Abstract

A wavefront sensor has been used to measure the Kerr nonlinear focal shift of a high intensity ultrashort pulse beam in a focusing beam geometry while accounting for the effects of plasma-defocusing. It is shown that plasma-defocusing plays a major role in the nonlinear focusing dynamics and that measurements of Kerr nonlinearity and ionization are coupled. Furthermore, this coupled effect leads to a novel way that measures the laser ionization rates in air under atmospheric conditions as well as Kerr nonlinearity. The measured nonlinear index n2 compares well with values found in the literature and the measured ionization rates could be successfully benchmarked to the model developed by Perelomov, Popov, and Terentev (PPT model) [Sov. Phys. JETP 50, 1393 (1966)].

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  4. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
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  5. M. Ghotbi, P. Trabs, and M. Beutler, “Generation of high-energy, sub-20-fs pulses in the deep ultraviolet by using spectral broadening during filamentation in argon,” Opt. Lett. 36, 463–465 (2011).
    [CrossRef] [PubMed]
  6. A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
    [CrossRef]
  7. C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
    [CrossRef]
  8. V. Fedorov and V. Kandidov, “Interaction/laser radiation with matter filamentation of laser pulses with different wavelengths in air,” Laser Phys. 18, 1530–1538 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  21. A. M. Perelomov, V. S. Popov, and M. V. Terentev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 50, 1393–1409 (1966).
  22. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (5)

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

M. Ghotbi, P. Trabs, and M. Beutler, “Generation of high-energy, sub-20-fs pulses in the deep ultraviolet by using spectral broadening during filamentation in argon,” Opt. Lett. 36, 463–465 (2011).
[CrossRef] [PubMed]

P. Whalen, J. V. Moloney, and M. Kolesik, “Self-focusing collapse distance in ultrashort pulses and measurement of nonlinear index,” Opt. Lett. 36, 2542–2544 (2011).
[CrossRef] [PubMed]

2010 (2)

2008 (1)

V. Fedorov and V. Kandidov, “Interaction/laser radiation with matter filamentation of laser pulses with different wavelengths in air,” Laser Phys. 18, 1530–1538 (2008).
[CrossRef]

2007 (2)

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

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

2006 (1)

2005 (1)

2002 (1)

J. Schwarz and J. C. Diels, “Long distance propagation of UV filaments,” J. Mod. Opt. 49, 2583–2597 (2002).
[CrossRef]

2001 (1)

J. Schwarz and J. C. Diels, “Analytical solution for UV filaments,” Phys. Rev. A 65013806 (2001).
[CrossRef]

2000 (1)

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B: Lasers Opt. 71, 877–879 (2000).
[CrossRef]

1999 (1)

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999).
[CrossRef]

1997 (1)

1995 (1)

1989 (1)

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

1966 (2)

A. M. Perelomov, V. S. Popov, and M. V. Terentev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 50, 1393–1409 (1966).

P. R. Yurii, “Breakdown and heating of gases under the influence of a laser beam,” Sov. Phys. Usp. 8, 650–673 (1966).
[CrossRef]

1965 (1)

L. V. Keldysh, “Ionization in field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

Ališauskas, S.

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

Baltuška, A.

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

Bergé, L.

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

R. Nuter and L. Bergé, “Pulse chirping and ionization of O2 molecules for the filamentation of femtosecond laser pulses in air,” J. Opt. Soc. Am. B 23, 874–884 (2006).
[CrossRef]

Bethge, J.

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

Beutler, M.

Börzsönyi, Á

Braun, A.

Brée, C.

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

Bukin, O.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Bykova, E.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Chin, S.

Chin, S. L.

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B: Lasers Opt. 71, 877–879 (2000).
[CrossRef]

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999).
[CrossRef]

Couairon, A.

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

Demircan, A.

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

Diels, J. C.

J. Schwarz and J. C. Diels, “Long distance propagation of UV filaments,” J. Mod. Opt. 49, 2583–2597 (2002).
[CrossRef]

J. Schwarz and J. C. Diels, “Analytical solution for UV filaments,” Phys. Rev. A 65013806 (2001).
[CrossRef]

Du, D.

Faucher, O.

Fedorov, V.

V. Fedorov and V. Kandidov, “Interaction/laser radiation with matter filamentation of laser pulses with different wavelengths in air,” Laser Phys. 18, 1530–1538 (2008).
[CrossRef]

Franco, M. A.

Geints, Y.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Ghotbi, M.

Golik, S.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Grillon, G.

Heiner, Z.

Hellwarth, R. W.

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

Henesian, M. A.

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

Hertz, E.

Ilyin, A.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Kabanov, A.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Kalashnikov, M. P.

Kandidov, V.

