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States of light encoding multiple polarizations - vector beams - offer unique capabilities in metrology and communication. However, their practical application is limited by the lack of methods for measuring many polarizations in a scalable and compact way. Here we demonstrate polarimetry of vector beams in a single shot without any polarization optics. We map the beam polarization content into a spatial intensity distribution through light scattering and exploit supervised learning for single-shot measurements of multiple polarizations. We characterize structured light encoding up to nine polarizations with accuracy beyond 95% on each Stokes parameter. The method also allows us to classify beams with an unknown number of polarization modes, a functionality missing in conventional techniques. Our findings enable a fast and compact polarimeter for polarization-structured light, a general tool that may radically impact optical devices for sensing, imaging, and computing.

We experimentally and numerically explore the role of dimensionality in multiple (three or more) soliton fusion supported by nonreciprocal energy exchange. Three-soliton fusion into an intense wave is found when an extra dimension, with no broken inversion symmetry, is involved. The phenomenon is observed for 2+1D spatial waves in photorefractive crystals, where solitons are supported by a spatially local saturated Kerr-like self-focusing and fusion is driven by the leading nonlocal correction, the spatial analog of the nonlinear Raman effect.

The race to heuristically solve non-deterministic polynomial-time (NP) problems through efficient methods is ongoing. Recently, optics was demonstrated as a promising tool to find the ground state of a spin-glass Ising Hamiltonian, which represents an archetypal NP problem. However, achieving completely programmable spin couplings in these large-scale optical Ising simulators remains an open challenge. Here, by exploiting the knowledge of the transmission matrix of a random medium, we experimentally demonstrate the possibility of controlling the couplings of a fully connected Ising spin system. By further tailoring the input wavefront we showcase the possibility of modifying the Ising Hamiltonian both by accounting for an external magnetic field and by controlling the number of degenerate ground states and their properties and probabilities. Our results represent a relevant step toward the realisation of fully-programmable Ising machines on thin optical platforms, capable of solving complex spin-glass Hamiltonians on a large scale., 13 pages, 9 figures

Modern machine learning applications require huge artificial networks demanding in computational power and memory. Light-based platforms promise ultra-fast and energy-efficient hardware, which may help in realizing next-generation data processing devices. However, current photonic networks are limited by the number of input-output nodes that can be processed in a single shot. This restricted network capacity prevents their application to relevant large-scale problems such as natural language processing. Here, we realize a photonic processor with a capacity exceeding $1.5 \times 10^{10}$ optical nodes, more than one order of magnitude larger than any previous implementation, which enables photonic large-scale text encoding and classification. By exploiting the full three-dimensional structure of the optical field propagating in free space, we overcome the interpolation threshold and reach the over-parametrized region of machine learning, a condition that allows high-performance natural language processing with a minimal fraction of training points. Our results provide a novel solution to scale-up light-driven computing and open the route to photonic language processing., Comment: 12 pages, 4 figures, 1 table

