We develop a general theory of spatial solitons in a liquid crystalline medium exhibiting a nonlinearity with an arbitrary degree of effective nonlocality. The model accounts the observability of accessible solitons and establishes an important link with parametric solitons.

We report on the observation and quantitative assessment of self-trapped pulsating beams in a highly non-local nonlinear regime. The experiments were conducted in nematic liquid crystals and allow a meaningful comparison with the prediction of a scalar theory in the perturbative limit, while addressing the need for beyond-paraxial analytical treatments.

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We investigate the formation of collisionless shocks along the spatial profile of a Gaussian laser beam propagating in nonlocal nonlinear media. For defocusing nonlinearity the shock survives the smoothing effect of the nonlocal response, though its dynamics is qualitatively affected by the latter, whereas for focusing nonlinearity it dominates over filamentation. The patterns observed in a thermal defocusing medium are interpreted in the framework of our theory.

In certain materials, the spontaneous spreading of a laser beam (owing to diffraction) can be compensated for by the interplay of optical intensity and material nonlinearity. The resulting non-diffracting beams are called 'spatial solitons' (refs 1-3), and they have been observed in various bulk media. In nematic liquid crystals, solitons can be produced at milliwatt power levels and have been investigated for both practical applications and as a means of exploring fundamental aspects of light interactions with soft matter. Spatial solitons effectively operate as waveguides, and so can be considered as a means of channelling optical information along the self-sustaining filament. But actual steering of these solitons within the medium has proved more problematic, being limited to tilts of just a fraction of a degree. Here we report the results of an experimental and theoretical investigation of voltage-controlled 'walk-off' and steering of self-localized light in nematic liquid crystals. We find not only that the propagation direction of individual spatial solitons can be tuned by several degrees, but also that an array of direction-tunable solitons can be generated by modulation instability. Such control capabilities might find application in reconfigurable optical interconnects, optical tweezers and optical surgical techniques.

Quantum and classical physics can be used for mathematical computations that are hard to tackle by conventional electronics. Very recently, optical Ising machines have been demonstrated for computing the minima of spin Hamiltonians, paving the way to new ultra-fast hardware for machine learning. However, the proposed systems are either tricky to scale or involve a limited number of spins. We design and experimentally demonstrate a large-scale optical Ising machine based on a simple setup with a spatial light modulator. By encoding the spin variables in a binary phase modulation of the field, we show that light propagation can be tailored to minimize an Ising Hamiltonian with spin couplings set by input amplitude modulation and a feedback scheme. We realize configurations with thousands of spins that settle in the ground state in a low-temperature ferromagnetic-like phase with all-to-all and tunable pairwise interactions. Our results open the route to classical and quantum photonic Ising machines that exploit light spatial degrees of freedom for parallel processing of a vast number of spins with programmable couplings.A large number of internal states characterizes complex systems from biology to social science. The fact that the number of these states grows exponentially with the system size hampers large-scale computational possibilities. Complex optimization problems involving these models are in many cases classified as NP-hard and cannot be tackled efficiently by standard computing architectures. A broad class of such computationally intractable problems maps to the search of the ground state of a classical system of interacting spins: the minimization of an Ising Hamiltonian with specific spin couplings [1-3].Growing research interest is emerging towards physical and artificial systems that evolve according to an Ising Hamiltonian and enable to find the optimal combinatorial solution by the ground state observed in the experiment. Quantum and classical Ising systems have been realized by trapped atoms [4,5], single photons [6], superconducting circuits [7], electromechanical modes [8], nanomagnets [9] and polariton condensates [10]. In optics, spin-glass dynamics have been observed in random lasers [11, 12], multimodal cavities [13, 14] , coupled laser lattices [15], beam filamentation [16] and nonlinear wave propagation in disordered media [17]. These photonic systems host thousands of optical spins, but the spin variables are not easy to access and controlling their interaction is challenging.Novel photonic platforms with numerous and easily accessible spins are particularly relevant for computation. Optical computing machines offer high-speed and parallelization. Various authors reported coherent Ising machines based on time-multiplexed optical parametric oscillators finding approximate solutions to optimization problems with several nodes [18][19][20][21][22][23][24]. Others proposed nanophotonic circuits to implement any small-scale spin systems directly on a programmable chip [25][26][27]. Matrix opera...

The discovery of the spontaneous mode-locking of lasers, i.e., the synchronous oscillation of electromagnetic modes in a cavity, has been a milestone of photonics allowing the realization of oscillators delivering ultra-short pulses. This process is so far known to occur only in standard ordered lasers with meter size length and only in the presence of a specific device (the saturable absorber). Here we demonstrate that mode-locking can spontaneously arise also in random lasers composed by micronsized laser resonances dwelling in intrinsically disordered, self-assembled clusters of nanometer-sized particles. Moreover by engineering a novel mode-selective pumping mechanism we show that it is possible to continuously drive the system from a configuration in which the various excited electromagnetic modes oscillate in the form of several, weakly interacting, resonances to a collective strongly interacting regime. By realizing the smallest mode-locking device ever fabricated, we open the way to novel generation of miniaturized and all-optically controlled light sources.Random lasers[1] (RLs) are made by disordered highly scattering materials able to amplify light when externally pumped. The simultaneous presence of structural disorder and nonlinearity makes these devices a fertile ground to connect photonics with advanced theoretical paradigms[2] like chaos [3], non Gaussian statistics[4], complexity [5] and also the physics of Bose Einstein condensation [6]. Historically there has been a bridge in the RL interpretation. In pioneering experiments a smooth, single-peaked emission was produced by pumping finely ground laser crystals [7], or titania particles dispersed in a dye-doped solution [8,9]. This phenomenon has been dubbed RL with incoherent feedback (IFRL) because it may be explained in the framework of the diffusion approximation [10] that neglects interference and treats light rays as the trajectories of random walking particles. However this theoretical framework does not explain another kind of RL exhibitting subnanometre sharp spectral peaks [11][12][13] associated with high-Q resonances[14-17] and labeled resonant feedback random laser(RFRL).Standard multimode lasers without disorder and characterized by equispaced resonances may be driven to a synchronous regime through the so called mode-locking transition [18,19], which so far has only been shown to occur spontaneously in the presence of a saturable absorber and allows to generate ultra-short light pulses [20,21]. We show that the same transition occurs in RLs and allows to lock modes of a RFRL casting its emission in the typical IFRL spectrum and demonstrating the inherently coherent nature of the random lasing phenomenon.The system we consider is an isolated micrometer sized cluster of titania nanoparticles immersed in a rhodamine dye solution (see supplementary information (SI)). In our novel setup we use the amplified spontaneous emission (ASE) from the surrounding dye to pump the cluster. The the ASE areas are defined by shaping the beam of an e...

Spin-glass theory is one of the leading paradigms of complex physics and describes condensed matter, neural networks and biological systems, ultracold atoms, random photonics and many other research fields. According to this theory, identical systems under identical conditions may reach different states. This effect is known as replica symmetry breaking and is revealed by the shape of the probability distribution function of an order parameter named the Parisi overlap. However, a direct experimental evidence in any field of research is still missing. Here we investigate pulse-to-pulse fluctuations in random lasers, we introduce and measure the analogue of the Parisi overlap in independent experimental realizations of the same disordered sample, and we find that the distribution function yields evidence of a transition to a glassy light phase compatible with a replica symmetry breaking.