Abstract

A high-power laser beam propagating through a dielectric in the presence of fluctuations is subject to diffraction, dissipation, and optical Kerr nonlinearity. A method of moments was applied to a stochastic, nonlinear enveloped wave equation to analyze the evolution of the long-term spot radius. For propagation in atmospheric turbulence described by a Kolmogorov-von Kármán spectral density, the analysis was benchmarked against field experiments in the low-power limit and compared with simulation results in the high-power regime. Dissipation reduced the effect of self-focusing and led to chromatic aberration.

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References

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    [Crossref]
  2. J. P. Palastro, J. Peñano, W. Nelson, G. DiComo, M. Helle, L. A. Johnson, and B. Hafizi, Opt. Express 24, 18817 (2016).
    [Crossref]
  3. J. Peñano, J. P. Palastro, M. H. Helle, G. P. DiComo, and B. Hafizi, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. Lett, submitted for publication.
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    [Crossref]
  5. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1971).
    [Crossref]
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    [Crossref]
  7. M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
    [Crossref]
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    [Crossref]
  9. V. I. Tatarskiĭ, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).
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    [Crossref]
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    [Crossref]
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2016 (1)

2014 (1)

2005 (2)

G. Fibich, S. Eisenmann, B. Ilan, Y. Ehrlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, Opt. Express 13, 5897 (2005).
[Crossref]

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

2002 (2)

P. Sprangle, J. R. Peñano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
[Crossref]

G. Gbur and E. Wolf, J. Opt. Soc. Am. A 19, 1592 (2002).
[Crossref]

1998 (1)

1996 (1)

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

1986 (1)

J. J. Rasmussen and K. Rypdal, Phys. Scripta 33, 481 (1986).
[Crossref]

1980 (1)

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

1975 (2)

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[Crossref]

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

1971 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1971).
[Crossref]

1967 (1)

Z. I. Feizulin and Y. A. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[Crossref]

Acharya, P. K.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Adler-Golden, S. M.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Anderson, G. P.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Andrews, L. C.

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

Bally, J.

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Berk, A.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Bernstein, L. S.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Billawala, Y.

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Chetwynd, J. H.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Christiansen, P. L.

Cooley, T. W.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

DiComo, G.

DiComo, G. P.

J. Peñano, J. P. Palastro, M. H. Helle, G. P. DiComo, and B. Hafizi, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. Lett, submitted for publication.

Ehrlich, Y.

Eisenmann, S.

Fante, R. L.

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

Feizulin, Z. I.

Z. I. Feizulin and Y. A. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[Crossref]

Fibich, G.

G. Fibich, S. Eisenmann, B. Ilan, Y. Ehrlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, Opt. Express 13, 5897 (2005).
[Crossref]

G. Fibich, in Self-focusing: Past and Present, R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. (Springer, 2009).

Fox, M.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Fraenkel, M.

Gaeta, A. L.

Gaididei, Y. B.

Gardner, J. A.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Gbur, G.

Goldman, M. V.

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Hafizi, B.

J. P. Palastro, J. Peñano, W. Nelson, G. DiComo, M. Helle, L. A. Johnson, and B. Hafizi, Opt. Express 24, 18817 (2016).
[Crossref]

J. Peñano, B. Hafizi, A. Ting, and M. Helle, J. Opt. Soc. Am. B 31, 963 (2014).
[Crossref]

P. Sprangle, J. R. Peñano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
[Crossref]

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

J. Peñano, J. P. Palastro, M. H. Helle, G. P. DiComo, and B. Hafizi, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. Lett, submitted for publication.

Helle, M.

Helle, M. H.

J. Peñano, J. P. Palastro, M. H. Helle, G. P. DiComo, and B. Hafizi, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. Lett, submitted for publication.

Henis, Z.

Hoke, M. L.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Ilan, B.

Johnson, L. A.

Kravtsov, Y. A.

Z. I. Feizulin and Y. A. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[Crossref]

Lee, J.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Lloyd, J. P.

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Lockwood, R. B.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Loewenstein, R. F.

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Marburger, J. H.

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[Crossref]

Mrozek, F.

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Muratov, L.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Nelson, W.

Palastro, J. P.

J. P. Palastro, J. Peñano, W. Nelson, G. DiComo, M. Helle, L. A. Johnson, and B. Hafizi, Opt. Express 24, 18817 (2016).
[Crossref]

J. Peñano, J. P. Palastro, M. H. Helle, G. P. DiComo, and B. Hafizi, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. Lett, submitted for publication.

Peñano, J.

J. P. Palastro, J. Peñano, W. Nelson, G. DiComo, M. Helle, L. A. Johnson, and B. Hafizi, Opt. Express 24, 18817 (2016).
[Crossref]

J. Peñano, B. Hafizi, A. Ting, and M. Helle, J. Opt. Soc. Am. B 31, 963 (2014).
[Crossref]

J. Peñano, J. P. Palastro, M. H. Helle, G. P. DiComo, and B. Hafizi, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. Lett, submitted for publication.

Peñano, J. R.

P. Sprangle, J. R. Peñano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
[Crossref]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1971).
[Crossref]

Phillips, R. L.

