Rest-frame analysis of rotating metamaterials, gyroscopes, and century-old problems in number theory [Invited]

Abstract

We discuss the electrodynamics of slowly rotating metamaterials as observed in their rest frame of reference, using first order polarizability theory. A formulation governing the response of an arbitrary array of scatterers to excitation under rotation is provided and used to explore the rotation footprint properties, with applications to non-reciprocal dynamics, rotation sensors and optical gyroscopes. The metamaterial sensitivity to rotation is rigorously defined, and the associated physical mechanisms are exposed. These can be intimately related to two century-old problems in number theory: the no-three-in-line problem (N3IL), and the Heilbronn triangle problem. New arrays, base on Erdős solution to the former, are proposed and their sensitivity to rotation is explored. It is shown that structures inspired by Erdős solution may achieve rotation sensitivities that outperform that of the Sagnac loop gyroscope. The average and peak performances of ensembles of random arrays are also explored. The effect of signal noise on the rotation sensitivity is studied. It is shown that the additional degrees of freedom suggested by metamaterial approach to rotation sensing can be used to minimize the negative effect of signal noise on the smallest detectable rotation rate. Furthermore, we show that the systematic N3IL constructions inspired by Erdős encapsulates most of the significant factors leading to enhanced rotation-sensitive metamaterials.