Hierarchical mean-field T operator bounds on electromagnetic scattering: Upper bounds on near-field radiative Purcell enhancement

We present a general framework, based on Lagrange duality, for computing physical bounds on a wide array of electromagnetic scattering problems. Namely, we show that, via projections into increasingly localized spatial clusters, the central equality of scattering theory - the definition of the T operator - can be used to generate a hierarchy of increasingly accurate mean-field approximations (enforcing local power conservation) that naturally complement the standard design problem of optimizing some objective with respect to structural degrees of freedom. Utilizing the systematic control over the spatial extent of local violations of physics offered by the approach, proof-of-concept application to maximizing radiative Purcell enhancement for a dipolar current source in the vicinity of a structured medium, an effect central to many sensing and quantum technologies, yields bounds that are often more than an order of magnitude tighter than past results, highlighting the need for a theory capable of accurately handling differing domain and field-localization length scales. Similar to related domain decomposition and multigrid notions, analogous constructions are possible in any branch of wave physics, providing a unified approach for investigating fundamental limits.