Abstract

We use a phase-retrieval cross-correlation technique to analyze the spatiotemporal field evolution of self-focused ultrashort pulses. The technique features a new phase-retrieval algorithm based on functional differentiation. Its sensitivity, rapid convergence, and temporal nonreciprocality enable reliable three-dimensional waveform reconstruction. At less than the critical power, the experiments verify conventional description of self-focusing and give a direct proof of the Kerr-lens mode-locking mechanism. In contrast, for pulses with peak power much more than the critical power, nearly uniform self-focusing and quasi-stable single-filament trapping to a universal beam diameter were observed. The trapping can be explained by the saturation of the nonlinear refractive-index change at Δn7×10-5. The saturation is verified by an independent cross-polarization modulation measurement.

© 2000 Optical Society of America

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    [CrossRef]
  65. 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]
  66. M. Wittmann and A. Penzkofer, “Spectral superbroadening of femtosecond laser pulses,” Opt. Commun. 126, 308–317 (1996).
    [CrossRef]

1999 (4)

A. Brodeur and S. L. Chin, “Ultrafast white-light continuum generation and self-focusing in transparent condensed media,” J. Opt. Soc. Am. B 16, 637–650 (1999).
[CrossRef]

J. W. Nicholson, J. Jasapara, W. Rudolph, F. G. Omenetto, and A. J. Taylor, “Full-field characterization of femtosecond pulses by spectrum and cross-correlation measurements,” Opt. Lett. 24, 1774–1776 (1999).
[CrossRef]

S. Henz and J. Herrmann, “Self-channeling and pulse shortening of femtosecond pulses in multiphoton-ionized dispersive dielectric solids,” Phys. Rev. A 59, 2528–2531 (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]

1998 (8)

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

A. L. Gaeta, “Catatrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (1998).
[CrossRef]

S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (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]

S. Linden, H. Giessen, and J. Kuhl, “XFROG—a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, 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, 120–122 (1998).
[CrossRef]

T.-H. Her, R. J. Finlay, C. Wu, S. Deliwala, and E. Mazur, “Microstructuring of silicon with femtosecond laser pulses,” Appl. Phys. Lett. 73, 1673–1675 (1998).
[CrossRef]

1997 (3)

1996 (3)

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach,” Phys. Rev. E 53, 4092–4097 (1996).
[CrossRef]

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359 (1996).
[CrossRef] [PubMed]

M. Wittmann and A. Penzkofer, “Spectral superbroadening of femtosecond laser pulses,” Opt. Commun. 126, 308–317 (1996).
[CrossRef]

1995 (3)

1994 (7)

1993 (4)

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993).
[CrossRef] [PubMed]

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

1992 (4)

1991 (5)

1990 (2)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

T. R. Gosnell, A. J. Taylor, and D. P. Greene, “Supercontinuum generation at 248 nm using high-pressure gases,” Opt. Lett. 15, 130–132 (1990).
[CrossRef] [PubMed]

1988 (2)

1978 (1)

M. J. Weber, D. Milam, and W. L. Smith, “Nonlinear refractive index of glasses and crystals,” Opt. Eng. 17, 463–469 (1978).
[CrossRef]

1976 (1)

W. L. Smith and J. H. Bechtel, “Laser-induced breakdown and nonlinear refractive index measurements in phosphate glasses, lanthanum beryllate, and Al2O3,” Appl. Phys. Lett. 28, 606–607 (1976).
[CrossRef]

1975 (2)

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Dielectric-breakdown threshold and nonlinear-refractive-index measurements with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[CrossRef]

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975), and the references therein.
[CrossRef]

1974 (1)

N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron. QE-10, 375–387 (1974).
[CrossRef]

1969 (1)

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

1968 (2)

J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
[CrossRef]

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Acioli, L. H.

Akhmediev, N.

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

Akhmediev, N. N.

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[CrossRef] [PubMed]

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Antonetti, A.

Audebert, P.

Bechtel, J. H.

W. L. Smith and J. H. Bechtel, “Laser-induced breakdown and nonlinear refractive index measurements in phosphate glasses, lanthanum beryllate, and Al2O3,” Appl. Phys. Lett. 28, 606–607 (1976).
[CrossRef]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Dielectric-breakdown threshold and nonlinear-refractive-index measurements with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[CrossRef]

Bloembergen, N.