We use giant optical refraction (GR) to perform, for the first time, real-time imaging of ferroelectric volume cluster dynamics [1] . Experiments are carried out in biased KTN:Li kept below its room-temperature Curie point. The analysis of polarization transmission images through the sample versus electric field indicate a low-dimensional fractal percolation $\mathcal{D} = 1.65$ , followed by a second high-dimensional percolation $\mathcal{D} > 2$ , from a highly organized super-crystal (SC) state to a disordered ferroelectric cluster state ( Fig. 1 ). Results help shed light on a basic still open question: is there a general class of dynamics at the microscopic scale that characterizes the transition of a system from one macroscopic state to another? While continuous phase-transitions are macroscopically characterized by universal power laws that can be deduced from scale-invariance in their statistical field description, there is to date no evidence of an analogous universality in the microscopic details of how the transition actually occurs, i.e., in its underlying space-time dynamics. The issue has been previously investigated in ferroelectrics where the transition is from a non-polar or paraelectric crystal state to a polar ferroelectric one. One candidate microscopic mechanism driving the transition is polar cluster percolation [2] , a mechanism that, being self-similar, has intruiguing and hereto unexplored similarities to statistical scale-invariance. Percolation is triggered by changes in the average cluster size and can be activated by changes in temperature, pressure and external electric fields. Direct evidence of percolation dynamics in a ferroelectric transition in its full 3D geometry was previously unavailable. This is because while ferroelectric clusters can be optically imaged using crossed polarizers, percolation occurs for clusters whose size can range from tens of nanometers to hundreds of micrometers in a full macroscopic volume, so that direct wide-area detection is generally inaccessible. The discovery of a ferroelectric super-crystal state opens up an entirely new scenario [3] . The specific 3D geometry of the SC, with its spatially ordered and periodic birefringent polar clusters and its giant optical refraction [4] make direct crossed-polarizer high-resolution 3D orthographic projections possible ( Fig 1 ). Findings shed light on the nature and susceptibility of ferroelectric supercrystals. They suggest a specific role for organized 3D polarization structures below the Curie temperature which can have profound repercussions on the development of future energy and information storing technology [5] .

We demonstrate second-harmonic generation (SHG) in a KTN:Li ferroelectric perovskite that manifests giant refraction (n ⪢ 1) [1] . When light is focused on the vortex or anti-vortex cores of an underlying ferroelectric super-crystal (SC) [2] , [3] , enhanced response allows conversion to occur through bulk nonlinear Cherenkov radiation even for highly focused non-phase-matched beams. The result is that wavelength conversion has a wide spectral acceptance of more than 100 nm in the near infrared, an ultra-wide angular acceptance of up to 40°, has no polarization selectivity, and occurs even in the presence of strong transverse wave-vector mismatch (see ± Fig. 1 ). The underlying physical mechanism is associated to the nonlinear response of the SC state, a 3D mesh of ferroelectric polarization that spontaneously forms below the Curie point ( Fig 1a ). In the SC lattice, each site is the core of a 3D vortex or anti-vortex topological defect along which light experiences broadband giant optical refraction and with it, a greatly enhanced nonlinear susceptibility. Results open up new applicative scenarios in the realm of frequency conversion and parametric amplification, the fundamental ingredients for a wide family of applications, including light sources, detection, optical processing, and quantum-state-generation [4] . Reduced constraints on launch angle can considerably reduce alignment requirements in nonlinear-based light sources, while increased tolerances in wavelength and polarization can support multiple simultaneous nonlinear processes with specific impact, for example, in the conversion of infrared images to the visible spectrum. In comparison, conventional wavelength conversion schemes are based on a cumulative wave interaction and constructive interference, a configuration that requires phase-matching and places constraints on polarization, wavelength, and direction of propagation of the interacting waves, apart from requiring highly specific crystal fabrication and poling geometries.

We observe chaotic optical wave dynamics characterized by erratic energy transfer and soliton annihilation and creation in the aftermath of a three-soliton collision in a photorefractive crystal. Irregular dynamics are found to be mediated by the nonlinear Raman effect, a coherent interaction that leads to nonreciprocal soliton energy exchange. Results extend the analogy between solitons and particles to the emergence of chaos in three-body physics and provide new insight into the origin of the irregular dynamics that accompany extreme and rogue waves.

We perform percolation analysis of crossed-polarizer transmission images in a biased nanodisordered bulk KTN:Li perovskite. Two distinct percolative transitions are identified at two electric field thresholds. The low-field transition involves a directional fractal chain of dimension $\mathcal{D}=1.65$, while the high-field transition has a dimension $\mathcal{D}g2$. Direct cluster imaging in the volume is achieved using high-resolution orthographic 3D projections based on giant refraction. Percolation is attributed to a full-3D domain reorientation that mediates the transition from a ferroelectric supercrystal state to a disordered domain mosaic.

We study theoretically neural networks embedding a nonlinear wave as a computing reservoir. We demonstrate interpolation of large datasets and Boolean logic. We discuss the existence of a critical nonlinearity for learning.

