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

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2009 (10)

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

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]

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]

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]

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]

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

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[CrossRef]

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]

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]

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]

2008 (10)

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

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]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008).
[CrossRef] [PubMed]

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]

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]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008).
[CrossRef] [PubMed]

G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. 40(7), 930–935 (2008).
[CrossRef]

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]

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]

2007 (5)

2006 (2)

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]

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]

2005 (4)

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]

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]

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

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

2004 (5)

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]

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]

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
[CrossRef]

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 21(12), 2375–2381 (2004).
[CrossRef]

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]

2003 (1)

S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003).
[CrossRef]

2002 (5)

2001 (5)

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

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

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]

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]

2000 (3)

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[CrossRef]

R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 (2000).
[CrossRef] [PubMed]

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]

1999 (2)

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]

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

1998 (1)

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]

1997 (1)

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]

1995 (1)

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]

1993 (1)

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

1992 (1)

1991 (1)

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]

1990 (1)

1988 (1)

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

1987 (1)

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]

1986 (1)

1985 (1)

1984 (1)

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]

1983 (1)

G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 (1983).

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

1980 (2)

1979 (1)

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

1978 (1)

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]

1973 (1)

Agarwal, G. S.

Alavinejad, M.

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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).
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T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
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J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

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Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
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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).
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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).
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J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
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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).
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V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
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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).
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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).
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Kuga, T.

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).
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J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

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Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

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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).
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Lu, X.

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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).
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K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

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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).
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Mayo, S. C.

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).
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Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
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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).
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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 21(12), 2375–2381 (2004).
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Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
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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).
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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).
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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).
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G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 (1983).

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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).
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G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 (2002).
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V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
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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).
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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).
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T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
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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).
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Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
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Ricklin, J. C.

Rubinsztein-Dunlop, H.

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).
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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).
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Shiokawa, N.

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).
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K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

Siegman, A. E.

Simon, R.

Stabinis, A.

S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003).
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V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[CrossRef]

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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).
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A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
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Takenaka, T.

Tamm, C.

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).
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van Dijk, T.

Visser, T. D.

Wang, F.

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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).
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Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).

Wang, T.

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

Wang, Z.

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).
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Weiss, C. O.

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).
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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).
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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).
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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).
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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]

Yokota, M.

Young, C. Y.

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).
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Zhang, T.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
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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]

Zhang, Z.

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]

Zhao, C.

Zhao, D.

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Normalized intensity distribution (cross line v = 0) of a partially coherent standard LG beam for different values of the initial coherence width σ g with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m

Fig. 4
Fig. 4

Effective beam sizes of partially coherent standard LG beams and elegant LG beams versus the propagation distance z in free space for different values of mode orders p and l and coherence width σ g

Fig. 2
Fig. 2

Normalized intensity distribution (cross line v = 0) of a partially coherent elegant LG beam for different values of the initial coherence width σ g with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m

Fig. 3
Fig. 3

Effective beam sizes of partially coherent standard LG beam and elegant LG beam versus the propagation distance z in free space for different values of the initial coherence width

Equations (19)

