Abstract

The scintillation properties of astigmatic dark hollow beams (DHBs) in weak atmospheric turbulence were investigated in detail. An explicit expression for the on-axis scintillation index of an astigmatic DHB propagating in weak atmospheric turbulence was derived. It was found that the scintillation index value of an astigmatic DHB with suitable astigmatism (i.e., ratio of the beam waist size in the x direction to that in the y direction), dark size, beam waist size, and wavelength can be smaller than that of a stigmatic DHB and that of stigmatic and astigmatic flat-topped, annular, and Gaussian beams in weak atmospheric turbulence particularly at long propagation ranges. Our results will be useful in long-distance free-space optical communications.

© 2008 Optical Society of America

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2008 (3)

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

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83-87 (2008).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87-92 (2008).
[CrossRef]

2007 (8)

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillation index of incoherent general type beams in turbulence,” Atmos. Oceanic Opt. 20, 1105-1109 (2007).

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Y. Cai, C. Chen, and F. Wang, “Modified hollow Gaussian beam and its paraxial propagation,” Opt. Commun. 278, 34-41 (2007).
[CrossRef]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32, 3179-3181 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076-2078 (2007).
[CrossRef] [PubMed]

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

2006 (11)

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B 23, 1398-1407 (2006).
[CrossRef]

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

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

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

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
[CrossRef]

Y. Cai and L. Zhang, “Propagation of a hollow Gaussian beam through a paraxial misaligned optical system,” Opt. Commun. 265, 607-615 (2006).
[CrossRef]

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

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23, 1410-1418 (2006).
[CrossRef]

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

2005 (3)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere,” Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

2004 (4)

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

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21, 1290-1299 (2004).
[CrossRef]

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media, in Free-Space Laser Communication and Active Laser Illumination III, D.G.Voelz and J.C.Ricklin, eds. (SPIE, 2004).

2003 (2)

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

2002 (1)

2001 (2)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

2000 (2)

1997 (3)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Atomic funnel with evanescent light,” Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

1996 (2)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

1995 (1)

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

1994 (2)

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (1)

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1979 (1)

1978 (2)

1969 (1)

Ahmad, M. A.

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271-291 (2001).
[CrossRef]

Anderson, D. Z.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media, in Free-Space Laser Communication and Active Laser Illumination III, D.G.Voelz and J.C.Ricklin, eds. (SPIE, 2004).

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Arnaud, J. A.

Atewart, B. W.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Banakh, V. A.

Baykal, Y.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87-92 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillation index of incoherent general type beams in turbulence,” Atmos. Oceanic Opt. 20, 1105-1109 (2007).

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere,” Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21, 1290-1299 (2004).
[CrossRef]

Cai, Y.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87-92 (2008).
[CrossRef]

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillation index of incoherent general type beams in turbulence,” Atmos. Oceanic Opt. 20, 1105-1109 (2007).

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32, 3179-3181 (2007).
[CrossRef] [PubMed]

Y. Cai, C. Chen, and F. Wang, “Modified hollow Gaussian beam and its paraxial propagation,” Opt. Commun. 278, 34-41 (2007).
[CrossRef]

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

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B 23, 1398-1407 (2006).
[CrossRef]

Y. Cai and L. Zhang, “Propagation of a hollow Gaussian beam through a paraxial misaligned optical system,” Opt. Commun. 265, 607-615 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23, 1410-1418 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

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

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
[CrossRef]

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

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

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

Chakraborty, R.

Chavez-Cerda, S.

Chen, C.

Y. Cai, C. Chen, and F. Wang, “Modified hollow Gaussian beam and its paraxial propagation,” Opt. Commun. 278, 34-41 (2007).
[CrossRef]

Chen, Y.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87-92 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Choi, K.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Churnside, J. H.

Clifford, S. F.

Cochetti, F.

Consortini, A.

Cornell, E. A.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Deng, D.

Eyyuboglu, H. T.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87-92 (2008).
[CrossRef]

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

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillation index of incoherent general type beams in turbulence,” Atmos. Oceanic Opt. 20, 1105-1109 (2007).

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere,” Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

Fan, Z.

Fenichel, H.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Gao, W.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

Ge, D.

