Abstract

Numerical simulations of self-focusing laser pulses obtained via a slowly evolving wave approach (modified nonlinear Schrödinger equation) are compared with published experimental results in fused silica as well as with experimental results in air and fused silica obtained in our laboratory. The mathematical model includes group-velocity dispersion and third-order dispersion, optical shock, and both instantaneous Kerr and delayed Raman nonlinearities as well as a perfectly matched layer absorbing boundary condition. Second-harmonic frequency-resolved optical gating data taken after 10.91 m of propagation allow a direct comparison between experimental and computational envelopes at a number of pulse energies. FWHM measurements of the spectral width, intensity autocorrelation duration, and pulse radius at distances of 4.35, 10.91, 17.47, and 22.73 m provide a view of model fidelity across both energy and propagation distance variables. The magnitude of the nonlinear refractive index, n2, is inferred.

© 2004 Optical Society of America

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  1. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).
  2. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
  3. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  4. 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, 73–75 (1995).
    [CrossRef] [PubMed]
  5. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale, Optics and Photonics, 1st ed. (Academic, San Diego, 1996).
  6. A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
    [CrossRef]
  7. A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
    [CrossRef]
  8. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
    [CrossRef]
  9. A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
    [CrossRef] [PubMed]
  10. J. Moloney, M. Kolesik, M. Mlejnek, and E. Wright, “Femtosecond self-guided atmospheric light strings,” Chaos 10, 559–569 (2000).
    [CrossRef]
  11. N. Aközbek, M. Scalora, C. Bowden, and S. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001).
    [CrossRef]
  12. M. R. Junnarkar, “Short pulse propagation in tight focusing conditions,” Opt. Commun. 195, 273–292 (2001).
    [CrossRef]
  13. A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
    [CrossRef]
  14. A. L. Gaeta, “Nonlinear propagation and continuum generation in microstructured optical fibers,” Opt. Lett. 27, 924–926 (2002).
    [CrossRef]
  15. M. Kolesik, J. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
    [CrossRef]
  16. N. Aközbek, C. M. Bowden, and S. L. Chin, “Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation,” J. Mod. Opt. 49, 475–486 (2002).
    [CrossRef]
  17. S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
    [CrossRef]
  18. A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
    [CrossRef] [PubMed]
  19. R. W. Boyd, Nonlinear Optics (Elsevier Science, New York, 1992).
  20. P. Sprangle, J. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002).
    [CrossRef]
  21. A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
    [CrossRef]
  22. M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “Recurrent femtosecond pulse collapse in air due to plasma generation: numerical results,” Math. Comput. Simul. 56, 563–570 (2001).
    [CrossRef]
  23. A. C. Bernstein, “Measurements of ultrashort pulses self-focusing in air,” Ph.D. thesis (University of New Mexico, Albuquerque, New Mexico, 2004).
  24. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [CrossRef]
  25. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [CrossRef]
  26. G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics, 3rd ed. (Academic, San Diego, 2001).
  27. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  28. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
    [CrossRef]
  29. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  30. J. R. Goates, J. B. Ott, and E. A. Butler, General Chemistry: Theory and Description (Harcourt Brace Jovanovitch, New York, 1981).
  31. A. Talebpour, J. Yang, and S. 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]
  32. 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]
  33. D. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
    [CrossRef]
  34. T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
    [CrossRef] [PubMed]
  35. T. Pitts and J. Greenleaf, “Three-dimensional optical measurement of instantaneous pressure,” J. Acoust. Soc. Am. 108, 2873–2883 (2000).
    [CrossRef]
  36. P. Robert and C. Weast, eds., “General physical constants,” in Handbook of Chemistry and Physics, 65th ed. (CRC Press, Boca Raton, Fla., 1984–1985), p. E-359. Sellmeier coefficients for air are at 30 °C and 76 cm Hg.
  37. Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41, 4318–4324 (2002).
    [CrossRef] [PubMed]
  38. B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  39. M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “A dynamic spatial replenishment scenario for femtosecond pulses propagating in air—a route to optical turbulence?” Laser Phys. 10, 107–110 (2000).
  40. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Vol. 163 of Applied Mathematical Sciences (Springer-Verlag, New York, 1999).
  41. M. Feit and J. J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]

