Brian Edwards

The challenge of scaling digital computing motivates innovation, especially through the evolution of physical systems that mimic neural networks and combinatorial optimization problems. Light is a hyperefficient information carrier, and if efficient interactions with it could be uncovered, then direct information processing would become far more feasible. We harness an ensemble of hundreds of Kerr microresonator solitons and implement an analog feedback network to create an Ising machine with fully programmable all-to-all interactions. By increasing the feedback for self, on-diagonal interactions, each soliton exhibits a universal spin-like bifurcation, and using this palette of interactions, we solve the canonical Boolean satisfiability problem (SAT). The combination of uniform soliton interactions and the compatibility of our Ising machine with high-speed data interconnects enables rapid and precise solutions of complex SAT problems. The well-established theoretical properties of Kerr solitons bound the trade-off of optical power and time use by the machine at ~0.15 milliwatts per soliton and 1 microsecond for a single feedback step. We performed >10,000 trials on more than 100 randomly generated SAT instances to evaluate the Ising machine, demonstrating the potential to exceed the performance of benchmark digital SAT solvers. Our work highlights the convergence of optical nonlinearity, ultralow loss photonics, and optoelectronic circuits for computation-acceleration tasks.

We introduce a mechanism that can both hold and amplify electromagnetic waves by rapidly changing the permittivity of the medium during the wave travel from a positive to a dispersionless (i.e. non-Foster) negative value and then back again. The underlying physics behind this phenomenon is theoretically explored by considering plane wave and Gaussian pulse propagation in an unbounded medium. Interestingly, we show that a rapid positive-to-negative temporal change of ε(t) causes the propagation of the wave to stop (observed by a frozen phase in time) while the amplitude of the frozen field exponentially grows. Stepping the permittivity back to the original (or a new) positive value will cause the wave to thaw and resume propagation with the original (or the new) frequency, respectively. We numerically study the case of dipole radiation in such time-varying non-Foster structures. As a possible implementation, we propose a parallel plate waveguide platform loaded with time-dependent media emulating parallel lumped non-Foster negative capacitors. Such non-Foster time-varying structures may open new venues in controlling and manipulating wave-matter interaction.

The ability to perform mathematical computations using metastructures is an emergent paradigm that carries the potential of wave-based analog computing to the realm of near-speed-of-light, low-loss, compact devices. We theoretically introduce and experimentally verify the concept of a reconfigurable metastructure that performs analog complex mathematical computations using electromagnetic waves. Reconfigurable, RF-based components endow our device with the ability to perform stationary and non-stationary iterative algorithms. After demonstrating matrix inversion (stationary problem), we use the machine to tackle two major non-stationary problems: root finding with Newton’s method and inverse design (constrained optimization) via the Lagrange multiplier method. The platform enables possible avenues for wave-based, analog computations for general linear algebraic problems and beyond in compact, ultrafast, and parallelized ways.

Structured optical materials create new computing paradigms using photons, with transformative impact on various fields, including machine learning, computer vision, imaging, telecommunications, and sensing. This Perspective sheds light on the potential of free-space optical systems based on engineered surfaces for advancing optical computing. Manipulating light in unprecedented ways, emerging structured surfaces enable all-optical implementation of various mathematical functions and machine learning tasks. Diffractive networks, in particular, bring deep-learning principles into the design and operation of free-space optical systems to create new functionalities. Metasurfaces consisting of deeply subwavelength units are achieving exotic optical responses that provide independent control over different properties of light and can bring major advances in computational throughput and data-transfer bandwidth of free-space optical processors. Unlike integrated photonics-based optoelectronic systems that demand preprocessed inputs, free-space optical processors have direct access to all the optical degrees of freedom that carry information about an input scene/object without needing digital recovery or preprocessing of information. To realize the full potential of free-space optical computing architectures, diffractive surfaces and metasurfaces need to advance symbiotically and co-evolve in their designs, 3D fabrication/integration, cascadability, and computing accuracy to serve the needs of next-generation machine vision, computational imaging, mathematical computing, and telecommunication technologies.

Performing analog computations with metastructures is an emerging wave-based paradigm for solving mathematical problems. For such devices, one major challenge is their reconfigurability, especially without the need for a priori mathematical computations or computationally-intensive optimization. Their equation-solving capabilities are applied only to matrices with special spectral (eigenvalue) distribution. Here we report the theory and design of wave-based metastructures using tunable elements capable of solving integral/differential equations in a fully-reconfigurable fashion. We consider two architectures: the Miller architecture, which requires the singular-value decomposition, and an alternative intuitive direct-complex-matrix (DCM) architecture introduced here, which does not require a priori mathematical decomposition. As examples, we demonstrate, using system-level simulation tools, the solutions of integral and differential equations. We then expand the matrix inverting capabilities of both architectures toward evaluating the generalized Moore–Penrose matrix inversion. Therefore, we provide evidence that metadevices can implement generalized matrix inversions and act as the basis for the gradient descent method for solutions to a wide variety of problems. Finally, a general upper bound of the solution convergence time reveals the rich potential that such metadevices can offer for stationary iterative schemes.

In the search for improved computational capabilities, conventional microelectronic computers are facing various problems arising from the miniaturization and concentration of active electronics. Therefore, researchers have explored wave systems, such as photonic or quantum devices, for solving mathematical problems at higher speeds and larger capacities. However, previous devices have not fully exploited the linearity of the wave equation, which as we show here, allows for the simultaneous parallel solution of several independent mathematical problems within the same device. Here we demonstrate that a transmissive cavity filled with a judiciously tailored dielectric distribution and embedded in a multi-frequency feedback loop can calculate the solutions of a number of mathematical problems simultaneously. We design, build, and test a computing structure at microwave frequencies that solves two independent integral equations with any two arbitrary inputs and also provide numerical results for the calculation of the inverse of four 5 x 5 matrices.

The process of diffusion is central to the ever increasing entropic state of the universe and is fundamental in many branches of science and engineering. Although non-reciprocal metamaterials are well developed for wave systems, the studies of diffusive metamaterials have been limited by their characteristic spatial inversion symmetry and time inversion antisymmetry. Here, we achieve large spatial asymmetric diffusion characteristics inside a metamaterial whose material parameters are space- and time-modulated. Inside such a spatiotemporal metamaterial, diffusion occurs as if the material had an intrinsic flow velocity, whose direction is dictated by the relative phase between the modulations of the conductivity and capacity. This creates dramatic out-of-equilibrium concentrations and depletions, which we demonstrate experimentally for the diffusion of electric charges in a one-dimensional electrical system composed of an array of space-time-modulated variable capacitors and switches. These results may offer exciting possibilities in various fields, including electronics, thermal management, chemical mixing, etc.

We present a metasurface-based platform that solves Fredholm integral equations of the second kind for free-space radiation. An inverse-designed metagrating is coupled to a semitransparent mirror providing feedback in order to perform an analog version of the Neumann series.

Solving equations with waves: Signal processing of light waves can be used to represent certain mathematical functions and to perform computational tasks on signals or images in an analog fashion. Such processing typically requires complex systems of bulk optical elements such as lenses, filters, and mirrors. Mohammadi Estakhri et al. demonstrate that specially designed nanophotonic structures can take input waveforms encoded as complex mathematical functions, manipulate them, and provide an output that is the integral of the functions. The results, demonstrated for microwaves, provide a route to develop chip-based analog optical computers and computing elements. Science , this issue p. 1333