

5. | C. Tamm and C. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B |

6. | M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna V, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A |

7. | Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B |

8. | T. Hasegawa and T. Shimizu, “Frequency-doubled Hermite-Gaussian beam and the mode conversion to the Laguerre–Gaussian beam,” Opt. Commun. |

9. | H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. |

10. | C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. |

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

12. | T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. |

13. | J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B |

14. | Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D |

15. | X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. |

16. | D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Opt. Lett. |

17. | J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. |

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23. | Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B |

24. | G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. |

25. | C. J. R. Sheppard, “Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams,” Opt. Express |

26. | A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. |

27. | T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A |

28. | E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A |

29. | S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. |

30. | M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A |

31. | R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A |

32. | M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. |

33. | Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. |

34. | Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A |

35. | Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. |

36. | A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. |

37. | Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express |

38. | J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implication for free-space laser communication,” J. Opt. Soc. Am. A |

39. | Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. |

40. | M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A |

41. | Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express |

42. | C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express |

43. | G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. |

44. | C. Cheng, W. Liu, and W. Gui, “Diffraction halo function of partially coherent speckle photography,” Appl. Opt. |

45. | T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. |

46. | Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

47. | Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express |

48. | Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. |

49. | 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 |

50. | X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A |

51. | X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. |

52. | H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. |

53. | F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. |

54. | 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. |

55. | H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. |

56. | M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. |

57. | G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express |

58. | S. A. Ponamorenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. |

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

60. | S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. |

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

62. | T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. |

63. | Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. |

64. | K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. |

65. | E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. |

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

67. | F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. |

68. | A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. |

69. | 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 |

70. | E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A |

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

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

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

74. | W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. |

75. | Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(140.3300) Lasers and laser optics : Laser beam shaping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: September 23, 2009

Revised Manuscript: November 3, 2009

Manuscript Accepted: November 13, 2009

Published: November 23, 2009

**Citation**

Fei Wang, Yangjian Cai, and Olga Korotkova, "Partially coherent standard and elegant Laguerre-Gaussian beams of all orders," Opt. Express **17**, 22366-22379 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22366

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### References

- A. E. Siegman, Lasers (Mill Valley, CA: University Science Books, 1986)
- N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25(7), 1642–1651 (2008). [CrossRef]
- A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13(13), 4952–4962 (2005). [CrossRef] [PubMed]
- C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38(11), 5960–5963 (1988). [CrossRef] [PubMed]
- C. Tamm and C. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1034–1038 (1990). [CrossRef]
- M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991). [CrossRef] [PubMed]
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