Rest-frame electrodynamics and polarizability theory for rotating two-dimensional structures and particle arrays

An electrodynamic (ED) theory for 2D particle arrays that undergo a slow rigid rotation $\Omega$ and observed in their rest frame of reference $R^{\Omega }$ is developed. The theory holds for both transverse electric (TE) and transverse magnetic (TM) excitations. Analytical and numerical techniques are applied, in conjunction of a suitable Greens function, to study the effect of rotation on electrically small 2D scatterers such as dielectric cylinders of arbitrary isotropic material and cross-section geometry, and to derive the polarizability matrix in $R^{\Omega }$ and the corresponding discrete dipole formulation. The formulation is then used for a preliminary study of particle arrays and tested against full wave numerical solutions of rotating structures that hold in $R^{\Omega }$. The basic physical mechanisms that govern the rotation footprint on the array response are explored and discussed, and their ramifications on TE and TM cases are contrasted. These mechanisms hold in general and are not idiosyncratic to 2D. It is shown that the ED is mainly due to the excitation of a large number of Sagnac loops created inside the structure as a result of the multiple scattering events between the array inclusions. However, the collective interactions of a large number of loop exhibit a net dynamics that is far more intricate than that of a single Sagnac loop, and even contradicts it in some aspects. The rotation footprint on the collective ED properties of a 2D periodic structure such as stop-band location, induced nonreciprocity, and the role of the structure symmetries in these effects are explored.