Abstract

The characteristics of high-power partially coherent laser beams propagating upwards in the turbulent atmosphere are studied, where the principal features of diffraction, nonlinear self-focusing and turbulence are considered. Based on the “thin window” model, the analytical propagation formulae are derived by using the quadratic approximation of the nonlinear phase shift. It is found that the turbulence effect plays an important role in beam propagation characteristics. But the turbulence and self-focusing effects can be suppressed by increasing the laser elevation. Furthermore, the influence of laser elevation on the turbulence effect is stronger than that on the self-focusing effect, and influence of laser elevation on the self-focusing effect is stronger than that on the diffraction effect. In particular, the optimal focal length and wavelength are proposed to decrease the beam spot size on the target.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2020 (1)

2019 (4)

2018 (2)

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Y. Zhang, X. Ji, H. Zhang, X. Li, T. Wang, H. Wang, and Y. Deng, “Self-focusing and group-velocity dispersion of pulsed laser beams in the inhomogeneous atmosphere,” Opt. Express 26(11), 14617–14625 (2018).
[Crossref]

2017 (2)

J. Peñano, J. P. Palastro, B. Hafizi B, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

B. Hafizi, J. R. Peñano, J. P. Palastro, R. P. Fischer, and G. DiComo, “Laser beam self-focusing in turbulent dissipative media,” Opt. Lett. 42(2), 298–301 (2017).
[Crossref]

2016 (1)

I. A. Vaseva, M. P. Fedoruk, A. M. Rubenchik, and S. K. Turitsyn, “Light self-focusing in the atmosphere: thin window model,” Sci. Rep. 6(1), 30697 (2016).
[Crossref]

2014 (1)

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “The effect of self-focusing on laser space-debris cleaning,” Light: Sci. Appl. 3(4), e159 (2014).
[Crossref]

2013 (1)

2011 (2)

X. Chu, “Evolution of an airy beam in turbulence,” Opt. Lett. 36(14), 2701–2703 (2011).
[Crossref]

R. M. Tao, L. Si, Y. X. Ma, Y. C. Zou, and P. Zhou, “Propagation of high-power partially coherent fibre laser beams in a real environment,” Chin. Phys. B 20(9), 094208 (2011).
[Crossref]

2009 (2)

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref]

2007 (1)

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

2003 (1)

2002 (1)

2000 (1)

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana 55(3), 423–431 (2000).
[Crossref]

1996 (1)

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

1979 (1)

1978 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics. (Academic Press, 1995), Vol. II, Chap. 2.

Albrecht, G.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Amarande, S.

Baton, S.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Baykal, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

Berthe, L.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Bonnal, C.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Boustie, M.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Boyer, S. A. E.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Brambrink, E.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Cai, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

Chen, L.

Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

Chevalier, J.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Chu, X.

Dan, Y.

Deng, Y.

DiComo, G.

DiComo, G. P.

J. Peñano, J. P. Palastro, B. Hafizi B, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

Dogariu, A.

Eyyuboglu, H. T.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

Fan, X.

Fedoruk, M. P.

I. A. Vaseva, M. P. Fedoruk, A. M. Rubenchik, and S. K. Turitsyn, “Light self-focusing in the atmosphere: thin window model,” Sci. Rep. 6(1), 30697 (2016).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “The effect of self-focusing on laser space-debris cleaning,” Light: Sci. Appl. 3(4), e159 (2014).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref]

Fischer, R. P.

Friedman, H.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Gavel, D.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Gbur, G.

George, E. V.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Hafizi, B.

Hafizi B, B.

J. Peñano, J. P. Palastro, B. Hafizi B, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

Helle, M. H.

J. Peñano, J. P. Palastro, B. Hafizi B, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

Ho, C.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Ji, X.

Kaul, S. S.

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana 55(3), 423–431 (2000).
[Crossref]

Korotkova, O.

Leader, J. C.

Leonard, M.

