Interpretable-machine-learning-enabled discretized antireflective multilayer design outperforms traditional continuous graded-index coatings

Designing deep-ultraviolet antireflective coatings remains elusive due to the limited refractive index range of available transparent materials and high sensitivity of performance to nanometer-scale thickness deviations. Here, we revisit the antireflective coating problem as a discrete optimization task, employing an active learning scheme based on second-order factorization machines with binary, quaternary, and octal refractive index bases. These discrete designs were benchmarked against analytical double-/tri-layer and idealized graded-index coatings. The double- and tri-layer coatings with predetermined thicknesses and refractive indices yielded average angular reflectance values of 0.24% and 0.28%, respectively. In contrast, the best quaternary design achieved an exceptionally low reflectance of 0.05% at a total thickness of only 170 nm, outperforming an ideal air-to-substrate graded-index profile at the equivalent thickness (0.07%). To assess fabrication tolerance, we conducted Monte Carlo simulations incorporating layer thickness variations up to ±50%. These simulations revealed that at a 150 nm total thickness, the mean reflectance of the binary design can rise to 5.6%, whereas the octal design remains near 1%, indicating that richer index bases enhance the robustness to fabrication perturbations. We further introduced a phasor-based interpretation to illustrate the justification of second-order factorization machines in navigating the complex design space and identifying near-global optima. These findings highlight the promise of a discrete optimization framework in machine learning to surpass analytical multilayer design rules, enabling fast, interpretable, and fabrication-tolerance inverse design of high-performance multilayer coatings.