Abstract

The Rytov theory for the propagation of Helmholtz-Gauss (HzG) beams in turbulent atmosphere is presented. We derive expressions for the first and second-order normalized Born approximations, the second-order moments, and the transverse intensity pattern of the HzG beams at any arbitrary propagation distance. The general formulation is applied to study the propagation of several special cases of the HzG beams, in particular, the Bessel-Gauss and Mathieu-Gauss beams and their pure nondiffracting counterparts, the Bessel and Mathieu beams. For numerical purposes, we assume the standard Kolmogorov distribution to model the power spectrum of the atmospheric fluctuations.

© 2007 Optical Society of America

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  5. Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,"Opt. Commun. 271, 509-516 (2007).
    [CrossRef]
  6. H. T. Eyyuboglu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
    [CrossRef] [PubMed]
  7. H. T. Eyyuboglu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005).
    [CrossRef] [PubMed]
  8. H. T. Eyyuboglu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,"Opt. Commun. 245, 37-47 (2005).
    [CrossRef]
  9. H. T. Eyyuboglu, "Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere," J. Opt. Soc. Am. A 22, 1527-1535 (2005).
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  10. 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).
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    [CrossRef]
  16. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
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  17. Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstructionof a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
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    [CrossRef]
  27. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Propagation of Helmholtz-Gauss beams in absorbing and gain media," J. Opt. Soc. Am. A,  23, 1994-2001 (2006).
    [CrossRef]
  28. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006).
    [CrossRef] [PubMed]
  29. M. A. Bandres and J. C. Gutiérrez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005).
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  30. Raul I. Hernandez-Aranda, J. C . Gutiérrez-Vega, Manuel Guizar-Sicairos, and Miguel A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express 14, 8974-8988 (2006).
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    [CrossRef]
  32. J. C. Gutiérrez-Vega and M. A. Bandres, "On the normalization of the Mathieu-Gauss optical beams," J. Opt. Soc. Am. A,  24, 215-220 (2007).
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    [CrossRef]
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    [CrossRef]
  38. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express,  14, 4182-4187 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]

2007 (5)

2006 (10)

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Shaping soliton properties in Mathieu lattices," Opt. Lett. 31, 238-240 (2006).
[CrossRef] [PubMed]

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

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express,  14, 4182-4187 (2006).
[CrossRef] [PubMed]

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Propagation of Helmholtz-Gauss beams in absorbing and gain media," J. Opt. Soc. Am. A,  23, 1994-2001 (2006).
[CrossRef]

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006).
[CrossRef] [PubMed]

C. Arpali, C. Yazicioglu, H. Eyyuboglu, S. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

Raul I. Hernandez-Aranda, J. C . Gutiérrez-Vega, Manuel Guizar-Sicairos, and Miguel A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express 14, 8974-8988 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,"Opt. Commun. 264, 25-34 (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]

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

2005 (6)

2004 (3)

2003 (1)

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003).
[CrossRef]

2002 (4)

Z. Bouchal, "Resistance of nondiffracting vortex beam against amplitude and phase perturbations," Opt. Commun. 210, 155-164 (2002).
[CrossRef]

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

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[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 (2002).
[CrossRef]

2000 (2)

1999 (1)

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstructionof a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

1987 (1)

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

Allison, I.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

Altay, S.

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,"Opt. Commun. 264, 25-34 (2006).
[CrossRef]

Arpali, C.

Arpali, S.

Aruga, T.

Bandres, M. A.

Bandres, Miguel A.

Bandrés, M. A.

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

Baykal, Y.

Belafhal, A.

A. Chafiq, Z. Hricha, A. Belafhal, "Paraxial propagation of Mathieu beams through an apertured ABCD optical system," Opt. Commun. 253,223-230 (2005).
[CrossRef]

Bouchal, Z.

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003).
[CrossRef]

Z. Bouchal, "Resistance of nondiffracting vortex beam against amplitude and phase perturbations," Opt. Commun. 210, 155-164 (2002).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstructionof a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Cai, Y.

Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,"Opt. Commun. 271, 509-516 (2007).
[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, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006).
[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 (2002).
[CrossRef]

Chafiq, A.

A. Chafiq, Z. Hricha, A. Belafhal, "Paraxial propagation of Mathieu beams through an apertured ABCD optical system," Opt. Commun. 253,223-230 (2005).
[CrossRef]

Chávez-Cerda, S.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
[CrossRef] [PubMed]

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

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

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstructionof a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Courtial, J.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

Dartora, C. A.

Dholakia, K.

Egorov, A. A.

Eyyuboglu, H.

Eyyuboglu, H. T.

H. T. Eyyuboglu, "Propagation of higher order Bessel-Gaussian beams in turbulence," Appl. Phys. B,  88, 259-265 (2007).
[CrossRef]

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,"Opt. Commun. 264, 25-34 (2006).
[CrossRef]

H. T. Eyyuboglu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,"Opt. Commun. 245, 37-47 (2005).
[CrossRef]

H. T. Eyyuboglu, "Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere," J. Opt. Soc. Am. A 22, 1527-1535 (2005).
[CrossRef]

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

H. T. Eyyuboglu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Frehlich, R.

R. Frehlich, "Simulation of laser propagation in a turbulent atmosphere,"Appl. Opt. 30, 393-397 (2000).
[CrossRef]

Gbur, G.

Ge, D.

Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,"Opt. Commun. 271, 509-516 (2007).
[CrossRef]

Gilchrest, Y. V.

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

Gori, F.

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

Guattari, G.

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

Guizar-Sicairos, M.

Guizar-Sicairos, Manuel

Gutiérrez-Vega, C.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

Gutiérrez-Vega, J. C

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega and M. A. Bandres, "On the normalization of the Mathieu-Gauss optical beams," J. Opt. Soc. Am. A,  24, 215-220 (2007).
[CrossRef]

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006).
[CrossRef] [PubMed]

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Propagation of Helmholtz-Gauss beams in absorbing and gain media," J. Opt. Soc. Am. A,  23, 1994-2001 (2006).
[CrossRef]

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express,  14, 4182-4187 (2006).
[CrossRef] [PubMed]

M. A. Bandres and J. C. Gutiérrez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
[CrossRef]

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
[CrossRef] [PubMed]

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

He, S.

Hernandez-Aranda, Raul I.

Hernández-Figueroa, H. E.

Hricha, Z.

A. Chafiq, Z. Hricha, A. Belafhal, "Paraxial propagation of Mathieu beams through an apertured ABCD optical system," Opt. Commun. 253,223-230 (2005).
[CrossRef]

Iturbe-Castillo, M. D.

Julio, G.H.C.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

Kartashov, Y. V.

Korotkova, O.

Li, R.

Li, S. W.

López-Mariscal, C.

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express,  14, 4182-4187 (2006).
[CrossRef] [PubMed]

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

Macon, B. R.

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

MacVicar, I.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

Milne, G.

New, G.H.C.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

O’Neil, A.T.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

Padgett, M.J.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

Padovani, C.

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

Takabe, M.

Torner, L.

Vysloukh, V. A.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstructionof a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Yazicioglu, C.

Yoshikado, S.

Young, C. Y.

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

Appl. Opt. (3)

Appl. Phys. B (1)

H. T. Eyyuboglu, "Propagation of higher order Bessel-Gaussian beams in turbulence," Appl. Phys. B,  88, 259-265 (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]

Czech. J. Phys. (1)

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Commun. (7)

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,"Opt. Commun. 264, 25-34 (2006).
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Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,"Opt. Commun. 271, 509-516 (2007).
[CrossRef]

H. T. Eyyuboglu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,"Opt. Commun. 245, 37-47 (2005).
[CrossRef]

Z. Bouchal, "Resistance of nondiffracting vortex beam against amplitude and phase perturbations," Opt. Commun. 210, 155-164 (2002).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstructionof a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
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F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
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A. Chafiq, Z. Hricha, A. Belafhal, "Paraxial propagation of Mathieu beams through an apertured ABCD optical system," Opt. Commun. 253,223-230 (2005).
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Opt. Eng. (2)

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

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

Opt. Express (5)

Opt. Lett. (6)

Other (5)

L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

M. Abramowitz, and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

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

O. Korotkova and G. Gbur, "Propagation of beams with any spectral, coherence and polarization properties in turbulent atmosphere," Proc. of SPIE 6457, 64570J1-64570J12 (2007).

