Negative Refractive Index in a Two-Dimensional Nonlinear Rotator Lattice

Acoustic metamaterials achieving negative index refraction usually operate linearly over a narrowband of frequency and consist of complex unit cell structures incorporating resonators. In this paper, we propose and analyze a simple, non-resonant, nonlinear rotator lattice structure which can be configured with either a positive or negative index of refraction over a broadband frequency range. The system’s frequency-dependent transmission is studied analytically via a reduced model along the interface of positive and negative refractive index lattices. Results for energy transmission are compared to those obtained using direct numerical simulation and close agreement is documented for small amplitude waves. For larger amplitude waves, a multiple scales analysis approach is used to show that the nonlinearity of the lattice shifts the system’s band structure, inducing amplitude-dependent transmission. For the studied system, the transmission decreases as we increase the incident wave amplitude, agreeing qualitatively with results from direct numerical simulation. At large-enough amplitudes, near the interface the wave amplitude decreases rapidly. As the wave travels further into the media, the amplitude drops, causing the nonlinear effect to decline as well. This decaying envelope does not result in a zero transmission in the far field, as expected from linear theory, and instead, the nonlinearity of the proposed rotator lattice prevents the far-field transmitted wave from surpassing a specific threshold amplitude, regardless of the incident wave. This finding may serve as an inspiration for designing nonlinear wave saturators.