RAPID COMMUNICATIONS PHYSICAL REVIEW A 84, 021807(R) (2011) Stationary nonlinear Airy beams A. Lotti, 1,3 D. Faccio, 1,2,* A. Couairon, 3 D. G. Papazoglou, 4,5 P. Panagiotopoulos, 4 D. Abdollahpour, 4,6 and S. Tzortzakis 4 1 Dipartimento di Fisica e Matematica, Universit` a del’Insubria, Via Valleggio 11, I-22100 Como, Italy 2 School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom 3 Centre de Physique Th´ eorique, CNRS, ´ Ecole Polytechnique, F-91128 Palaiseau, France 4 Institute of Electronic Structure and Laser (IESL), Foundation for Research and Technology, Hellas (FORTH), P.O. Box 1527, GR-71110 Heraklion, Greece 5 Materials Science and Technology Department, University of Crete, GR-71003 Heraklion, Greece 6 Physics Department, University of Crete, GR-71003 Heraklion, Greece (Received 7 March 2011; published 22 August 2011) We demonstrate the existence of an additional class of stationary accelerating Airy wave forms that exist in the presence of third-order (Kerr) nonlinearity and nonlinear losses. Numerical simulations and experiments, in agreement with the analytical model, highlight how these stationary solutions sustain the nonlinear evolution of Airy beams. The generic nature of the Airy solution allows extension of these results to other settings, and a variety of applications are suggested. DOI: 10.1103/PhysRevA.84.021807 PACS number(s): 42.25.−p, 03.50.−z, 42.65.Jx Introduction. Airy beams are a well-known family of stationary freely accelerating wave forms. Originally proposed in the context of quantum mechanics as a nonspreading solution to the Schr ¨ odinger equation for free particles [1], they were later proposed as optical wave packets with finite energy content [2,3]. The finite-energy Airy beam is characterized by a main intensity lobe that decays exponentially to zero on one side and decays with damped oscillations on the other. The interest for these beams lies in the fact that, if they have a sufficiently wide apodization, the main intensity lobe propagates free of diffraction while bending in the direction transverse to propagation or accelerating along the propagation direction if the temporal profile of the pulse is Airy shaped [4]. The ballisticlike properties of the Airy beam [5] lend it to particular applications such as optically mediated particle clearing [6] or generation of curved plasma filaments [7]. Recently a demonstration of light bullets using Airy cube wave packets (Airy in space and time) has also been reported [8]. Alongside the linear properties of Airy beams, nonlinear propagation of high-intensity Airy beams has also attracted attention [9,10]. It has been noted that upon increasing the Airy peak intensity the beam may either break up and emit a series of tangential emissions [11] or exhibit shrinking and modification of the Airy profile even below the critical threshold power for self-focusing [12,13]. Notably, Giannini et al. first described temporal self-accelerating solitons in Kerr media [14]. In this Rapid Communication we demonstrate the existence of stationary Airy-like solutions in the presence of third-order Kerr nonlinearity of any sign (i.e., focusing or defocusing) and, most importantly, even in the presence of nonlinear losses (NLLs). We perform an analytical analysis that describes the shape and main features of one-dimensional nonlinear Airy wave packets, i.e., monochromatic beams that exhibit a curved trajectory. The Kerr nonlinearity is shown to lead to a compression of the Airy lobes (for a focusing nonlinearity) and nonlinear losses lead to an imbalance of the incoming energy * d.faccio@hw.ac.uk flux toward the main lobe which in turn induces a reduction in the contrast of the Airy oscillations. This finding is then verified in numerical simulations and experiments that show the spontaneous emergence of the main features of stationary nonlinear Airy beams. Analytical description. We consider the propagation of a monochromatic beam of frequency ω 0 in one spatial dimension. The electric field E (x,z,t ) is decomposed into carrier and envelope as E (x,z,t ) = E(x,z) exp(−iωt + ik 0 z), where k 0 = ω 0 n 0 /c is the modulus of the wave vector at ω 0 and n 0 = n(ω 0 ) is the value of the refractive index at ω 0 . In the presence of nonlinearity, such as the Kerr effect and multiphoton absorption, propagation may be described by the nonlinear Schr¨ odinger equation for the complex envelope of the field: ∂E ∂z = i 2k 0 ∂ 2 E ∂x 2 + ik 0 n 2 n 0 |E| 2 E − β (K) 2 |E| 2K−2 E, (1) where the nonlinear Kerr modification of the refractive index is δn = n 2 |E| 2 , while K and β (K)  0 are the order and the coefficient of multiphoton absorption, respectively. In the case of linear propagation, Eq. (1) admits the Airy beam solution E = Ai(y ) exp[iφ L (y,ζ )], whose intensity profile is invariant in the uniformly accelerated reference system defined by the normalized coordinates ζ = z/k 0 w 2 0 , y = x/w 0 − ζ 2 /4, with φ L (y,ζ ) ≡ yζ/2 + ζ 3 /24 and w 0 a typical length scale so that the acceleration or curvature is given by 1/2k 2 0 w 3 0 . We are interested in finding stationary nonlinear solutions to Eq. (1), in the above sense (invariant in the accelerated reference system), with boundary conditions compatible with the shape and properties of Airy beams, whose asymptotic behavior as y → ±∞ reads [15] Ai(y ) ∼|yπ 2 | −1/4 sin(|ρ |+ π/4) for y → −∞, (2) Ai(y ) ∼ (yπ 2 ) −1/4 2 exp(−|ρ |) for y → +∞, (3) where ρ = (2/3)sgn(y )|y | 3/2 . We therefore impose the con- straints of a weakly localized tail toward y → −∞ and an exponentially decaying tail toward y → +∞. Solutions, 021807-1 1050-2947/2011/84(2)/021807(4) ©2011 American Physical Society