M. Leonard and E. Wolf, Optical coherence and quantum optics (Cambridge university, 1995).

Li, Q.

H. Wang, X. Ji, Y. Deng, X. Li, T. Wang, H. Yu, and Q. Li, “Effect of spatial coherence on laser space-debris removal in the inhomogeneous atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 235, 244–249 (2019).
[Crossref]

Li, X.

Ma, Y. X.

R. M. Tao, L. Si, Y. X. Ma, Y. C. Zou, and P. Zhou, “Propagation of high-power partially coherent fibre laser beams in a real environment,” Chin. Phys. B 20(9), 094208 (2011).
[Crossref]

Masson, F.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Mei, Z.

Michaelis, M. M.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Murray, J.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Palastro, J. P.

J. Peñano, J. P. Palastro, B. Hafizi B, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

B. Hafizi, J. R. Peñano, J. P. Palastro, R. P. Fischer, and G. DiComo, “Laser beam self-focusing in turbulent dissipative media,” Opt. Lett. 42(2), 298–301 (2017).
[Crossref]

Peñano, J.

J. Peñano, J. P. Palastro, B. Hafizi B, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

Peñano, J. R.

Phipps, C. R.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

C. R. Phipps, Laser Ablation Propulsion and Its Applications in Space (Springer Cham, 2018), p. 217–246.

Plonus, M. A.

Priedhorsky, W.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Reilly, J. P.

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
[Crossref]

Rubenchik, A. M.

I. A. Vaseva, M. P. Fedoruk, A. M. Rubenchik, and S. K. Turitsyn, “Light self-focusing in the atmosphere: thin window model,” Sci. Rep. 6(1), 30697 (2016).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “The effect of self-focusing on laser space-debris cleaning,” Light: Sci. Appl. 3(4), e159 (2014).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref]

Saini, N. S.

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana 55(3), 423–431 (2000).
[Crossref]

Schchepakina, E.

Schneider, M.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Si, L.

R. M. Tao, L. Si, Y. X. Ma, Y. C. Zou, and P. Zhou, “Propagation of high-power partially coherent fibre laser beams in a real environment,” Chin. Phys. B 20(9), 094208 (2011).
[Crossref]

Singh, T.

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana 55(3), 423–431 (2000).
[Crossref]

Tao, R. M.

R. M. Tao, L. Si, Y. X. Ma, Y. C. Zou, and P. Zhou, “Propagation of high-power partially coherent fibre laser beams in a real environment,” Chin. Phys. B 20(9), 094208 (2011).
[Crossref]

Turitsyn, S. K.

I. A. Vaseva, M. P. Fedoruk, A. M. Rubenchik, and S. K. Turitsyn, “Light self-focusing in the atmosphere: thin window model,” Sci. Rep. 6(1), 30697 (2016).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “The effect of self-focusing on laser space-debris cleaning,” Light: Sci. Appl. 3(4), e159 (2014).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref]

Vaseva, I. A.

I. A. Vaseva, M. P. Fedoruk, A. M. Rubenchik, and S. K. Turitsyn, “Light self-focusing in the atmosphere: thin window model,” Sci. Rep. 6(1), 30697 (2016).
[Crossref]

Videau, L.

C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Wang, H.

Wang, S. C. H.

Wang, T.

H. Wang, X. Ji, Y. Deng, X. Li, T. Wang, H. Yu, and Q. Li, “Effect of spatial coherence on laser space-debris removal in the inhomogeneous atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 235, 244–249 (2019).
[Crossref]

Y. Zhang, X. Ji, H. Zhang, X. Li, T. Wang, H. Wang, and Y. Deng, “Self-focusing and group-velocity dispersion of pulsed laser beams in the inhomogeneous atmosphere,” Opt. Express 26(11), 14617–14625 (2018).
[Crossref]

Wolf, E.

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
[Crossref]

M. Leonard and E. Wolf, Optical coherence and quantum optics (Cambridge university, 1995).

Yu, H.

