The Art of Finding the Optimal Scattering Center(s)

Abstract

The truncated spatial multipolar spectra enable efficient approximate solutions to acoustic, quantum-mechanical, and electromagnetic problems. In photonics, the efficient multipole description of a general emitter or scatterer with controlled accuracy is complicated by the ambiguity in choosing the multipole expansion center—the multipole terms depend on the position of the expansion center and therefore are not unique. This study solves this fundamental problem by finding the optimal scattering centers for which the spatial multipole spectrum becomes unique. These optimal positions are derived separately for the electric and magnetic multipoles by minimizing the norms of the poloidal quadrupoles, employing the long-wave approximation (LWA) ansatz. The ultimate positions are verified with idealized discrete emitters and realistic scatterers. The optimal multipoles, including the toroidal terms, are calculated for several distinct scatterers; their utility for fast, low-cost numerical schemes is discussed. The number of optimal magnetic scattering centers, defined by the multiplicity of the problem, can serve as a new topological metric of a given emitter or scatterer. This finding hints at potential relations between nanoscale optomechanics and topological photonics. Expansion of the work beyond the LWA is possible, with the promise of more general foundational concepts for electrodynamics, acoustics, and quantum mechanics.