V. Fedorov and V. Kandidov, “Interaction/laser radiation with matter filamentation of laser pulses with different wavelengths in air,” Laser Phys. 18, 1530–1538 (2008).
[CrossRef]

Kasparian, J.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B: Lasers Opt. 71, 877–879 (2000).
[CrossRef]

Keldysh, L. V.

L. V. Keldysh, “Ionization in field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999).

Kolesik, M.

Korn, G.

Kovács, A. P.

Lavorel, B.

Liu, W.

Liu, X.

Loriot, V.

Matvienko, G.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Moloney, J. V.

Mourou, G.

Mücke, O. D.

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

Mysyrowicz, A.

Nibbering, E. T. J.

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–660 (1997).
[CrossRef]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

R. Nuter and L. Bergé, “Pulse chirping and ionization of O2 molecules for the filamentation of femtosecond laser pulses in air,” J. Opt. Soc. Am. B 23, 874–884 (2006).
[CrossRef]

Oshlakov, V.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Osvay, K.

Pennington, D. M.

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

Perelomov, A. M.

A. M. Perelomov, V. S. Popov, and M. V. Terentev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 50, 1393–1409 (1966).

Popov, V. S.

A. M. Perelomov, V. S. Popov, and M. V. Terentev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 50, 1393–1409 (1966).

Prade, B. S.

Pugžlys, A.

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

Sauerbrey, R.

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B: Lasers Opt. 71, 877–879 (2000).
[CrossRef]

Schwarz, J.

J. Schwarz and J. C. Diels, “Long distance propagation of UV filaments,” J. Mod. Opt. 49, 2583–2597 (2002).
[CrossRef]

J. Schwarz and J. C. Diels, “Analytical solution for UV filaments,” Phys. Rev. A 65013806 (2001).
[CrossRef]

Skupin, S.

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

Sokolova, E.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Squier, J.

Steinmeyer, G.

C. Brée, A. Demircan, J. Bethge, E. T. J. Nibbering, S. Skupin, L. Bergé, and G. Steinmeyer, “Filamentary pulse self-compression:the impact of the cell windows,” Phys. Rev. A 83, 043803 (2011).
[CrossRef]

Talebpour, A.

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999).
[CrossRef]

Terentev, M. V.

A. M. Perelomov, V. S. Popov, and M. V. Terentev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 50, 1393–1409 (1966).

Trabs, P.

Voronin, A. A.

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

Whalen, P.

Wolf, J.-P.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

Yang, J.

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999).
[CrossRef]

Yurii, P. R.

P. R. Yurii, “Breakdown and heating of gases under the influence of a laser beam,” Sov. Phys. Usp. 8, 650–673 (1966).
[CrossRef]

Zemlyanov, A.

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

Zheltikov, A. M.

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

Appl. Phys. B: Lasers Opt. (1)

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B: Lasers Opt. 71, 877–879 (2000).
[CrossRef]

Atmos. Oceanic Opt. (1)

O. Bukin, E. Bykova, Y. Geints, S. Golik, A. Zemlyanov, A. Ilyin, A. Kabanov, G. Matvienko, V. Oshlakov, and E. Sokolova, “Filamentation of a sharply focused ultrashort laser pulse at wavelengths of 800 and 400 nm: measurements of the nonlinear index of air refraction,” Atmos. Oceanic Opt. 24, 417–424 (2011).
[CrossRef]

J. Mod. Opt. (1)

J. Schwarz and J. C. Diels, “Long distance propagation of UV filaments,” J. Mod. Opt. 49, 2583–2597 (2002).
[CrossRef]

J. Opt. Soc. Am. B (2)

Laser Phys. (1)

V. Fedorov and V. Kandidov, “Interaction/laser radiation with matter filamentation of laser pulses with different wavelengths in air,” Laser Phys. 18, 1530–1538 (2008).
[CrossRef]

Opt. Commun. (1)

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rep. (1)

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

Phys. Rev. A (4)

A. A. Voronin, S. Ališauskas, O. D. Mücke, A. Pugžlys, A. Baltuška, and A. M. Zheltikov, “High-energy-throughput pulse compression by off-axis group-delay compensation in a laser-induced filament,” Phys. Rev. A 84, 023832 (2011).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

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[CrossRef]

Sov. Phys. JETP (2)

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[CrossRef]

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ZEMAX, “Software for Optical System Design,” http://www.zemax.com/

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Figures (6)

Fig. 1
Fig. 1

Setup to measure the self-focus shift using a wavefront sensor (WFS). The blue lines represent the low peak power case while red lines represent powers sufficient for self-focusing.

