Fundamental Limits to Performance of Photonic Nanostructures via Hierarchical Mean-Field Constraints: Upper Bounds on Scattering, Absorption, and Purcell Enhancement

We present a general method for computing fundamental limits on photonic design objectives that apply to arbitrary structures dependent only on the material the structure is made of and its maximum footprint. From an algebraic analysis of the scattering properties of Maxwell’s equations, we introduce spatially localized constraints on the polarization field that can be organized as a hierarchy of resolutions, reminiscent of mean-field theories and multigrid/multi-scale methods. These constraints are leveraged via Lagrangian duality to produce upper bounds on far-field scattering/absorption cross-sections and near-field radiative enhancement, relevant to applications in single-photon extraction, photovoltaics, LED design, and Raman scattering. Far field bounds exhibit a transition from volume scaling in the quasistatic limit to area scaling in the geometric optics limit. Near field bounds show a transition from non-resonant to resonant enhancement with increasing device footprint and elucidate the limiting role of both material and radiative loss on the Purcell factor. Topology optimization results are presented, which in many cases approach within an order of magnitude of the bounds.