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We experimentally demonstrate multilayer neural networks using ultrafast nonlinear optics, to perform audio and image classification. The proposed framework for constructing and training neural networks is general and applicable to other complex non-linear systems.

Photonic simulators using synthetic frequency dimensions have enabled flexible experimental analogues of condensed-matter systems, realizing phenomena that are impractical to observe in real-space systems. However, to date such photonic simulators have been limited to small systems suffering from finite-size effects. Here, we present an analog simulator capable of simulating large 2D and 3D lattices, as well as lattices with non-planar connectivity, including a tree lattice that serves as a toy model in quantum gravity. Our demonstration is enabled by the broad bandwidth achievable in photonics, allowing our simulator to realize lattices with over 100,000 lattice sites. We explore these large lattices in a wide range of previously inaccessible regimes by using a novel method to excite arbitrary states. Our work establishes the scalability and flexibility of programmable simulators based on synthetic frequency dimensions in the optical domain. We anticipate that future extensions of this platform will leverage advances in high-bandwidth optoelectronics to support simulations of dynamic, non-equilibrium phases at the scale of millions of lattice sites, and Kerr-frequency-comb technology to simulate models with higher-order interactions, ultimately in regimes and at scales inaccessible to both digital computers and realizable materials.

Here, we include the data and code needed to reproduce the figures in "Programmable large-scale simulation of lattices with photonic synthetic frequency dimensions". The data and codes include all that's needed to reproduce the figures in the main-text as well as the supplementary. The code and data is organized such that all of the data and code to reproduce each figure are in a stand-alone folder. In each folder, code for analysis and plotting is found along with data. Additionally, there is a "lib" folder that includes auxillary data processing code, and a "bin" folder, that contains a repository for all simulations. Some of the calculations require specific versions of Python packages. These can be found in the requirements.txt file.

We realize efficient full-quantum simulations of pulse propagation in highly nonlinear waveguides using matrix product states. As a demonstration, we study the quan-tum dynamics of an optical soliton, highlighting the emergence of non-Gaussian quantum features.

We demonstrate a multilayer optical-neural-network image encoder realized using optoelectronic, optical-to-optical nonlinear activations. We show that nonlinear preprocessing in the optical domain can enable machine-vision systems that operate faster and more sensitively.

We have shown that nonlinear pulse propagation can be designed to directly, all-optically implement machine learning calculations. Building from proof-of-concept experiments, we analyze opportunities for ultrafast, ultra-efficient nonlinear optical neuromorphic systems and smart sensors.

New Frontiers in Optical Technology The proliferation of optical sensors and user interfaces offers opportunities for deeper integration of sensing, processing, and data transmission.

The overall goal of photonics research is to understand and control light in new and richer ways to facilitate new and richer applications. Many major developments to this end have relied on nonlinear optical techniques, such as lasing, mode-locking, and parametric downconversion, to enable applications based on the interactions of coherent light with matter. These processes often involve nonlinear interactions between photonic and material degrees of freedom spanning multiple spatiotemporal scales. While great progress has been made with relatively simple optimizations, such as maximizing single-mode coherence or peak intensity alone, the ultimate achievement of coherent light engineering is complete, multidimensional control of light–light and light–matter interactions through tailored construction of complex optical fields and systems that exploit all of light’s degrees of freedom. This capability is now within sight, due to advances in telecommunications, computing, algorithms, and modeling. Control of highly multimode optical fields and processes also facilitates quantitative and qualitative advances in optical imaging, sensing, communication, and information processing since these applications directly depend on our ability to detect, encode, and manipulate information in as many optical degrees of freedom as possible. Today, these applications are increasingly being enhanced or enabled by both multimode engineering and nonlinearity. Here, we provide a brief overview of multimode nonlinear photonics, focusing primarily on spatiotemporal nonlinear wave propagation and, in particular, on promising future directions and routes to applications. We conclude with an overview of emerging processes and methodologies that will enable complex, coherent nonlinear photonic devices with many degrees of freedom.

We report an experimental demonstration of an optical neural network performing image classification with high accuracy using less than a single photon per scalar multiplication, validating theoretical predictions about quantum-limited performance of ONNs.

We experimentally demonstrate an optical neural network whose fundamental building blocks are controllable ultrafast quadratic nonlinear pulse-propagation processes. The network is trained in situ by a new backpropagation algorithm that generalizes to arbitrary physical systems.

We experimentally demonstrate deep nonlinear optical neural networks using a universal algorithm for backpropagating through arbitrary physical input-output transformations. Ultrafast second harmonic generation and other diverse processes are trained to perform image and audio classification.

