Substrate optimization with the adjoint method and layered medium Green’s functions

In recent years, the photonics community has shown increasing interest in the inverse design of photonic components and devices using the adjoint method (AM) due to its efficient gradient computation and suitability for large parameter and continuous design spaces. This work focuses on substrate optimization to maximize light transmission or field enhancement at specific locations using layered medium Green’s functions (LMGFs). We first provide a numerical formulation for calculating two-dimensional (2D) LMGFs, leveraging their efficiency for fixed sources and observation points parallel to layer interfaces. We then present a step-by-step implementation of the AM for substrate optimization using LMGFs. Through numerical studies, we verify the field enhancement achieved with AM-designed substrates using a frequency-domain solver. We compare the results of AM with particle swarm optimization (PSO) for two optimization problems, demonstrating that AM not only generates realistic designs with smooth permittivity profiles but also achieves inverse design more efficiently than PSO. The AM designs are easier to fabricate and require significantly less computational effort due to the efficient gradient computation inherent in the method. This study underscores the advantages of AM in designing photonic devices with continuous parameter spaces.