Abstract

Based on the second-order and the higher-order moments, analytical expressions for the beam propagation factors of a Lorentz–Gauss vortex beam with l=1 have been derived, and analytical propagation expressions for the kurtosis parameters of a Lorentz–Gauss vortex beam with l=1 through a paraxial and real ABCD optical system have also been presented. The M2 factor is determined by the parameters a and b and decreases with increasing the parameter a or b. The M2 factor is validated to be larger than 2. The kurtosis parameters depend on the diffraction-free ranges of the Lorentz part, the parameters a and b, and the ratio A/B. The kurtosis parameters of a Lorentz–Gauss vortex beam propagating in free space are demonstrated in different reference planes. In the far field, the kurtosis parameter K decreases with increasing one of the parameters a and b. Upon propagation, the kurtosis parameter K first decreases, then increases, and finally tends to a saturated value. In any case, the kurtosis parameter K is larger than 2. This research is beneficial to optical trapping, guiding, and manipulation of microscopic particles and atoms using Lorentz–Gauss vortex beams.

© 2014 Optical Society of America

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  1. A. Naqwi and F. Durst, “Focus of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).
    [CrossRef]
  2. J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
    [CrossRef]
  3. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
    [CrossRef]
  4. G. Zhou, “Focal shift of focused truncated Lorentz–Gauss beam,” J. Opt. Soc. Am. A 25, 2594–2599 (2008).
    [CrossRef]
  5. G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
    [CrossRef]
  6. A. Torre, “Wigner distribution function of Lorentz–Gauss beams: a note,” Appl. Phys. B 109, 671–681 (2012).
    [CrossRef]
  7. C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
    [CrossRef]
  8. G. Zhou, “Fractional Fourier transform of Lorentz–Gauss beams,” J. Opt. Soc. Am. A 26, 350–355 (2009).
    [CrossRef]
  9. W. Du, C. Zhao, and Y. Cai, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Lasers Eng. 49, 25–31 (2011).
    [CrossRef]
  10. G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere,” Opt. Express 18, 726–731 (2010).
    [CrossRef]
  11. H. P. Zheng, R. Chen, and C. H. R. Ooi, “Self-focusing dynamics of Lorentz–Gaussian beams in Kerr media,” Lasers Eng. 24, 345–354 (2013).
  12. A. Keshavarz and G. Honarasa, “Propagation of Lorentz–Gaussian beams in strongly nonlocal nonlinear media,” Commun. Theor. Phys. 61, 241–245 (2014).
    [CrossRef]
  13. Q. Sun, A. Li, K. Zhou, Z. Liu, G. Fang, and S. Liu, “Virtual source for rotational symmetric Lorentz–Gaussian beam,” Chin. Opt. Lett. 10, 062601 (2012).
    [CrossRef]
  14. Y. Jiang, K. Huang, and X. Lu, “Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle,” Opt. Express 19, 9708–9713 (2011).
    [CrossRef]
  15. G. Zhou, “Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system,” Opt. Express 18, 4637–4643 (2010).
    [CrossRef]
  16. H. T. Eyyuboğlu, “Partially coherent Lorentz–Gaussian beam and its scintillations,” Appl. Phys. B 103, 755–762 (2011).
    [CrossRef]
  17. C. Zhao and Y. Cai, “Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere,” J. Mod. Opt. 58, 810–818 (2011).
    [CrossRef]
  18. Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291, 19–25 (2013).
    [CrossRef]
  19. G. Zhou, X. Wang, and X. Chu, “Fractional Fourier transform of Lorentz–Gauss vortex beams,” Sci. China-Phys. Mech. Astron. 56, 1487–1494 (2013).
    [CrossRef]
  20. F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
    [CrossRef]
  21. Y. Ni and G. Zhou, “Nonparaxial propagation of Lorentz–Gauss vortex beams in uniaxial crystals orthogonal to the optical axis,” Appl. Phys. B 108, 883–890 (2012).
    [CrossRef]
  22. G. Zhou and G. Ru, “Propagation of a Lorentz–Gauss vortex beam in a turbulent atmosphere,” PIER 143, 143–163 (2013).
    [CrossRef]
  23. G. Zhou and G. Ru, “Angular momentum density of a linearly polarized Lorentz–Gauss vortex beam,” Opt. Commun. 313, 157–169 (2014).
    [CrossRef]
  24. R. Martínez-Herrero and P. M. Mejías, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef]
  25. R. Martínez-Herrero, P. M. Mejías, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
    [CrossRef]
  26. Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381–1386 (2005).
    [CrossRef]
  27. D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352–356 (2005).
    [CrossRef]
  28. G. Nemes and J. Serna, “Laser beam characterization with use of second order moments: an overview,” OSA TOPS 17, 200–207 (1998).
  29. B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719–723 (2000).
  30. B. D. Bock, Multivariate Statistical Method in Behavioral Research (McGraw-Hill, 1975).
  31. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
    [CrossRef]
  32. G. Zhou and X. Chu, “M2-factor of a partially coherent Lorentz–Gauss beam in a turbulent atmosphere,” Appl. Phys. B 100, 909–915 (2010).
    [CrossRef]