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Dielectric-breakdown threshold and nonlinear-refractive-index measurements with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[CrossRef]

N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron. QE-10, 375–387 (1974).
[CrossRef]

Bor, Z.

Braun, A.

Brodeur, A.

Cerullo, G.

G. Cerullo, S. De Silvestri, and V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Chen, S.

Chen, Y.

Chernev, P.

Chi, S.

Chien, C. Y.

Chin, S. L.

Chiron, A.

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]

S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (1998).
[CrossRef]

Dawes, E.

J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
[CrossRef]

Dawes, E. L.

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

De Silvestri, S.

G. Cerullo, S. De Silvestri, and V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Deliwala, S.

T.-H. Her, R. J. Finlay, C. Wu, S. Deliwala, and E. Mazur, “Microstructuring of silicon with femtosecond laser pulses,” Appl. Phys. Lett. 73, 1673–1675 (1998).
[CrossRef]

DeLong, K. W.

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]

S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (1998).
[CrossRef]

Du, D.

Dymott, M. J. P.

Eaton, H. K.

Fallies, F.

Feit, M. D.

Ferguson, A. I.

Fibich, G.

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359 (1996).
[CrossRef] [PubMed]

Finlay, R. J.

T.-H. Her, R. J. Finlay, C. Wu, S. Deliwala, and E. Mazur, “Microstructuring of silicon with femtosecond laser pulses,” Appl. Phys. Lett. 73, 1673–1675 (1998).
[CrossRef]

Fittinghoff, D. N.

Fleck Jr., J. A.

Franco, M. A.

Fujimoto, J. G.

Gaeta, A. L.

Gauthier, J. C.

Geindre, J. P.

Giessen, H.

S. Linden, H. Giessen, and J. Kuhl, “XFROG—a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Glezer, E. N.

E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. 71, 882–884 (1997).
[CrossRef]

E. N. Glezer, C. B. Schaffer, N. Nishimura, and E. Mazur, “Minimally disruptive laser-induced breakdown in water,” Opt. Lett. 22, 1817–1819 (1997).
[CrossRef]

Gosnell, T. R.

Greene, D. P.

Grillon, G.

Guo, Q.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Hamoniaux, G.

Haus, H. A.

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Heatley, D. R.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Henz, S.

S. Henz and J. Herrmann, “Self-channeling and pulse shortening of femtosecond pulses in multiphoton-ionized dispersive dielectric solids,” Phys. Rev. A 59, 2528–2531 (1999).
[CrossRef]

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach,” Phys. Rev. E 53, 4092–4097 (1996).
[CrossRef]

Her, T.-H.

T.-H. Her, R. J. Finlay, C. Wu, S. Deliwala, and E. Mazur, “Microstructuring of silicon with femtosecond laser pulses,” Appl. Phys. Lett. 73, 1673–1675 (1998).
[CrossRef]

Herrmann, J.

S. Henz and J. Herrmann, “Self-channeling and pulse shortening of femtosecond pulses in multiphoton-ionized dispersive dielectric solids,” Phys. Rev. A 59, 2528–2531 (1999).
[CrossRef]

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach,” Phys. Rev. E 53, 4092–4097 (1996).
[CrossRef]

Huang, D.

Hunter, J.

Iaconis, C.

Ilkov, F. A.

Ippen, E. P.

Jasapara, J.

Kandidov, V. P.

Kane, D. J.

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Kohler, B.

Korn, G.

Kosareva, O. G.

Kuhl, J.

S. Linden, H. Giessen, and J. Kuhl, “XFROG—a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Ladouceur, F.

Lamouroux, B.

Lange, H. R.

Linden, S.

S. Linden, H. Giessen, and J. Kuhl, “XFROG—a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Liu, X.

Luther, G. G.

Magni, V.

G. Cerullo, S. De Silvestri, and V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Marburger, J. H.

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
[CrossRef]

Mazur, E.

T.-H. Her, R. J. Finlay, C. Wu, S. Deliwala, and E. Mazur, “Microstructuring of silicon with femtosecond laser pulses,” Appl. Phys. Lett. 73, 1673–1675 (1998).
[CrossRef]

E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. 71, 882–884 (1997).
[CrossRef]

E. N. Glezer, C. B. Schaffer, N. Nishimura, and E. Mazur, “Minimally disruptive laser-induced breakdown in water,” Opt. Lett. 22, 1817–1819 (1997).
[CrossRef]

Milam, D.