Many developments in science and engineering depend on tackling complex optimizations on large scales. The challenge motivates intense search for specific computing hardware that takes advantage from quantum features, nonlinear dynamics, or photonics. A paradigmatic optimization problem is finding low-energy states in classical spin systems with fully-random interactions. To date no alternative computing platform can address such spin-glass problems on a large scale. Here we propose and realize an optical scalable spin-glass simulator based on spatial light modulation and multiple light scattering. By tailoring optical transmission through a disordered medium, we optically accelerate the computation of the ground state of large spin networks with all-to-all random couplings. Scaling of the operation time with the problem size demonstrates optical advantage over conventional computing. Our results point out optical vector-matrix multiplication as a tool for spin-glass problems and provide a general route towards large-scale computing that exploits speed, parallelism and coherence of light., 9 pages, 4 figures

Dispersive waves (DWs) radiation in optical fibers allow to excite frequencies in spectral regions that are otherwise difficult to access. However, the resonant emission is weak and phase matching inhibits its shaping, making it ineffective in many applications. Here, we numerically investigated that DWs generated from high-energy pulses propagating in nonlinear optical fibers can be controlled through spectral phase modulation. Using tunable asymmetric and oscillatory temporal pulse shapes obtained by cubic and quadratic spectral phase modulation, we excite a multiple emission process that strongly enhances the energy and peaks of DWs. Successive collisions between the shed soliton and multiple secondary peaks of the asymmetric pulse give the mechanism to stimulate various resonant frequencies. The whole process can be precisely controlled by simply adjusting the spectral phase modulation structure without the need to change the fiber length. This radiation is either reduced or enhanced by the Raman effect depending on whether the third-order dispersion is positive or negative. Our results provide a novel strategy to manipulate DWs emission and particularly to boost the DWs energy in optical fibers that may be relevant in the development of novel electromagnetic sources for broadband supercontinuum and on-chip frequency-comb generation.

We review our results on Ising machines by spatial light modulation. We report on their performance in simulating spin glasses and solving combinatorial optimization problems. We discuss different annealing strategies and recent developments.

We demonstrate experimentally in biased photorefractive crystals that collisions between random-amplitude optical spatial solitons produce long-tailed statistics from input Gaussian fluctuations. The effect is mediated by Raman nonlocal corrections to Kerr self-focusing that turn soliton–soliton interaction into a Maxwell demon for the output wave amplitude.

Optical neural networks process information at the speed of light and are energetically efficient. Photonic artificial intelligence allows speech recognition, image classification, and Ising machines. Modern machine learning paradigms, as extreme learning machines, reveal that disordered and biological materials may realize optical neural networks with thousands of nodes trained only at the input and at the readout. May we use living matter for machine learning? Here, we employ living three-dimensional tumor brain models to demonstrate a random optical learning machine (ROM) for the investigation of glioblastoma. The tumor spheroid act as a computational reservoir. The ROM detects cancer morphodynamics by laser-induced hyperthermia, quantifies chemotherapy, and cell metabolism. The ROM is a sensitive noninvasive smart probe for cytotoxicity assay and enables real-time investigation of tumor dynamics. We hence design and demonstrate a novel bio-hardware for optical computing and the study of light/complex matter interaction. Can living systems function as artificial neural networks for biophysical applications? Here, the authors show that living tumor spheroids can be employed as random optical learning machines and used to investigate cancer morphodynamics and quantify the effect of chemotherapy.