2014

G. Zhou and G. Ru, “Angular momentum density of a linearly polarized Lorentz–Gauss vortex beam,” Opt. Commun. 313, 157–169 (2014).
[CrossRef]

A. Keshavarz and G. Honarasa, “Propagation of Lorentz–Gaussian beams in strongly nonlocal nonlinear media,” Commun. Theor. Phys. 61, 241–245 (2014).
[CrossRef]

2013

H. P. Zheng, R. Chen, and C. H. R. Ooi, “Self-focusing dynamics of Lorentz–Gaussian beams in Kerr media,” Lasers Eng. 24, 345–354 (2013).

G. Zhou and G. Ru, “Propagation of a Lorentz–Gauss vortex beam in a turbulent atmosphere,” PIER 143, 143–163 (2013).
[CrossRef]

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291, 19–25 (2013).
[CrossRef]

G. Zhou, X. Wang, and X. Chu, “Fractional Fourier transform of Lorentz–Gauss vortex beams,” Sci. China-Phys. Mech. Astron. 56, 1487–1494 (2013).
[CrossRef]

F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
[CrossRef]

2012

Y. Ni and G. Zhou, “Nonparaxial propagation of Lorentz–Gauss vortex beams in uniaxial crystals orthogonal to the optical axis,” Appl. Phys. B 108, 883–890 (2012).
[CrossRef]

A. Torre, “Wigner distribution function of Lorentz–Gauss beams: a note,” Appl. Phys. B 109, 671–681 (2012).
[CrossRef]

Q. Sun, A. Li, K. Zhou, Z. Liu, G. Fang, and S. Liu, “Virtual source for rotational symmetric Lorentz–Gaussian beam,” Chin. Opt. Lett. 10, 062601 (2012).
[CrossRef]

2011

Y. Jiang, K. Huang, and X. Lu, “Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle,” Opt. Express 19, 9708–9713 (2011).
[CrossRef]

W. Du, C. Zhao, and Y. Cai, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Lasers Eng. 49, 25–31 (2011).
[CrossRef]

H. T. Eyyuboğlu, “Partially coherent Lorentz–Gaussian beam and its scintillations,” Appl. Phys. B 103, 755–762 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere,” J. Mod. Opt. 58, 810–818 (2011).
[CrossRef]

2010

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere,” Opt. Express 18, 726–731 (2010).
[CrossRef]

G. Zhou, “Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system,” Opt. Express 18, 4637–4643 (2010).
[CrossRef]

G. Zhou and X. Chu, “M2-factor of a partially coherent Lorentz–Gauss beam in a turbulent atmosphere,” Appl. Phys. B 100, 909–915 (2010).
[CrossRef]

2009

G. Zhou, “Fractional Fourier transform of Lorentz–Gauss beams,” J. Opt. Soc. Am. A 26, 350–355 (2009).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[CrossRef]

2008

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz–Gauss beam,” J. Opt. Soc. Am. A 25, 2594–2599 (2008).
[CrossRef]

2006

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

2005

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352–356 (2005).
[CrossRef]

Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381–1386 (2005).
[CrossRef]

2000

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719–723 (2000).

1998

G. Nemes and J. Serna, “Laser beam characterization with use of second order moments: an overview,” OSA TOPS 17, 200–207 (1998).

