Abstract

A radial phased-locked (PL) Lorentz beam array provides an appropriate theoretical model to describe a coherent diode laser array, which is an efficient radiation source for high-power beaming use. The propagation of a radial PL Lorentz beam array in turbulent atmosphere is investigated. Based on the extended Huygens-Fresnel integral and some mathematical techniques, analytical formulae for the average intensity and the effective beam size of a radial PL Lorentz beam array are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a radial PL Lorentz beam array in turbulent atmosphere are numerically calculated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a radial PL Lorentz beam array in turbulent atmosphere are discussed in detail.

© 2011 OSA

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2011

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (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(9), 810–818 (2011).
[CrossRef]

H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(6), 1016–1021 (2011).
[CrossRef] [PubMed]

2010

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
[CrossRef] [PubMed]

P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010).
[CrossRef]

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010).
[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(5), 375–384 (2010).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010).
[CrossRef]

2009

L. Wang and W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam array,” J. Opt. A, Pure Appl. Opt. 11(6), 065703 (2009).
[CrossRef]

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[CrossRef]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[CrossRef] [PubMed]

S. P. Ng and P. B. Phua, “Coherent polarization locking of a diode emitter array,” Opt. Lett. 34(13), 2042–2044 (2009).
[CrossRef] [PubMed]

X. Li and X. Ji, “Angular spread and directionality of the Hermite-Gaussian array beam propagating through atmospheric turbulence,” Appl. Opt. 48(22), 4338–4347 (2009).
[CrossRef] [PubMed]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
[CrossRef]

2008

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3579 (2008).
[CrossRef]

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[CrossRef]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[CrossRef] [PubMed]

K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16(26), 21315–21320 (2008).
[CrossRef] [PubMed]

2007

O. El Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A, 68240A-8 (2007).
[CrossRef]

2006

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31(5), 568–570 (2006).
[CrossRef] [PubMed]

2005

1990

1989

J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55(6), 531–533 (1989).
[CrossRef]

1980

1979

1976

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

1975

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

Baykal, Y.

Baykal, Y. K.

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010).
[CrossRef]

Brennan, T. M.

J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55(6), 531–533 (1989).
[CrossRef]

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(9), 810–818 (2011).
[CrossRef]

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010).
[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(5), 375–384 (2010).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[CrossRef] [PubMed]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31(5), 568–570 (2006).
[CrossRef] [PubMed]

Carter, W. H.

Chen, F.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

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, 68240A-8 (2007).
[CrossRef]

Chen, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

Dang, A.

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, 68240A-8 (2007).
[CrossRef]

Du, X.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[CrossRef]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[CrossRef]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[CrossRef] [PubMed]

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

Durst, F.

Evans, W. A. B.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

Eyyuboglu, H. T.

Gawhary, O. E.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

Gawhary, O. El

O. El Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

Guo, H.

Hammons, B. E.

J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55(6), 531–533 (1989).
[CrossRef]

He, S.

Hobimer, J. P.

J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55(6), 531–533 (1989).
[CrossRef]

Ji, X.

Li, J.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

Li, X.

Liu, Z.

P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010).
[CrossRef]

Luo, B.

Ma, H.

P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

Ma, Y.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010).
[CrossRef]

Myers, D. R.

J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55(6), 531–533 (1989).
[CrossRef]

Naqwi, A.

Ng, S. P.

Ou, B.

Phua, P. B.

Plonus, M. A.

Qu, J.

Schmidt, P. P.

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

Severini, S.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

O. El Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

Tang, H.

Torre, A.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

Wang, F.

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010).
[CrossRef]

Wang, L.

L. Wang and W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam array,” J. Opt. A, Pure Appl. Opt. 11(6), 065703 (2009).
[CrossRef]

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[CrossRef]

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[CrossRef]

Wang, S. C. H.

Wang, X.

P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010).
[CrossRef]

Wang, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

Xin, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

Xu, S.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

Xu, X.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

Xu, Y.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

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J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A, 68240A-8 (2007).
[CrossRef]

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[CrossRef]

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[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(5), 375–384 (2010).
[CrossRef]

Zhao, D.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

Zheng, J.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

Zheng, W.

L. Wang and W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam array,” J. Opt. A, Pure Appl. Opt. 11(6), 065703 (2009).
[CrossRef]

Zheng, X.

Zhou, G.

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010).
[CrossRef]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
[CrossRef]

K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16(26), 21315–21320 (2008).
[CrossRef] [PubMed]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3579 (2008).
[CrossRef]

Zhou, M.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
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[CrossRef]

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P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010).
[CrossRef]

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L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[CrossRef]

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[CrossRef]

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[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3579 (2008).
[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(5), 375–384 (2010).
[CrossRef]

J. Opt.

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

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[CrossRef]

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[CrossRef]

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[CrossRef]

Opt. Express

Opt. Laser Technol.

