Abstract

The propagation of high peak-power laser beams in real atmospheres will be affected at long range by both linear and nonlinear effects contained therein. Arguably, J. H. Marburger is associated with the mathematical characterization of this phenomenon. This paper provides a validated set of engineering equations for characterizing the self-focusing distance from a laser beam propagating through non-turbulent air with, and without, loss as well as three source configurations: (1) no lens, (2) converging lens and (3) diverging lens. The validation was done against wave-optics simulation results. Some validated equations follow Marburger completely, but others do not, requiring modification of the original theory. Our results can provide a guide for numerical simulations and field experiments.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

Thomas Karr, Larry B. Stotts, Jason A. Tellez, Jason D. Schmidt, and Justin D. Mansell, "Engineering equations for characterizing nonlinear laser intensity propagation in air with loss: erratum," Opt. Express 26, 8417-8417 (2018)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-26-7-8417

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References

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2017 (2)

J. Peñano, J. P. Palastro, B. Hafizi, 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] [PubMed]

2016 (1)

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

2015 (2)

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths,” Opt. Lett. 40(24), 5794–5797 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (1)

2008 (1)

A. A. Zemlyanov and Y. E. Geints, “Zonal model of nonstationary self-focusing of femtosecond laser radiation in air: effective beam characteristics evolution,” Eur. Phys. J. D 42(2), 349–357 (2008).
[Crossref]

2007 (2)

A. A. Zemlyanov and Y. E. Geints, “Evolution of Effective Characteristics of Laser Beam of Femtosecond Duration upon Self-Action in a Gas Medium,” Opt. Spectrosc. 104(5), 772–783 (2007).
[Crossref]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]

2005 (1)

1991 (1)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30(8), 1228 (1991).
[Crossref]

1984 (1)

T. R. Taha and M. J. Ablowitz, “Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations. II. Numerical, Nonlinear Schrodinger Equation,” J. Comput. Phys. 55(2), 203–230 (1984).
[Crossref]

1970 (1)

V. I. Talanov, “Focusing of Light in Cubic Media,” Sov. Phys. JETP 11, 199 (1970).

1969 (1)

E. L. Dawes and J. H. Marburger, “Computer Studies in Self-Focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

1966 (1)

W. Kaiser, A. Laubereau, M. Maier, and J. A. Giordmaine, “Self-Focusing of Optical Beams in Absorbing Media,” Phys. Lett. 22(1), 60–62 (1966).
[Crossref]

1965 (1)

P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[Crossref]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beans,” Phys. Rev. Lett. 13, 479 (1964).

Ablowitz, M. J.

T. R. Taha and M. J. Ablowitz, “Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations. II. Numerical, Nonlinear Schrodinger Equation,” J. Comput. Phys. 55(2), 203–230 (1984).
[Crossref]

Bergé, L.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Borchert, H.

Chateauneuf, M.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beans,” Phys. Rev. Lett. 13, 479 (1964).

Chin, S.

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]

Daigle, J. F.

Dawes, E. L.

E. L. Dawes and J. H. Marburger, “Computer Studies in Self-Focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

Dergachev, A. A.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

DiComo, G.

DiComo, G. P.

J. Peñano, J. P. Palastro, B. Hafizi, 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]

Diener, K.

Dubois, J.

Durand, M.

Durécu, A.

Fischer, R. P.

Fleury, D.

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beans,” Phys. Rev. Lett. 13, 479 (1964).

Geints, Y. E.

A. A. Zemlyanov and Y. E. Geints, “Zonal model of nonstationary self-focusing of femtosecond laser radiation in air: effective beam characteristics evolution,” Eur. Phys. J. D 42(2), 349–357 (2008).
[Crossref]

A. A. Zemlyanov and Y. E. Geints, “Evolution of Effective Characteristics of Laser Beam of Femtosecond Duration upon Self-Action in a Gas Medium,” Opt. Spectrosc. 104(5), 772–783 (2007).
[Crossref]

Giordmaine, J. A.

W. Kaiser, A. Laubereau, M. Maier, and J. A. Giordmaine, “Self-Focusing of Optical Beams in Absorbing Media,” Phys. Lett. 22(1), 60–62 (1966).
[Crossref]

Goldberg, V. N.

V. N. Goldberg, V. I. Talanov, and R. K. Irm, Izv. Vysshikh Uc’hebn. Zavedenii Radiofiz.10, 674 (1967).

Hafizi, B.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30(8), 1228 (1991).
[Crossref]

Helle, M.

Helle, M. H.

J. Peñano, J. P. Palastro, B. Hafizi, 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]

Houard, A.

