The development of inverse design, where computational optimization techniques are used to design devices based on certain specifications, has led to the discovery of many compact, nonintuitive structures with superior performance. Among various methods, large-scale, gradient-based optimization techniques have been one of the most important ways to design a structure containing a vast number of degrees of freedom. These techniques are made possible by the adjoint method, in which the gradient of an objective function with respect to all design degrees of freedom can be computed using only two full-field simulations. However, this approach has so far mostly been applied to linear photonic devices. Here, we present an extension of this method to modeling nonlinear devices in the frequency domain, with the nonlinear response directly included in the gradient computation. As illustrations, we use the method to devise compact photonic switches in a Kerr nonlinear material, in which low-power and high-power pulses are routed in different directions. Our technique may lead to the development of novel compact nonlinear photonic devices.

Analog machine learning hardware platforms promise to be faster and more energy-efficient than their digital counterparts. Wave physics, as found in acoustics and optics, is a natural candidate for building analog processors for time-varying signals. Here we identify a mapping between the dynamics of wave physics, and the computation in recurrent neural networks. This mapping indicates that physical wave systems can be trained to learn complex features in temporal data, using standard training techniques for neural networks. As a demonstration, we show that an inverse-designed inhomogeneous medium can perform vowel classification on raw audio signals as their waveforms scatter and propagate through it, achieving performance comparable to a standard digital implementation of a recurrent neural network. These findings pave the way for a new class of analog machine learning platforms, capable of fast and efficient processing of information in its native domain.

We introduce an electro-optic hardware platform for nonlinear activation functions in optical neural networks. The optical-to-optical nonlinearity operates by converting a small portion of the input optical signal into an analog electric signal, which is used to intensity-modulate the original optical signal with no reduction in processing speed. Our scheme allows for complete nonlinear on-off contrast in transmission at relatively low optical power thresholds and eliminates the requirement of having additional optical sources between each layer of the network. Moreover, the activation function is reconfigurable via electrical bias, allowing it to be programmed or trained to synthesize a variety of nonlinear responses. Using numerical simulations, we demonstrate that this activation function significantly improves the expressiveness of optical neural networks, allowing them to perform well on two benchmark machine learning tasks: learning a multi-input exclusive-OR (XOR) logic function and classification of images of handwritten numbers from the MNIST dataset. The addition of the nonlinear activation function improves test accuracy on the MNIST task from 85% to 94%.

Metamaterials with an effective zero refractive index associated to their electromagnetic response are sought for a number of applications in communications and nonlinear optics. A promising way that this can be achieved in all-dielectric photonic crystals is through the design of a Dirac cone at zero Bloch wave-vector in the photonic band structure. In the optical frequency range, the natural way to implement this design is through the use of a photonic crystal slab. In the existing implementation, however, the zero-index photonic modes also radiate strongly into the environment due to intrinsic symmetry properties. This has resulted in large losses in recent experimental realizations of this zero-index paradigm. Here, we propose a photonic crystal slab with zero-index modes which are also symmetry-protected bound states in the continuum. Our approach thus eliminates the associated radiation loss. This could enable, for the first time, large-scale integration of zero-index materials in photonic devices.

Gradient-based inverse design in photonics has already achieved remarkable results in designing small-footprint, high-performance optical devices. The adjoint variable method, which allows for the efficient computation of gradients, has played a major role in this success. However, gradient-based optimization has not yet been applied to the mode-expansion methods that are the most common approaches to studying periodic optical structures such as photonic crystals. This is because, in such simulations, the adjoint variable method cannot be defined as explicitly as in standard finite-difference or finite-element time-or frequency-domain methods. Here, we overcome this gap through the use of automatic differentiation, which is a generalization of the adjoint variable method to arbitrary computational graphs. We implement the planewave expansion and the guided-mode expansion methods using an automatic differentiation library, and we show that the gradient of any simulation output can be computed efficiently and in parallel, with respect to all input parameters. We then use this implementation to optimize the dispersion of a photonic crystal waveguide, and the quality factor of an ultrasmall cavity in a lithium niobate slab. This extends photonic inverse design to an entirely new class of simulations, and more broadly highlights the importance that automatic differentiation could play in the future for tracking and optimizing complicated physical models.