Optical-Acoustic Mode Coupling in a Hierarchical Nonlinear Acoustic Metamaterial

Locally resonant nonlinear acoustic metamaterials can provide low-frequency control and bandwidth extension, but many designs are narrow-band, have limited tunability and robustness, and face low-frequency–thickness trade-offs. To address these limitations, this work investigates wave propagation in a one-dimensional inward configuration hierarchical metamaterial with weakly nonlinearity using an approximate analytical method of Lindstedt–Poincar´e and numerical simulations; the analytical model captures the acoustic and first-optical branches, while the secondoptical branch is characterized numerically. Spectro-spatial analysis reveals coupling between the acoustic and optical branches, which broadens the effective bandwidth. From the dispersion relation, we observe that chain hardening/softening primarily upshifts/downshifts the acoustic and second-optical branches, whereas resonator nonlinearity mainly tunes the optical branches; the first-optical branch is largely insensitive to chain nonlinearity. Consistent with these shifts, under hardening nonlinearity, the chain generates solitary-like and dispersive packets, whereas with softening, propagation is primarily dispersive. Finally, two passive, time-invariant diode architectures are designed that exploit hierarchy-enabled optical–acoustic mode coupling to achieve nonreciprocal waveguiding, with an explicit efficiency–isolation trade-off. These results identify hierarchy-enabled optic–acoustic coupling as a practical passive route to broadband wave control and direction-biased devices, with potential for energy transfer and sensing.