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E p l ( r , φ ; 0 ) = ( q r ω 0 ) l L p l ( q 2 r 2 ω 0 2 ) exp ( r 2 ω 0 2 ) exp ( i l φ ) ,
e i l φ ρ l L p l ( ρ 2 ) = ( 1 ) p 2 2 p + l p ! m = 0 p n = 0 l i n ( p m ) ( l n ) H 2 m + l n ( x ) H 2 p 2 m + n ( y ) ,
E p l ( x , y ; 0 ) = ( 1 ) p 2 2 p + l p ! m = 0 p s = 0 l i s ( p m ) ( l s ) H 2 m + l s ( q x ω 0 ) H 2 p 2 m + s ( q y ω 0 ) exp ( x 2 + y 2 ω 0 2 ) .
W ( x 1 , y 1 , x 2 , y 2 ; 0 ) = I ( x 1 , y 1 ; 0 ) I ( x 2 , y 2 ; 0 ) g ( x 1 x 2 ; y 1 y 2 ; 0 ) ,
g ( x 1 x 2 ; y 1 y 2 ; 0 ) = exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 2 ] ,
I ( x , y ; 0 ) = | ( 1 ) p 2 2 p + l p ! m = 0 p s = 0 l i s ( p m ) ( l s ) H 2 m + l s ( q x ω 0 ) H 2 p 2 m + s ( q y ω 0 ) exp ( x 2 + y 2 ω 0 2 ) | 2 .
W ( x 1 , y 1 , x 2 , y 2 ; 0 ) = 1 2 4 p + 2 l ( p ! ) 2 m = 0 p n = 0 l h = 0 p s = 0 l ( i n ) * i s ( p m ) ( l n ) ( p h ) ( l s )                                 × H 2 m + l n ( q x 1 ω 0 ) H 2 h + l s ( q x 2 ω 0 ) H 2 p 2 m + n ( q y 1 ω 0 ) H 2 p 2 h + s ( q y 2 ω 0 )                                 × exp ( x 1 2 + y 1 2 + x 2 2 + y 2 2 ω 0 2 ) exp ( ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 2 ) .
W ( u 1 , v 1 , u 2 , v 2 , z ) = ( 1 λ | B | ) 2 W ( x 1 , y 1 , x 2 , y 2 ; 0 )            × exp [ i k 2 B * ( A * x 1 2 2 x 1 u 1 + D * u 1 2 ) i k 2 B * ( A * y 1 2 2 y 1 v 1 + D * v 1 2 ) ]            × exp [ i k 2 B ( A x 2 2 2 x 2 u 2 + D u 2 2 ) + i k 2 B ( A y 2 2 2 y 2 v 2 + D y 2 2 ) ] d x 1 d x 2 d y 1 d y 2 ,
W ( u 1 , v 1 , u 2 , v 2 , z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 m = 0 p n = 0 l h = 0 p s = 0 l ( i n ) * i s ( p m ) ( l n ) ( p h ) ( l s )                             H 2 m + l n ( q x 1 ω 0 ) H 2 h + l s ( q x 2 ω 0 ) exp ( x 1 2 + x 2 2 ω 0 2 ) exp ( ( x 1 x 2 ) 2 2 σ g 2 )                               × exp [ i k 2 B * ( A * x 1 2 2 x 1 u 1 + D * u 1 2 ) + i k 2 B ( A x 2 2 2 x 2 u 2 + D u 2 2 ) ] d x 1 d x 2                           H 2 p 2 m + n ( q y 1 ω 0 ) H 2 p 2 h + s ( q y 2 ω 0 ) exp ( y 1 2 + y 2 2 ω 0 2 ) exp ( ( y 1 y 2 ) 2 2 σ g 2 )                             × exp [ i k 2 B * ( A * y 1 2 2 y 1 v 1 + D * v 1 2 ) + i k 2 B ( A y 2 2 2 y 2 v 2 + D y 2 2 ) ] d y 1 d y 2 .
W ( u 1 , u 2 , v 1 , v 2 , z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 π 2 M 1 M 2 ( 1 2 M 2 q 2 2 M 1 M 2 ω 0 2 ) ( 2 p + l ) / 2 ( 2 q ω 0 ) 2 p + l                          × exp [ i k D * 2 B * u 1 2 + i k D 2 B u 2 2 ) ] exp ( k 2 u 2 2 4 M 1 B 2 ) exp [ k 2 4 M 2 ( u 1 B * u 2 2 M 1 σ g 2 B ) 2 ]                          × exp [ i k D * 2 B * v 1 2 + i k D 2 B v 2 2 ) ] exp ( k 2 