Wavelength rigidly fixes the diffraction that distorts waves during propagation, and poses fundamental limits to imaging, microscopy and communication. This distortion can be avoided by using waveguides or nonlinearity to produce solitons. In both cases, however, diffraction is only compensated, so the wavelength still imposes rigid laws on wave shape, size and soliton intensity(1-14). Nonlinearity, in turn, can introduce new spatial scales. In principle, if one is able to identify a nonlinearity that introduces an intensity-independent scale that cancels the wavelength, 'scale-free' propagation can occur. In this regime, diffraction ceases, and waveforms will naturally propagate without distortion, forming solitons of any size and intensity, even arbitrarily low. Here we provide the first experimental evidence of scale-free optical propagation in supercooled copper-doped KTN:Li, a recently developed out-of-equilibrium ferroelectric(15-17). This demonstrates that diffraction can be cancelled, and not merely compensated, thus leading to a completely new paradigm for ultraresolved imaging and microscopy

One of the most controversial phenomena in nonlinear dynamics is the reappearance of initial conditions. Celebrated as the Fermi-Pasta-Ulam-Tsingou problem, the attempt to understand how these recurrences form during the complex evolution that leads to equilibrium has deeply influenced the entire development of nonlinear science. The enigma is rendered even more intriguing by the fact that integrable models predict recurrence as exact solutions, but the difficulties involved in upholding integrability for a sufficiently long dynamic has not allowed a quantitative experimental validation. In natural processes, coupling with the environment rapidly leads to thermalization, and finding nonlinear multimodal systems presenting multiple returns is a long-standing open challenge. Here, we report the observation of more than three Fermi-Pasta-Ulam-Tsingou recurrences for nonlinear optical spatial waves and demonstrate the control of the recurrent behavior through the phase and amplitude of the initial field. The recurrence period and phase shift are found to be in remarkable agreement with the exact recurrent solution of the nonlinear Schrödinger equation, while the recurrent behavior disappears as integrability is lost. These results identify the origin of the recurrence in the integrability of the underlying dynamics and allow us to achieve one of the basic aspirations of nonlinear dynamics: the reconstruction, after several return cycles, of the exact initial condition of the system, ultimately proving that the complex evolution can be accurately predicted in experimental conditions.

Rogue waves are observed as light propagates in the extreme nonlinear regime that occurs when a photorefractive ferroelectric crystal is undergoing a structural phase transition. The transmitted spatial light distribution contains bright localized spots of anomalously large intensity that follow a signature long-tail statistics that disappears as the nonlinearity is weakened. The isolated wave events form as out-of-equilibrium response and disorder enhance the Kerr-saturated nonlinearity at the critical point. Self-similarity associable to the individual observed filaments and numerical simulations of the generalized nonlinear Schrödinger equation suggests that dynamics of soliton fusions and scale invariance can microscopically play an important role in the observed rogue intensities and statistics.