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

Ghosh, A.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic Press, 2000).

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gutierrez-Vega, J. C.

He, S.

Heckenberg, N. R.

Hill, R. J.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271-291 (2001).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978), Vol. 2.

Ito, H.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Atomic funnel with evanescent light,” Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Iturbe-Castillo, M. D.

Jhe, W.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Atomic funnel with evanescent light,” Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Kim, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

Kogelink, H.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Lee, H. S.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Lee, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

Lee, K. I.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Li, J.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Li, Y.

Lin, J.

Lin, Q.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

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

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

Littman, M. G.

Liu, J.

Liu, S.

Liu, Z.

Lu, X.

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

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

Marksteiner, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

McDuff, R.

Mei, Z.

Mironov, V. L.

Montgomery, D.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Nakata, T.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Noh, H.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

Ohtsu, M.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Atomic funnel with evanescent light,” Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Renn, M. J.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Rolston, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic Press, 2000).

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

Sakaki, K.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Atomic funnel with evanescent light,” Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Savage, C. M.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Smith, C. P.

Tian, G.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Vdovin, O.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Vetelino, F. E. S.

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media, in Free-Space Laser Communication and Active Laser Illumination III, D.G.Voelz and J.C.Ricklin, eds. (SPIE, 2004).

Wang, F.

Y. Cai, C. Chen, and F. Wang, “Modified hollow Gaussian beam and its paraxial propagation,” Opt. Commun. 278, 34-41 (2007).
[CrossRef]

Wang, X.

Wang, Y.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

White, A. G.

Wieman, C. E.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Wu, J.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Wu, Y. K.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Xu, S.

Yin, J.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

Yu, H.

Zhang, L.

Y. Cai and L. Zhang, “Propagation of a hollow Gaussian beam through a paraxial misaligned optical system,” Opt. Commun. 265, 607-615 (2006).
[CrossRef]

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B 23, 1398-1407 (2006).
[CrossRef]

Zhao, D.

Zhao, H.

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

Zoller, P.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. B (2)

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87-92 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Atmos. Oceanic Opt. (1)

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillation index of incoherent general type beams in turbulence,” Atmos. Oceanic Opt. 20, 1105-1109 (2007).

J. Opt. A, Pure Appl. Opt. (2)

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (2)

Opt. Commun. (5)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138, 287-292 (1997).
[CrossRef]

Y. Cai, C. Chen, and F. Wang, “Modified hollow Gaussian beam and its paraxial propagation,” Opt. Commun. 278, 34-41 (2007).
[CrossRef]

Y. Cai and L. Zhang, “Propagation of a hollow Gaussian beam through a paraxial misaligned optical system,” Opt. Commun. 265, 607-615 (2006).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

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

Opt. Lett. (10)

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493-1495 (2000).
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer generated holograms,” Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

X. Wang and M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18, 767-768 (1993).
[CrossRef] [PubMed]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32, 3179-3181 (2007).
[CrossRef] [PubMed]

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

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076-2078 (2007).
[CrossRef] [PubMed]

V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirchhoff method in problems of space-limited optical-beam propagation in turbulent atmosphere,” Opt. Lett. 4, 259-261 (1979).
[CrossRef] [PubMed]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

Phys. Lett. A (2)

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

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

Phys. Rev. A (3)

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Atomic funnel with evanescent light,” Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

Phys. Rev. Lett. (5)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, andH. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guiding by evanescent waves in micro-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Waves Random Media (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271-291 (2001).
[CrossRef]

Other (6)

