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

TL;DR

This study uses real-space imaging to directly observe how symmetry influences nonlinear energy transfer between vibrational modes in a microresonator, revealing that spatial symmetry governs modal coupling.

Contribution

The paper introduces a real-space imaging technique to visualize nonlinear modal energy routing and demonstrates symmetry as a key factor in modal interactions.

Findings

Energy transfer is suppressed unless modes share the same spatial symmetry.

Intermodal coupling depends on a symmetry-determined modal-overlap integral.

Symmetry governs nonlinear modal coupling more than spectral proximity.

Abstract

Nonlinear energy exchange between vibrational modes underlies phenomena ranging from internal resonance to wave mixing, yet modal interactions are typically inferred from frequency-domain signatures rather than directly observed in space. Here, we present real-space imaging of nonlinear modal energy routing in a near-mirror-symmetric microelectromechanical resonator using phase-locked multi-harmonic stroboscopic interferometry. By reconstructing the spatial eigenmode content of individual harmonic components, we directly resolve the energy transfer pathway between interacting modes. Our measurements reveal that nonlinear energy exchange is not governed by frequency proximity alone. Even when harmonic frequencies lie closer to an opposite-symmetry mode, energy transfer remains strongly suppressed unless the interacting modes share identical spatial symmetry. A reduced two-mode model incorporating geometric nonlinearity shows that the intermodal coupling terms factorize into a single symmetry-determined modal-overlap integral, establishing spatial parity as the fundamental admissibility condition for nonlinear coherent energy exchange. These results demonstrate that symmetry, rather than spectral detuning alone, governs nonlinear modal coupling and introduce real-space nonlinear modal imaging as a general framework for controlling energy flow in nonlinear wave systems.