“…Archi-tectures possessing impressive field-transformation capabilities have already been demonstrated, from free-space (grating) couplers [33,34] to beam steering [35,36] and polarization control [37,38], suggesting that large-scale optimization methods may allow scattering attributes to be tailored to a far greater degree than what has been seen in past intuition-based designs. Simultaneously, ramifying from the core ideas of Lagrange duality and interpreting physical relations as optimization constraints expounded below [39][40][41][42], a string of recent articles on improved bounds for scattering phenomena (including radiative heat transfer [32], absorbed power [43], scattered power [44], and Purcell enhancement [45]) have shown that, in some cases, only modest improvements over standard designs are even hypothetically attainable [44][45][46][47][48][49][50][51][52][53].…”