We experimentally investigate the interplay between spatial shock waves and the degree of disorder during nonlinear optical propagation in a thermal defocusing medium. We characterize the way the shock point is affected by the amount of disorder and scales with wave amplitude. Evidence for the existence of a phase diagram in terms of nonlinearity and amount of randomness is reported. The results are in quantitative agreement with a theoretical approach based on the hydrodynamic approximation. PACS numbers:Laser beams propagating in nonlinear media undergo severe distortions as the power is increased: spreading due to diffraction can be progressively reduced through self narrowing, up to the generation of solitons [1,2] and dissipative and dispersive shock waves (SWs) [3][4][5][6][7][8], thus fostering the formation of a variety of nonlinear waves. The way these are affected by disorder is a leading mainstream of modern research [9][10][11][12]. Attention is given to the competition between strongly nonlinear and coherent phenomena, and their frustration due to randomness and scattering; recent theoretical investigations deal with general frameworks described by "phase-diagrams" in terms of the two parameters: the amount of nonlinearity and of disorder [13]. However, no direct experimental nonlinearity-disorder phase diagram has been reported.The case of SWs is specifically relevant [14][15][16], as they represent a strongly nonlinear and coherent oscillation (the undular bore) [8,[17][18][19][20] and are expected to be strongly affected (and eventually inhibited) by disorder, at variance, e.g., with solitons, which can survive a certain amount of randomness (see, e.g., [21,22]). This leads to the direct opposition between the two effects: on one hand increasing the nonlinearity favors the shock formation, on the other hand disorder-induced scattering limits this phenomenon. This is relevant in colloidal systems [8,[23][24][25][26][27] where disorder is unavoidable, as well as in out-of-equilibrium photorefractive nonlinearities [28], optical fibers [5,29], and also in Bose-Einstein condensation [30][31][32] and acoustics [33].In this Letter, we report on the direct experimental evidence of the competition between SWs and disorder, and support our experiments by a theoretical model based on the hydrodynamical approximation. We measure the first phase-diagram (where the order parameter is the position of the formation of the shock) for nonlinear waves in terms of disorder and nonlinearity, and characterize the scaling laws for the random SWs formation and propagation. Experiment -We use dispersions of silica spheres of diameter 1µm in 0.1mM aqueous solutions of RhodamineB displaying a thermal defocusing effect due to partial light absorption [14,16,34,35]. To vary the degree of disorder several silica concentrations c are prepared, ranging from 0.005 w/w to 0.03 w/w, in units of weight of silica particles over suspension weight. A continuous-wave laser at wavelength λ=532 nm is focused on the input facet of the sa...

We report the first observation of spatial one-dimensional photorefractive screening solitons in centrosymmetric media and compare the experimental results with recent theoretical predictions. We find good qualitative agreement with theory. © 1998 Optical Society of America OCIS codes: 190.5330, 230.6120. Photorefractive spatial solitons have been a subject of intense study over the past few years. They have been predicted and observed in the quasi-steady-state regime, 1,2 in photovoltaic materials, 3,4 in the screening configuration, 5 -9 and in photorefractive semiconductors. 10More-complicated phenomena have also been reported, giving rise to intriguing observations, such as self-trapping of incoherent light beams. 11All these phenomena have been observed in noncentrosymmetric materials, in which soliton formation is governed by a change in refractive index that is due to the linear electro-optic response to an internal photoinduced space-charge field. Recently, spatial screening solitons of a different nature that should exist in photorefractive centrosymmetric materials were predicted. 12 We report the f irst observation of such solitons and compare our experimental results with the theoretical predictions.Centrosymmetric screening solitons arise from the index change produced by the quadratic electro-optic response to a photoinduced internal f ield. The f ield has, in this case, the double role both of polarizing the crystalline structure and of distorting the electronic polarization. In centrosymmetric crystals the change in refractive index is proportional to the square of the polarization ͑1͞Dn͒ ij g ijkl P k P l and is expressed by Dn 2͑1͞2͒n b 3 g eff e 0 2 ͑e r 2 1͒ 2 E 2 , where E is the internal field, g eff is the effective quadratic electrooptic coeff icient, and n b is the background refractive index and it is assumed that the (dc) polarization is in the linear regime, i.e., P e 0 ͑e r 2 1͒E.Our experiments are performed in potassium lithium tantalate niobate 13 (KLTN), which is treated to have a first-order ferroelectric -paraelectric phase transition slightly below room temperature. Working at room temperature enables one to operate in a centrosymmetric phase close to that transition, thereby enhancing the electro-optic response, 13 making centrosymmetric soliton observation possible with moderate electric fields. In Fig. 1 we show e r as a function of temperature and observe the large increase of e r at the vicinity of the ferroelectric -paraelectric transition (which occurs at ϳ12 ± C). Because Dn scales with ͑e r 2 1͒ 2 , operation at temperatures slightly above the Curie temperature results in an increase of the quadratic electro-optic response. In the specif ic case of KLTN, g eff is positive and thus only bright solitons can be observed; i.e., in the screening regime KLTN is a self-focusing medium. 12Bright centrosymmetric screening solitons in ͑1 1 1͒ D obey the wave equationwhere u͑j͒ is the soliton amplitude normalized to the square root of the sum of background and dark irradianc...