Abstract

We analyze and simulate the propagation of laser beams with powers larger than the nonlinear self-focusing power through atmospheric turbulence. Turbulence-induced filamentation is theoretically described in terms of whole-beam self-focusing and transverse modulational instability. We describe a numerical simulation method to model nonlinear focusing in turbulence. We find good agreement between our simulation and a previously published laboratory-scale experiment. Simulations are then performed to calculate probability distributions for filament onset and wander over kilometer distances, and are compared with theory. The theoretical model can be made to agree with the simulations for physically meaningful choices of a few fitting parameters. We also examine the use of focusing optics to control filamentation range of modulationally unstable beams.

© 2014 Optical Society of America

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2014

P. Sprangle, B. Hafizi, H. Milchberg, G. Nusinovich, and A. Zigler, “Active remote detection of radioactivity based on electromagnetic signatures,” Phys. Plasmas 21, 013103 (2014).
[CrossRef]

2011

2009

G. Franssen, H. Schleijpen, J. Van den Heuvel, H. Buersing, B. Eberle, and D. Walter, “Femtosecond lasers for countermeasure applications,” Proc. SPIE 7483, 748309 (2009).
[CrossRef]

P. 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

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

2007

R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

2006

2005

2004

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

2003

2002

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002).
[CrossRef]

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

2000

1999

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29, 911 (1999).
[CrossRef]

1995

W. A. Coles, J. P. Filice, R. G. Frehlich, and M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995).
[CrossRef]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

1990

1988

1976

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

1971

1966

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Ackermann, R.

R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J.-P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31, 86–88 (2006).
[CrossRef]

Agrawal, G. P.

Alexeev, I.

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Vol. PM99.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), Vol. PM152.

Angel, S. M.

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Biegert, J.

Bourdon, P.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Boyd, R.

R. Boyd, Nonlinear Optics (Academic, 2003).

Briscoe, E.

Brodeur, A.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29, 911 (1999).
[CrossRef]

Buersing, H.

G. Franssen, H. Schleijpen, J. Van den Heuvel, H. Buersing, B. Eberle, and D. Walter, “Femtosecond lasers for countermeasure applications,” Proc. SPIE 7483, 748309 (2009).
[CrossRef]

Buser, R. G.

Carter, J. C.

Chin, S.

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

Chin, S. L.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29, 911 (1999).
[CrossRef]

Coles, W. A.

Colston, B. W.

Couairon, A.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Dantus, M.

Diels, J.-C.

Durécu, A.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Eberle, B.

G. Franssen, H. Schleijpen, J. Van den Heuvel, H. Buersing, B. Eberle, and D. Walter, “Femtosecond lasers for countermeasure applications,” Proc. SPIE 7483, 748309 (2009).
[CrossRef]

Eisenmann, S.

Erlich, Y.

Feit, M.

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

Fibich, G.

Filice, J. P.

Fischer, R.

P. 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]

Flatté, S. M.

Fleck, J. A.

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

Fleury, B.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Fraenkel, M.

Franco, M.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Franssen, G.

G. Franssen, H. Schleijpen, J. Van den Heuvel, H. Buersing, B. Eberle, and D. Walter, “Femtosecond lasers for countermeasure applications,” Proc. SPIE 7483, 748309 (2009).
[CrossRef]

Frehlich, R. G.

Fried, D. L.

Gaeta, A.

Goode, S. R.

Gord, J. R.

Gordon, D.

Gordon, D. F.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

Hafizi, B.

P. Sprangle, B. Hafizi, H. Milchberg, G. Nusinovich, and A. Zigler, “Active remote detection of radioactivity based on electromagnetic signatures,” Phys. Plasmas 21, 013103 (2014).
[CrossRef]

P. 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. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002).
[CrossRef]

Henis, Z.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Vol. PM99.

Houard, A.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Hubbard, R.

Ilan, B.

Kandidov, V.

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

Kandidov, V. P.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29, 911 (1999).
[CrossRef]

Kapetanakos, C. A.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

Kasparian, J.

R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J.-P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31, 86–88 (2006).
[CrossRef]

Kosareva, O.

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

Kosareva, O. G.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29, 911 (1999).
[CrossRef]

Lascoux, N.

R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

Lombard, L.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Lozovoy, V. V.

Martin, J. M.

Méjean, G.

Michau, V.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Milchberg, H.

P. Sprangle, B. Hafizi, H. Milchberg, G. Nusinovich, and A. Zigler, “Active remote detection of radioactivity based on electromagnetic signatures,” Phys. Plasmas 21, 013103 (2014).
[CrossRef]

Milonni, P. W.