D.C. Cowan, J. Recolons, L.C. Andrews, C.Y. Young, Atmospheric Propagation III, Proc. SPIE 6215, 62150 B-1- 62150B-10 (2006).

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Figures (5)

Fig. 1.
Fig. 1.

Plot of the ensemble average of the on-axis intensity for a pure Gaussian beam, a zeroth-order pure nondiffracting Bessel beam, and a zeroth-order Bessel-Gauss beam, each propagated both in free space and a turbulent atmosphere

Fig. 2.
Fig. 2.

Plot of the ensemble average of the intensity versus the radial distance from the origin, for the zeroth-order nondiffracting Bessel beam and zeroth-order Bessel- Gauss beam. The intensity profiles correspond to several propagation distances L=0,zmax /4,zmax /2,3zmax /4

Fig. 3.
Fig. 3.

Plot of the ensemble average of the intensity versus the radial distance from the origin, for the zeroth-order nondiffracting Bessel beam, zeroth-order Bessel-Gauss beam, and a pure Gaussian beam. The propagation distance for all of them was L=3zmax /4, under the same turbulence conditions.

Fig. 4.
Fig. 4.

Plot of the ensemble average of the intensity versus the radial distance from the origin, for the third-order Bessel-Gauss beam, for several propagation distances L=0,zmax /4,zmax /2,3zmax /4.

Fig. 5.
Fig. 5.

Plot of the ensemble average of the intensity versus the radial distance from the origin, for the second-order Mathieu-Gauss beam along the positive x and y axes, for several propagation distances L=0,zmax /4,zmax /2,3zmax /4.

Equations (64)