Ionin, A. A.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Irm, R. K.

V. N. Goldberg, V. I. Talanov, and R. K. Irm, Izv. Vysshikh Uc’hebn. Zavedenii Radiofiz.10, 674 (1967).

Kaiser, W.

W. Kaiser, A. Laubereau, M. Maier, and J. A. Giordmaine, “Self-Focusing of Optical Beams in Absorbing Media,” Phys. Lett. 22(1), 60–62 (1966).
[Crossref]

Kandidov, V. P.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Kelley, P. L.

P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[Crossref]

Kurz, H. G.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Laubereau, A.

W. Kaiser, A. Laubereau, M. Maier, and J. A. Giordmaine, “Self-Focusing of Optical Beams in Absorbing Media,” Phys. Lett. 22(1), 60–62 (1966).
[Crossref]

Liu, W.

Maier, M.

W. Kaiser, A. Laubereau, M. Maier, and J. A. Giordmaine, “Self-Focusing of Optical Beams in Absorbing Media,” Phys. Lett. 22(1), 60–62 (1966).
[Crossref]

Marburger, J. H.

E. L. Dawes and J. H. Marburger, “Computer Studies in Self-Focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

Milchberg, H. M.

Mokrousova, D. V.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Moreau, B.

Mysyrowicz, A.

Palastro, J. P.

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] [PubMed]

J. Peñano, J. P. Palastro, B. Hafizi, 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.

J. Peñano, J. P. Palastro, B. Hafizi, 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]

J. Peñano, B. Hafizi, A. Ting, and M. Helle, “Theoretical and numerical investigation of filament onset distance in atmospheric turbulence,” J. Opt. Soc. Am. B 31(5), 963 (2014).
[Crossref]

Peñano, J. R.

Polynkin, P.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Prade, B.

Said, A. A.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30(8), 1228 (1991).
[Crossref]

Schmitt, R.

Schmitt-Sody, A.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Seleznev, L. V.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30(8), 1228 (1991).
[Crossref]

Shlenov, S. A.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Shustikova, A. P.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Sinitsyn, D. V.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Skupin, S.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30(8), 1228 (1991).
[Crossref]

Sunchugasheva, E. S.

S. A. Shlenov, A. A. Dergachev, A. A. Ionin, V. P. Kandidov, D. V. Mokrousova, L. V. Seleznev, D. V. Sinitsyn, E. S. Sunchugasheva, and A. P. Shustikova, “Femtosecond laser filament and plasma channels in focused beam in air,” Proc. SPIE 9447, 944717 (2015).
[Crossref]

Taha, T. R.

T. R. Taha and M. J. Ablowitz, “Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations. II. Numerical, Nonlinear Schrodinger Equation,” J. Comput. Phys. 55(2), 203–230 (1984).
[Crossref]

Talanov, V. I.

V. I. Talanov, “Focusing of Light in Cubic Media,” Sov. Phys. JETP 11, 199 (1970).

V. N. Goldberg, V. I. Talanov, and R. K. Irm, Izv. Vysshikh Uc’hebn. Zavedenii Radiofiz.10, 674 (1967).

Théberge, F.

Ting, A.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beans,” Phys. Rev. Lett. 13, 479 (1964).

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30(8), 1228 (1991).
[Crossref]

Vasseur, O.

Wahlstrand, J. K.

Zahedpour, S.

Zemlyanov, A. A.

A. A. Zemlyanov and Y. E. Geints, “Zonal model of nonstationary self-focusing of femtosecond laser radiation in air: effective beam characteristics evolution,” Eur. Phys. J. D 42(2), 349–357 (2008).
[Crossref]

A. A. Zemlyanov and Y. E. Geints, “Evolution of Effective Characteristics of Laser Beam of Femtosecond Duration upon Self-Action in a Gas Medium,” Opt. Spectrosc. 104(5), 772–783 (2007).
[Crossref]

Eur. Phys. J. D (1)

A. A. Zemlyanov and Y. E. Geints, “Zonal model of nonstationary self-focusing of femtosecond laser radiation in air: effective beam characteristics evolution,” Eur. Phys. J. D 42(2), 349–357 (2008).
[Crossref]

J. Comput. Phys. (1)

T. R. Taha and M. J. Ablowitz, “Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations. II. Numerical, Nonlinear Schrodinger Equation,” J. Comput. Phys. 55(2), 203–230 (1984).
[Crossref]

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

New J. Phys. (1)

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Opt. Eng. (1)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30(8), 1228 (1991).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Opt. Spectrosc. (1)

