Multimodal Tuning of an Axisymmetric Resonator

This paper reports an approach for the simultaneous elimination of the modal frequency differences within two pairs of modes in an axisymmetric resonator. Fabricated devices exhibit frequency detuning, which can be eliminated by strategically mass loading the resonator. Each pair of modes responds to the mass loading differently so models are developed to predict the postloading frequency differences. The models are incorporated into a search procedure to select deposition sites that simultaneously reduce the modal frequency difference within each pair. The proposed approach is successfully implemented on a resonator whose modal frequency differences are reduced below 200 mHz from an initial frequency difference of 23.5 Hz for a pair of modes at 13.8 kHz, and a 2.4 Hz difference for another pair of modes at 24.3 kHz.