v 2 2 4 M 1 B 2 ) exp [ k 2 4 M 2 ( v 1 B * v 2 2 M 1 σ g 2 B ) 2 ]                          × m = 0 p n = 0 l c 1 = 0 [ ( 2 m + l n ) / 2 ] e 1 = 0 [ ( 2 p 2 m + n ) / 2 ] h = 0 p s = 0 l d = 0 2 h + l s c 2 = 0 [ d / 2 ] d 1 = 0 2 p 2 h + s e 2 = 0 [ d 1 / 2 ] ( i n ) * i s ( p m ) ( l n )                          × ( p h ) ( l s ) ( 2 h + l s d ) ( 2 p 2 h + s d 1 ) ( 1 ) c 1 + c 2 + e 1 + e 2 ( 2 m + l n ) ! c 1 ! ( 2 m + l n 2 c 1 ) !                          × d ! c 2 ! ( d 2 c 2 ) ! ( 2 p 2 m + n ) ! e 1 ! ( 2 p 2 m + n 2 e 1 ) ! d 1 ! e 2 ! ( d 1 2 e 2 ) ! ( 2 i ) 2 c 1 + 2 c 2 + 2 e 1 + 2 e 2 d d 1 2 p l                          × ( 1 M 2 ) d + d 1 2 c 1 2 c 2 2 e 1 2 e 2 ( 2 q ω 0 ) 2 c 1 2 e 1 ( 2 q 2 σ g 2 M 1 2 ω 0 2 q 2 M 1 ) d + d 1 2 c 2 2 e 2                          × H 2 h + l s d ( i q k u 2 2 B M 1 2 ω 0 2 q 2 M 1 ) H 2 m + l n + d 2 c 1 2 c 2 ( k u 2 4 M 1 M 2 σ g 2 B k u 1 2 M 2 B * )                          × H 2 p 2 h + s d 1 ( i q k v 2 2 B M 1 2 ω 0 2 q 2 M 1 ) H 2 p 2 m + n + d 1 2 e 1 2 e 2 ( k v 2 4 M 1 M 2 σ g 2 B k v 1 2 M 2 B * ) ,
exp [ ( x y ) 2 ] H n ( a x ) d x = π ( 1 a 2 ) n 2 H n ( a y ( 1 a 2 ) 1 / 2 )
x n exp [ ( x β ) 2 ] d x = ( 2 i ) n π H n ( i β ) ,
H n ( x + y ) = 1 2 n / 2 k = 0 n ( n k ) H k ( 2 x ) H n k ( 2 y ) ,
H n ( x 1 ) = m = 0 [ n / 2 ] ( 1 ) m n ! m ! ( n 2 m ) ! ( 2 x 1 ) n 2 m .
W s z ( z ) = 2 s 2 I ( x , y , z ) d x d y I ( x , y , z ) d x d y       ( s = x , y ) .
W x z = W y z = A 1 ( z ) A 2 ( z )
A 1 ( z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 π 3 M 1 M 2 M 3 ( 1 2 M 2 q 2 2 M 1 M 2 ω 0 2 ) ( 2 p + l ) / 2 ( 2 q ω 0 ) 2 p + l           × m = 0 p n = 0 l c 1 = 0 [ ( 2 m + l n ) / 2 ] e 1 = 0 [ ( 2 p 2 m + n ) / 2 ] h = 0 p s = 0 l d = 0 2 h + l s c 2 = 0 [ d / 2 ] d 1 = 0 2 p 2 h + s e 2 = 0 [ d 1 / 2 ] f 1 = 0 [ ( 2 h + l s d ) / 2 ]           × f 2 = 0 [ ( 2 m + l n + d 2 c 1 2 c 2 ) / 2 ] g 1 = 0 [ ( 2 p 2 h + s d 1 ) / 2 ] g 2 = 0 [ ( 2 p 2 m + n + d 1 2 e 1 2 c 2 ) / 2 ] ( i n ) * i s ( p m ) ( l n ) ( p h )           × ( l s ) ( 2 h + l s d ) ( 2 p 2 h + s d 1 ) ( 1 ) c 1 + c 2 + e 1 + e 2 + f 1 + f 2 + g 1 + g 2 ( 2 m + l n ) ! c 1 ! ( 2 m + l n 2 c 1 ) ! d ! c 2 ! ( d 2 c 2 ) !           × ( 2 p 2 m + n ) ! e 1 ! ( 2 p 2 m + n 2 e 1 ) ! d 1 ! e 2 ! ( d 1 2 e 2 ) ! ( 2 m + l n + d 2 c 1 2 c 2 ) ! f 2 ! ( 2 m + l n + d 2 c 1 2 c 2 2 f 2 ) !           × ( 2 h + l s d ) ! f 1 ! ( 2 h + l s d 2 f 1 ) ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 ) ! g 2 ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ) ! ( 2 p 2 h + s d 1 ) ! g 1 ! ( 2 p 2 h + s d 1 2 g 1 ) !           × ( 2 i ) 4 c 1 + 4 c 2 + 4 e 1 + 4 e 2 2 f 1 2 f 2 2 g 1 2 g 2 d d 1 6 p 3 l 2 ( 1 M 2 ) d + d 1 2 c 1 2 c 2 2 e 1 2 e 2 ( 2 q ω 0 ) 2 c 1 2 e 1           × ( 2 q 2 σ g 2 M 1 2 ω 0 2 q 2 M 1 ) d + d 1 2 c 2 2 e 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 h + l s d 2 f 1           × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 m + l n + d 2 c 1 2 c 2 2 f 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 p 2 h + s d 1 2 g 1           × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ( 1 M 3 ) 2 p + l c 1 c 2 f 1 f 2 e 1 e 2 g 1 g 2 + 1           × H 2 h + 2 m + 2 l s n 2 c 1 2 c 2 2 f 1 2 f 2 + 2 ( 0 ) H 4 p 2 h 2 m + s + n 2 e 1 2 e 2 2 g 1 2 g 2 ( 0 ) ,                                        
A 2 ( z ) = ( 1 λ | B | ) 2 1 2 4 p + 2 l ( p ! ) 2 π 3 M 1 M 2 M 3 ( 1 2 M 2 q 2 2 M 1 M 2 ω 0 2 ) ( 2 p + l ) / 2 ( 2 q ω 0 ) 2 p + l                  × m = 0 p n = 0 l c 1 = 0 [ ( 2 m + l n ) / 2 ] e 1 = 0 [ ( 2 p 2 m + n ) / 2 ] h = 0 p s = 0 l d = 0 2 h + l s c 2 = 0 [ d / 2 ] d 1 = 0 2 p 2 h + s e 2 = 0 [ d 1 / 2 ] f 1 = 0 [ ( 2 h + l s d ) / 2 ]              × f 2 = 0 [ ( 2 m + l n + d 2 c 1 2 c 2 ) / 2 ] g 1 = 0 [ ( 2 p 2 h + s d 1 ) / 2 ] g 2 = 0 [ ( 2 p 2 m + n + d 1 2 e 1 2 c 2 ) / 2 ] ( i n ) * i s ( p m ) ( l n ) ( p h ) ( l s )             × ( 2 h + l s d ) ( 2 p 2 h + s d 1 ) ( 1 ) c 1 + c 2 + e 1 + e 2 + f 1 + f 2 + g 1 + g 2 ( 2 m + l n ) ! c 1 ! ( 2 m + l n 2 c 1 ) ! d ! c 2 ! ( d 2 c 2 ) !             × ( 2 p 2 m + n ) ! e 1 ! ( 2 p 2 m + n 2 e 1 ) ! d 1 ! e 2 ! ( d 1 2 e 2 ) ! ( 2 m + l n + d 2 c 1 2 c 2 ) ! f 2 ! ( 2 m + l n + d 2 c 1 2 c 2 2 f 2 ) !             × ( 2 h + l s d ) ! f 1 ! ( 2 h + l s d 2 f 1 ) ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 ) ! g 2 ! ( 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ) ! ( 2 p 2 h + s d 1 ) ! g 1 ! ( 2 p 2 h + s d 1 2 g 1 ) !             × ( 2 i ) 4 c 1 + 4 c 2 + 4 e 1 + 4 e 2 + 2 g 1 + 2 g 2 + 2 f 1 + 2 f 2 d d 1 6 p 3 l ( 1 M 2 ) d + d 1 2 c 1 2 c 2 2 e 1 2 e 2 ( 2 q ω 0 ) 2 c 1 2 e 1             × ( 2 q 2 σ g 2 M 1 2 ω 0 2 q 2 M 1 ) d + d 1 2 c 2 2 e 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 h + l s d 2 f 1            × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 m + l n + d 2 c 1 2 c 2 2 f 2 ( 2 i q k 2 B M 1 2 ω 0 2 q 2 M 1 ) 2 p 2 h + s d 1 2 g 1            × ( k 2 M 1 M 2 σ g 2 B k M 2 B * ) 2 p 2 m + n + d 1 2 e 1 2 e 2 2 g 2 ( 1 M 3 ) 2 p + l e 1 e 2 g 1 g 2 c 1 c 2 f 1 f 2            × H 2 h + 2 m + 2 l s n 2 c 1 2 c 2 2 f 1 2 f 2 ( 0 ) H 4 p 2 h 2 m + s + n 2 e 1 2 e 2 2 g 1 2 g 2 ( 0 ) ,                                              
( A B C D ) = ( 1 z 0 1 ) .

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