Schrödinger's Red Beyond 65,000 Pixel-Per-Inch by Multipolar Interaction in Freeform Meta-Atom Through Efficient Neural Optimizer

Freeform nanostructures have the potential to support complex resonances and their interactions, which are crucial for achieving desired spectral responses. However, the design optimization of such structures is nontrivial and computationally intensive. Furthermore, the current "black box" design approaches for freeform nanostructures often neglect the underlying physics. Here, a hybrid data-efficient neural optimizer for resonant nanostructures by combining a reinforcement learning algorithm and Powell's local optimization technique is presented. As a case study, silicon nanostructures with a highly-saturated red color are designed and experimentally demonstrated. Specifically, color coordinates of (0.677, 0.304) in the International Commission on Illumination (CIE) chromaticity diagram - close to the ideal Schrödinger's red, with polarization independence, high reflectance (>85%), and a large viewing angle (i.e., up to ± 25°) is achieved. The remarkable performance is attributed to underlying generalized multipolar interferences within each nanostructure rather than the collective array effects. Based on that, pixel size down to ≈400 nm, corresponding to a printing resolution of 65000 pixels per inch is demonstrated. Moreover, the proposed design model requires only ≈300 iterations to effectively search a thirteen-dimensional (13D) design space - an order of magnitude more efficient than the previously reported approaches. The work significantly extends the free-form optical design toolbox for high-performance flat-optical components and metadevices.