Adiabatic limit for scattering-free waveguiding in space-graded arrays of micro-resonators

We report on the dynamics of graded metamaterials in the context of micro-electromechanical systems (MEMS). Graded metamaterials are known to exhibit space confinement of mechanical energy - or rainbow trapping - in response to a tailored wave speed reduction. Such a behavior, in turn, is controlled by a gradual variation of the resonant characteristics of an array of resonators dressed on a host structure which, as a result, exhibits arbitrarily slow waves. The paper shows quantitatively that there is an adiabatic limit for the rate of change of the grading. Below this limit, the rainbow effect takes place with a negligible scattering of energy, which is otherwise detrimental to practical applications such as harvesting, sensing, and communication. The paper also paves new ways to manipulate wave motion in MEMS through elastic metamaterials.