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R = ( r , z ) , r = ( x , y ) = ( r cos θ , r sin θ ) ,
K = ( k , k z ) , k = ( k x , k y ) = ( k cos φ , k sin φ ) ,
[ 2 x 2 + 2 y 2 + 2 z 2 + K 2 n 2 ( R ) ] U ( R ) = 0 ,
n 2 ( R ) = [ 1 + n 1 ( R ) ] 2 1 + 2 n 1 ( R ) , n 1 ( R ) 1 ,
U ( r , L ) = U 0 ( r , L ) + U 1 ( r , L ) + U 2 ( r , L ) + ,
= U 0 ( r , L ) exp [ Ψ 1 ( r , L ) + Ψ 2 ( r , L ) + ] ,
U m ( r , L ) = K 2 2 π 0 L d z d 2 s exp [ iK ( L z ) + iK s r 2 2 ( L z ) ] n 1 ( s , z ) L z U m 1 ( s , z ) ,
U ¯ m ( r , L ) U m ( r , L ) U 0 ( r , L ) , m = 1 , 2 , 3 , ,
ψ 1 = ln [ 1 + U ¯ 1 ( r , L ) ] U ¯ 1 ( r , L ) , U ¯ 1 ( r , L ) 1 ,
Ψ 2 = U ¯ 2 ( r , L ) 1 2 U ¯ 1 2 ( r , L ) .
n 1 ( s , z ) = d ν ( κ , z ) exp ( i κ · s ) ,
κ = ( κ x , κ y ) , κ = κ ,
u 0 ( R ) = exp ( i k 0 2 z 2 K μ ) exp ( iKz ) μ exp ( r 2 μ w 0 2 ) exp ( i k φ · r μ ) ,
μ = μ ( z ) = 1 + i z z R ,
k φ = ( k 0 cos φ , k 0 sin φ ) , k 0 = k φ ,
U 0 ( R ) = A ( φ ) u 0 ( R ) d φ .
u ¯ 1 ( r , L ) = K 2 2 π u 0 ( r , L ) 0 L d z d 2 s exp [ i K s r 2 2 ( L z ) ]
× [ 1 μ exp ( s 2 μ w 0 2 ) exp ( i σ z μ ) exp ( i k φ · s μ ) ]    [ 1 L z d ν ( κ , z ) exp ( i κ · s ) ] ,
u ¯ 1 ( r , L ) = K 2 2 π u 0 ( r , L ) 0 L d ν d ν ( κ , z ) exp ( i σ z μ ) ( L z ) μ exp [ i K r 2 2 ( L z ) ]
× d 2 s exp ( a s 2 ) exp ( i q · s ) ,
a a ( z , L ) L i z R μ w 0 2 ( L z ) ,
q q ( κ , z , k φ , r , L ) κ K r L z + k φ μ , q = q ,
d 2 s exp ( a s 2 ) exp ( i q · s ) = π a exp ( q 2 4 a ) .
u ¯ 1 ( r , L ) = K 2 2 u 0 ( r , L ) 0 L d z d ν ( κ , z ) [ 1 a μ ( L z ) ]
× exp ( i σ z μ ) exp [ i K r 2 2 ( L z ) ] exp ( q 2 4 a ) ,
u ¯ 2 ( r , L ) = K 4 w 0 2 4 π u 0 ( r , L ) 0 L d z 0 z d z d 2 s d ν ( κ , z ) d ν ( κ , z )
× exp [ iKr 2 2 ( L z ) ] exp ( i σ z μ ) ( L z ) ( z iz R ) exp [ i K ( L z ) s 2 2 ( L z ) ( z z ) ] exp [ i ( κ K r L z ) · s ] exp ( q 2 4 a ) ,
u ¯ 2 ( r , L ) = K 4 w 0 2 4 u 0 ( r , L ) 0 L d z 0 z d z d ν ( κ , z ) d ν ( κ , z )
× exp [ iKr 2 2 ( L 2 ) ] exp ( i σ z μ ) ( L z ) ( z iz R ) b exp { [ z R ( z z ) 2 K ( z iz R ) ] ( κ 2 μ + k 0 2 μ ' + 2 κ · k φ ) } exp ( p 2 4 b ) ,
b b ( z ) K ( z R + iL ) 2 ( L z ) ( z iz R ) ,
p p ( κ , r , L , z , z , κ , k φ ) κ k r L z + ( z iz R z iz R ) ( κ + k φ μ ) , p 2 = p 2 .
I ( r , L ) = U * ( r , L ) U ( r , L ) = π π d φ π π d χ A * ( φ ) A ( χ ) u φ * ( r , L ) u χ ( r , L ) ,
u φ * ( r , L ) u χ ( r , L ) = u 0 , φ * ( r , L ) u 0 , χ ( r , L )
× exp [ Ψ 1 , φ * ( r , L ) + Ψ 2 , φ * ( r , L ) + Ψ 1 , χ ( r , L ) + Ψ 2 , χ ( r , L ) ] .
