Abstract

Numerical simulations of nonlinear pulse propagation in air show an initial pulse formed, absorbed by plasma generation, and subsequently replenished by power from the trailing edge. Here we show that this scenario implies that the length scale for filament propagation is relatively insensitive to the peak input power beyond the threshold for filament generation.

© 1999 Optical Society of America

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References

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  1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
    [Crossref] [PubMed]
  2. Xin Miao Zhao, P. Rambo, and J.-C. Diels. Filamentation of femtosecond uv pulses in air. In QELS 95. Summaries of Papers Presented at the Quantum Electronics and Laser Science Conference. Technical Digest Series Conference Edition, volume 16, pages 178–9, 1995.
  3. Xin Miao Zhao, J.-C. Diels, Cai Yi Wang, and J. M. Elizondo. Femtosecond ultraviolet laser pulse induced lightning discharges in gases. Quantum Electron., QE-31, 599–612, 1995.
  4. E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
    [Crossref] [PubMed]
  5. A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
    [Crossref] [PubMed]
  6. L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.
  7. M. Mlejnek, E. M. Wright, and J. V. Moloney. Dynamic spatial replenishment of femtosecond pulses propagating in air. Opt. Lett., 23, 382–4, 1998.
    [Crossref]
  8. O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin. Conical emission from laser-plasma interactions in the filamentation of powerful ultrashort laser pulses in air. Opt. Lett., 22, 1332–4, 1997.
    [Crossref]
  9. H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
    [Crossref]
  10. M. Mlejnek, E. M. Wright, and J. V. Moloney. Moving-focus versus self-waveguiding model for long distance propagation of femtosecond pulses in air. To be published in Quantum Electron. (1999).

Braun, A.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
[Crossref] [PubMed]

Brodeur, A.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
[Crossref] [PubMed]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin. Conical emission from laser-plasma interactions in the filamentation of powerful ultrashort laser pulses in air. Opt. Lett., 22, 1332–4, 1997.
[Crossref]

Chien, C. Y.

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin. Conical emission from laser-plasma interactions in the filamentation of powerful ultrashort laser pulses in air. Opt. Lett., 22, 1332–4, 1997.
[Crossref]

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
[Crossref] [PubMed]

Chin, S. L.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
[Crossref] [PubMed]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin. Conical emission from laser-plasma interactions in the filamentation of powerful ultrashort laser pulses in air. Opt. Lett., 22, 1332–4, 1997.
[Crossref]

Chiron, A.

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Curley, P. F.

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

Diels, J.-C.

Xin Miao Zhao, J.-C. Diels, Cai Yi Wang, and J. M. Elizondo. Femtosecond ultraviolet laser pulse induced lightning discharges in gases. Quantum Electron., QE-31, 599–612, 1995.

Xin Miao Zhao, P. Rambo, and J.-C. Diels. Filamentation of femtosecond uv pulses in air. In QELS 95. Summaries of Papers Presented at the Quantum Electronics and Laser Science Conference. Technical Digest Series Conference Edition, volume 16, pages 178–9, 1995.

Du, D.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
[Crossref] [PubMed]

Elizondo, J. M.

Xin Miao Zhao, J.-C. Diels, Cai Yi Wang, and J. M. Elizondo. Femtosecond ultraviolet laser pulse induced lightning discharges in gases. Quantum Electron., QE-31, 599–612, 1995.

Franco, M. A.

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Grillon, G.

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

Ilkov, F. A.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
[Crossref] [PubMed]

Kandidov, V. P.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
[Crossref] [PubMed]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin. Conical emission from laser-plasma interactions in the filamentation of powerful ultrashort laser pulses in air. Opt. Lett., 22, 1332–4, 1997.
[Crossref]

Korn, G.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
[Crossref] [PubMed]

Kosareva, O. G.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
[Crossref] [PubMed]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin. Conical emission from laser-plasma interactions in the filamentation of powerful ultrashort laser pulses in air. Opt. Lett., 22, 1332–4, 1997.
[Crossref]

Lameuroux, B.

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Lange, H. R.

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Liu, X.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
[Crossref] [PubMed]

Mlejnek, M.

M. Mlejnek, E. M. Wright, and J. V. Moloney. Dynamic spatial replenishment of femtosecond pulses propagating in air. Opt. Lett., 23, 382–4, 1998.
[Crossref]

M. Mlejnek, E. M. Wright, and J. V. Moloney. Moving-focus versus self-waveguiding model for long distance propagation of femtosecond pulses in air. To be published in Quantum Electron. (1999).

Moloney, J. V.

M. Mlejnek, E. M. Wright, and J. V. Moloney. Moving-focus versus self-waveguiding model for long distance propagation of femtosecond pulses in air. To be published in Quantum Electron. (1999).

M. Mlejnek, E. M. Wright, and J. V. Moloney. Dynamic spatial replenishment of femtosecond pulses propagating in air. Opt. Lett., 23, 382–4, 1998.
[Crossref]

Mourou, G.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
[Crossref] [PubMed]

Mysyrowicz, A.

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Mysy-rowicz, A.

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

Nibbering, E. T. J.

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Nieder-meier, S.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Nikolov, S.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Prade, B. S.

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Rairoux, P.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Rambo, P.

Xin Miao Zhao, P. Rambo, and J.-C. Diels. Filamentation of femtosecond uv pulses in air. In QELS 95. Summaries of Papers Presented at the Quantum Electronics and Laser Science Conference. Technical Digest Series Conference Edition, volume 16, pages 178–9, 1995.