A. A. Zemlyanov and Y. E. Geints, “Evolution of Effective Characteristics of Laser Beam of Femtosecond Duration upon Self-Action in a Gas Medium,” Opt. Spectrosc. 104(5), 772–783 (2007).
[Crossref]

Phys. Lett. (1)

W. Kaiser, A. Laubereau, M. Maier, and J. A. Giordmaine, “Self-Focusing of Optical Beams in Absorbing Media,” Phys. Lett. 22(1), 60–62 (1966).
[Crossref]

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[Crossref]

Phys. Rev. (1)

E. L. Dawes and J. H. Marburger, “Computer Studies in Self-Focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

Phys. Rev. A (1)

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Figures (13)

Fig. 1
Fig. 1 Normalized intensity versus normalized propagation distance as a function of P peak / P crit
Fig. 2
Fig. 2 Exponent parameter γ as a function of P peak / P crit
Fig. 3
Fig. 3 Normalized Intensity versus normalized propagation distance z * as a function of P peak / P crit
Fig. 4
Fig. 4 Comparison of modified self-focusing distance derived from Eq. (17) and computer simulation results versus telescope power for various values
Fig. 5
Fig. 5 Normalized intensity versus normalized propagation distance z * for various value of normalized extinction coefficient α * for P peak / P crit =333
Fig. 6
Fig. 6 Sea-Level volume extinction coefficients α versus wavelength as a function of atmospheric visibility
Fig. 7
Fig. 7 Comparison of Eq. (23) and computer simulation results versus telescope power for various values of P peak / P crit for α=0.1k m 1
Fig. 8
Fig. 8 Comparison of Eq. (23) and computer simulation results versus telescope power for various values of P peak / P crit for α=0.3k m 1
Fig. 9
Fig. 9 Comparison of Eq. (23) and computer simulation results versus telescope power for various values of P peak / P crit for α=0.5k m 1
Fig. 10
Fig. 10 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of P peak / P crit for α=0.1k m 1
Fig. 11
Fig. 11 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of P peak / P crit for α=0.3k m 1
Fig. 12
Fig. 12 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of P peak / P crit for α=0.5k m 1
Fig. 13
Fig. 13 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of P peak / P crit for α=1.0k m 1

Equations (25)

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1 ρ * ρ * ( ρ * E 0 * ρ * )+i z * E 0 * + k 2 a 2 ε 2 2 ε 0 | E 0 * | E 0 * 2 =0
ρ * =ρ/a
z * =z/ 2k a 2
E 0 * =ka ε 2 2 ε 0 E 0 .
P crit 1.4884 λ 0 2 ( c n 0 128 n 2 )
λ 0 =2π ε 0 k 1
n 2 = ε 2 /2 .
P crit 3.77 λ 2 8π n 0 n 2 ,
z f = 0.367 z r ( P peak / P crit 0.852 ) 2 0.0219
z r =k a 2
I( z ) I( 0 ) = [ 1 ( z z f ) 2 ] γ ,
γ= n=0 8 a n ( log 10 [ P peak / P crit ] ) n
{ a n ;n=0,...,8 }={ 0.23805,2.4559,10.027,3.0896,20.010,18.467, 3.3566,4.608,0.30433 }
1 z f * = 1 f + 1 z f
(z/ z r ) 2 +[ 1 ( z/ z f * ) 2 ] .
1 ρ *2 2 E 0 * ϕ 2 + 1 ρ * ρ * ( ρ * E 0 * ρ * )+i z * E 0 * + k 2 a 2 ε 2 2 ε 0 | E 0 * | E 0 * 2 =0
d a 4 dz = 4P k 2 P 1 zexp{ αz },
a 4 = a 0 4 { 1( 2 α 2 z f 2 )[ 1( 1+αz )exp{ αz } ] },
ζ 0 =f/ ( z f +f ) ,
z f * = z f f/ ( z f +f ) = z f ζ 0 .
d a 4 dζ = 4P k 2 P 1 ζexp{ αζ }.
d a 4 dz = 4P k 2 P 1 ζ 0 2 zexp{ αz/ ζ 0 },
a 4 = a 0 4 { 1( 2 α 2 z f 2 )[ 1( 1+αz/ ζ 0 )exp{ αz/ ζ 0 } ] },
α v 3.912 z v ( λ c 0.55μm ) q q={ 1.6, z v >50km 1.3,6km z v 50km 0.585 z v 1/3 , z v <6km
a 4 = a 0 4 { 1( 2 α 2 z f ' 2 )[ 1( 1+αz )exp{ αz } ] },

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