We study artificial neural networks with nonlinear waves as a computing reservoir. We discuss universality and the conditions to learn a dataset in terms of output channels and nonlinearity. A feed-forward three-layer model, with an encoding input layer, a wave layer, and a decoding readout, behaves as a conventional neural network in approximating mathematical functions, real-world datasets, and universal Boolean gates. The rank of the transmission matrix has a fundamental role in assessing the learning abilities of the wave. For a given set of training points, a threshold nonlinearity for universal interpolation exists. When considering the nonlinear Schroedinger equation, the use of highly nonlinear regimes implies that solitons, rogue, and shock waves do have a leading role in training and computing. Our results may enable the realization of novel machine learning devices by using diverse physical systems, as nonlinear optics, hydrodynamics, polaritonics, and Bose-Einstein condensates. The application of these concepts to photonics opens the way to a large class of accelerators and new computational paradigms. In complex wave systems, as multimodal fibers, integrated optical circuits, random, topological devices, and metasurfaces, nonlinear waves can be employed to perform computation and solve complex combinatorial optimization., 5 pages, 4 figures

Ising machines are novel computing devices for the energy minimization of Ising models. These combinatorial optimization problems are of paramount importance for science and technology, but remain difficult to tackle on large scale by conventional electronics. Recently, various photonics-based Ising machines demonstrated ultra-fast computing of Ising ground state by data processing through multiple temporal or spatial optical channels. Experimental noise acts as a detrimental effect in many of these devices. On the contrary, we here demonstrate that an optimal noise level enhances the performance of spatial-photonic Ising machines on frustrated spin problems. By controlling the error rate at the detection, we introduce a noisy-feedback mechanism in an Ising machine based on spatial light modulation. We investigate the device performance on systems with hundreds of individually-addressable spins with all-to-all couplings and we found an increased success probability at a specific noise level. The optimal noise amplitude depends on graph properties and size, thus indicating an additional tunable parameter helpful in exploring complex energy landscapes and in avoiding trapping into local minima. The result points out noise as a resource for optical computing. This concept, which also holds in different nanophotonic neural networks, may be crucial in developing novel hardware with optics-enabled parallel architecture for large-scale optimizations., Comment: 7 pages, 4 figures

We report the observation of strong nonreciprocal soliton amplification mediated by a Raman-scattering-like effect in single isolated collisions. A pump soliton is found to lose the greater part of its energy to a signal soliton, irrespective of the pump-signal relative amplitude. The result is an efficient rectifying mechanism able to accumulate energy into extreme waves. Experiments are carried out through photorefractive soliton two-wave mixing with a gain coefficient of up to 80 ${\mathrm{cm}}^{\ensuremath{-}1}$ that emerges in conditions of nonlinear response, leading to the formation of rogue waves.

Dispersive shock waves in thermal optical media belong to the third-order nonlinear phenomena, whose intrinsic irreversibility is described by time asymmetric quantum mechanics. Recent studies demonstrated that nonlocal wave breaking evolves in an exponentially decaying dynamics ruled by the reversed harmonic oscillator, namely, the simplest irreversible quantum system in the rigged Hilbert spaces. The generalization of this theory to more complex scenarios is still an open question. In this work, we use a thermal third-order medium with an unprecedented giant Kerr coefficient, the M-Cresol/Nylon mixed solution, to access an extremely-nonlinear highly-nonlocal regime and realize anisotropic shock waves. We prove that a superposition of the Gamow vectors in an ad hoc rigged Hilbert space describes the nonlinear beam propagation beyond the shock point. Specifically, the resulting rigged Hilbert space is a tensorial product between the reversed and the standard harmonic oscillators spaces. The anisotropy turns out from the interaction of trapping and antitrapping potentials in perpendicular directions. Our work opens the way to a complete description of novel intriguing shock phenomena, and those mediated by extreme nonlinearities., 6 pages, 5 figures

Physical systems that evolve according to an Ising Hamiltonian are currently attracting an enormous research interest as novel computing architectures for solving optimization problems that cannot be tackled efficiently on large scales by conventional hardware. In particular, it has been recently demonstrated that photonic platforms such as coherent Ising machines [1,2] and nanophotonic circuits [3] can find the ground state of a classical system of spins with programmable interactions. However, all the proposed optical machines either involve a limited number of spins or fail to properly solve large-scale systems. Although spatial optical waves can present spin-glass dynamics [4] and matrix operations can also be performed by spatially shaped fields [5], the use of spatial optical modulation to solve Ising spin models has remained unexplored. Here, we exploit the spatial degree of freedom of light to realize an optical Ising machine with an unprecedented number of interacting spins [6].