M. J. Weber, D. Milam, and W. L. Smith, “Nonlinear refractive index of glasses and crystals,” Opt. Eng. 17, 463–469 (1978).
[CrossRef]

Mitchell, D. J.

Moloney, J. V.

Mourou, G.

Müller, A.

Mysyrowicz, A.

Negus, D. K.

Newell, A. C.

Nibbering, E. T. J.

Nicholson, J. W.

Nishimura, N.

Nishioka, H.

Odajima, W.

Omenetto, F. G.

Paye, J.

Penzkofer, A.

M. Wittmann and A. Penzkofer, “Spectral superbroadening of femtosecond laser pulses,” Opt. Commun. 126, 308–317 (1996).
[CrossRef]

Petrov, V.

Piché, M.

Poladian, L.

Prade, B. S.

Ramaswamy, M.

Ranka, J. K.

Reed, M. K.

Ripoche, J.-F.

Rothenberg, J. E.

Rousse, A.

Rudolph, W.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Salin, F.

Santos, A. Dos

Schaffer, C. B.

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Shen, Y. R.

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975), and the references therein.
[CrossRef]

Smith, W. L.

M. J. Weber, D. Milam, and W. L. Smith, “Nonlinear refractive index of glasses and crystals,” Opt. Eng. 17, 463–469 (1978).
[CrossRef]

W. L. Smith and J. H. Bechtel, “Laser-induced breakdown and nonlinear refractive index measurements in phosphate glasses, lanthanum beryllate, and Al2O3,” Appl. Phys. Lett. 28, 606–607 (1976).
[CrossRef]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Dielectric-breakdown threshold and nonlinear-refractive-index measurements with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[CrossRef]

Snyder, A. W.

Soto-Crespo, J. M.

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[CrossRef] [PubMed]

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Squier, J.

Steiner-Shepard, M. K.

Szabó, G.

Takuma, H.

Taylor, A. J.

Trebino, R.

Ueda, K.

Ulman, M.

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]

van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Wagner, W. G.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Walmsley, I. A.

Wang, J.

Weber, M. J.

M. J. Weber, D. Milam, and W. L. Smith, “Nonlinear refractive index of glasses and crystals,” Opt. Eng. 17, 463–469 (1978).
[CrossRef]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

White, W. E.

Wilson, K.

Wittmann, M.

M. Wittmann and A. Penzkofer, “Spectral superbroadening of femtosecond laser pulses,” Opt. Commun. 126, 308–317 (1996).
[CrossRef]

Wright, E. M.

G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862–864 (1994).
[CrossRef] [PubMed]

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[CrossRef] [PubMed]

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Wu, C.

T.-H. Her, R. J. Finlay, C. Wu, S. Deliwala, and E. Mazur, “Microstructuring of silicon with femtosecond laser pulses,” Appl. Phys. Lett. 73, 1673–1675 (1998).
[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]

S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (1998).
[CrossRef]

Appl. Phys. Lett. (3)

E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. 71, 882–884 (1997).
[CrossRef]

T.-H. Her, R. J. Finlay, C. Wu, S. Deliwala, and E. Mazur, “Microstructuring of silicon with femtosecond laser pulses,” Appl. Phys. Lett. 73, 1673–1675 (1998).
[CrossRef]

W. L. Smith and J. H. Bechtel, “Laser-induced breakdown and nonlinear refractive index measurements in phosphate glasses, lanthanum beryllate, and Al2O3,” Appl. Phys. Lett. 28, 606–607 (1976).
[CrossRef]

IEEE J. Quantum Electron. (2)

N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron. QE-10, 375–387 (1974).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

M. Wittmann and A. Penzkofer, “Spectral superbroadening of femtosecond laser pulses,” Opt. Commun. 126, 308–317 (1996).
[CrossRef]

Opt. Eng. (1)

M. J. Weber, D. Milam, and W. L. Smith, “Nonlinear refractive index of glasses and crystals,” Opt. Eng. 17, 463–469 (1978).
[CrossRef]

Opt. Lett. (27)

F. Salin, J. Squier, and M. Piché, “Mode locking of Ti:Al2O3 lasers and self-focusing: a Gaussian approximation,” Opt. Lett. 16, 1674–1676 (1991).
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S. Chen and J. Wang, “Self-starting issues of passive self-focusing mode locking,” Opt. Lett. 16, 1689–1691 (1991).
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G. Cerullo, S. De Silvestri, and V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

