Optical analog signal processing has been gaining significant attention as a way to overcome speed and energy limitations of digital techniques. Metasurfaces offer a promising avenue towards this goal, due to their efficient manipulation of optical signals over deeply subwavelength volumes. To date, metasurfaces have been proposed to transform signals in the spatial domain, e.g., for beam steering, focusing or holography, for which angulardependent responses, or nonlocality, are unwanted features that must be avoided or mitigated. Here, we show that the metasurface nonlocality can be engineered to enable signal manipulation in the momentum domain over an ultrathin platform. We explore nonlocal metasurfaces performing basic mathematical operations, paving the way towards fast and power efficient ultrathin devices for edge detection and optical image processing.

Image processing and edge detection are at the core of several newly emerging technologies, such as augmented reality, autonomous driving, and more generally object recognition. Image processing is typically performed digitally using integrated electronic circuits and algorithms, implying fundamental size and speed limitations, as well as significant power needs. On the other hand, it can also be performed in a low-power analog fashion using Fourier optics, requiring, however, bulky optical components. Here, we introduce dielectric metasurfaces that perform optical image edge detection in the analog domain using a subwavelength geometry that can be readily integrated with detectors. The metasurface is composed of a suitably engineered array of nanobeams designed to perform either first- or second-order spatial differentiation. We experimentally demonstrate the second-derivative operation on an input image, showing the potential of all-optical edge detection using a silicon metasurface geometry working at a numerical aperture as large as 0.35.

Optical analog computing using metasurfaces has been the subject of numerous studies, aimed at implementing highly efficient and ultrafast image processing in a compact device. The proposed approaches to date have shown limitations in terms of spatial resolution, overall efficiency, polarization and azimuthal angular dependence. Here, we present the design of a polarizationinsensitive metasurface with tailored nonlocality based on a Fano resonant response, enabling both odd-and even-order analog mathematical operations on an incoming image. The metasurface is formed by a single-layered triangular lattice of holes in a suspended silicon membrane, which induces a strong nonlocal response in the transverse spatial frequency spectrum. Our paper provides a path to realize highly efficient optical metasurfaces performing isotropic and polarizationinsensitive edge detection on an arbitrary 2D optical image.