Abstract
In various subdisciplines of optics and photonics, Mie theory has been serving as a fundamental language and playing 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 a global Mie scattering theory to explore topological invariants for the characterization 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.
Highlights
Conventional studies based on Mie theory concentrate on properties that can be roughly divided into two categories: overall integrated ones such as total absorption, scattering, and extinction cross sections; and local ones such as differential scattering cross sections and angular polarization distributions
As for such continuous vector field on the momentum sphere, the Poincaré−Hopf theorem[18,19] requires that there must be isolated directions where there is no scattering. Those directions correspond to singularities of vector fields, and for each singularity, an integer Poincaré index can be assigned
Our work here has addressed a fundamental problem: For light scattering by arbitrary obstacles, is it possible to define globally invariant properties to characterize any scattering that is dependent on neither the scattering bodies nor the incident waves? Through a cornerstone theorem of global differential geometry, the Poincare−́ Hopf theorem, we show that for arbitrary finite obstacle scatterings in a homogeneous background, there must be isolated singular directions where the scattering is either zero or circularly polarized
Summary
The seminal problem of light scattering by particles and the associated Mie theory has been pervasive in every subject of photonics, laying the foundation for studies and applications in optics and physics[1−3] and those in many other interdisciplinary fields including even biology and medicine.[4,5] Conventional studies based on Mie theory concentrate on properties that can be roughly divided into two categories: overall integrated ones such as total absorption, scattering, and extinction cross sections; and local ones such as differential scattering cross sections and angular polarization distributions. Besides the rapid progress relying on Mie theory in various directions, photonics, at the same time, has gained great momentum through incorporating novel topological concepts.[14,15] New topology-related ideas from condensed matter physics and other branches of physics have rendered extra degree of freedom for manipulations of light−matter interactions, through comprehensive exploitations of topological properties that are globally bounded.[14,15] For the classical scenario of light scattering by arbitrary obstacles, at first glance, the identification of globally invariant properties seems to be out of reach This is due to the fact that both the aforementioned overall and the local scattering properties are intrinsically dependent on geometric and optical parameters of the specific scattering bodies, their distribution patterns, and the cross interactions.[1,2] those scattering features are dependent on the morphologies
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