Optimal Poloidal Multipole Expansion Centres

Resonant optical response is one of the key effects in nanophotonic structures. The characterization of the fundamental resonance metrics, quality factor, electric or magnetic field localization, polarization, and chirality, is often built on a multipolar decomposition of the fields at the resonance and employs either the exact spherical or simplified poloidal multipole descriptions. The poloidal Cartesian decompositions can enable efficient numerical schemes and differentiable solvers that are desirable in modern optimization frameworks. Unfortunately, multipolar decompositions of any kind depend on the choice of their origin, and multipoles become non-unique. Despite their simpler form, the poloidal multipole expansions are also complicated by the ambiguity in choosing their expansion centers. Their multipole sets are also not unique and may require optimization to obtain efficient spatial spectra. We address this issue by deriving the optimal scattering centers where the poloidal multipolar spectra become unique. The optimal positions are defined separately for electric and magnetic components by minimizing the norms of the poloidal quadrupolar terms. We verify this approach with simplified tests and more realistic cases. We also explore the connection between the number of optimal magnetic centers and the multiplicity of the cost function roots.