Fig. 2
Fig. 2

(a) Flat reference/initial wavefront recorded for the f/120 geometry at a beam energy of 0.3 mJ. The associated “defocus” term was measured at R 0 2 = 0.005 waves. (b) Wave-front curvature recorded at 5.4 mJ energy with R 0 2 = 0.05 waves “defocus” for the same experimental configuration as (a).

Fig. 3
Fig. 3

Plot of focal shift Δ versus laser input energy. The scatter plots show experimental data for f/120 and f/40 focusing geometries. The solid lines represent theoretical fits based on the pulse propagation equation [Eq. (9)] (see below) and the dashed lines depict the theoretical behavior without ionization present.

Fig. 4
Fig. 4

Plot of the modeled laser beam waist versus propagation distance for the f/120 focusing geometry at various laser input energies. It can be seen that the waist moves towards the input optic as the laser beam energy increases. The geometrical focus is marked by the black dashed line. This is consistent with the experimental observation in Fig. 3. One can also see that the beam waist initially decreases as a function of self-focusing and later increases as plasma de-focusing sets in. The model parameters were as follows:wi = 6.2 mm, τ = 540 fs, λ = 1054 nm, NO2 = 4.6 × 1024 m−3 (accounting for 20% of O2 in the atmosphere and 85% of atmospheric pressure at the altitude of Albuquerque, NM), σ(K=11) = 3 × 10−191m22W−11s−1, and n2 = 2.6 × 10−23 m2/W.

Fig. 5
Fig. 5

Plot of the laser intensity versus input energy (right axis). Experimental data are shown for comparison for the same energy range (left).

Fig. 6
Fig. 6

Ionization rate versus intensity for oxygen and nitrogen. The dashed, vertical green line depicts the transition region between MPI and tunneling ionization, where the Keldysh parameter γ = 1 for λ = 1054 nm.

Tables (1)

Tables Icon

Table 1 Comparison of various values of n2 found in the literature. All values are in units of 10−23m2/W and correspond to atmospheric pressure. Note, reference [17] uses a 20 fs beam chirped to 200 fs.

Equations (17)

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1 f 2 = 1 f 2 + Δ + 1 R + d Δ = 1 1 f 2 1 R + d f 2 .
R = a 2 4 R 0 2 λ ,
z s f = π w i 2 λ ( P P crit 1 ) 1 / 2 ,
2 w ( z , t ) z 2 = 4 k 2 w ( z , t ) 3 ( 1 P ( t ) P crit ) + 2 K ( K + 1 ) 2 n 0 N crit w ( z , t ) × N e ( z , t ) ,
d N e ( z , t ) d t = ( N O 2 N e ( z , t ) ) σ ( K ) I ( z , t ) K ,
d N e ( z , t ) d t = N O 2 σ ( K ) I ( z , t ) K .
σ ( K ) = ω l ( W O 2 h ¯ ω l ) 1.5 × ( e 2 4 c ɛ 0 m e ω l 2 W O 2 ) K × e K = 3.2 × 10 181 m 22 W 11 s 1 .
N e ( z ) = N O 2 [ 1 Exp ( σ ( K ) I 0 K ( w i w ( z ) ) 2 K τ ) ] ,
2 w ( z ) z 2 = 4 k 2 w ( z ) 3 ( 1 P 0 P crit ) + 2 K N O 2 ( 1 e σ ( K ) I 0 K ( w i w ( z ) ) 2 K τ ) ( K + 1 ) 2 n 0 N crit w ( z ) ,
R ( E ) = 6 π e E H 2 m e W H | c n * , l * | 2 f ( l , m ) W i 2 W H A m ( γ ) ( 2 E 0 E 1 + γ 2 ) 2 n * | m | 3 / 2 exp ( 2 E 0 3 E g ( γ ) ) ,
| c n * , l * | 2 = 2 2 n * n * Γ ( n * + l * + 1 ) Γ ( n * l * ) ,
f ( l , m ) = ( 2 l + 1 ) ( l + | m | ) ! 2 | m | ( | m | ) ! ( l | m | ) ! , with f ( 0 , 0 ) = 1.
A m ( γ ) = 4 3 π 1 | m | ! γ 2 1 + γ 2 κ > ν + exp ( ( κ ν ) α ( γ ) ) Φ m ( ( κ ν ) β ( γ ) ) ,
Φ m ( x ) = x 2 | m | + 1 2 0 1 e x 2 t t | m | 1 t d t = e x 2 0 x ( x 2 y 2 ) | m | e y 2 d y .
α ( γ ) = 2 ( ( sinh 1 γ ) γ 1 + γ 2 ) ,
β ( γ ) = 2 γ 1 + γ 2 ,
g ( γ ) = 3 2 γ [ ( 1 + 1 2 γ 2 ) ( sinh 1 γ ) 1 + γ 2 2 γ ] .

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