Abstract

Visualization of the energy density flux gives a unique insight into the propagation properties of complex ultrashort pulses. This analysis, formerly relegated to numerical investigations, is here shown to be an invaluable experimental diagnostic tool. By retrieving the spatio-temporal amplitude and phase we experimentally obtain the energy density flux within complex ultrashort pulses generated by filamentation in a nonlinear Kerr medium.

©2009 Optical Society of America

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Ultrashort laser pulse filamentation from spontaneous X Wave formation in air

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
    [Crossref]
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2008 (8)

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77, 043814 (2008).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt.Express 16, 9411–9416 (2008).
[Crossref] [PubMed]

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

J. Extermann, L. Bonacina, F. Courvoisier, D. Kiselev, Y. Mugnier, R. Le Dantec, Christine Galez, and J. P. Wolf, “Nano-FROG: Frequency Resolved Optical Gating by a nanometric object” Opt. Express 16, 10405–10411 (2008).
[Crossref] [PubMed]

P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16, 13663–13675 (2008).
[Crossref] [PubMed]

F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
[Crossref] [PubMed]

2007 (3)

A. Zaïr, A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15, 5394–5405 (2007).
[Crossref] [PubMed]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

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

2006 (5)

2005 (2)

D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862–869 (2005).
[Crossref]

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

2004 (4)

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic Nonlinear X waves for Femtosecond Pulse Propagation in Water,” Phys. Rev. Lett. 92, 2539011 (2004).
[Crossref]

P. Gabolde and R. Trebino, “Self-referenced measurement of the complete electric field of ultrashort pulses,” Opt. Express 12, 4423–4429 (2004).
[Crossref] [PubMed]

2003 (3)

A.C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

1998 (1)

E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[Crossref]

1982 (1)

1959 (1)

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. 74, 269–280 (1959).
[Crossref]

1953 (1)

H. S. Green and E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. LXVI 66, 1129–1137 (1953).
[Crossref]

Akturk, S.

Alfano, R. R.

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt.Express 16, 9411–9416 (2008).
[Crossref] [PubMed]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Averchi, A.

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

Biegert, J.

Blasi, G.

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

Bonacina, L.

Bonaretti, F.

Bowlan, P.

Boyd, R.

R. Boyd, Nonlinear Optics (Academic Press, New York, 2002).

Bragheri, F.

F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
[Crossref] [PubMed]

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. 96, 1939011 (2006).
[Crossref]

D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862–869 (2005).
[Crossref]

Bramati, A.

Broderick, N. G. R.

Christodoulides, D. N.

Clerici, M.

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
[Crossref] [PubMed]

Conti, C.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

Couairon, A.

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77, 043814 (2008).
[Crossref]

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

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

A. Zaïr, A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15, 5394–5405 (2007).
[Crossref] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. 96, 1939011 (2006).
[Crossref]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E. 73, 016608 (2006).
[Crossref]

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

Courvoisier, F.

Danielius, R.

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

Degiorgio, V.

Dorrer, C.

Dubietis, A.

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. 96, 1939011 (2006).
[Crossref]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E. 73, 016608 (2006).
[Crossref]

D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862–869 (2005).
[Crossref]

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

El-Ganainy, R.

Extermann, J.

Faccio, D.

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77, 043814 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
[Crossref] [PubMed]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E. 73, 016608 (2006).
[Crossref]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. 96, 1939011 (2006).
[Crossref]

D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862–869 (2005).
[Crossref]

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

Fienup, J. R.

Franco, M.

Fuchs, U.

Gabolde, P.

Gaizauskas, E.

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E. 73, 016608 (2006).
[Crossref]

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

Galez, Christine

Gallmann, L.

Green, H. S.

H. S. Green and E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. LXVI 66, 1129–1137 (1953).
[Crossref]

Grier, D. G.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Guandalini, A.

Henin, S.

Holler, M.

Jedrkiewicz, O.

F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
[Crossref] [PubMed]

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

Keller, U.

Kiselev, D.

Kolesik, M.

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic Nonlinear X waves for Femtosecond Pulse Propagation in Water,” Phys. Rev. Lett. 92, 2539011 (2004).
[Crossref]

Kosik, E. M.

Kucinskas, E.

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

Le Dantec, R.

Liberale, C.

Lotti, A.

Matijosius, A.

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862–869 (2005).
[Crossref]

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

McGresham, K.

Minardi, S.

D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862–869 (2005).
[Crossref]

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

Mokhov, S.

Moloney, J. V.

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic Nonlinear X waves for Femtosecond Pulse Propagation in Water,” Phys. Rev. Lett. 92, 2539011 (2004).
[Crossref]

Mugnier, Y.

Mysyrowicz, A.

Oughstun, K. E.

Papazoglou, D.

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

Peacock, A.C.

Piskarskas, A.

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

Piskarskas, R.

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

Polesana, P.

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77, 043814 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

Porras, M. A.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. 96, 1939011 (2006).
[Crossref]

Potenza, M. A. C.

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

Recami, E.

E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[Crossref]

Roichman, Y.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Salerno, D.

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

Schapper, F.

Shreenath, A.

Steinmeyer, G.

Stibenz, G.

Sun, B.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Sztul, H. I.

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt.Express 16, 9411–9416 (2008).
[Crossref] [PubMed]

Tamosauskas, G.

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

Tartara, L.

Trapani, P. Di

F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
[Crossref] [PubMed]

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77, 043814 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
[Crossref]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E. 73, 016608 (2006).
[Crossref]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. 96, 1939011 (2006).
[Crossref]

D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862–869 (2005).
[Crossref]

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

Trebino, R.