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43(3), 506–514 (2011).
[CrossRef]

Opt. Lett.

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, 68240A-8 (2007).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a radial PL Lorentz beam array in the source plane.

Fig. 2
Fig. 2

Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 2mm, R = 3mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 350zr.

Fig. 6
Fig. 6

Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 8, w0x = w0y = 2mm, R = 3mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 350zr.

Fig. 3
Fig. 3

Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 2mm, R = 3mm, and Cn2 = 10−15m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 1500zr.

Fig. 4
Fig. 4

Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 1mm, R = 3mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 1500zr.

Fig. 5
Fig. 5

Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 2mm, R = 5mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 200zr.

Fig. 7
Fig. 7

The effective beam size in the x-direction of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere. N = 6 and R = 3mm (a) Cn2 = 10−14m-2/3. w0x and w0y take different values. (b) w0x = w0y = 2mm, and Cn2 takes different value.

Fig. 8
Fig. 8

The effective beam size in the x-direction of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere. w0x = w0y = 2mm and Cn2 = 10−14m-2/3. (a) N = 6, and R takes different values. (b) R = 3mm, and N takes different value.

Tables (1)

Tables Icon

Table 1 Value of the weight coefficient σ2m.

Equations (23)

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E n ( r 0 ,0)= 1 w 0x w 0y [1+ ( x 0 a nx ) 2 / w 0x 2 ][1+ ( y 0 a ny ) 2 / w 0y 2 ] exp(i φ n ),
a nx =Rcos φ n , a ny =Rsin φ n , φ n =n φ 0 =2nπ/N, n=1,2,3,,N,
E( r 0 ,0)= n=1 N E n ( r 0 ,0) .
E( r 0 ,0)= π 2 w 0x w 0y n=1 N m 1 =0 M m 2 =0 M σ 2 m 1 σ 2 m 2 H 2 m 1 ( x 0 w 0x ) H 2 m 2 ( y 0 w 0y )exp( x 0 2 2 w 0x 2 y 0 2 2 w 0y 2 +i φ n ),
x 0 = x 0 a nx , y 0 = y 0 a ny .
E(r,z)= ik 2πz E( r 0 ,0 )exp[ ik 2z ( r 0 r) 2 +ψ( r 0 ,r) ]d x 0 d y 0 ,
<I(r,z)>= k 2 4 π 2 z 2 E( r 01 ,0 ) E ( r 02 ,0)exp [ ik 2z ( r 01 r) 2 + ik 2z ( r 02 r) 2 ] ×<exp[ψ( r 01 ,r)+ ψ ( r 02 ,r)]>d r 01 d r 02 ,
<exp[ψ( r 01 ,r)+ ψ ( r 02 ,r)]>=exp[ ( r 01 r 02 ) 2 ρ 0 2 ],
H 2 m 1 (x) exp[ (xy) 2 /Ω]dx= πΩ (1Ω) m 1 H 2 m 1 [y (1Ω) 1/2 ],
H 2 m 1 (x)= l=0 m 1 (1) l (2 m 1 )! l!(2 m 1 2l)! (2x) 2 m 1 2l ,
(x+y) 2 m 1 = u=0 2 m 1 ( 2 m 1 u ) x 2 m 1 u y u ,
x 2t exp(b x 2 +2cx)dx=(2t)! π b ( c b ) 2t exp( c 2 b ) s=0 t 1 s!(2t2s)! ( b 4 c 2 ) s ,