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media, in Free-Space Laser Communication and Active Laser Illumination III, D.G.Voelz and J.C.Ricklin, eds. (SPIE, 2004).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic Press, 2000).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978), Vol. 2.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Contour graph for the normalized irradiance of an astigmatic DHB and crossline ( y = 0 ) for different values of N, q, w 0 x , and w 0 y . (a) N = 1 , q = 0 , w 0 x = w 0 y = 1 mm (stigmatic Gaussian beam); (b) N = 1 , q = 0 , w 0 x = 1.5 mm , w 0 y = 1 mm (astigmatic Gaussian beam); (c) N = 5 , q = 0 , w 0 x = w 0 y = 1 mm (stigmatic flat-topped beam); (d) N = 5 , q = 0 , w 0 x = 1.5 mm , w 0 y = 1 mm (astigmatic flat-topped beam); (e) N = 1 , q = 0.3 , w 0 x = w 0 y = 1 mm (stigmatic annular beam); (f) N = 1 , q = 0.3 , w 0 x = 1.5 mm , w 0 y = 1 mm (astigmatic annular beam); (g) N = 10 , q = 0.3 , w 0 x = w 0 y = 1 mm (stigmatic DHB); (h) N = 10 , q = 0.3 , w 0 x = 1.5 mm , w 0 y = 1 mm (astigmatic DHB).

Fig. 2
Fig. 2

Variation of the scintillation index of an astigmatic DHB against propagation length L for different values of N, q, w 0 x , and w 0 y in weak atmospheric turbulence with C n 2 = 10 15 m 2 3 , l 0 = 5 mm , and λ = 1.55 μ m . (a) a 1 , N = 1 , w 0 x = w 0 y = 1 cm , q = 0 (stigmatic Gaussian beam); a 2 , N = 1 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0 (astigmatic Gaussian beam); a 3 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB). (b) b 1 , N = 1 , w 0 x = w 0 y = 1 cm , q = 0 (stigmatic Gaussian beam); b 2 , N = 3 , w 0 x = w 0 y = 1 cm , q = 0 (stigmatic flat-topped beam); b 3 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0 (astigmatic flat-topped beam); b 4 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB). (c) c 1 , N = 1 , w 0 x = w 0 y = 1 cm , q = 0 (stigmatic Gaussian beam); c 2 , N = 1 , w 0 x = w 0 y = 1 cm , q = 0.9 (stigmatic annular beam); c 3 , N = 1 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic annular beam); c 4 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB). (d) d 1 , N = 1 , w 0 x = w 0 y = 1 cm , q = 0 (stigmatic Gaussian beam); d 2 , N = 3 , w 0 x = w 0 y = 1 cm , q = 0.9 (stigmatic DHB); d 3 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB).

Fig. 3
Fig. 3

Variation of the scintillation index of an astigmatic DHB against propagation length L for different values of the inner scale ( l 0 ) of the turbulence with C n 2 = 10 15 m 2 3 , N = 2 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.3 , and λ = 1.55 μ m .

Fig. 4
Fig. 4

Dependence of scintillation index of an astigmatic DHB on the beam order N, astigmatism w 0 x w 0 y and beam waist size w 0 y for different values of q and l 0 at L = 5 km with C n 2 = 10 15 m 2 3 and λ = 1.55 μ m .

Fig. 5
Fig. 5

Variation of scintillation index m 2 against wavelength λ of an astigmatic DHB at L = 3 km for different values of N, q, w 0 x , and w 0 y with l 0 = 5 mm and C n 2 = 10 15 m 2 3 . (a) a 1 , N = 1 , w 0 x = w 0 y = 1 cm , q = 0 (stigmatic Gaussian beam); a 2 , N = 1 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0 (astigmatic Gaussian beam); a 3 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB). (b) b 1 , N = 3 , w 0 x = w 0 y = 1 cm , q = 0 (stigmatic flat-topped beam); b 2 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0 (astigmatic flat-topped beam); b 3 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB). (c) c 1 , N = 1 , w 0 x = w 0 y = 1 cm , q = 0.9 (stigmatic annular beam); c 2 , N = 1 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic annular beam); c 3 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB). (d) d 1 , N = 3 , w 0 x = w 0 y = 1 cm , q = 0.9 (stigmatic DHB); d 2 , N = 3 , w 0 x = 1.5 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB); d 3 , N = 3 , w 0 x = 2 cm , w 0 y = 1 cm , q = 0.9 (astigmatic DHB).