Controlling nonlinear optical processes is a significant challenge in photonics. Shock waves, rogue waves, and solitons are widespread, from optics to hydrodynamics. Intense research is dedicated to advanced techniques for tailoring extreme waves and finding the conditions to induce transitions from one kind of wave to another.

Celebrated as the Fermi-Pasta-Ulam-Tsingou (FPUT) problem, the reappearance of initial conditions in unstable and chaotic systems is one of the most controversial phenomena in nonlinear dynamics. Integrable models predict recurrence as exact solutions [1], but the difficulties involved in upholding integrability for a long dynamics has not allowed a quantitative experimental validation. Evidences of the recurrence of states have been reported from deep water waves [2] to optical fibers [3]. However, the observation of the FPUT dynamics as predicted by exact solutions of an underlying integrable model remains an open challenge. Here, we report the observation of the FPUT recurrence in spatial nonlinear optics and provide evidence that the recurrent behavior is ruled by the exact solution of the Nonlinear Schrodinger Equation [4].

Optical fabrication of waveguides in a volume is limited by diffraction in the writing beams. We demonstrate the use of nondiffracting waves in the form of Bessel beams to fabricate scalable optical wiring through direct writing in a photosensitive volume. Experiments are performed in paraelectric potassium-lithium-tantalate-niobate (KLTN), where writing occurs through photogenerated space charge while guiding and electro-optic functionality are supported by the quadratic electro-optic effect. The method allows components to be integrated sequentially without interfering with each other during fabrication, an intrinsic superposition property that is used to realize single, double, and multiple waveguides, and 1×2, 1×3, and 1×4 splitters, and electrically controlled optical switching.

Optical neuromorphic computing processes information at the speed of light, but requires a careful design and fabrication of the deep layers, which strongly hampers the development of large-scale photonic learning machines [1,2]. New paradigms, as reservoir computing [3], suggest that brain-inspired complex systems such as disordered and biological materials may realize artificial neural networks with thousands of computational nodes trained only at the input and at the readout.

We introduce the topological control, based on correspondences between phases and genus of toroidal surfaces associated with nonlinear Schroedinger equation. We prove it experimentally and report observations of controlled transitions from shock to rogue waves.

We propose and experimentally demonstrate the use of spatial light modulation for calculating the ground state of an Ising Hamiltonian. We realize configurations with thousands of interacting spins that settle in a low-temperature ferromagnetic-like phase.

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.

Recently P.G. Grinevich and P.M. Santini suggested simple approximate formulas for solving periodic Cauchy problem for the focusing Nonlnear Schrodinger Equation (NLS) under assumption that one starts from a small perturbation of the unstable condensate and the number of unstable modes is not too large. With the help of these results, optical experiments in photorefractive crystal where conducted, in which the second and third return of anomalous waves was observed. A good agreement between the experimental data and the predictions made on the basis of the NLS theory was obtained. P.G. Grinevich was supported by the Russian Science Foundation grant No. 18-11-00316.

We investigate Airy-soliton interaction in a nonlinear fiber with Raman effect. We find that Airy solitons may fuse upon interaction at a position that can be controlled by a proper engineering of the Airy tail direction. This control allows us to generate Airy solitons with varying deceleration. At variance with the case of two solitons interaction, Raman-induced soliton self-frequency shift (SSFS) is strongly enhanced when the leading soliton is replaced with the accelerating Airy pulse and slightly suppressed for the decelerating one. These notable features are ascribed to the unique properties of asymmetrical Airy pulses with a switchable direction of the oscillatory tails. We show the way these processes are uncovered unambiguously by cross-correlation frequency resolved optical gating. We also investigate the impact of chirp imposed on the input pulse on the SSFS dynamics. Our results not only provide a new way to manipulate the SSFS, but may help to improve the control of soliton fusion events during supercontinuum generation, optical rogue waves and giant dispersive waves formation. (C) 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

