Wide-angle deep ultraviolet antireflective multilayers via discrete-to-continuous optimization

To date, various optimization algorithms have been used to design non-intuitive photonic structures with unconventional optical performance. Good training datasets facilitate the optimization process, particularly when an objective function has a non-convex shape containing multiple local optima in a continuous parametric space. Herein, we developed a discrete-to-continuous optimization algorithm and confirmed its validity by designing and fabricating deep-ultraviolet antireflective MgF₂/LaF₃ multilayers. For discrete optimization, a multilayer was encoded into a binary vector with multiple bits; a 10 nm thick MgF₂ or LaF₃ layer was assigned a binary digit of 0 or 1, respectively. Using the binary-based training datasets, a factorization machine formulated a surrogate function, which discovered the ground binary vector representing a near-optimal figure of merit. Then, the figure of merit was refined through continuous optimization (e.g., using an interior-point method) of the ground binary vector. MgF₂/LaF₃ multilayers with a variety of bit levels were created to attain a minimum average angular (0°-45°) reflectance at 193 nm. A MgF₂/LaF₃ multilayer optimized at ten bits (i.e., a total thickness of approximately 100 nm) yielded an average reflectance of 0.2%, which agreed well with the experimental results. Moreover, an integrated ray-wave optics simulation predicted that a single CaF₂ plano-convex lens coated with the optimized multilayer could exhibit a transmittance of 99.7%. The developed optimization approach will be widely applicable to any photonic structures that can represent a binary vector with multiple bits, such as microwave metasurfaces, in addition to being useful for producing ideal optical multilayers.