2003 (3)

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
[CrossRef] [PubMed]

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

2002 (6)

Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct measurement of the group refractive index of air with interferometry between adjacent femtosecond pulses,” Appl. Opt. 41, 4318–4324 (2002).
[CrossRef] [PubMed]

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

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

A. L. Gaeta, “Nonlinear propagation and continuum generation in microstructured optical fibers,” Opt. Lett. 27, 924–926 (2002).
[CrossRef]

M. Kolesik, J. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

N. Aközbek, C. M. Bowden, and S. L. Chin, “Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation,” J. Mod. Opt. 49, 475–486 (2002).
[CrossRef]

2001 (3)

N. Aközbek, M. Scalora, C. Bowden, and S. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001).
[CrossRef]

M. R. Junnarkar, “Short pulse propagation in tight focusing conditions,” Opt. Commun. 195, 273–292 (2001).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “Recurrent femtosecond pulse collapse in air due to plasma generation: numerical results,” Math. Comput. Simul. 56, 563–570 (2001).
[CrossRef]

2000 (5)

T. Pitts and J. Greenleaf, “Three-dimensional optical measurement of instantaneous pressure,” J. Acoust. Soc. Am. 108, 2873–2883 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “A dynamic spatial replenishment scenario for femtosecond pulses propagating in air—a route to optical turbulence?” Laser Phys. 10, 107–110 (2000).

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[CrossRef] [PubMed]

J. Moloney, M. Kolesik, M. Mlejnek, and E. Wright, “Femtosecond self-guided atmospheric light strings,” Chaos 10, 559–569 (2000).
[CrossRef]

1999 (4)

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

D. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
[CrossRef]

A. Talebpour, J. Yang, and S. 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]

1998 (2)

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
[CrossRef]

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

1997 (3)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[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]

1995 (1)

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1988 (1)

1975 (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

1966 (1)

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Aközbek, N.

N. Aközbek, C. M. Bowden, and S. L. Chin, “Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation,” J. Mod. Opt. 49, 475–486 (2002).
[CrossRef]

N. Aközbek, M. Scalora, C. Bowden, and S. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001).
[CrossRef]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bernstein, A. C.

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
[CrossRef] [PubMed]

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

Bonnaud, G.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Bowden, C.

N. Aközbek, M. Scalora, C. Bowden, and S. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001).
[CrossRef]

Bowden, C. M.

N. Aközbek, C. M. Bowden, and S. L. Chin, “Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation,” J. Mod. Opt. 49, 475–486 (2002).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Braun, A.

Cameron, S. M.

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
[CrossRef] [PubMed]

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

Chin, S.

N. Aközbek, M. Scalora, C. Bowden, and S. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001).
[CrossRef]

A. Talebpour, J. Yang, and S. 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]

Chin, S. L.

N. Aközbek, C. M. Bowden, and S. L. Chin, “Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation,” J. Mod. Opt. 49, 475–486 (2002).
[CrossRef]

Chiron, A.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Clement, T. S.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Diddams, S. A.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

Diels, J. C.

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
[CrossRef] [PubMed]

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

Du, D.

Edlén, B.

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Etrich, C.

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

Feit, M.

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Fleck, J. J. A.

Franco, M.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Franco, M. A.

Gaeta, A. L.

Greenleaf, J.

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

T. Pitts and J. Greenleaf, “Three-dimensional optical measurement of instantaneous pressure,” J. Acoust. Soc. Am. 108, 2873–2883 (2000).
[CrossRef]

Grillon, G.

Hafizi, B.

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

Junnarkar, M. R.

M. R. Junnarkar, “Short pulse propagation in tight focusing conditions,” Opt. Commun. 195, 273–292 (2001).
[CrossRef]

Kane, D.

D. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
[CrossRef]

Kane, D. J.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Kolesik, M.

M. Kolesik, J. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “Recurrent femtosecond pulse collapse in air due to plasma generation: numerical results,” Math. Comput. Simul. 56, 563–570 (2001).
[CrossRef]

J. Moloney, M. Kolesik, M. Mlejnek, and E. Wright, “Femtosecond self-guided atmospheric light strings,” Chaos 10, 559–569 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “A dynamic spatial replenishment scenario for femtosecond pulses propagating in air—a route to optical turbulence?” Laser Phys. 10, 107–110 (2000).

Korn, G.

Krausz, F.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Lamouroux, B.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Lange, R.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Lederer, F.

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

Leine, L.

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

Liu, X.

Luk, T. S.

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
[CrossRef] [PubMed]

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Matsumoto, H.

McPherson, A.

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
[CrossRef] [PubMed]

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

Michaelis, D.

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

Minoshima, K.

Mlejnek, M.

M. Kolesik, J. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “Recurrent femtosecond pulse collapse in air due to plasma generation: numerical results,” Math. Comput. Simul. 56, 563–570 (2001).
[CrossRef]

J. Moloney, M. Kolesik, M. Mlejnek, and E. Wright, “Femtosecond self-guided atmospheric light strings,” Chaos 10, 559–569 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “A dynamic spatial replenishment scenario for femtosecond pulses propagating in air—a route to optical turbulence?” Laser Phys. 10, 107–110 (2000).

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
[CrossRef]

Moloney, J.

M. Kolesik, J. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “Recurrent femtosecond pulse collapse in air due to plasma generation: numerical results,” Math. Comput. Simul. 56, 563–570 (2001).
[CrossRef]

J. Moloney, M. Kolesik, M. Mlejnek, and E. Wright, “Femtosecond self-guided atmospheric light strings,” Chaos 10, 559–569 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “A dynamic spatial replenishment scenario for femtosecond pulses propagating in air—a route to optical turbulence?” Laser Phys. 10, 107–110 (2000).

Moloney, J. V.

Mourou, G.

Mysyrowicz, A.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[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]

Nelson, T. R.

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003).
[CrossRef] [PubMed]

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

Nibbering, E. T. J.

Peñano, J.

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

Peschel, U.

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

Pitts, T.

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

T. Pitts and J. Greenleaf, “Three-dimensional optical measurement of instantaneous pressure,” J. Acoust. Soc. Am. 108, 2873–2883 (2000).
[CrossRef]

Prade, B.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Prade, B. S.

Riazuelo, G.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Ripoche, J.-F.

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Scalora, M.

N. Aközbek, M. Scalora, C. Bowden, and S. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001).
[CrossRef]

Skupin, S.

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

Sprangle, P.

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

Squier, J.

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Talebpour, A.

A. Talebpour, J. Yang, and S. 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]

Trebino, R.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Van Engen, A. G.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

Wright, E.

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “Recurrent femtosecond pulse collapse in air due to plasma generation: numerical results,” Math. Comput. Simul. 56, 563–570 (2001).
[CrossRef]

J. Moloney, M. Kolesik, M. Mlejnek, and E. Wright, “Femtosecond self-guided atmospheric light strings,” Chaos 10, 559–569 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “A dynamic spatial replenishment scenario for femtosecond pulses propagating in air—a route to optical turbulence?” Laser Phys. 10, 107–110 (2000).

Wright, E. M.

Yamaoka, Y.

Yang, J.