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

Potter, D.

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Rasmussen, J. J.

J. J. Rasmussen and K. Rypdal, Phys. Scripta 33, 481 (1986).
[Crossref]

Rypdal, K.

J. J. Rasmussen and K. Rypdal, Phys. Scripta 33, 481 (1986).
[Crossref]

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Sprangle, P.

P. Sprangle, J. R. Peñano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
[Crossref]

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1971).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskiĭ, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).

Theil, D.

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Ting, A.

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1971).
[Crossref]

Wolf, E.

Zigler, A.

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Express (2)

Opt. Lett. (1)

Phys. Fluids (1)

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Phys. Rev. E (1)

P. Sprangle, J. R. Peñano, and B. Hafizi, Phys. Rev. E 66, 046418 (2002).
[Crossref]

Phys. Scripta (1)

J. J. Rasmussen and K. Rypdal, Phys. Scripta 33, 481 (1986).
[Crossref]

Proc. IEEE (1)

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

Proc. SPIE (1)

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, Proc. SPIE 5655, 88 (2005).
[Crossref]

Prog. Quantum Electron. (1)

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[Crossref]

Publ. Astron. Soc. Aust. (1)

J. Bally, D. Theil, Y. Billawala, D. Potter, R. F. Loewenstein, F. Mrozek, and J. P. Lloyd, Publ. Astron. Soc. Aust. 13, 22 (1996).
[Crossref]

Radiophys. Quantum Electron. (2)

Z. I. Feizulin and Y. A. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
[Crossref]

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1971).
[Crossref]

Other (4)

J. Peñano, J. P. Palastro, M. H. Helle, G. P. DiComo, and B. Hafizi, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. Lett, submitted for publication.

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

V. I. Tatarskiĭ, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).

G. Fibich, in Self-focusing: Past and Present, R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. (Springer, 2009).

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Figures (3)

Fig. 1.
Fig. 1. Long-term spot radius versus refractive index structure constant for a low-power beam (i.e., in the absence of self-focusing). Black circles are data points from field experiments at the Air Force Research Laboratory, NM. Orange circles were obtained using HELCAP full-scale simulations. The analytical results are shown in (i) dashed green, obtained by using Eq. (37) of Ref. [10], and (ii) blue, obtained using Eq. (3b) here.
Fig. 2.
Fig. 2. Long-term spot radius versus propagation distance for a beam focused at 8 km. Blue curves are the results of HELCAP simulations, and the red curves were generated using Eq. (3b). Dashed curves represent cases where P / P NL = 0.8 , while solid curves represent linear propagation; that is, P / P NL 0 . Black dots denote the spatial coherence radius computed in HELCAP. In panel (a) C n 2 = 10 15    m 2 / 3 , and in panel (b) C n 2 = 10 16    m 2 / 3 .
Fig. 3.
Fig. 3. Long-term spot radius versus propagation distance for P / P NL = 0.8 and wavelengths λ = 1    μm (blue, solid), λ = 3    μm (green, short dashed), and λ = 5    μm (red, long dashed). (a) Extinction rate γ = 0 and (b) extinction rate γ = 1    km 1 . A beam with initial spot radius w = 10    cm was focused by a lens with focal length | F | = 8    km in air with C n 2 = 10 15    m 2 / 3 .

Tables (1)

Tables Icon

Table 1. Minimum Spot Radius at Focal Point for Three Wavelengths, Comparing the Low-Power Case with the High-Power Case, in the Presence of Extinction γ = 1    km 1

Equations (8)

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i k 0 z E ( r , z ) = [ 1 2 2 n ¯ 2 ( r , z ) | E ( r , z ) | 2 + η ( r , z ) i k 0 γ / 2 ] E ( r , z ) ,
( d / d z + γ ) N ( z ) = 0 ,
d 2 R L 2 ( z ) / d z 2 = 4 H ( z ) / k 0 2 N ( z ) ,
( d / d z + γ ) H ( z ) = [ N ( z ) / 2 k 0 ] 2 C η ( r , z ) | r = 0 + γ c n 0 d 2 r Γ 4 ( r , r ; r , r , z ) n ¯ 2 ( r , z ) / 16 π ,
d 2 R L 2 ( z ) / d z 2 = 4 H ( z ) e γ z / k 0 2 N 0 ,
( d / d z + γ ) H ( z ) / N 0 = ( e γ z / 2 k 0 ) 2 C η ( r , z ) | r = 0 + γ n ¯ 2 N 0 e 2 γ z / 4 π R L 2 ( z ) .
R L 2 ( z ) = w 2 2 ( 1 + z F ) 2 + 2 z 2 k 0 2 w 2 ( 1 P P NL ) + C strong z 3 , for    r 0 ( z ) 0 or z 1 / C n 2 k 0 2 0 5 / 3 ,
R L 2 ( z ) = w 2 2 ( 1 + z F ) 2 + 2 z 2 k 0 2 w 2 ( 1 P P NL ) + C weak ( z ) z 2 , for    r 0 ( z ) 0 or z 1 / C n 2 k 0 2 0 5 / 3 ,

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