Analysis of spurious vibrations in mesa-shaped AT-cut quartz plates

The effect of mesa structures on the coupled vibrations of AT-cut quartz plates is analyzed based on two-dimensional equations of motion. The mesa structures are used to obtain effective energy trapping and to separate the thickness-shear (TS) and thickness-flexural (TF) modes spatially. First, the effect of the bi-mesa structure is analyzed. The calculated displacement field confirms that the TS mode is confined to the mesa portion and the TF mode is localized in the thinner portion. The coupling strength estimated from the mode chart decreases to about one-fifth that of conventional resonators. Although the coupling reduction between the TS and TF modes can be obtained when using the piano-mesa structure, the asymmetric geometry of the structure increases the coupling between the TS and extensional modes