M. J. P. Dymott and A. I. Ferguson, “Self-mode-locked diode-pumped Cr:LiSAF laser,” Opt. Lett. 19, 1988–1990 (1994).
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D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
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[CrossRef]

J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993).
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K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19, 2152–2154 (1994).
[CrossRef] [PubMed]

G. Szabó, Z. Bor, and A. Müller, “Phase-sensitive single-pulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 13, 746–748 (1988).
[CrossRef] [PubMed]

J. P. Geindre, P. Audebert, A. Rousse, F. Fallies, J. C. Gauthier, A. Mysyrowicz, A. Dos Santos, G. Hamoniaux, and A. Antonetti, “Frequency-domain interferometor for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma,” Opt. Lett. 19, 1997–1999 (1994).
[CrossRef] [PubMed]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[CrossRef]

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]

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, 304–306 (1997).
[CrossRef] [PubMed]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, 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, 120–122 (1998).
[CrossRef]

E. N. Glezer, C. B. Schaffer, N. Nishimura, and E. Mazur, “Minimally disruptive laser-induced breakdown in water,” Opt. Lett. 22, 1817–1819 (1997).
[CrossRef]

T. R. Gosnell, A. J. Taylor, and D. P. Greene, “Supercontinuum generation at 248 nm using high-pressure gases,” Opt. Lett. 15, 130–132 (1990).
[CrossRef] [PubMed]

M. K. Reed, M. K. Steiner-Shepard, and D. K. Negus, “Widely tunable femtosecond optical parametric amplifier at 250 kHz with a Ti:sapphire regenerative amplifier,” Opt. Lett. 19, 1855–1857 (1994).
[CrossRef] [PubMed]

H. Nishioka, W. Odajima, K. Ueda, and H. Takuma, “Ultrabroadband flat continuum generation in multichannel propagation of terrawatt Ti:sapphire laser pulses,” Opt. Lett. 20, 2505–2507 (1995).
[CrossRef] [PubMed]

S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1600 (1995).
[CrossRef] [PubMed]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (1998).
[CrossRef]

Y. Chen, “Self-trapped light in saturable nonlinear media,” Opt. Lett. 16, 4–6 (1991).
[CrossRef] [PubMed]

A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, “Self-induced optical fibers: spatial solitary waves,” Opt. Lett. 16, 21–23 (1991).
[CrossRef] [PubMed]

P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. 17, 172–174 (1992).
[CrossRef] [PubMed]

J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. 17, 583–585 (1992).
[CrossRef] [PubMed]

G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862–864 (1994).
[CrossRef] [PubMed]

Phys. Rev. (2)

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

Phys. Rev. A (4)

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

S. Henz and J. Herrmann, “Self-channeling and pulse shortening of femtosecond pulses in multiphoton-ionized dispersive dielectric solids,” Phys. Rev. A 59, 2528–2531 (1999).
[CrossRef]

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[CrossRef] [PubMed]

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. B (1)

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Dielectric-breakdown threshold and nonlinear-refractive-index measurements with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[CrossRef]

Phys. Rev. E (1)

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach,” Phys. Rev. E 53, 4092–4097 (1996).
[CrossRef]

Phys. Rev. Lett. (5)

J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
[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, “Catatrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (1998).
[CrossRef]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

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[CrossRef] [PubMed]

Phys. Status Solidi B (1)

S. Linden, H. Giessen, and J. Kuhl, “XFROG—a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119–124 (1998).
[CrossRef]

Prog. Quantum Electron. (1)

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975), and the references therein.
[CrossRef]

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X. Liu and G. Mourou, “Ultrashort laser pulses tackle precision machining,” Laser Focus World (August 1997), pp. 101–118.

L. Spinelli, B. Couillaud, N. Goldblatt, and D. K. Negus, “Starting and generation of sub-100-fs pulses in Ti:Al2O3 by self-focusing,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

J. Wang, T.-W. Yau, and C.-H. Lee, “Time-resolved study on self-focusing of ultrashort pulses,” in The 36th Seminar on Science and Technology–Femtosecond Science and Technology (Interchange Asociation, Tokyo, 1998), pp. 72–82.

T.-W. Yau, C.-H. Lee, and J. Wang, “Spatial-temporal field distribution of self-focused femtosecond pulses,” in Ultrafast Phenomena XI (Springer, Berlin, 1998), pp. 106–108.

