Global Mie Scattering

TL;DR

This paper introduces a global Mie scattering theory that identifies topological invariants in scattering patterns, revealing universal properties independent of obstacle shape or incident light polarization.

Contribution

It proposes a novel topological framework for Mie scattering that uncovers invariant directions and associated indices, extending the understanding of scattering beyond local properties.

Findings

Existence of directions with zero or circular polarization in scattered light.

Assignment of half-integer topological indices to singular scattering directions.

Sum of indices is a global topological invariant equal to 2.

Abstract

In various subdisciplines of optics and photonics, Mie theory has been serving as a fundamental language and play indispensable roles widely. Conventional studies related to Mie scattering largely focus on local properties such as differential cross sections and angular polarization distributions. Though spatially integrated features of total cross sections in terms of both scattering and absorption are routine for investigations, they are intrinsically dependent on the specific morphologies of both the scattering bodies and the incident waves, consequently manifesting no sign of global invariance. Here we propose global Mie scattering theory to explore topological invariants for characterizations of scatterings by any obstacles of arbitrarily structured or polarized coherent light. It is revealed that, independent of distributions and interactions among the scattering bodies of arbitrary geometric and optical parameters, in the far field inevitably there are directions where the scatterings are either zero or circularly polarized. Furthermore, for each such singular direction we can assign a half-integer index and the index sum of all those directions are bounded to be a global topological invariant of $2$. The global Mie theory we propose, which is mathematically simple but conceptually penetrating, can render new perspectives for light scattering and topological photonics in both linear and nonlinear regimes, and would potentially shed new light on the scattering of acoustic and matter waves of various forms.