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Nonlinear X waves

X waves are non-spreading spatio-temporal wave-packets first introduced in linear optics and acoustics. Here we discuss their important role in the context of self-focusing, nonlinear optics and Bose-Einstein condensation. Self-focusing can balance at once diffractive beam spreading and temporal pulse broadening. induced by group-velocity dispersion (GVD). However, the observation of the outcome of such balance, i.e. spatio-temporal solitons known as light bullets, has been elusive for different reasons among which material requirement (anomalous GVD) and stability issues. Recently, a new perspective in this field was opened by the observation, carried out by Di Trapani and co-workers [1], that spatio-temporal trapping can occur in a quadratic medium that mimics self-focusing also in the normal GVD regime. In this regime, solitons cease to exist, and trapping can be interpreted in terms of new entities, namely nonlinear X waves, wavepackets that, while travelling at light group speed, are constituted by a localized (hot) central spot and slowly decaying (cold) conical tails that match the linear spatio-temporal dispersive properties of the material [2,3]. X waves can be seen as suitable non-monochromatic generalizations of Bessel non-diffracting (or Durnin) beams, and have been proposed and widely studied theoretically and experimentally in the last decade in the context of linear wave propagation in optics and acoustics. In particular, they are localized solutions of Maxwell equations that can propagate in vacuum or dielectric (thus dispersive) materials and have attracted a great deal of attention for their superluminal features [4]. Quite independently from linear studies, it was realized that X waves are a paradigm that can explain the observed formation of nearly diffractionand dispersion-less wave-packets occurring at high intensities in normally dispersive bulk samples (where we recall that self-focusing is expected to enforce temporal broadening and causes pulse splitting). Theoretically, such trapping phenomenon is supported by the fact that the most common envelope models for parametric wave-mixing interactions (e.g., Kerr self-action, second-harmonic generation, etc.), valid in the paraxial and slowly-varying regime, possess X wave propagation-invariant solutions. Two of such wave-packets, obtained for the standard 1+3D nonlinear Schroedinger (NLS) equation with self-focusing Kerr-type nonlinearity, are displayed in Fig. 1 and show the essential features of weak localization associated with the slow radial decay (1/r) characteristic of Bessel beams. Figure 1: Profile of (a) fundamental (b) higher-order X wave solutions in a Kerr medium Besides the shape and the different decay rate, nonlinear X waves differ from solitons also because they have a finite linear (low intensity) limit. In this sense they can be regarded as a continuation of linear paraxial X waves in dispersive media. The nonlinearity, however, plays a central role in the dynamics, driving the reshape of a conventional (ball-shaped) laser beam and pulse into an X wave. This can be shown to occur through the exponential QFL4-3-INV