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 Alberto Parola, Daniele Faccio, Paolo Di Trapani University of Insubria, Como, Italy Arnaud Couairon Centre de Physique Théorique, CNRS, Palaiseau, France 1

1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media ultrashort pulse TW/cm^2, 100 fs self-focusing collapsefilamentation 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 2

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

ultrashort pulse TW/cm^2, 100 fs self-focusing collapsefilamentation 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) 1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media 4

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

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

Unstable modes: Y-waves up-shifted axial emission 2. From spatial self-focusing to the birth of spatiotemporal dynamics at collapse:Y-waves up-shifted conical emission down-shifted conical emission down-shifted axial emission first Y-wave second Y-wave 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). superluminalsubluminal GVM = This establishes a casual connection between pre-collapse and post-collapse dynamics 7

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

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

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

3. Filament spatiotemporal dynamics: wave-modes Transversal dispersion curve of WMs X-wave: Longitudinal wave number: Transversal wave number: water  fs/  m water  fs/  m 11

 0 K  pump  idler signal intense, ST localized, pump wave (Y-wave)  k +  k (self-phase modulation), v =1/k´ 00gNL0 k +  k 0NL k +  k 0NL k i s k i 2 weak, noncollinear, idler and signal waves at  and – , propagation constants k = k(  and  k = k(-  s k +  k 0NL k +  k 0NL k i s k 1) axial phase matching k + k = 2k + 2  k z,i z,s 0 NL z 2) group matching k - k 2  = 1 v z,i z,s 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 3. Filament spatiotemporal dynamics: wave-modes 12 (two photons)

 0 K  pump  idler 1) axial phase matching 2) group matching k + k = 2k + 2  k z,i z,s 0 NL k = (k +  k ) +  / v z, i,s 0 NL 0 g linear with frequency k - k 2  = 1 v z,i z,s g 0 Transversal dispersion curve of a WM K = k (  [(k –  ) + (k’ –  )(  )] i,s signal  k NL 0  k’ – 1/v 0 g The ST frequencies preferentially amplified by a split-off pump pulse are just those forming wave-mode The wave-mode travels at the same group velocity as the split-off pump pulse For two split-off pump pulses, two WMs are amplified, each one accompanying each split-off pulse Filament spatiotemporal dynamics: X-waves

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. 14