1995

1993

1992

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

1990

Arias, M.

Bock, B. D.

B. D. Bock, Multivariate Statistical Method in Behavioral Research (McGraw-Hill, 1975).

Cai, Y.

C. Zhao and Y. Cai, “Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere,” J. Mod. Opt. 58, 810–818 (2011).
[CrossRef]

W. Du, C. Zhao, and Y. Cai, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Lasers Eng. 49, 25–31 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

Chen, R.

H. P. Zheng, R. Chen, and C. H. R. Ooi, “Self-focusing dynamics of Lorentz–Gaussian beams in Kerr media,” Lasers Eng. 24, 345–354 (2013).

Chen, T.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Chu, X.

G. Zhou, X. Wang, and X. Chu, “Fractional Fourier transform of Lorentz–Gauss vortex beams,” Sci. China-Phys. Mech. Astron. 56, 1487–1494 (2013).
[CrossRef]

G. Zhou and X. Chu, “M2-factor of a partially coherent Lorentz–Gauss beam in a turbulent atmosphere,” Appl. Phys. B 100, 909–915 (2010).
[CrossRef]

G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere,” Opt. Express 18, 726–731 (2010).
[CrossRef]

Deng, D.

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352–356 (2005).
[CrossRef]

Ding, G.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Du, W.

W. Du, C. Zhao, and Y. Cai, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Lasers Eng. 49, 25–31 (2011).
[CrossRef]

Durst, F.

Eyyuboglu, H. T.

H. T. Eyyuboğlu, “Partially coherent Lorentz–Gaussian beam and its scintillations,” Appl. Phys. B 103, 755–762 (2011).
[CrossRef]

Fang, G.

Gao, X.

F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
[CrossRef]

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

Honarasa, G.

A. Keshavarz and G. Honarasa, “Propagation of Lorentz–Gaussian beams in strongly nonlocal nonlinear media,” Commun. Theor. Phys. 61, 241–245 (2014).
[CrossRef]

Huang, K.

Jiang, Y.

Keshavarz, A.

A. Keshavarz and G. Honarasa, “Propagation of Lorentz–Gaussian beams in strongly nonlocal nonlinear media,” Commun. Theor. Phys. 61, 241–245 (2014).
[CrossRef]

Li, A.

Liu, S.

Liu, Z.

Lu, X.

Lü, B.

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719–723 (2000).

Ma, H.

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719–723 (2000).

Martínez-Herrero, R.

Mei, Z.

Mejías, P. M.

Naqwi, A.

Nemes, G.

G. Nemes and J. Serna, “Laser beam characterization with use of second order moments: an overview,” OSA TOPS 17, 200–207 (1998).

Ni, Y.

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291, 19–25 (2013).
[CrossRef]

Y. Ni and G. Zhou, “Nonparaxial propagation of Lorentz–Gauss vortex beams in uniaxial crystals orthogonal to the optical axis,” Appl. Phys. B 108, 883–890 (2012).
[CrossRef]

Ooi, C. H. R.

H. P. Zheng, R. Chen, and C. H. R. Ooi, “Self-focusing dynamics of Lorentz–Gaussian beams in Kerr media,” Lasers Eng. 24, 345–354 (2013).

Ru, G.

G. Zhou and G. Ru, “Angular momentum density of a linearly polarized Lorentz–Gauss vortex beam,” Opt. Commun. 313, 157–169 (2014).
[CrossRef]

G. Zhou and G. Ru, “Propagation of a Lorentz–Gauss vortex beam in a turbulent atmosphere,” PIER 143, 143–163 (2013).
[CrossRef]

Rui, F.

F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
[CrossRef]

Serna, J.

G. Nemes and J. Serna, “Laser beam characterization with use of second order moments: an overview,” OSA TOPS 17, 200–207 (1998).

Severini, S.

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

Sun, Q.

Ting, M.

F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
[CrossRef]

Torre, A.

A. Torre, “Wigner distribution function of Lorentz–Gauss beams: a note,” Appl. Phys. B 109, 671–681 (2012).
[CrossRef]

Wang, X.

