Real-space imaging reveals symmetry-selected nonlinear energy routing in a mechanical resonator

Abstract

Nonlinear energy exchange between vibrational modes underlies phenomena ranging from internal resonance and wave mixing to frequency-comb generation, yet modal interactions are typically inferred from spectra rather than directly observed in space. Here, we image nonlinear modal energy routing in a nearly mirror-symmetric microelectromechanical resonator using phase-locked multi-harmonic stroboscopic interferometry. By reconstructing the spatial eigenmode content of individual harmonics, we show that harmonics generated by a driven mode can be carried by distinct spatial eigenmodes, directly resolving spatial pathways of nonlinear energy transfer. Our measurements further reveal that this modal routing persists away from integer frequency matching: in the off-resonant regime, generated harmonic components are dominated by eigenmodes sharing the driven mode's mirror parity, whereas spectrally closer opposite-parity modes remain strongly suppressed. A nonlinear modal framework based on geometric nonlinearity shows that the relevant cubic coupling coefficients factorize into symmetry-dependent modal-overlap integrals, identifying mirror parity as the selection rule for nonlinear modal interaction. This work identifies spatial symmetry as a design parameter for nonlinear energy routing and provides a route to symmetry-engineered control of energy flow in multimode nonlinear wave systems.