Equations (14)

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u N ( x , y , 0 ) = n = 1 N ( 1 ) n 1 N ( N n ) [ exp ( x 2 w n x 2 y 2 w n y 2 ) exp ( x 2 w n q x 2 y 2 w n q y 2 ) ] ,
u N FS ( p , z ) = k exp ( i k z ) 2 π i z u N ( r , 0 ) exp [ i k 2 z ( p r ) 2 ] d x d y ,
u FS ( p , z ) = exp ( i k z + i k 2 z p 2 ) n = 1 N ( 1 ) n 1 N ( N n ) [ 1 ( 1 + 2 i z k w n x 2 ) ( 1 + 2 i z k w n y 2 ) exp ( k 2 w n x 2 p x 2 4 z 2 2 i k z w n x 2 k 2 w n y 2 p y 2 4 z 2 2 i k z w n y 2 ) 1 ( 1 + 2 i z k w n q x 2 ) ( 1 + 2 i z k w n q y 2 ) exp ( k 2 w n q x 2 p x 2 4 z 2 2 i k z w n q x 2 k 2 w n q y 2 p y 2 4 z 2 2 i k z w n q y 2 ) ] ,
m 2 = 4 B χ ( p = 0 , L ) = 4 π 0 L d z 1 d κ x d κ y [ Re ( H H ) + Re ( H * H ) ] ϕ n ( κ ) ,
H ( p = 0 , L , κ x , κ y , z 1 ) = k 2 2 π ( L z 1 ) u FS ( 0 , L ) d p 1 x d p 1 y u FS ( p 1 , z 1 ) exp ( i κ x p 1 x + i κ y p 1 y ) exp [ i k ( L z 1 ) + i k p 1 2 2 ( L z 1 ) ] ,
H ( p = 0 , L , κ x , κ y , z 1 ) = i k exp ( i k L ) u FS ( 0 , L ) n = 1 N ( 1 ) n 1 N ( N n ) [ 1 a n x a n y exp ( b n x κ x 2 + b n y κ y 2 ) 1 a n q x a n q y exp ( b n q x κ x 2 + b n q y κ y 2 ) ] ,
a n x = ( 2 i L k w n x 2 + 1 ) ,
b n x = ( L z 1 ) ( 2 z 1 i k w n x 2 ) 2 k ( 2 i L + k w n x 2 ) ,
ϕ n ( κ ) = 0.033 C n 2 κ 11 3 exp ( κ 2 κ l 2 ) [ 1 + 1.802 ( κ κ l ) 0.254 ( κ κ l ) 7 6 ] ,
m 2 = 0.132 π 2 C n 2 Re { k 2 exp ( 2 i k L ) [ u FS ( 0 , L ) ] 2 n = 1 N m = 1 N ( 1 ) n + m N 2 ( N n ) ( N m ) ) [ ( 1 a n x a n y a m x a m y m 1 2 1 a n x a n y a m q x a m q y m 2 2 1 a n q x a n q y a m x a m y m 3 2 + 1 a n q x a n q y a m q x a m q y m 4 2 ) ] + n = 1 N m = 1 N ( 1 ) n + m N 2 ( N n ) ( N m ) k 2 u FS ( 0 , L ) 2 × ( 1 a n x a n y a m x * a m y * m 5 2 1 a n x a n y a m q x * a m q y * m 6 2 1 a n q x a n q y a m x * a m y * m 7 2 + 1 a n q x a n q y a m q x * a m q y * m 8 2 ) } ,
m r 2 = Γ ( 5 6 ) 0 L ( α r ) 5 6 F ( 5 12 , 1 12 ; 1 ; β r 2 α r 2 ) d z 1 + 1.802 κ l Γ ( 1 3 ) 0 L ( α r ) 1 3 F ( 1 6 , 1 3 ; 1 ; β r 2 α r 2 ) d z 1 0.254 κ l 7 6 Γ ( 1 4 ) 0 L ( α r ) 1 4 F ( 1 8 , 3 8 ; 1 ; β r 2 α r 2 ) d z 1 ,
( r = 1 , 2 , , 8 ) ,
α r = 0.5 ( c r + d r ) 1 κ l 2 , β r = 0.5 ( c r d r ) , ( r = 1 , 2 , , 8 ) .
0 e α x x μ 1 J ν ( β x ) d x = ( β 2 α ) ν Γ ( μ + ν ) α μ Γ ( ν + 1 ) F ( μ + ν 2 , μ + ν + 1 2 ; ν + 1 ; β 2 α 2 ) .

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