Zhang, B.

Zhang, H.

Zhang, Y.

Zhou, P.

R. M. Tao, L. Si, Y. X. Ma, Y. C. Zou, and P. Zhou, “Propagation of high-power partially coherent fibre laser beams in a real environment,” Chin. Phys. B 20(9), 094208 (2011).
[Crossref]

Zou, Y. C.

R. M. Tao, L. Si, Y. X. Ma, Y. C. Zou, and P. Zhou, “Propagation of high-power partially coherent fibre laser beams in a real environment,” Chin. Phys. B 20(9), 094208 (2011).
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C. R. Phipps, C. Bonnal, F. Masson, M. Boustie, L. Berthe, M. Schneider, S. Baton, E. Brambrink, J. Chevalier, L. Videau, and S. A. E. Boyer, “Transfers from Earth to LEO and LEO to interplanetary space using lasers,” Acta Astronaut. 146, 92–102 (2018).
[Crossref]

Appl. Phys. B (1)

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[Crossref]

Chin. Phys. B (1)

R. M. Tao, L. Si, Y. X. Ma, Y. C. Zou, and P. Zhou, “Propagation of high-power partially coherent fibre laser beams in a real environment,” Chin. Phys. B 20(9), 094208 (2011).
[Crossref]

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H. Wang, X. Ji, Y. Deng, X. Li, T. Wang, H. Yu, and Q. Li, “Effect of spatial coherence on laser space-debris removal in the inhomogeneous atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 235, 244–249 (2019).
[Crossref]

Laser Part. Beams (1)

C. R. Phipps, G. Albrecht, H. Friedman, D. Gavel, E. V. George, J. Murray, C. Ho, W. Priedhorsky, M. M. Michaelis, and J. P. Reilly, “ORION: Clearing near-Earth space debris using a 20-kW, 530-nm, Earth-based, repetitively pulsed laser,” Laser Part. Beams 14(1), 1–44 (1996).
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A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “The effect of self-focusing on laser space-debris cleaning,” Light: Sci. Appl. 3(4), e159 (2014).
[Crossref]

Opt. Express (4)

Opt. Lett. (6)

Phys. Rev. A (1)

J. Peñano, J. P. Palastro, B. Hafizi B, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

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A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref]

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T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana 55(3), 423–431 (2000).
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Sci. Rep. (1)

I. A. Vaseva, M. P. Fedoruk, A. M. Rubenchik, and S. K. Turitsyn, “Light self-focusing in the atmosphere: thin window model,” Sci. Rep. 6(1), 30697 (2016).
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Figures (13)

Fig. 1.
Fig. 1. Diagram of the laser beam propagation in the turbulent atmosphere.
Fig. 2.
Fig. 2. Confirmation of the validity of the approximation of Eq. (3).
Fig. 3.
Fig. 3. Confirmation of the validity of Eq. (19). Beam width wtar on the target versus the relative power P/PcrGs, h1=0, σ0→∞, ρ0→∞.
Fig. 4.
Fig. 4. (a) the beam width w, and (b) the spatial coherence width σ versus the propagation distance z, P=1000PcrGs, C0=1.7×10−16 m-2/3, h1=0.
Fig. 5.
Fig. 5. Beam width w versus the propagation distance z for different effects, P=1000PcrGs, C0=1.7×10−16 m-2/3, σ0=1 m, h1=0.
Fig. 6.
Fig. 6. Beam width w versus the propagation distance z for different laser elevation h1, P=1000PcrGs, C0=1.7×10−16 m-2/3, σ0=1 m.
Fig. 7.
Fig. 7. Beam width wtar on the target versus P/PcrGs and log(C0), σ0=3 m. (a) h1=0; (b) h1=6 km.
Fig. 8.
Fig. 8. Beam width wtar on the target versus σ0 and log(C0), P=1000PcrGs. (a) h1=0; (b) h1=6 km.
Fig. 9.
Fig. 9. Beam width wtar on the target versus P/PcrGs and σ0, C0=1.7×10−16m-2/3. (a) h1=0; (b) h1=6 km.
Fig. 10.
Fig. 10. Beam width wtar on the target versus the focal length f, P=500PcrGs, C0=1.7×10−16 m-2/3, σ0=1 m, and the target altitude z=500 km.
Fig. 11.
Fig. 11. Optimal focal length fopt versus the relative beam power P/PcrGs, σ0=1 m.
Fig. 12.
Fig. 12. Beam width on the target versus the wavelength λ, P=1000PcrGs, C0=1.7×10−16m-2/3, σ0=1 m.
Fig. 13.
Fig. 13. Optimal wavelength λopt versus (a) log(C0) and (b) σ0.