G. Zhou, X. Wang, and X. Chu, “Fractional Fourier transform of Lorentz–Gauss vortex beams,” Sci. China-Phys. Mech. Astron. 56, 1487–1494 (2013).
[CrossRef]

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Yang, J.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Yuan, X.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Zhang, D.

F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
[CrossRef]

Zhao, C.

W. Du, C. Zhao, and Y. Cai, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Lasers Eng. 49, 25–31 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere,” J. Mod. Opt. 58, 810–818 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

Zhao, D.

Zheng, H. P.

H. P. Zheng, R. Chen, and C. H. R. Ooi, “Self-focusing dynamics of Lorentz–Gaussian beams in Kerr media,” Lasers Eng. 24, 345–354 (2013).

Zhou, G.

G. Zhou and G. Ru, “Angular momentum density of a linearly polarized Lorentz–Gauss vortex beam,” Opt. Commun. 313, 157–169 (2014).
[CrossRef]

G. Zhou, X. Wang, and X. Chu, “Fractional Fourier transform of Lorentz–Gauss vortex beams,” Sci. China-Phys. Mech. Astron. 56, 1487–1494 (2013).
[CrossRef]

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291, 19–25 (2013).
[CrossRef]

G. Zhou and G. Ru, “Propagation of a Lorentz–Gauss vortex beam in a turbulent atmosphere,” PIER 143, 143–163 (2013).
[CrossRef]

Y. Ni and G. Zhou, “Nonparaxial propagation of Lorentz–Gauss vortex beams in uniaxial crystals orthogonal to the optical axis,” Appl. Phys. B 108, 883–890 (2012).
[CrossRef]

G. Zhou and X. Chu, “M2-factor of a partially coherent Lorentz–Gauss beam in a turbulent atmosphere,” Appl. Phys. B 100, 909–915 (2010).
[CrossRef]

G. Zhou, “Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system,” Opt. Express 18, 4637–4643 (2010).
[CrossRef]

G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere,” Opt. Express 18, 726–731 (2010).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of Lorentz–Gauss beams,” J. Opt. Soc. Am. A 26, 350–355 (2009).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz–Gauss beam,” J. Opt. Soc. Am. A 25, 2594–2599 (2008).
[CrossRef]

Zhou, K.

Zhuang, S.

F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
[CrossRef]

Appl. Opt.

Appl. Phys. B

G. Zhou and X. Chu, “M2-factor of a partially coherent Lorentz–Gauss beam in a turbulent atmosphere,” Appl. Phys. B 100, 909–915 (2010).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[CrossRef]

A. Torre, “Wigner distribution function of Lorentz–Gauss beams: a note,” Appl. Phys. B 109, 671–681 (2012).
[CrossRef]

H. T. Eyyuboğlu, “Partially coherent Lorentz–Gaussian beam and its scintillations,” Appl. Phys. B 103, 755–762 (2011).
[CrossRef]

Y. Ni and G. Zhou, “Nonparaxial propagation of Lorentz–Gauss vortex beams in uniaxial crystals orthogonal to the optical axis,” Appl. Phys. B 108, 883–890 (2012).
[CrossRef]

Chin. Opt. Lett.

Commun. Theor. Phys.

A. Keshavarz and G. Honarasa, “Propagation of Lorentz–Gaussian beams in strongly nonlocal nonlinear media,” Commun. Theor. Phys. 61, 241–245 (2014).
[CrossRef]

J. Mod. Opt.

C. Zhao and Y. Cai, “Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere,” J. Mod. Opt. 58, 810–818 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719–723 (2000).

J. Opt. A

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Lasers Eng.

H. P. Zheng, R. Chen, and C. H. R. Ooi, “Self-focusing dynamics of Lorentz–Gaussian beams in Kerr media,” Lasers Eng. 24, 345–354 (2013).

Opt. Commun.

G. Zhou and G. Ru, “Angular momentum density of a linearly polarized Lorentz–Gauss vortex beam,” Opt. Commun. 313, 157–169 (2014).
[CrossRef]

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291, 19–25 (2013).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

W. Du, C. Zhao, and Y. Cai, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Lasers Eng. 49, 25–31 (2011).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Optik

F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, “Focusing of linearly polarized Loretnz–Gauss beam with one optical vortex,” Optik 124, 2969–2973 (2013).
[CrossRef]

OSA TOPS

G. Nemes and J. Serna, “Laser beam characterization with use of second order moments: an overview,” OSA TOPS 17, 200–207 (1998).