Trillo, S.

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

Trull, J.

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

Tzortzakis, S.

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

Valiulis, G.

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
[Crossref]

Varanavicius, A.

D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E 72, 037601 (2005).
[Crossref]

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
[Crossref]

Walmsley, I. A.

Wolf, E.

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. 74, 269–280 (1959).
[Crossref]

H. S. Green and E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. LXVI 66, 1129–1137 (1953).
[Crossref]

Wolf, J. P.

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic Nonlinear X waves for Femtosecond Pulse Propagation in Water,” Phys. Rev. Lett. 92, 2539011 (2004).
[Crossref]

Wyatt, A. S.

Zaïr, A.

Zeitner, U. D.

Appl. Opt. (1)

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

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

Opt. Commun. (1)

M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. Di Trapani, “Three dimensional imaging of short-pulses,” Opt. Commun. 229, 381–390 (2004).
[Crossref]

Opt. Express (9)

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13079–13095 (2007).
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S. Mokhov, R. El-Ganainy, and D. N. Christodoulides, “Power circulation via negative energy-flux wormholes in optical nanowaveguides” Opt. Express 14, 3255–3262 (2006).
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A.C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003).
[Crossref] [PubMed]

P. Gabolde and R. Trebino, “Self-referenced measurement of the complete electric field of ultrashort pulses,” Opt. Express 12, 4423–4429 (2004).
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P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral inter-ferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in realtime,” Opt. Express 14, 11892–11900 (2006).
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A. Zaïr, A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15, 5394–5405 (2007).
[Crossref] [PubMed]

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
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J. Extermann, L. Bonacina, F. Courvoisier, D. Kiselev, Y. Mugnier, R. Le Dantec, Christine Galez, and J. P. Wolf, “Nano-FROG: Frequency Resolved Optical Gating by a nanometric object” Opt. Express 16, 10405–10411 (2008).
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P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16, 13663–13675 (2008).
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Opt. Lett. (3)

Opt.Express (1)

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Phys. Rev. A (1)

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77, 043814 (2008).
[Crossref]

Phys. Rev. A. (1)

D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser pulse collapse in Kerr media: the role of group velocity dispersion and X wave formation,” Phys. Rev. A. 78, 033826 (2008).
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Phys. Rev. E (2)

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, “Spatiotemporal three-dimensional mapping of nonlinear X waves,” Phys. Rev. E 69, 026607 (2004).
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D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. 96, 1939011 (2006).
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P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 93904 (2003).
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C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull,“Nonlinear electromagnetic X Waves, Phys. Rev. Lett. 90, 170406 (2003).
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Figures (4)

Fig. 1.
Fig. 1. (a) Experimental layout for the 3D tomography measurement: DL - delay line, F -focusing lens, KS - Kerr sample, T - imaging telescope, BBO - nonlinear BBO crystal for sum-frequency between laser pulse and 20 fs NOPA gate pulse, IL - imaging lens, CCD - imaging plane with CCD camera. (b), the experimentally retrieved space-time intensity profile of the laser filament (40 mm in water). (c) Experimental layout for the angularly resolved spectrum with (d), the experimentally measured spectrum for the same filament pulse shown in (b). IF - Fourier lens, S - imaging spectrometer.

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Fig. 2.
Fig. 2. Diagram of the Gerchberg-Saxton algorithm applied to the data in order to retrieve the pulse spatiotemporal amplitude and phase profile. An initial guess for the phase profile is superimposed on the measured 3D intensity profile. The spatiotemporal far-field spectrum is obtained by a Fourier transform whose amplitude is substituted with the measured one. An inverse Fourier transform gives a complex valued 3D near field profile in which the amplitude is replaced by the measurement. This process is iterated until convergence is obtained. The graph shows the the mean quadratic deviation between the retrieved and measured near-field intensity profiles over 400 steps.

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Fig. 3.
Fig. 3. Experimentally retrieved energy density flux profiles for an 800 nm, 160 fs laser pulse with a power 11 times the critical power for self-focusing in water. Blue indicates a negative (opposite to the axis direction), Red indicates a positive (same as the axis direction) flux. (a), (c), (e) and (g) show the normalized r-t distribution of transverse flux for the input focused pulse, for the pulse after 0 mm, 10 mm, 20 mm and 40 mm of nonlinear propagation in water, respectively. (b), (d), (f) and (h), for the same conditions, show the longitudinal fluxes. The grey contour plots show the pulse space-time intensity profile over 2 decades (0.4 decade spacing between each contour line). In (e), “L”, “T” and “W” indicate the leading peak, the trailing peak and the central wings, respectively.

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Fig. 4.
Fig. 4. Numerically simulated laser filament pulse and transverse flux after 40 mm of water. The input power and focusing conditions are chosen to match the experiment. The grey contour plots show the pulse space-time intensity profile over 4 decades (0.8 decade spacing between each contour line). “L”, “T” and “W” indicate the leading peak, the trailing peak and the central wings, respectively. The main experimental features are reproduced, in particular the marked X-shaped flux associated to the main leading intensity peak.

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Equations (3)

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

∂u∂z=i2k0∇⊥2u−ik0″∂2u2∂t2+N(u)
Jr(r,t)=12ik0[u(r,t)*∂u(r,t)∂r−u(r,t)∂u(r,t)*∂r]
Jr(r,t)=k0″2i[u(r,t)*∂u(r,t)∂r−u(r,t)∂u(r,t)*∂r]

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