Frequency shift of a nanowaveguide resonator driven by the tunable optical gradient force

A nanowaveguide resonator driven by a tunable optical gradient force can easily enter into the nonlinear oscillation regime, where the resonance frequency will shift. In this work, a continuum elastic model of the optoresonator is presented and solved analytically using the method of multiple scales. The effects of the optical gradient force on the resonance frequency and dynamic behavior are investigated. The results theoretically figure out why and when the nonlinear behavior of spring softening and spring hardening can occur. It is shown that the nonlinear phenomenon of spring softening is generally more dominant than the hardening effect when the optical gradient force is strong. However, the nonlinear cubic mechanical stiffness of the waveguide makes the dynamic behavior of spring softening dominant when the optical force is not strong enough. Based on the logical derivation of the closed-form solution, it can be found that the decrease of resonance frequency is due to the bias term, which is inherent in the nature of the tunable optical gradient force. Additionally, the complex variations of the resonance frequency and maximum vibration amplitude with different waveguide widths, lengths, and initial gaps are investigated and discussed. The proposed solutions are also verified with the reported experimental results.