We present a computational framework for efficient optimization-based "inverse design" of large-area "metasurfaces" (subwavelength-patterned surfaces) for applications such as multiwavelength/multi-angle optimizations, and demultiplexers. To optimize surfaces that can be thousands of wavelengths in diameter, with thousands (or millions) of parameters, the key is a fast approximate solver for the scattered field. We employ a "locally periodic" approximation in which the scattering problem is approximated by a composition of periodic scattering problems from each unit cell of the surface, and validate it against brute-force Maxwell solutions. This is an extension of ideas in previous metasurface designs, but with greatly increased flexibility, e.g. to automatically balance tradeoffs between multiple frequencies or to optimize a photonic device given only partial information about the desired field. Our approach even extends beyond the metasurface regime to non-subwavelength structures where additional diffracted orders must be included (but the period is not large enough to apply scalar diffraction theory).

We demonstrate optimization of optical metasurfaces over 10 5 -10 6 degrees of freedom in two and three dimensions, 100-1000+ wavelengths (λ) in diameter, with 100+ parameters per λ 2 . In particular, we show how topology optimization, with one degree of freedom per high-resolution "pixel," can be extended to large areas with the help of a locally periodic approximation that was previously only used for a few parameters per λ 2 . In this way, we can computationally discover completely unexpected metasurface designs for challenging multi-frequency, multi-angle problems, including designs for fully coupled multi-layer structures with arbitrary per-layer patterns. Unlike typical metasurface designs based on subwavelength unit cells, our approach can discover both sub-and suprawavelength patterns and can obtain both the near and far fields.

Meta-optics has achieved major breakthroughs in the past decade; however, conventional forward design faces challenges as functionality complexity and device size scale up. Inverse design aims at optimizing meta-optics design but has been currently limited by expensive brute-force numerical solvers to small devices, which are also difficult to realize experimentally. Here, we present a general inverse-design framework for aperiodic large-scale (20k × 20k λ2) complex meta-optics in three dimensions, which alleviates computational cost for both simulation and optimization via a fast approximate solver and an adjoint method, respectively. Our framework naturally accounts for fabrication constraints via a surrogate model. In experiments, we demonstrate aberration-corrected metalenses working in the visible with high numerical aperture, poly-chromatic focusing, and large diameter up to the centimeter scale. Such large-scale meta-optics opens a new paradigm for applications, and we demonstrate its potential for future virtual-reality platforms by using a meta-eyepiece and a laser back-illuminated micro-Liquid Crystal Display.

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . Inverse design arises in a variety of areas in engineering such as acoustic, mechanics, thermal/electronic transport, electromagnetism, and optics. Topology optimization is an important form of inverse design, where one optimizes a designed geometry to achieve targeted properties parameterized by the materials at every point in a design region. This optimization is challenging, because it has a very high dimensionality and is usually constrained by partial differential equations (PDEs) and additional inequalities. Here, we propose a new deep learning method---physics-informed neural networks with hard constraints (hPINNs)---for solving topology optimization. hPINN leverages the recent development of PINNs for solving PDEs, and thus does not require a large dataset (generated by numerical PDE solvers) for training. However, all the constraints in PINNs are soft constraints, and hence we impose hard constraints by using the penalty method and the augmented Lagrangian method. We demonstrate the effectiveness of hPINN for a holography problem in optics and a fluid problem of Stokes flow. We achieve the same objective as conventional PDE-constrained optimization methods based on adjoint methods and numerical PDE solvers, but find that the design obtained from hPINN is often smoother for problems whose solution is not unique. Moreover, the implementation of inverse design with hPINN can be easier than that of conventional methods because it exploits the extensive deep-learning software infrastructure.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . inverse design, topology optimization, partial differential equations, physicsinformed neural networks, penalty method, augmented Lagrangian method \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 35R30, 65K10, 68T20 \bfD \bfO \bfI .

Extended depth of focus (EDOF) lenses are important for various applications in computational imaging and microscopy. In addition to enabling novel functionalities, EDOF lenses can alleviate the need for stringent alignment requirements for imaging systems. Existing EDOF lenses, however, are often inefficient or produce an asymmetric point spread function (PSF) that blurs images. Inverse design of nanophotonics, including metasurfaces, has generated strong interest in recent years owing to its potential for generating exotic and innovative optical elements, which are generally difficult to model intuitively. Using adjoint optimization-based inverse electromagnetic design, in this paper, we designed a cylindrical metasurface lens operating at ~ 625nm with a depth of focus exceeding that of an ordinary lens. We validated our design by nanofabrication and optical characterization of silicon nitride metasurface lenses (with lateral dimension of 66.66 ) with three different focal lengths (66.66 , 100 , 133.33 ). The focusing efficiencies of the fabricated extended depth of focus metasurface lenses are similar to those of traditional metalenses. Main Text