Equations (24)

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ϕ = k h 1 z 1 n 2 ( z ) I 0 ( r ) d z = k n 20 h 0 2 P π w 0 2 [ exp ( h 1 h 0 ) exp ( z 1 h 0 ) ] exp ( 2 r 2 w 0 2 ) ,
c W ( r 01 , r 02 , h 1 ) = 2 P π w 0 2 exp ( r 01 2 + r 02 2 w 0 2 ) exp [ ( r 01 r 02 ) 2 2 σ 0 2 ] × exp [ i k ( r 01 2 r 02 2 ) 2 f ] exp [ i ( ϕ 1 ϕ 2 ) ] ,
exp ( 2 r 2 w 0 2 ) ( 1 + ξ 1 r w 0 + ξ 2 r 2 w 0 2 ) ,
W ( r 1 , r 2 , h 1 ) = 2 P π w 0 2 exp ( r 1 2 + r 2 2 w 0 2 ) exp [ ( r 1 r 2 ) 2 2 σ 0 2 ] × exp [ i k ( r 1 2 r 2 2 ) 2 R 0 ] exp [ i 2 P k n 20 h 0 ξ 1 ( r 1 r 2 ) π w 0 3 ] ,
f s = π w 0 4 / 4 ξ 2 n 20 P h 0 [ exp ( h 1 / h 0 ) exp ( z 1 / h 0 ) ] .
W ( r 1 , r 2 , z ) = [ k 2 π ( z h 1 ) ] 2 d 2 r 01 d 2 r 02 W ( r 01 , r 02 , h 1 ) × exp { i k 2 ( z h 1 ) [ ( r 1 r 01 ) 2 ( r 2 r 02 ) 2 ] } exp [ ψ ( r 01 , r 1 ) + ψ ( r 02 , r 2 ) ] m ,
exp [ ψ ( r 01 , r 1 ) + ψ ( r 02 , r 2 ) ] m exp { 1 ρ 0 2 [ ( r 01 r 02 ) 2 + ( r 01 r 02 ) ( r 1 r 2 ) + ( r 1 r 2 ) 2 ] } ,
C n 2 ( h ) = 8.148 × 10 56 V 2 h 10 exp ( h / 1000 ) + 2.7 × 10 16 exp ( h / 1500 ) + C 0 exp ( h / 100 ) ,
u 0 = r 01 + r 02 2 , v 0 = r 02 r 01 ,
u = r 1 + r 2 2 , v = r 2 r 1 ,
W ( u 0 , v 0 , h 1 ) = 2 P π w 0 2 exp ( 2 u 0 2 w 0 2 ) exp ( v 0 2 2 w 0 2 ) exp ( v 0 2 2 σ 0 2 ) × exp ( i k u 0 v 0 R 0 ) exp ( i 2 ξ 1 k P n 20 h 0 v 0 π w 0 3 ) ,
W ( u , v , z ) = [ k 2 π ( z h 1 ) ] 2 d 2 u 0 d 2 v 0 W ( u 0 , v 0 , h 1 ) exp [ i k ( z h 1 ) ( v 0 v ) ( u u 0 ) ] × exp [ 1 ρ 0 2 ( v 0 2 + v 0 v + v 2 ) ] .