A. Talebpour, J. Yang, and S. 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]

Zozulya, A. A.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

A. C. Bernstein, T. S. Luk, T. R. Nelson, A. McPherson, J. C. Diels, and S. M. Cameron, “Asymmetric ultra-short pulse splitting measured in air using FROG,” Appl. Phys. B 75, 119–122 (2002).
[CrossRef]

Chaos (1)

J. Moloney, M. Kolesik, M. Mlejnek, and E. Wright, “Femtosecond self-guided atmospheric light strings,” Chaos 10, 559–569 (2000).
[CrossRef]

Eur. Phys. J. D (1)

A. Chiron, B. Lamouroux, R. Lange, J.-F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (1)

T. Pitts and J. Greenleaf, “Three-dimensional optical measurement of instantaneous pressure,” J. Acoust. Soc. Am. 108, 2873–2883 (2000).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Mod. Opt. (1)

N. Aközbek, C. M. Bowden, and S. L. Chin, “Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation,” J. Mod. Opt. 49, 475–486 (2002).
[CrossRef]

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

Laser Phys. (1)

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “A dynamic spatial replenishment scenario for femtosecond pulses propagating in air—a route to optical turbulence?” Laser Phys. 10, 107–110 (2000).

Math. Comput. Simul. (1)

M. Mlejnek, M. Kolesik, E. Wright, and J. Moloney, “Recurrent femtosecond pulse collapse in air due to plasma generation: numerical results,” Math. Comput. Simul. 56, 563–570 (2001).
[CrossRef]

Metrologia (1)

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Opt. Commun. (3)

A. Talebpour, J. Yang, and S. 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]

N. Aközbek, M. Scalora, C. Bowden, and S. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001).
[CrossRef]

M. R. Junnarkar, “Short pulse propagation in tight focusing conditions,” Opt. Commun. 195, 273–292 (2001).
[CrossRef]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

S. Skupin, U. Peschel, C. Etrich, L. Leine, F. Lederer, and D. Michaelis, “Simulation of femtosecond pulse propagation in air,” Opt. Quantum Electron. 35, 573–582 (2003).
[CrossRef]

Phys. Rev. A (1)

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (4)

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

M. Kolesik, J. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Rev. Sci. Instrum. (1)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Other (10)

G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics, 3rd ed. (Academic, San Diego, 2001).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

A. C. Bernstein, “Measurements of ultrashort pulses self-focusing in air,” Ph.D. thesis (University of New Mexico, Albuquerque, New Mexico, 2004).

J. R. Goates, J. B. Ott, and E. A. Butler, General Chemistry: Theory and Description (Harcourt Brace Jovanovitch, New York, 1981).

P. Robert and C. Weast, eds., “General physical constants,” in Handbook of Chemistry and Physics, 65th ed. (CRC Press, Boca Raton, Fla., 1984–1985), p. E-359. Sellmeier coefficients for air are at 30 °C and 76 cm Hg.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale, Optics and Photonics, 1st ed. (Academic, San Diego, 1996).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

R. W. Boyd, Nonlinear Optics (Elsevier Science, New York, 1992).

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Vol. 163 of Applied Mathematical Sciences (Springer-Verlag, New York, 1999).

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Figures (9)

Fig. 1
Fig. 1

Comparison of our numerical model with published fused-silica data of Ref. 21. (a) Model temporal envelope intensity and phase. (b) Model spectral intensity. (c) Experimental temporal envelope intensity and phase measured via the FROG technique. (d) Experimental spectral intensity.

Fig. 2
Fig. 2

Comparison between the numerical model and the spectral data taken at a propagation distance of 1 cm in fused silica. (a), (b), (c), and (d) compare normalized spectral density obtained from the numerical model with experimental data at 22, 55, 85, and 110 µJ, respectively.

Fig. 3
Fig. 3

Determination of initial spatial intensity and phase distribution. (a) After determination of the initial wave-front shape via phase retrieval, a beam with wavelength equal to that of the pulse carrier frequency was numerically propagated along the system optical axis. The FWHM transverse beam diameter compares well with the experimental pulse beam diameter (squares). A radially averaged beam derived from these data was also propagated along the beam path (with almost no discernible difference in transverse beam dimension). A Gaussian beam with the same FWHM diameter is plotted for comparison. Note the significant differences in beam spread during propagation. The location of the compressor used in the nonlinear propagation experiments is also indicated on the plot. Propagation distances in Figs. 46, 8 and 9 reference this point as zero (the start of nonlinear propagation). (b) The radially averaged beam intensity is plotted along with the intensity profile for a Gaussian beam with the same FWHM diameter. The radially averaged retrieved phase is plotted as well. The phase structure beyond 3 mm is believed to be an artifact of a soft apodizing pinhole in the experiment setup.