A landmark of statistical mechanics, spin-glass theory describes critical phenomena in disordered systems that range from condensed matter to biophysics and social dynamics. The most fascinating concept is the breaking of replica symmetry: identical copies of the randomly interacting system that manifest completely different dynamics. Replica symmetry breaking has been predicted in nonlinear wave propagation, including Bose-Einstein condensates and optics, but never observed. Here, we report the experimental evidence of replica symmetry breaking in optical wave propagation The findings demonstrate that nonlinear propagation can manifest features typical of spin-glasses and provide a novel platform for testing so-far unexplored fundamental physical theories for complex systems.

We have demonstrated experimentally and numerically that a Bessel-Beam propagating in a strongly self-focusing medium undergoes relevant dynamics along the propagation axis that ultimately cause it to self-trap into a periodic breather.

In principle, materials with a broadband giant index of refraction (n > 10) overcome chromatic aberration and shrink the diffraction limit down to the nanoscale, allowing new opportunities for nanoscopic imaging1. They also open alternative avenues for the management of light to improve the performance of photovoltaic cells2. Recent advances have demonstrated the feasibility of a giant refractive index in metamaterials at microwave and terahertz frequencies3,4, but the highest reported broadband index of refraction in the visible is n 26 across the entire visible spectrum and demonstrate its behaviour using white-light and laser refraction and diffraction experiments. The sample, a solid-solution K0.997Ta0.64Nb0.36:Li0.003 (KTN:Li) perovskite6–12, has a naturally occurring room-temperature phase that propagates visible light along its normal axis without significant diffraction or chromatic dispersion, irrespective of beam size, intensity and angle of incidence. When cooled, the ferroelectric perovskite KTN:Li is found to possess a giant refractive index of n > 26 in the visible region.

We report the observation of replica symmetry breaking in optical wave propagation, a phenomenon that emerges from the interplay of disorder and nonlinearity and demonstrates a spin-glass phase for light.

We predict and observe Bessel-beam optical self-trapping in a strongly self-focusing medium. Nonlinearity leads to solutions with a signature pulsing along the propagation direction, a breather that we observe in a photorefractive potassium-lithium-tantalate-niobate crystal. The nonlinear construct is formed as an interference pattern locks together with a diffracting wave.

We investigate the mutual interaction of two spatially-separated Airy beams in the nonlinear Schrodinger equation with the fractional Laplacian. Depending on the beam separation (d), relative phase and Levy index (α), we observed an anomalous attraction or repulsion between the Airy beams. Anomalous attraction leads to a single breather soliton with a period that grows exponentially as α increases. In this region of the parameter space, we identify a crossover between two asymmetric regimes: as the Levy index exceeds a critical value α c, the period of breather soliton for d>0 is orders of magnitude larger than for d

We investigate the evolution of the state of polarization of light propagating through bulk depoled composite ferroelectrics below the Curie temperature. In contrast to standard depoled ferroelectrics, where random birefringence causes depolarization and scattering, light is observed to suffer varying degrees of depolarization and remains fully polarized when linearly polarized along the crystal principal axes. The effect is found to be supported by the formation of polarized speckles organized into a spatial lattice and occurs as the ferroelectric settles into a spontaneous super-crystal, a three-dimensional coherent mosaic of ferroelectric clusters. The polarization lattices gradually disappear as the ferroelectric state reduces to a disordered distribution of polar nanoregions above the critical point.

We report the direct observation of the onset of turbulence in propagating one-dimensional optical waves. The transition occurs as the disordered hosting material passes from being linear to one with extreme nonlinearity. As the response grows, increased wave interaction causes a modulational unstable quasihomogeneous flow to be superseded by a chaotic and spatially incoherent one. Statistical analysis of high-resolution wave behavior in the turbulent regime unveils the emergence of concomitant rogue waves. The transition, observed in a photorefractive ferroelectric crystal, introduces a new and rich experimental setting for the study of optical wave turbulence and information transport in conditions dominated by large fluctuations and extreme nonlinearity.