W ( u , v , z ) = 2 P π w 0 2 [ k 2 π ( z h 1 ) ] 2 π w 0 2 2 π α exp ( i k z h 1 v u ) exp ( 1 ρ 0 2 v 2 ) exp [ k 2 w 0 2 8 ( z h 1 ) 2 v 2 ] × exp ( β 2 4 α v 2 ) exp [ k 2 4 α ( z h 1 ) 2 u 2 ] exp [ i k β 2 α ( z h 1 ) u v ] × exp ( k 2 n 20 2 h 0 2 P 2 ξ 1 2 α π 2 w 0 6 ) exp ( i ξ 1 k n 20 h 0 C 1 P β v α π w 0 3 ) exp [ ξ 1 k 2 n 20 h 0 P u α π w 0 3 ( z h 1 ) ] ,
α = k 2 w 0 2 8 ( z h 1 ) 2 ( 1 z h 1 R 0 ) 2 + 1 ε 2 ,
β = k 2 w 0 2 4 ( z h 1 ) 2 ( 1 z h 1 R 0 ) + 1 ρ 0 2 ,
1 ε 2 = 1 2 w 0 2  +  1 2 σ 0 2 + 1 ρ 0 2 .
W ( r 1 , r 2 , z ) = 2 P π w 0 2 k 2 w 0 2 8 α ( z h 1 ) 2 exp ( ξ 1 2 k 2 n 20 2 P 2 h 2 α π 2 w 0 6 ) exp [ k 2 ( r 1 2 + r 2 2 ) 8 α ( z h 1 ) 2 ] × exp { [ k 2 16 α ( z h 1 ) 2 1 ρ 0 2 k 2 w 0 2 8 ( z h 1 ) 2 + β 2 4 α ] ( r 1 r 2 ) 2 } × exp { [ β 2 α ( z h 1 ) 1 z h 1 ] i k ( r 1 2 r 2 2 ) 2 } exp [ i ξ 1 k n 20 h 0 P β ( r 1 r 2 ) α π w 0 3 ] × exp [ ξ 1 k 2 n 20 h 0 P α π w 0 3 ( z h 1 ) ( r 1 + r 2 ) ] .
I ( r , z ) = 2 P π w 0 2 k 2 w 0 2 8 α ( z h 1 ) 2 exp ( ξ 1 2 k 2 n 20 2 h 0 2 P 2 α π 2 w 0 6 ) × exp [ k 2 r 2 4 α ( z h 1 ) 2 ] exp [ 2 ξ 1 k 2 n 20 h 0 P r α π w 0 3 ( z h 1 ) ] .
w 2 ( z ) = 2 r 2 I ( r , z ) d 2 r I ( r , z ) d 2 r 8 α ( z h 1 ) 2 k 2  =  w 0 2 w 0 2 [ 2 ( z h 1 ) R 0 ( z h 1 R 0 ) 2 ] + ( 4 w 0 2 + 4 σ 0 2 + 8 ρ 0 2 ) ( z h 1 k ) 2 .
μ ( r 1 , r 2 , z ) = W ( r 1 , r 2 , z ) W ( r 1 , r 1 , z ) W ( r 2 , r 2 , z ) .
μ ( r 1 , r 2 , z ) = exp [ ( r 1 r 2 ) 2 2 σ 2 ( z ) ] .
1 σ 2 ( z ) = k 2 8 α ( z h 1 ) 2 β 2 2 α + 2 ρ 0 2 + k 2 w 0 2 4 ( z h 1 ) 2 .
1 f opt = 1 z h 1 1 f s .
λ opt  =  2.67 π { 5 ( 1 w 0 2 + 1 σ 0 2 ) [ 1.46 h 1 z C n 2 ( 1 h z h 1 ) 5 / 3 d h ]  -  6 / 5 }  - 5 / 12 .

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