Fig. 4
Fig. 4

Comparison between the numerical model [(a), (c), (e), (g)] and the axial temporal profiles derived from FROG data [(b), (d), (f), (h)] taken at a propagation distance of 10.91 m. Figure pairs (a) and (b), (c) and (d), (e) and (f), and (g) and (h) compare temporal envelope intensity (solid curves) and phase (dashed curves) obtained from the numerical model with experimental data at 2.0, 1.4, 1.0, and 0.7 mJ, respectively. Note the progression toward a doubly peaked temporal structure with concave down phase function in both the model and the experiment.

Fig. 5
Fig. 5

Comparison between the numerical model [(a), (c), (e), (g)] and the axial spectral intensity and phase profiles derived from FROG data [(b), (d), (f), (h)] taken at a propagation distance of 10.91 m. Figure pairs (a) and (b), (c) and (d), (e) and (f), and (g) and (h) compare spectral envelope intensity (solid curves) and phase (dashed curves) obtained from the numerical model with experimental data at 2.0, 1.4, 1.0, and 0.7 mJ, respectively. Note the shift of the most significant spectral feature toward longer wavelengths with increasing pulse energy. In the numerical model this is a direct result of the Raman effect.

Fig. 6
Fig. 6

Comparison between experimental and computational model FWHM intensity autocorrelation pulse widths at distances of 4.35, 10.91, 17.47, and 22.73 m for a range of energies as high as 2.0 mJ.

Fig. 7
Fig. 7

Comparison between experimental and computational model FWHM spectral intensity widths at distances of 4.35, 10.91, 17.47, and 22.73 m for a range of energies as high as 2.0 mJ.

Fig. 8
Fig. 8

Comparison between experimental and computational model integrated 10–90 spectral intensity widths at distances of 4.35, 10.91, 17.47, and 22.73 m for a range of energies as high as 2.0 mJ.

Fig. 9
Fig. 9

Comparison between experimental and computational transverse beam diameter (FWHM) at distances of 4.35, 10.91, 17.47, and 22.73 m for a range of energies as high as 2.0 mJ.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

(z2+2)E(r, t)-1c2t2-tdt(t-t)E(r, t)=μ0t2Pn(r, t),
k(ω)=n=0km(ω0)m!(ω-ω0)m
m=0βm+iαm/2m!(ω-ω0)m,
E(r, t)=A(r, z, t)exp[i(β0z-ω0t+ψ0)]+c.c.,
ω0=0ω|E(ω)|2dω0|E(ω)|2dω,
Pn(r, t)=B(r, z, t, A)exp[i(β0z-ω0t+ψ0)]+c.c.
|ξA|β0|A|,
|ηA|ω0|A|.
|1-β1c|=β0-ω0β1β01,
1=1ω0τ0,
3=β36τ03lD,
Nn=i2πlDn2|A0|2λ,
iza+1+i1t-12a-2t2a-i33t3a+Nn1+i1tga=0,
g(1-α)|a(t)|2+α-tdτf(t-τ)|a(t)|2,
f(t)=ncnӨ(t)Ω2 exp(-Γt/2)sin(Λt)/Λ,
Δχp=-e2Nem0ω2
ΔχpO2=-25×10-12,
ΔχpN2=-11.6×10-12,
Δn=n2+χpO2+χpN2-n=2×10-11,
Δn=In2=1012(W/cm2)2.8×10-19(cm2/W)=2.8×10-7,
n(λ)-1=10-7a+bλ210-8+cλ410-16.
abs(λ)dλ,

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