Morris, J.

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

Mysyrowicz, A.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Nusinovich, G.

P. Sprangle, B. Hafizi, H. Milchberg, G. Nusinovich, and A. Zigler, “Active remote detection of radioactivity based on electromagnetic signatures,” Phys. Plasmas 21, 013103 (2014).
[CrossRef]

Pearman, W.

Penano, J.

P. 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]

Peñano, J.

Peñano, J. R.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002).
[CrossRef]

Pender, J.

Pestov, D.

Petit, S.

S. Chin, A. Talebpour, J. Yang, S. Petit, V. Kandidov, O. Kosareva, and M. 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, 2nd ed. (SPIE, 2005), Vol. PM152.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Vol. PM99.

Prade, B.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Robert, C.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Roy, S.

Rubel, G.

Salame, R.

R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

Salmon, E.

R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J.-P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31, 86–88 (2006).
[CrossRef]

Scaffidi, J.

Schleijpen, H.

G. Franssen, H. Schleijpen, J. Van den Heuvel, H. Buersing, B. Eberle, and D. Walter, “Femtosecond lasers for countermeasure applications,” Proc. SPIE 7483, 748309 (2009).
[CrossRef]

Sprangle, P.

P. Sprangle, B. Hafizi, H. Milchberg, G. Nusinovich, and A. Zigler, “Active remote detection of radioactivity based on electromagnetic signatures,” Phys. Plasmas 21, 013103 (2014).
[CrossRef]

P. 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]

A. Ting, I. Alexeev, D. Gordon, E. Briscoe, J. Peñano, R. Hubbard, P. Sprangle, and G. Rubel, “Remote atmospheric breakdown for standoff detection by using an intense short laser pulse,” Appl. Opt. 44, 5315–5320 (2005).
[CrossRef]

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002).
[CrossRef]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Talebpour, A.

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

Tamarov, M.

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

Tamarov, M. P.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29, 911 (1999).
[CrossRef]

Ting, A.

P. 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]

A. Ting, I. Alexeev, D. Gordon, E. Briscoe, J. Peñano, R. Hubbard, P. Sprangle, and G. Rubel, “Remote atmospheric breakdown for standoff detection by using an intense short laser pulse,” Appl. Opt. 44, 5315–5320 (2005).
[CrossRef]

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

Van den Heuvel, J.

G. Franssen, H. Schleijpen, J. Van den Heuvel, H. Buersing, B. Eberle, and D. Walter, “Femtosecond lasers for countermeasure applications,” Proc. SPIE 7483, 748309 (2009).
[CrossRef]

Vasseur, O.

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Walter, D.

G. Franssen, H. Schleijpen, J. Van den Heuvel, H. Buersing, B. Eberle, and D. Walter, “Femtosecond lasers for countermeasure applications,” Proc. SPIE 7483, 748309 (2009).
[CrossRef]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Wolf, J.-P.

R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
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R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J.-P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31, 86–88 (2006).
[CrossRef]

Wrzesinski, P. J.

Yadlowsky, M.

Yang, J.

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

Yu, J.

Zigler, A.

P. Sprangle, B. Hafizi, H. Milchberg, G. Nusinovich, and A. Zigler, “Active remote detection of radioactivity based on electromagnetic signatures,” Phys. Plasmas 21, 013103 (2014).
[CrossRef]

G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. Gaeta, and A. Zigler, “Self-focusing distance of very high power laser pulses,” Opt. Express 13, 5897–5903 (2005).
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[CrossRef]

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[CrossRef]

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R. Salame, N. Lascoux, E. Salmon, R. Ackermann, J. Kasparian, and J.-P. Wolf, “Propagation of laser filaments through an extended turbulent region,” Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

IEEE J. Quantum Electron.

P. 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]

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J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865–2874 (2004).
[CrossRef]

P. Sprangle, B. Hafizi, H. Milchberg, G. Nusinovich, and A. Zigler, “Active remote detection of radioactivity based on electromagnetic signatures,” Phys. Plasmas 21, 013103 (2014).
[CrossRef]

Phys. Rev. A

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78, 033804 (2008).
[CrossRef]

Phys. Rev. E

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Figures (12)

Fig. 1.
Fig. 1.

MI growth rate Γ (solid curves) and number of MI e-foldings N (dashed curves) as a function of propagation distance z. Laser parameters are P=20PNL, R0=11.5cm. The self-focusing distance is 5 km in the absence of turbulence. Black curves represent turbulence index Cn2=2×1014m2/3; red curves denote Cn2=1013m2/3.