Phys. Lett. A

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352–356 (2005).
[CrossRef]

PIER

G. Zhou and G. Ru, “Propagation of a Lorentz–Gauss vortex beam in a turbulent atmosphere,” PIER 143, 143–163 (2013).
[CrossRef]

Proc. SPIE

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Sci. China-Phys. Mech. Astron.

G. Zhou, X. Wang, and X. Chu, “Fractional Fourier transform of Lorentz–Gauss vortex beams,” Sci. China-Phys. Mech. Astron. 56, 1487–1494 (2013).
[CrossRef]

Other

B. D. Bock, Multivariate Statistical Method in Behavioral Research (McGraw-Hill, 1975).

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Figures (6)

Fig. 1.
Fig. 1.

Mx2, My2, and M2 factors of the Lorentz–Gauss vortex beam as functions of the parameters a and b.

Fig. 2.
Fig. 2.

(a) Given the parameter a, the M2 factor of the Lorentz–Gauss vortex beam as a function of the parameter b. (b) Given the parameter b, the M2 factor of the Lorentz–Gauss vortex beam as a function of the parameter a.

Fig. 3.
Fig. 3.

Kx parameter of the Lorentz–Gauss vortex beam in the different reference planes as functions of the parameters a and b.

Fig. 4.
Fig. 4.

Ky parameter of the Lorentz–Gauss vortex beam in the different reference planes as functions of the parameters a and b.

Fig. 5.
Fig. 5.

K parameter of the Lorentz–Gauss vortex beam in the different reference planes as functions of the parameters a and b.

Fig. 6.
Fig. 6.

K parameter of the Lorentz–Gauss vortex beam as a function of the axial propagation distance.