I n 1 , n 2 (j,z)= kπ 4z 1 α 1j α 2j m 1 =0 M m 1 =0 M exp( β 1j 2 4 α 1j + β 2j 2 4 α 2j a n 2j 2 2 w 0j 2 a n 1j 2 ρ 0 2 + ik a n 1j 2 2z ik a n 1j j z ) σ 2 m 1 σ 2 m 1 × ( 1 1 α 1j ) m 1 l 1 =0 m 1 (1) l 1 (2 m 1 )! l 1 !(2 m 1 2 l 1 )! l 2 =0 2 m 1 2 l 1 ( 2 m 1 2 l 1 l 2 ) γ j 2 m 1 2 l 1 l 2 δ j l 2 l 3 =0 m 1 (1) l 3 (2 m 1 )! l 3 !(2 m 1 2 l 3 )! × l 4 =0 2 m 1 2 l 3 ( 2 m 1 2 l 3 l 4 ) ( 2 a n 2j w 0j ) 2 m 1 2 l 3 l 4 ( l 2 + l 4 )! s=0 [( l 2 + l 4 )/2] α 2j s l 2 l 4 β 2j l 2 + l 4 2s s!( l 2 + l 4 2s)! ,
α 1j = 1 2 + w 0j 2 ρ 0 2 ik w 0j 2 2z , α 2j = 1 2 + w 0j 2 ρ 0 2 + ik w 0j 2 2z w 0j 4 α 1j ρ 0 4 ,
β 1j = ik w 0j a n 1j z 2 w 0j a n 1j ρ 0 2 ik w 0j j z , β 2j = a n 2j w 0j + β w 1j 0j 2 α 1j ρ 0 2 + 2 w 0j a n 1j ρ 0 2 + ik w 0j j z ,
γ j = β 1j ( α 1j 2 α 1j ) 1/2 , δ j = w 0j 2 ( α 1j 2 α 1j ) 1/2 ρ 0 2 .
W jz = 2 j 2 <I(r,z)>dxdy <I(r,z)>dxdy .
W xz = 2 n 1 =1 N n 2 =1 N A n 1 , n 2 (x,z;2) A n 1 , n 2 (y,z;0)exp[i( φ n 1 φ n 2 )] n 1 =1 N n 2 =1 N A n 1 , n 2 (x,z;0) A n 1 , n 2 (y,z;0)exp[i( φ n 1 φ n 2 )] ,
W yz = 2 n 1 =1 N n 2 =1 N A n 1 , n 2 (x,z;0) A n 1 , n 2 (y,z;2)exp[i( φ n 1 φ n 2 )] n 1 =1 N n 2 =1 N A n 1 , n 2 (x,z;0) A n 1 , n 2 (y,z;0)exp[i( φ n 1 φ n 2 )] ,
A n 1 , n 2 (j,z;v)= m 1 =0 M m 1 =0 M exp( ξ j 2 4 α 1j + η j 2 4 α 2j a n 2j 2 2 w 0j 2 a n 1j 2 ρ 0 2 + ik a n 1j 2 2z ) σ 2 m 1 σ 2 m 1 ( 1 1 α 1j ) m 1 × l 1 =0 m 1 (1) l 1 (2 m 1 )! l 1 !(2 m 1 2 l 1 )! l 2 =0 2 m 1 2 l 1 ( 2 m 1 2 l 1 l 2 ) t 1 =0 2 m 1 2 l 1 l 2 ( 2 m 1 2 l 1 l 2 t 1 ) p j 2 m 1 2 l 1 l 2 t 1 × q j t 1 δ j l 2 l 3 =0 m 1 (1) l 3 (2 m 1 )! l 3 !(2 m 1 2 l 3 )! l 4 =0 2 m 1 2 l 3 ( 2 m 1 2 l 3 l 4 ) ( 2 a n 2j w 0j ) 2 m 1 2 l 3 l 4 ( l 2 + l 4 )! × s=0 [( l 2 + l 4 )/2] α 2j s l 2 l 4 s!( l 2 + l 4 2s)! t 2 =0 l 2 + l 4 2s ( l 2 + l 4 2s t 2 ) η j l 2 + l 4 2s t 2 μ j t 2 2 t 1 t 2 1-v ×exp( τ 2j 2 4 τ 1j ) τ 1j 1( t 1 + t 2 +v)/2 2 t 1 + t 2 +v { τ 1j [1+ (1) t 1 + t 2 ]Γ( t 1 + t 2 +1+v 2 ) × F 1 1 ( t 1 + t 2 +v 2 ; 1 2 ; τ 2j 2 4 τ 1j ) τ 2j [ (1) t 1 + t 2 1]Γ( t 1 + t 2 +2+v 2 ) × F 1 1 ( t 1 + t 2 +v1 2 ; 3 2 ; τ 2j 2 4 τ 1j ) }, v=0 or 2,
ξ j = ik w 0j a n 1j z 2 w 0j a n 1j ρ 0 2 , η j = a n 2j w 0j + ξ j w 0j 2 α 1j ρ 0 2 + 2 w 0j a n 1j ρ 0 2 , p j = ξ j ( α 1j 2 α 1j ) 1/2 ,
q j = ik w 0j z ( α 1j 2 α 1j ) 1/2 , μ j = ik w 0j z ( 1 w 0j 2 α 1j ρ 0 2 ),
τ 1j = k 2 w 0j 2 4 z 2 α 1j + k 2 w 0j 2 4 z 2 α 2j ( 1 w 0j 2 α 1j ρ 0 2 ) 2 , τ 2j = ik w 0j η j 2z α 2j ( 1 w 0j 2 α 1j ρ 0 2 ) ik w 0j ξ j 2z α 1j ik a n 1j z .

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