Abstract

We investigate, through simulation, the modifications to Bessel and Airy beams during propagation through atmospheric turbulence. We find that atmospheric turbulence disrupts the quasi-non-diffracting nature of Bessel and Airy beams when the transverse coherence length (Fried parameter) nears the initial aperture diameter or diagonal, respectively. The turbulence-induced transverse phase distortion limits the effectiveness of Bessel and Airy beams for applications requiring propagation over long distances in the turbulent atmosphere.

© 2014 Optical Society of America

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References

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  1. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  2. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  4. P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).
  5. R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39, 393–397 (2000).
    [CrossRef]
  6. S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
    [CrossRef]
  7. P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
    [CrossRef]
  8. C. C. Davis and I. I. Smolyaninov, “The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system,” Proc. SPIE 4489, 126–137 (2002).
    [CrossRef]
  9. A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues (Optical Society of America, 1998).
  10. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  11. P. Sprangle and B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991).
    [CrossRef]
  12. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
    [CrossRef]
  13. L. Gong, Y.-X. Ren, G.-S. Xue, Q.-C. Wang, J.-H. Zhou, M.-C. Zhong, Z.-Q. Wang, and Y.-M. Li, “Generation of nondiffracting Bessel beam using digital micromirror device,” Appl. Opt. 52, 4566–4575 (2013).
    [CrossRef]
  14. M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
    [CrossRef]
  15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  16. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef]
  17. X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21, 2154–2164 (2013).
    [CrossRef]
  18. P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. V. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733 (2008).
    [CrossRef]
  19. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
    [CrossRef]
  20. C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18, 1033–1039 (2009).
    [CrossRef]
  21. I. P. Lukin, “Coherence of a Bessel beam in a turbulent atmosphere,” Atmos. Oceanic Opt. 25, 328–337 (2012).
    [CrossRef]
  22. Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35, 3456–3458 (2010).
    [CrossRef]
  23. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011).
    [CrossRef]

2013

2012

I. P. Lukin, “Coherence of a Bessel beam in a turbulent atmosphere,” Atmos. Oceanic Opt. 25, 328–337 (2012).
[CrossRef]

2011

2010

2009

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18, 1033–1039 (2009).
[CrossRef]

P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

2008

2007

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef]

2002

C. C. Davis and I. I. Smolyaninov, “The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system,” Proc. SPIE 4489, 126–137 (2002).
[CrossRef]

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

2000

R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39, 393–397 (2000).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

1991

P. Sprangle and B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991).
[CrossRef]

1987

1976

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1975

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

Bandres, M. A.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

Bao-Suan, C.

C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18, 1033–1039 (2009).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Chin, S. L.

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Christodoulides, D. N.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Chu, X.

Davis, C. C.

C. C. Davis and I. I. Smolyaninov, “The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system,” Proc. SPIE 4489, 126–137 (2002).
[CrossRef]

Dholakia, K.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

Di Trapani, P.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Durnin, J.

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

Eyyuboglu, H. T.

Faccio, D.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fischer, R.

P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Frehlich, R.

Gbur, G.

Gong, L.

Greenfield, E.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

Gu, Y.

Hafizi, B.

P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

P. Sprangle and B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991).
[CrossRef]

Ji, G.

Ji, X.

Jia, X.

Ji-Xiong, P.

C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18, 1033–1039 (2009).
[CrossRef]

Kaminer, I.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

Kandidov, V. P.

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. V. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733 (2008).
[CrossRef]

Kosareva, O. G.

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Li, Y.-M.

Lukin, I. P.

I. P. Lukin, “Coherence of a Bessel beam in a turbulent atmosphere,” Atmos. Oceanic Opt. 25, 328–337 (2012).
[CrossRef]

Mills, M.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. V. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733 (2008).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Penano, J.

P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Petit, S.

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. V. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733 (2008).
[CrossRef]

Ren, Y.-X.

Roberts, A.

Rodríguez-Lara, B. M.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

Segev, M.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues (Optical Society of America, 1998).

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef]

Smolyaninov, I. I.

C. C. Davis and I. I. Smolyaninov, “The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system,” Proc. SPIE 4489, 126–137 (2002).
[CrossRef]

Sprangle, P.

P. Sprangle and B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991).
[CrossRef]

Sprangle, P. A.

P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Talebpour, A.

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Tamarov, M. P.

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Ting, A.

P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Wang, Q.-C.

Wang, Z.-Q.

Xue, G.-S.

Yang, J.

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Zhong, M.-C.

Zhou, J.-H.

Appl. Opt.