S. Linden, H. Giessen, and J. Kuhl, “XFROG: new method for amplitude and phase characterization of ultraweak ultrashort pulses,” in Conference on Lasers and Electro-Optics, Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper CThW5.

Y.-Y. Hwang, Y.-H. Kuo, C.-H. Lee, and J. Wang, in “Beam-profile evolution of trapped femtosecond pulses in Kerr media,” Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper CTuP22.

S. Gatz and J. Herrmann, “Soliton propagation in materials with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 2296–2302 (1991); “Propagation of optical beams and the properties of two-dimensional spatial solitons in media with a local saturable nonlinear refractive index,” J. Opt. Soc. Am. B 14, 1795–1806 (1997).
[CrossRef]

Y. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), Sect. 3.4.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 385–386.

This should not be confused with the fact that tightly focused pulses can create optical damage, as shown in Refs. 910111213. In our case the beam enters the medium at the beam waist; therefore without the Kerr effect the beam does not focus at all.

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Figures (10)

Fig. 1
Fig. 1

Phase-retrieval algorithm employed in this paper.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

Views of the front surface of the nonlinear crystal. (a) Side view: The sample beam, which crosses the reference beam on a horizontal line, is walked by a retroreflector moving along the vertical direction. (b) Top view: The correlation signal is generated within the overlapped region of the two pulses in the crystal. The spatial span of the signal beam corresponds to the relative temporal delay of sample pulses and reference pulses.

Fig. 4
Fig. 4

(a) Spatiotemporal profile of the pulse produced by a Kerr-lens mode-locked Ti:sapphire laser, shown by contours of the relative intensity in percentage. (b) Instantaneous beam diameters. The pulse center is focused more than the pulse wings by ∼5%. (c) Instantaneous beam diameters of the pulse after passing through a 5-cm single-mode optical fiber.

Fig. 5
Fig. 5

Instantaneous beam-diameter evolution of high-intensity self-focused pulses in sapphire; d is the propagation distance in the sapphire crystal, the pulse duration is 180 fs; and the peak power is 4.7 MW ( 2.2   P c ) .

Fig. 6
Fig. 6

Spatiotemporal profile evolution of femtosecond pulses with peak power P c . The beams were prefocused such that their waist was exactly at 0 mm (the input surface of the nonlinear medium). The pulse duration is 280 fs.

Fig. 7
Fig. 7

Beam-diameter evolution. After 4-mm propagation, for high-power pulses the spatial beam profile deviates more and more from a Gaussian profile, and the beam size becomes less accurately defined.

Fig. 8
Fig. 8

Experimental setup to measure the saturation of n 2 with polarization rotation.

Fig. 9
Fig. 9

The intensity-dependent change of refractive index.

Fig. 10
Fig. 10

FWHM diameters of the self-trapped beams at 810 nm and 630 nm calculated from the graded-index waveguide model. Circles denote experimental data at 810 nm measured by three-dimensional cross correlation. Squares denote experimental data at 630 nm measured in Ref. 39.

Equations (15)

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- | S ˜ ( ω ,   τ ) | 2 d τ = | a ˜ ( ω ) | 2 | a ˜ ( ω ) | 2 ,
Z auto = t , τ = 1 N | S ( t ,   τ ) - a ( t ) a ( t - τ ) | 2
Z cross = t , τ = 1 N | S ( t ,   τ ) - a ( t ) a r ( t - τ ) | 2 ,
a ( t ) = τ = 1 N S ( t ,   τ ) a r * ( t - τ ) E ,
n 2 = 12 π n 0   χ xxxx ( 3 ) .
P sig = χ xyxy ( 3 ) E pump * E pump E probe ,
I sig = k 0 2 4 n 0 2   | χ xyxy ( 3 ) | 2 | E pump | 4 I probe L 2 ,
χ xxxx ( 3 ) = χ xxyy ( 3 ) + χ xyyx ( 3 ) + χ xyxy ( 3 ) .
Δ n = 3 2 n 0   χ xyxy ( 3 ) | E pump | 2 .
Δ n = n 2 I 1 + I / I s ,
n ( r ) = n c [ 1 - α ( r / a ) 2 ] 1 / 2 r < a n 0 r a ,
a = 2 k 0 n c 2 - n 0 2 2 k 0 2 n 0 Δ n c ,
Δ n c = n c - n 0 = n 2 I 0 1 + I 0 / I s ,
a = 2 I s ( k 0 2 n 0 n 2 - π / P ) .
B = k 0 0 L Δ n n 0 d l ,

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