Ultra-short pulses propagating in nonlinear nanophotonic waveguides can simultaneously leverage both temporal and spatial field confinement, promising a route towards single-photon nonlinearities in an all-photonic platform. In this multimode quantum regime, however, faithful numerical simulations of pulse dynamics na\"ively require a representation of the state in an exponentially large Hilbert space. Here, we employ a time-domain, matrix product state (MPS) representation to enable efficient simulations by exploiting the entanglement structure of the system. In order to extract physical insight from these simulations, we develop an algorithm to unravel the MPS quantum state into constituent temporal supermodes, enabling, e.g., access to the phase-space portraits of arbitrary pulse waveforms. As a demonstration, we perform exact numerical simulations of a Kerr soliton in the quantum regime. We observe the development of non-classical Wigner-function negativity in the solitonic mode as well as quantum corrections to the semiclassical dynamics of the pulse. A similar analysis of $\chi^{(2)}$ simultons reveals a unique entanglement structure between the fundamental and second harmonic. Our approach is also readily compatible with quantum trajectory theory, allowing full quantum treatment of propagation loss and decoherence. We expect this work to establish the MPS technique as part of a unified engineering framework for the emerging field of broadband quantum photonics., Comment: 12 pages, 7 figures

We study the emergence of non-Gaussian quantum features in pulsed squeezed light generation with a mesoscopic number (i.e., dozens to hundreds) of pump photons. Due to the strong optical nonlinearities necessarily involved in this regime, squeezing occurs alongside significant pump depletion, compromising the predictions made by conventional semiclassical models for squeezing. Furthermore, nonlinear interactions among multiple frequency modes render the system dynamics exponentially intractable in na\"ive quantum models, requiring a more sophisticated modeling framework. To this end, we construct a nonlinear Gaussian approximation to the squeezing dynamics, defining a "Gaussian interaction frame" (GIF) in which non-Gaussian quantum dynamics can be isolated and concisely described using a few dominant (i.e., principal) supermodes. Numerical simulations of our model reveal non-Gaussian distortions of squeezing in the mesoscopic regime, largely associated with signal-pump entanglement. We argue that the state of the art in nonlinear nanophotonics is quickly approaching this regime, providing an all-optical platform for experimental studies of the semiclassical-to-quantum transition in a rich paradigm of coherent, multimode nonlinear dynamics. Mesoscopic pulsed squeezing thus provides an intriguing case study of the rapid rise in dynamic complexity associated with semiclassical-to-quantum crossover, which we view as a correlate of the emergence of new information-processing capacities in the quantum regime., Comment: The first two authors contributed equally to this work; 16 pages, 7 figures

We propose a measurement-free construction of a cubic phase gate based on a Kerr nonlinearity and Gaussian transformations. Experimental feasibility is discussed for pulsed nanophotonic waveguides where quantum states are encoded into quantum solitons.

We uncover the mechanisms behind a range of new 3D solitons in multimode lasers using an approach called attractor dissection. Experimental measurements allow us to decompose multimode states containing up to 30 million locked modes.

Building on the scientific understanding and technological infrastructure of single-mode fibers, multimode fibers are being explored as a means of adding new degrees of freedom to optical technologies such as telecommunications, fiber lasers, imaging, and measurement. Here, starting from a baseline of single-mode nonlinear fiber optics, we introduce the growing topic of multimode nonlinear fiber optics. We demonstrate a new numerical solution method for the system of equations that describes nonlinear multimode propagation, the generalized multimode nonlinear Schrodinger equation. This numerical solver is freely available, and includes a number of multimode fiber analysis tools. It features a significant parallel computing speed-up on modern graphical processing units, translating to orders-of-magnitude speed-up over the split-step Fourier method. We demonstrate its use with several examples in graded- and step-index multimode fibers. Finally, we discuss several key open directions and questions, whose answers could have significant scientific and technological impact., Comment: https://github.com/WiseLabAEP/GMMNLSE-Solver-FINAL

Unexpected multimode solitary waves can be formed spontaneously in hollow-core fibres, hinting at a vast world of exciting nonlinear optics, with applications for generating few-cycle, ultra-intense pulses.

We outline a theoretical framework to understand the multitude of new mode-locked states possible in multi-transverse mode resonators. Full-3D measurements of mode-locked states comprising roughly 30 million modes agree with theoretical expectations.