Fig. 2.
Fig. 2.

Number of MI power e-foldings, N, after 5 km of propagation versus Cn2 for P=10PNL (shorter dashed curve), P=30PNL (longer dashed curve), and P=50PNL (solid curve). Spot size is adjusted so that the self-focusing distance for all cases is 5 km in the absence of turbulence.

Fig. 3.
Fig. 3.

Cn2 threshold for filamentation versus location of turbulence screen for the laser parameters and propagation geometry of Ref. [14], as discussed in the text. Points represent cases of 10% (diamonds), 50% (squares), and 90% (triangle) filamentation probability.

Fig. 4.
Fig. 4.

(a) Ensemble of simulation results showing normalized fluence versus propagation distance for a laser beam with P=1.5PNL, R0=3.85cm, and Cn2=1015m2/3. The nonlinear focal length, LSF, in the absence of turbulence is 5 km. (b) Probability distribution of FOD for the same laser parameters and Cn2=1017m2/3 (green), Cn2=1016m2/3 (blue), and Cn2=1015m2/3 (red).

Fig. 5.
Fig. 5.

Cumulative probability of filamentation onset versus radius for the same laser parameters as Fig. 4 and Cn2=1017m2/3 (green), Cn2=1016m2/3 (blue), and Cn2=1015m2/3 (red).

Fig. 6.
Fig. 6.

(a) Filamentation onset probability versus propagation distance for a collimated Gaussian laser beam with P=10PNL and R0=9.3cm, for Cn2=1017m2/3 (green), Cn2=1016m2/3 (blue), Cn2=1015m2/3 (red), and Cn2=1014m2/3 (yellow). (b) Cumulative probability of filamentation versus radius for the same laser parameters as in panel (a). Colors denote the same values of Cn2 as in panel (a).

Fig. 7.
Fig. 7.

(a) Filamentation onset probability versus propagation distance for a collimated Gaussian laser beam with P=20PNL and R0=11.5cm, for Cn2=1017m2/3 (green), Cn2=1016m2/3 (blue), Cn2=1015m2/3 (red), and Cn2=1014m2/3 (yellow). (b) Cumulative probability of filamentation versus radius for the same laser parameters as in panel (a). Colors denote the same values of Cn2 as in panel (a).

Fig. 8.
Fig. 8.

Intensity contours for beams with (a) P=1.5PNL and (b) P=20PNL near the onset of filamentation for Cn2=1015m2/3. Peak intensity in panels (a) and (b) are 20 and 2.5GW/cm2, respectively.

Fig. 9.
Fig. 9.

FOD LFO (solid curves) for (a) P=1.5PNL and (b) P=20PNL versus Cn2. Also shown are the self-focusing distance LSF (shorter-dashed, increasing curve) and MI gain length LMI (longer-dashed, decreasing curve). The points denote LFO from simulations. In both (a) and (b), the FOD in the absence of turbulence is 5 km.

Fig. 10.
Fig. 10.

Cn,SF2 versus self-focusing length, LSF,0, for a beam with P=1.5PNL according to Eq. (2) (solid curve). LSF,0 is varied by changing the initial beam spot size, R0. Points represent values of Cn2, for a given LSF,0, at which the filamentation probability goes to zero in the simulations.

Fig. 11.
Fig. 11.

Cn,MI2 versus P/PNL for a beam with 5 km filamentation range in the absence of turbulence (solid curve). Points represent values of Cn2 at which the filamentation probability goes to zero in the simulations.

Fig. 12.
Fig. 12.

Filamentation onset probability of a focused beam with P=20PNL, R0=30cm for (a) fixed focal length (5.9 km), and (b) variable focal length. In both panels, colors represent various turbulence indices, i.e., Cn2=1017m2/3 (green), Cn2=1016m2/3 (blue), and Cn2=1015m2/3 (red). In panel (b), the lens focal lengths are 6.3 km (green), 7.4 km (blue), and 11 km (red).

Equations (6)

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Θ2PPNL(M2)2α(R0r0)β>0,
Cn,SF2=(λ3πR011/3)(0.184α1/β)5/3(ZR0LSF,0)1+103β,
N(z)=0z2Γ(z)dz.
LFO=(LMI2+(γLSF)2)1/2,
ZR0zA=[(iR02/4)2i(k02R02/2)δn+2i(P/PNL)]A,
Cn2=σ2(2R0)1/3/(2.91L),

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