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Miguel A. Porras Departamento de Física Aplicada. Universidad Politécnica de Madrid

1. On the role of conical waves in the self-focusing and filamentation of ultrashort pulses or a reinterpretation of the spatiotemporal dynamics of ultrashort pulses in Kerr media. Miguel A. Porras Departamento de Física Aplicada. Universidad Politécnica de Madrid.

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Miguel A. Porras Departamento de Física Aplicada. Universidad Politécnica de Madrid

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  1. 1 On the role of conical waves in the self-focusingand filamentation of ultrashort pulsesor a reinterpretation of the spatiotemporal dynamics of ultrashort pulses in Kerr media Miguel A. Porras Departamento de Física Aplicada. Universidad Politécnica de Madrid Alberto Parola, Daniele Faccio, Paolo Di Trapani University of Insubria, Como, Italy Arnaud Couairon Centre de Physique Théorique, CNRS, Palaiseau, France

  2. 2 1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media collapse filamentation self-focusing ultrashort pulse TW/cm^2, 100 fs nonlinear Kerr sample Self-focusing stage: fast nearly pure spatial dynamics self-similar compression towards a universal transversal profile: the Townes profile Collapse region: enormous intensities (hundreds of TW/cm^2) onset of higher-order nonlinear phenomena birth of a full-spatiotemporal dynamics

  3. 3 1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media collapse filamentation self-focusing ultrashort pulse TW/cm^2, 100 fs nonlinear Kerr sample Filamentary regime: Axial emission (spectral broadening, new temporal frequencies) Conical emission (new spatiotemporal frequencies) Pulse temporal splitting, narrowing and recombination Several self-refocusing cycles, etc Spatiotemporal spectrum CE time splitting and narrowing k 0 AE intensity transverse frequency recombination w w 0 time temporal frequency

  4. 4 1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media collapse filamentation self-focusing ultrashort pulse TW/cm^2, 100 fs nonlinear Kerr sample Spatial soliton ??? (balance Kerr focusing, plasma defocusing and diffraction) Axial emission (four-wave mixing amplification) Conical emission (Cerenkov radiation ???) Pulse splitting (arrest of collapse by GVD ???) Alternate, unified interpretation of the basic features of the whole ST dynamics from the self-focusing stage to the end of the filamentary regime in terms of conical waves (Y-waves and wave-modes)

  5. 2. From spatial self-focusing to the onset of spatiotemporal dynamics at collapse:Y-waves 5 spatial self-focusing collapse spatiotemporal dynamics Blowing-up Transversal Townes profile Incipient AE, CE and pulse splitting Townes beam monochromatic light beam a (r)exp(iaz) ground state of the cubic NLSE 0

  6. 2. From spatial selfocusing to the onset of spatiotemporal dynamics at collapse:Y-waves 6 • Spatiotemporal instability of the Townes profile Spatiotemporal perturbation TP • Two unstable modes: Y-waves Y-shaped spatiotemporal spectra of the 2 unstable modes exp. growing ST frequencies exp. growing ST frequencies exp. growing ST frequencies Townes beam frequency

  7. 2. From spatial self-focusing to the birth of spatiotemporal dynamics at collapse:Y-waves 7 • Unstable modes: Y-waves first Y-wave up-shifted conical emission second Y-wave down-shifted conical emission up-shifted axial emission down-shifted axial emission superluminal subluminal GVM = • The ST instabilityof the TP can then explain the main ST phenomena (CE, • AE and pulse splitting) observed immediately after collapse, from the features • of the self-focusing beam (self-focusing into a TP). • This establishes a casual connection between pre-collapse and post-collapse • dynamics

  8. 2. From spatial self-focusing to the birth of spatiotemporal dynamics at collapse:Y-waves 8 self-focusing collapse extremely localized and intense Townes profile extremely localized and intense Townes profile Growth of two Y-waves Incipient AE, CE and pulse splitting Growth of two Y-waves Incipient AE, CE and pulse splitting perturbations (higher-order, noise) seed ST instability perturbations (higher-order, noise) seed ST instability • Experimental observation of Y-waves upon collapse: • 527 nm, 200 fs, fused silica E= 2 mJ E= 3 mJ 15 cm 15 cm

  9. 3. Filament spatiotemporal dynamics: X-like wave-modes 9 filament Water, 200 fs, l=527 nm, 3 mJ D. Faccio, M.A. Porras, A. Dubietis, F. Bragheri, A. Couairon, P. Di Trapani, Phys. Rev. Lett. 96 (2006) 193901 M. Kolesik, E.M.Wright, and J.V. Moloney, Phys. Rev. Lett. 94 (2004) 253901 We interpret this conical emission as the manifestation of the Kerr-driven formation of two X wave-modes (one for each split-off pulse)

  10. 3. Filament spatiotemporal dynamics: X wave-modes 10 • Wave-modes: spatiotemporal localized waves that can propagate in a linear dispersive medium without any temporal dispersion broadening Pulsed (polychromatic) Bessel beam Bessel beam free parameters

  11. 3. Filament spatiotemporal dynamics: wave-modes 11 X-wave: Longitudinal wave number: Transversal wave number: Transversal dispersion curve of WMs water b=0, a= -0.025 fs/mm water b=0, a = 0.025 fs/mm

  12. 3. Filament spatiotemporal dynamics: wave-modes 12 K (w) • intense, ST localized, pump wave (Y-wave) • w , k +Dk (self-phase modulation), v =1/k´ idler 0 0 NL g 0 w-w -W pump W 0 • 2 weak, noncollinear, idler and signal waves at • W and –W, propagation constants k = k(W) and • k = k(-W) i s k k k k s s i i k + k = 2k + 2Dk z,i z,s 0 NL k + Dk k + Dk k + Dk k + Dk k - k 1 0 0 0 0 NL NL NL NL z,i z,s = 2(w-w ) v g 0 WM results from the parametric amplification of new frequencies by a pump wave (a split-off pulse) via a Kerr-driven degenerate FWM interaction signal (two photons) 1) axial phase matching z 2) group matching

  13. 3. Filament spatiotemporal dynamics: X-waves 13 2 2 K = k (w) + [(k –b) + (k’ –a)(w-w )] i,s 0 0 0 a = k’ – 1/v 0 g k + k = 2k + 2Dk z,i z,s 0 NL b = -Dk 0 NL • The wave-mode travels at the same group • velocity as the split-off pump pulse k - k 1 z,i z,s = 2(w-w ) v g 0 1) axial phase matching 2) group matching k = (k +Dk ) + (w-w ) / v linear with frequency z, i,s 0 NL 0 g Transversal dispersion curve of a WM • The ST frequencies preferentially amplified • by a split-off pump pulse are just • those forming wave-mode K (w) idler signal w-w -W pump W 0 • For two split-off pump pulses, two WMs • are amplified, each one accompanying each • split-off pulse

  14. 14 Conclusory remarks • We have introduced a interpretation of the spatiotemporal dynamics of • ultrashort pulses in self-focusing Kerr media in terms of conical waves. • In this interpretation the only essential nonlinearity is the Kerr nonlinearity. • Higher-order effects play a secondary role. • Conical waves (X-waves and Y-waves) are essentially linear waves, but • are nonlinearly created (by Kerr-driven instability degenerate FWM). • Y-waves can explain the excitation of the post-collapse filamentary regime • from self-similar self-focusing prior to collapse. • X-waves armonize the apparent stationarity of the filament with its actual • complex spatiotemporal dynamics. • Results are presented for media with normal dispersion but can be readily • rewritten for anomalous dispersion.

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