Mode-locking is a process in which different modes of an optical resonator establish, through nonlinear interactions, stable synchronization. This self-organization underlies light sources that enable many modern scientific applications, such as ultrafast and high-field optics and frequency combs. Despite this, mode-locking has almost exclusively referred to self-organization of light in a single dimension - time. Here we present a theoretical approach, attractor dissection, for understanding three-dimensional (3D) spatiotemporal mode-locking (STML). The key idea is to find, for each distinct type of 3D pulse, a specific, minimal reduced model, and thus to identify the important intracavity effects responsible for its formation and stability. An intuition for the results follows from the 'minimum loss principle,' the idea that a laser strives to find the configuration of intracavity light that minimizes loss (maximizes gain extraction). Through this approach, we identify and explain several distinct forms of STML. These novel phases of coherent laser light have no analogues in 1D and are supported by experimental measurements of the three-dimensional field, revealing STML states comprising more than $10^7$ cavity modes. Our results should facilitate the discovery and understanding of new higher-dimensional forms of coherent light which, in turn, may enable new applications.

We demonstrate a fiber source with the best performance from an ultrafast fiber oscillator to date. The ring-cavity Mamyshev oscillator produces 50-nJ and 40-fs pulses. The peak power is an order of magnitude higher than that of previous lasers with similar fiber mode area. This performance is achieved by designing the oscillator to support parabolic pulse formation which enables the management of unprecedented nonlinear phase shifts. Experimental results are limited by available pump power. Numerical simulations reveal key aspects of the pulse evolution, and realistically suggest that (after external compression) peak powers that approach 10 MW are possible from ordinary single-mode fiber. The combination of practical features such as environmental stability, established previously, with the performance described here make the Mamyshev oscillator extremely attractive for applications., Comment: 7 pages, 4 figures

Ultrafast fiber lasers could provide a compact, robust and cheaper alternative to solid-state sources, such as Ti:sapphire oscillators. If fiber sources can provide equal performance at a fraction of the cost, then many applications of ultrafast pulses and frequency combs will be poised for explosive adaption beyond research laboratories. For the past decade, a major impediment to this has been the reliance on nonlinear polarization rotation (NPR) to mode-lock the lasers. Without NPR, performance is usually insufficient. With NPR, the laser is environmentally unstable. In the past year, remarkable developments have been made based on the use of the Mamyshev regenerator, which simply combines the power-dependent spectral broadening of self-phase modulation with offset spectral filtering to achieve power-dependent transmission. A gain fiber is used to compensate for filtering losses. By arranging two Mamyshev regenerators with opposing spectral filter offsets in a loop, we created a Mamyshev oscillator. Polarization maintaining fibers are used for environmental stability. The power-dependent transmission means that continuous wave lasing is suppressed, and only pulsed operation is possible. This simple but significant change means that incredibly large nonlinear phase shifts can be tolerated. We leverage this large nonlinearity to create spectral bandwidth in a parabolic pulse. We find that the Mamyshev oscillator provides an order-of-magnitude better performance – 1 MW from standard single-mode fiber - than any previous design. Simulations suggest that still greater performance is achievable, and experiments in our laboratory show promise to 10 MW and beyond. Finally, self-starting Mamyshev oscillator results will be discussed.

Multimode fibers provide a means of scaling the peak power of ultrafast fiber lasers by orders of magnitude. While large mode area (LMA) fibers have been widely utilized in fiber amplifiers, these fibers often sacrifice practical benefits of fiber, such as flexibility, and increase system cost and complexity. In addition, the mode-field area of effectively single-mode LMA fibers is smaller than what can be achieved in multimode fibers. Recent work has shown that nonlinear interactions in multimode graded-index fiber can cause a highly multimode field to self-organize into the fundamental mode in a condensation-like process that is robust even with fiber perturbations or disorder. Building on this, we developed a series of Yb:fiber, mode-locked lasers utilizing normal-dispersion, multimode graded-index fiber. In experiments, we observe that the transition from continuous wave lasing to mode-locking is characterized by a beam cleaning process, whereby the highly multimode (speckled) beam of the continuous wave field transforms into a low-order mode beam. Remarkably, experiments and numerical simulations show that the pulses can consist not just the fundamental mode, but can even comprise multiple transverse modes. Our theoretical analysis shows this to be a consequence of a surprising kind of mode-locking – spatiotemporal mode-locking - which relies on strong intermode interactions and spatial filtering. Our initial experiments yield MW-power pulses after external compression, rivaling the best results with flexible LMA fibers. Meanwhile, simulations show that nearly GW peak powers should be possible, making spatiotemporal mode-locking extremely attractive for high-power ultrafast laser development.