Ripoche, J.-F.

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

Ronneberger, F.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Salin, F.

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

Sauerbrey, R.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Schillinger, H.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Squier, J.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
[Crossref] [PubMed]

Stein, B.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Wang, Cai Yi

Xin Miao Zhao, J.-C. Diels, Cai Yi Wang, and J. M. Elizondo. Femtosecond ultraviolet laser pulse induced lightning discharges in gases. Quantum Electron., QE-31, 599–612, 1995.

Wedekind, C.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Werner, C.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Wille, H.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Wöste, L.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

Wright, E. M.

M. Mlejnek, E. M. Wright, and J. V. Moloney. Dynamic spatial replenishment of femtosecond pulses propagating in air. Opt. Lett., 23, 382–4, 1998.
[Crossref]

M. Mlejnek, E. M. Wright, and J. V. Moloney. Moving-focus versus self-waveguiding model for long distance propagation of femtosecond pulses in air. To be published in Quantum Electron. (1999).

Zhao, Xin Miao

Xin Miao Zhao, P. Rambo, and J.-C. Diels. Filamentation of femtosecond uv pulses in air. In QELS 95. Summaries of Papers Presented at the Quantum Electronics and Laser Science Conference. Technical Digest Series Conference Edition, volume 16, pages 178–9, 1995.

Xin Miao Zhao, J.-C. Diels, Cai Yi Wang, and J. M. Elizondo. Femtosecond ultraviolet laser pulse induced lightning discharges in gases. Quantum Electron., QE-31, 599–612, 1995.

Other (10)

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou. Self-channeling of high-peak-power femtosecond laser pulses in air. Opt. Lett., 20, 73–5, 1995.
[Crossref] [PubMed]

Xin Miao Zhao, P. Rambo, and J.-C. Diels. Filamentation of femtosecond uv pulses in air. In QELS 95. Summaries of Papers Presented at the Quantum Electronics and Laser Science Conference. Technical Digest Series Conference Edition, volume 16, pages 178–9, 1995.

Xin Miao Zhao, J.-C. Diels, Cai Yi Wang, and J. M. Elizondo. Femtosecond ultraviolet laser pulse induced lightning discharges in gases. Quantum Electron., QE-31, 599–612, 1995.

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz. Conical emission from self-guided femtosecond pulses in air. Opt. Lett., 21, 62–4, 1996.
[Crossref] [PubMed]

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov. Moving focus in the propagation of ultrashort laser pulses in air. Opt. Lett., 22, 304–6, 1997.
[Crossref] [PubMed]

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Nieder-meier, F. Ronneberger, H. Schillinger, and R. Sauerbrey. Femtosecond atmospheric lamp. AT-Fachverlag, Stuttgart, Laser und Optoelectronik, 29, 51–3, 1997.

M. Mlejnek, E. M. Wright, and J. V. Moloney. Dynamic spatial replenishment of femtosecond pulses propagating in air. Opt. Lett., 23, 382–4, 1998.
[Crossref]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin. Conical emission from laser-plasma interactions in the filamentation of powerful ultrashort laser pulses in air. Opt. Lett., 22, 1332–4, 1997.
[Crossref]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lameuroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron. Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding? Opt. Lett., 23, 120–2, 1998.
[Crossref]

M. Mlejnek, E. M. Wright, and J. V. Moloney. Moving-focus versus self-waveguiding model for long distance propagation of femtosecond pulses in air. To be published in Quantum Electron. (1999).

Supplementary Material (4)

» Media 1: MOV (2232 KB)     
» Media 2: MOV (2356 KB)     
» Media 3: MOV (2558 KB)     
» Media 4: MOV (2690 KB)     

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Figures (6)

Figure 1.
Figure 1.

Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 4Pcr . The Rayleigh range was in this case ~ 2 m. (Dashed line … n 2 without the Raman contribution.)

Figure 2.
Figure 2.

Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 6Pcr . The Rayleigh range was in this case ~ 2 m. (Dashed line … n 2 without the Raman contribution.)

Figure 3.
Figure 3.

Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 6.5Pcr . The Rayleigh range was in this case ~ 2 m. Dashed line … n 2 without the Raman contribution.)

Figure 4.
Figure 4.

Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 7Pcr . The Rayleigh range was in this case ~ 2 m. (Dashed line … n 2 without the Raman contribution.)

Figure 5.
Figure 5.

The intensity of The field for the case shown in Fig. 1 (Raman response included) at the propagation distance of 1 m.

Figure 6.
Figure 6.

The intensity of the field for the case shown in Fig. 1 (Raman response included) at the propagation distance of 2.8 m.

Equations (5)

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ε z = i 2 k ( 2 r 2 + 1 r r ) ε ik 2 2 ε t 2 σ 2 ( 1 + iωτ ) ρε β ( K ) 2 ε 2 K 2 ε
+ i k 0 ( 1 f ) n 2 ε 2 ε + i k 0 f n 2 [ dt R ( t t ) ε ( t ) 2 ] ε ,
R ( t ) = θ ( t ) Ω 2 e Γ t / 2 sin ( Λ t ) Λ , Λ = Ω 2 Γ 2 / 4 ,
ρ t = 1 n b 2 σ E g ρ ε 2 + β ( K ) ε 2 K Kħω a ρ 2 .
ε ( r , 0 , t ) = 2 P in π ω 0 2 exp ( r 2 ω 0 2 t 2 t p 2 ) ,

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