Dispersion Engineering by Hybridizing the Back-Folded Soft Mode of Monomode Elastic Metamaterials with Stiff Acoustic Modes

In many cases, the hybridization of two or more excitation modes in solids has led to new and useful dispersion relations of waves. Well-studied examples are phonon polaritons, plasmon polaritons, particle-plasmon polaritons, cavity polaritons, and magnetic resonances at optical frequencies. In all of these cases, the lowest propagating mode couples to a finite-frequency localized resonance. Herein, the unusual metamaterial phonon dispersion relations arising from the hybridization of an ordinary acoustical phonon mode with a back-folded soft or easy phonon mode of a monomode elastic metamaterial are discussed. Conceptually, the single easy mode can have strictly zero wave velocity. In reality, its wave velocity is very much smaller than that of all other modes. Considering polymeric three-dimensional printed elastic monomode metamaterials at ultrasound frequencies, it is shown theoretically and experimentally that the resulting pronounced avoided crossing, with a frequency splitting comparable to the mid-frequency, leads to backward-wave behavior for the lowest band over a broad frequency range, conceptually at zero loss. The strong coupling between ordinary phonon modes and a back-folded soft phonon mode of a monomode metamaterial is suggested as a means to engineer the metamaterial dispersion relation of the lowest bands. The back-folding is accomplished by a superlattice or a set of separated beams. Unusual roton-like phonon bands serve as an example. Calculations and experiments are in excellent agreement. image