exp ( t ) exp [ t + 1 2 ( t 2 t 2 ) ] ,
u φ * ( r , L ) u χ ( r , L ) = u 0 , φ * ( r , L ) u 0 , χ ( r , L ) exp [ E φ ( 1 ) * ( r , L ) + E χ ( 1 ) ( r , L ) + E φ , χ ( 2 ) ( r , L ) ] ,
E j ( 1 ) ( r , L ) Ψ 2 , j * ( r , L ) + Ψ 1 , j * 2 ( r , L ) = u ¯ 2 , j * ( r , L ) , j = { φ , χ } ,
E φ , χ ( 2 ) ( r , L ) = Ψ 1 , φ * ( r , L ) Ψ 1 , χ ( r , L ) u ¯ 1 , φ * ( r , L ) u ¯ 1 , χ ( r , L ) ,
E j ( 1 ) ( r , L ) = 2 π 2 K 2 0 L d η 0 κ Φ n ( κ ) d κ , j = { φ , χ } ,
E φ , χ ( 2 ) ( r , L ) = 4 π 2 K 2 0 L d η 0 d κ Φ n ( κ ) κ exp [ z R ( L η ) 2 κ 2 K ( L 2 + z R 2 ) ] J 0 ( iQ κ ) ,
Q 2 z R ( L η ) r L 2 + z R 2 1 2 ( L η ) w 0 2 k φ , χ , Q = Q ,
k φ , χ k φ L + iz R + k χ L iz R .
I ( r , L ) = π π d φ π π d χ A * ( φ ) A ( χ ) u 0 , φ * ( r , L ) u 0 , χ ( r , L )
× exp { 2 E φ ( 1 ) ( r , L ) + E φ , χ ( 2 ) ( r , L ) } .
Φ n ( κ ) = T ( γ ) C n 2 κ γ 2 ,
T ( γ ) = Γ ( γ + 1 ) 4 π 2 sin [ π 2 ( γ 1 ) ] , γ = 5 3 ,
BG m ( r , θ ) = J m ( k 0 r ) exp ( im θ ) exp ( r 2 w 0 2 ) ,
MG mm ± ( r ) = C m Je m ( ξ , ε ) ce m ( η , ε ) ± i S m Jo m ( ξ , ε ) se m ( η , ε ) ,
x = h cosh ξ cos η , y = h sinh ξ sin η ,
E φ , χ ( 2 ) ( r , L ) = K 2 z R 2 ( L 2 + z R 2 ) u 0 , φ * ( r , L ) u 0 , χ ( r , L ) 0 L d z 0 L d z d ν * ( κ , z ) d ν ( κ , z )
× exp [ i σ ( z μ * z μ ) ] exp [ i Kr 2 2 ( 1 L z 1 L z ) ] exp { 1 4 ( q φ 2 a ( z , L ) ) * + q χ 2 a ( z , L ) ] } .
d ν * ( κ , z ) d ν ( κ , z ) = F n ( κ , z z ) δ ( κ κ ) d 2 κ d 2 κ ,
Φ n ( κ ) = 1 2 π d ξ F n ( κ x , κ y , ξ ) ,
0 L d z 0 L d z = 0 L d η 2 η L 2 η L d ξ .
E φ , χ ( 2 ) ( r , L ) = 2 π K 2 z R 2 ( L 2 + z R 2 ) u 0 , φ * ( r , L ) u 0 , χ ( r , L ) exp [ z R ( 2 σ L 2 + Kr 2 ) L 2 + z R 2 ] exp ( z R r · k φ χ )
L d η d 2 κ Φ n ( κ ) exp [ w 0 2 ( L η ) 2 κ 2 2 ( L 2 + z R 2 ) ] exp ( Q · κ ) ,
k φ χ k φ L + iz R + k χ L iz R ,
Q 2 z R ( L η ) r L 2 + z R 2 1 2 ( L η ) w 0 2 k φ χ .
E φ , χ ( 2 ) ( r , L ) = 4 π 2 K 2 0 L d η 0 d κ Φ n ( κ ) κ exp [ z R ( L η ) 2 κ 2 K ( L 2 + z R 2 ) ] J 0 ( iQ κ ) ,
E φ ( 1 ) ( r , L ) = K 4 w 0 2 4 u 0 , φ ( r , L ) 0 L d z 0 z d z d ν ( κ , z ) d ν ( κ , z ) 1 b ( L z ) ( z iz R )
× exp [ i Kr 2 2 ( L z ) ] exp [ i σ z μ ] exp [ ( z R ( z z ) 2 K ( z iz R ) ) ( κ 2 μ + k 0 2 μ + 2 κ · k φ ) ] exp ( p φ 2 4 b )
d ν ( κ , z ) d ν ( κ , z ) = d ν ( κ , z ) d ν * ( κ , z ) = F n ( κ , z z ) δ ( κ + κ ) d 2 κ d 2 κ .
0 L d z 0 z d z F n ( κ , z z ) = 1 2 0 L d η d ξ F n ( κ , ξ ) .
E φ ( 1 ) ( r , L ) = π K 2 0 L d η Φ n ( κ ) d 2 κ .

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