Appl. Phys.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Appl. Phys. B

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74, 67–76 (2002).
[CrossRef]

Atmos. Oceanic Opt.

I. P. Lukin, “Coherence of a Bessel beam in a turbulent atmosphere,” Atmos. Oceanic Opt. 25, 328–337 (2012).
[CrossRef]

Chin. Phys. B

C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18, 1033–1039 (2009).
[CrossRef]

IEEE J. Quantum Electron.

P. A. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photonics News

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photonics News 24(6), 30–37 (2013).
[CrossRef]

Phys. Rev. Lett.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

P. Sprangle and B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991).
[CrossRef]

Proc. IEEE

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Proc. SPIE

C. C. Davis and I. I. Smolyaninov, “The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system,” Proc. SPIE 4489, 126–137 (2002).
[CrossRef]

Science

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

Other

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues (Optical Society of America, 1998).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

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Figures (7)

Fig. 1.
Fig. 1.

Transverse profiles of a Gaussian beam with initial spot size w0=4.5cm after 6.4 km of propagation in (a) vacuum and (b) turbulence with Cn2=1×1015m2/3, and (c) averaged over 100 instances through turbulence.

Fig. 2.
Fig. 2.

Simulation results for a Gaussian beam. (a), (b) Comparison of the on-axis intensity in turbulence, red, and vacuum, blue, as a function of propagation distance for initial spot sizes of 1.8 and 4.5 cm, respectively. (c) Ratio of the RMS radius at one Rayleigh length to the initial spot size as a function of initial spot size. (d) Normalized on-axis intensity at one Rayleigh length as a function of initial spot size. In (a)–(d) the ratio of the Fried parameter to beam diameter, green, is plotted for reference.

Fig. 3.
Fig. 3.

Transverse profiles of a 15 ring Bessel beam hard-apertured at a radius of 23 cm after 6.4 km of propagation in (a) vacuum and (b) turbulence with Cn2=1×1015m2/3, and (c) averaged over 100 instances through turbulence.

Fig. 4.
Fig. 4.

Simulation results for a Bessel beam. (a), (b) Comparison of the on-axis intensity in turbulence, red, and vacuum, blue, as a function of propagation distance for initial aperture radii of 6.65 and 23 cm, respectively. (c) Ratio of the RMS radius at one diffraction length, LB, to the initial aperture radius as a function of initial aperture radius. (d) Normalized on-axis intensity at one diffraction length as a function of initial aperture radius. In (a)–(d) the ratio of the Fried parameter to beam diameter, green, is plotted for reference.

Fig. 5.
Fig. 5.

Power delivered to a circular aperture of radius 15 cm (top) and on-axis intensity (bottom) as a function of rings in a Bessel beam initially apertured at a radius of 15 cm after propagation distances of 1.6 km, blue, 4 km, red, and 6.4 km, green. The dashed and solid lines are results from propagation in vacuum and turbulence with Cn2=1×1015m2/3, respectively.

Fig. 6.
Fig. 6.

Transverse profiles of a 15 zero Airy beam with d=2×22cm=31.3cm after 6.4 km of propagation in (a) vacuum and (b) turbulence with Cn2=1×1015m2/3, and (c) averaged over 100 instances through turbulence.

Fig. 7.
Fig. 7.

Simulation results for an Airy beam. (a), (b) Comparison of the on-axis intensity in turbulence, red, and vacuum, blue, as a function of propagation distance for an aperture diagonal of 11.3 and 31.1 cm, respectively. (c) Ratio of the RMS radius at one diffraction length, LA, to the aperture diagonal as a function of aperture diagonal. (d) Normalized on-axis intensity at one diffraction as a function of aperture diagonal. In (a)–(d) the ratio of the Fried parameter to beam diagonal, green, is plotted for reference.

Equations (9)

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

E(r,t)=12E^(r,t)exp[i(kzω0t)]+c.c.,
[2+2ik0z]E^(r)eiδkz=2k02δnE^(r)eiδkz,
[2+2ik0z]E^(r)=2k02δnT(r)E^(r).
Φn(κ)=0.033Cn2e(κ0/2π)2(κ2+L02)11/6,
I(0,z)=12cε0(k02πz)2×E^(r,0)E^*(r,0)e12[ϕ(r,z)ϕ(r,z)]2drdr,
ϕ(r,z)=k00zδnT(r,z)dz.
ϕ(r,Δzs)=k0zz+ΔzsδnT(r,z)dz.
ϕ(r,Δzs)=(2πΔzs)1/2k0×dκeiκ·r[ar(κ)+iai(κ)]Φn1/2(κ,0),
wrms=(x2+y2)I(x,y)dxdyI(x,y)dxdy.

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