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

El(x0,y0,0)=w0xw0y(x0+iy0)l(w0x2+x02)(w0y2+y02)exp(x02+y02w02),
x0=x0|E1(x0,y0,0)|2dx0dy0|E1(x0,y0,0)|2dx0dy0=exp(2ρ02w02)x03+x0y02(w0x2+x02)2(w0y2+y02)2dx0dy0exp(2ρ02w02)x02+y02(w0x2+x02)2(w0y2+y02)2dx0dy0=0,
y0=y0|E1(x0,y0,0)|2dx0dy0|E1(x0,y0,0)|2dx0dy0=exp(2ρ02w02)x02y0+y03(w0x2+x02)2(w0y2+y02)2dx0dy0exp(2ρ02w02)x02+y02(w0x2+x02)2(w0y2+y02)2dx0dy0=0,
x02=(x0x0)2|E1(x0,y0,0)|2dx0dy0|E1(x0,y0,0)|2dx0dy0=w0x2T2(a)T0(b)+bT1(a)T1(b)aT1(a)T0(b)+bT0(a)T1(b),
y02=(y0y0)2|E1(x0,y0,0)|2dx0dy0|E1(x0,y0,0)|2dx0dy0=w0y2T0(a)T2(b)+aT1(a)T1(b)aT1(a)T0(b)+bT0(a)T1(b),
T0(ε)=(12ε)πexp(ε)erfc(ε)+2ε,
T1(ε)=(1+2ε)πexp(ε)erfc(ε)2ε,
T2(ε)=2(ε+1)ε(3ε+2ε2)πexp(ε)erfc(ε),
θx2=|E1(x0,y0,0)x0|2dx0dy0k2|E1(x0,y0,0)|2dx0dy0=aT3(a)T0(b)+bT4(a)T1(b)6k2w0x2[aT1(a)T0(b)+bT0(a)T1(b)],
θy2=|E1(x0,y0,0)y0|2dx0dy0k2|E1(x0,y0,0)|2dx0dy0=bT0(a)T3(b)+aT1(a)T4(b)6k2w0y2[aT1(a)T0(b)+bT0(a)T1(b)],
T3(ε)=(312ε30ε244ε3)πexp(ε)erfc(ε)+(6+8ε+44ε2)ε,
T4(ε)=(36ε24ε3)πexp(ε)erfc(ε)+(6+4ε+4ε2)ε.
x0θx=πik|E1(x0,y0,0)|2dx0dy0×{x0[E1(x0,y0,0)x0]*E1(x0,y0,0)x0E1(x0,y0,0)x0[E1(x0,y0,0)]*}dx0dy0=0,
y0θy=πik|E1(x0,y0,0)|2dx0dy0×{y0[E1(x0,y0,0)y0]*E1(x0,y0,0)y0E1(x0,y0,0)y0[E1(x0,y0,0)]*}dx0dy0=0,
Mx2=2k(x02θx2x0θx2)1/2=2kx02θx2,
My2=2k(y02θy2y0θy2)1/2=2ky02θy2.
M2=(Mx2My2)1/2.
Kx=x4x22,
xn=xn|E1(x,y,z)|2dxdy|E1(x,y,z)|2dxdy,n=2,4.
E1(x,y,z)=1iλBexp(ikz)E1(x0,y0,0)×exp{ik2B[Aρ022(x0x+yy0)+Dρ2]}dx0dy0=π24λBexp(ikCρ22A){w0x[Ex+(x)Ex(x)]×[Ey+(y)+Ey(y)]+iw0y[Ex+(x)+Ex(x)]×[Ey+(y)Ey(y)]},
Es±(s)=exp[kA2iB(w0s±isA)2]erfc[kA2iB(w0s±isA)],
sm=(As0+BΘs)m+δm4·3A2B2/k2,
Ks=(As0+BΘs)4+3A2B2/k2(As0+BΘs)22.
s0mΘsn=s0mE1(x0,y0,0)ns0nE1*(x0,y0,0)dx0dy02(ik)n|E1(x0,y0,0)|2dx0dy0+c.c.,
s0Θs=0,s03Θs=0,s0Θs3=0.
Ks=1[s02+(A/B)2Θs2]2[s04+3(A/B)2×(2s02Θs2+1/k2)+(A/B)4Θs4].
x04=w0x4T5(a)T0(b)+bT6(a)T1(b)aT1(a)T0(b)+bT0(a)T1(b),
y04=w0y4T0(a)T5(b)+aT1(a)T6(b)aT1(a)T0(b)+bT0(a)T1(b),
T5(ε)=(5+2ε)επexp(ε)erfc(ε)+(14ε2ε2)ε1/2,
T6(ε)=2(ε+1)ε1/2(3+2ε)πexp(ε)erfc(ε).
Θx2=aT7(a)T0(b)+bT4(a)T1(b)6k2w0x2[aT1(a)T0(b)+bT0(a)T1(b)],
Θy2=bT0(a)T7(b)+aT1(a)T4(b)6k2w0y2[aT1(a)T0(b)+bT0(a)T1(b)],
T7(ε)=(6+16ε+4ε2)ε(3+18ε2+4ε3)×πexp(ε)erfc(ε).
x02Θx2=aT8(a)T0(b)+bT9(a)T1(b)6k2[aT1(a)T0(b)+bT0(a)T1(b)],
y02Θy2=bT0(a)T8(b)+aT1(a)T9(b)6k2[aT1(a)T0(b)+bT0(a)T1(b)],
T8(ε)=(3+36ε+30ε2+4ε3)πexp(ε)erfc(ε)(24+28ε+4ε2)ε,
T9(ε)=(3+12ε+18ε2+4ε3)πexp(ε)erfc(ε)(6+16ε+4ε2)ε.
Θx4=aT10(a)T0(b)+bT11(a)T1(b)10k4w0x4[aT1(a)T0(b)+bT0(a)T1(b)],
Θy4=bT0(a)T10(b)+aT1(a)T11(b)10k4w0y4[aT1(a)T0(b)+bT0(a)T1(b)],
T10(ε)=(15+270ε1620ε23140ε3670ε44ε5)×πexp(ε)erfc(ε)+(30+560ε+2802ε2+672ε3+4ε4)ε,
T11(ε)=(15+20ε3+30ε4+4ε5)πexp(ε)erfc(ε)(30+20ε+80ε2+28ε3+4ε4)ε.
K=(KxKy)1/2.

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