Mode-locked oscillators are fantastic scientific instruments. Today, femtosecond pulses are now in demand for applications outside physics and chemistry laboratories, most notably for precision materials processing and nonlinear optical microscopy in biomedical laboratories and clinics. These applications require short pulses with high peak power and excellent beam quality. However, they also require devices that are cheap, reliable, and – crucially -- flexible. For example, synchronization of pulses with scanning optics is crucial for high efficiency machining and imaging alike. Lasers in materials processing are often required to exhibit pulse-on-demand operation in order to deposit energy only in specific locations. In biology, such behavior would enable region-of-interest monitoring, allowing e.g. ultrahigh resolution monitoring of a neuronal circuit’s dynamics. This kind of behavior is not compatible with the regular pulse train of a mode-locked oscillator. To this end, we seed an ultrafast fiber amplifier with a gain-switched diode. This a robust, integrated device that can be electronically triggered at virtually any repetition rate, or on-demand, yielding partially-coherent, ~10 ps duration pulses. Remarkably, a single Mamyshev regenerator (nonlinear spectral broadening in fiber followed by offset spectral filtering) allows isolation of a coherent component of the diode pulse. After subsequent nonlinear pulse shaping, amplification to the µJ-regime, and linear compression, we obtain transform-limited ~140-fs pulses with 13 MW peak power. As will be discussed, such a source cannot replace all mode-locked oscillators (e.g., for frequency comb applications), but is extremely attractive for the key mainstream applications of ultrashort pulses.

We find adiabatic four-wave mixing in optical fibers allows efficient, near-octave-spanning near-infrared to mid-infrared conversion. Simulations indicate several possible fiber platforms, extending one-to-one broadband frequency conversion both to high-repetition-rate and high-energy applications.

Using spatial light modulators, we demonstrate control of modal excitation and mode-resolved measurement of nonlinear pulse propagation in multimode fiber, and present a representative experiment in which we observe discrete Raman beam clean-up.

We demonstrate an environmentally stable, self-seeded fiber oscillator based on spectral reshaping and reamplification. The oscillator delivers 140 nJ, linearly chirped pulses with 110 nm spectral bandwidth that can be compressed externally to 65 fs.

We describe, in general terms, what spatiotemporal mode-locking is and how it happens. Then we describe several recent new developments, including qualitatively distinct kinds of 3D pulses and mode-locking physics.

We demonstrate a fiber oscillator that achieves 3 MW peak power, is easily started and is environmentally stable. The Mamyshev oscillator delivers 190-nJ pulses that can be compressed externally to 35 fs duration. Accurate numerical modeling of the gain medium provides insight into the behavior and performance of the device.

We demonstrate use of spatial light modulators to control modal excitation and make mode-resolved measurements of nonlinear pulse propagation in multimode fiber, and present a representative experiment wherein we observe discrete Raman beam clean-up.

We describe, through multiple theoretical and experimental realizations, spatiotemporal mode-locking (STML), the most general form of self-organization in optical oscillators. We discuss qualitatively new kinds of mode-locking, and routes to ultrahigh power, compact lasers.

Highly nonlinear effects are observed in graded-index multimode optical fibres. Multimode fibres are of interest for next-generation telecommunications systems and the construction of high-energy fibre lasers. However, relatively little work has explored nonlinear pulse propagation in multimode fibres. Here, we consider highly nonlinear ultrashort pulse propagation in the anomalous-dispersion regime of a graded-index multimode fibre. Low modal dispersion and strong nonlinear coupling between the fibre's many spatial modes result in interesting behaviour. We observe spatiotemporal effects reminiscent of nonlinear optics in bulk media—self-focusing and multiple filamentation1,2—at a fraction of the usual power. By adjusting the spatial initial conditions, we generate on-demand, megawatt, ultrashort pulses tunable between 1,550 and 2,200 nm; dispersive waves over one octave; intense combs of visible light; and a multi-octave-spanning supercontinuum. Our results indicate that multimode fibres present unique opportunities for observing new spatiotemporal dynamics and phenomena. They also enable the realization of a new type of tunable, broadband fibre source that could be useful for many applications.

We demonstrate a fiber system which amplifies and compresses pulses from a gain-switched diode. A Mamyshev regenerator shortens the pulses and improves their coherence, enabling subsequent amplification by parabolic pre-shaping. As a result, we are able to control nonlinear effects and generate nearly transform-limited, 140-fs pulses with 13-MW peak power---an order-of-magnitude improvement over previous gain-switched diode sources. Seeding with a gain-switched diode results in random fluctuations of 2% in the pulse energy, which future work using known techniques may ameliorate. Further development may allow such systems to compete directly with sources based on modelocked oscillators in some applications while enjoying unparalleled robustness and repetition rate control., 10 pages, 8 figures