Bounds as blueprints: towards optimal and accelerated photonic inverse design

Our ability to structure materials at the nanoscale has, and continues to, enable key advances in optical control. In pursuit of optimal photonic designs, substantial progress has been made on two complementary fronts: bottom-up structural optimizations (inverse design) discover complex high-performing structures but offer no guarantees of optimality; top-down field optimizations (convex relaxations) reveal fundamental performance limits but offer no guarantees that structures meeting the limits exist. We bridge the gap between these two parallel paradigms by introducing a ``verlan'' initialization method that exploits the encoded local and global wave information in duality-based convex relaxations to guide inverse design towards better-performing structures. We illustrate this technique via the challenging problem of Purcell enhancement, maximizing the power extracted from a small emitter in the vicinity of a photonic structure, where ill-conditioning and the presence of competing local maxima lead to sub-optimal designs for adjoint optimization. Structures discovered by our verlan method outperform standard (random) initializations by close to an order of magnitude and approach fundamental performance limits within a factor of two, highlighting the possibility of accessing significant untapped performance improvements.