Localization of acoustic bulk modes due to negative refraction in crystal resonators

A new possibility is suggested in this paper to create crystal resonators of bulk acoustic waves with focusing and mode localization due to negative refraction at a planar interface between single-crystal layers. Local concavities of the slowness surface of crystals are considered to be the reason for negative refraction. The parabolic equations are derived to describe the spatial distribution of the focused fields. These equations correspond to: (i) three bulk modes propagating close to the crystallographic axes in orthorhombic crystals, (ii) pure shear mode propagating in the vicinity of the X-axis in orthorhombic piezoelectric crystal of potassium niobate, (iii) radially polarized quasi-shear mode propagating nearly in the direction of the Z-axis in hexagonal crystals. It is found that azimuthally symmetric beams of quasi-shear waves in hexagonal crystals have a ring-like structure of the wave field with zero radial displacement on the Z-axis, so there is no focusing on the beam axis in this case. The solutions to the parabolic equations are then used to construct focused modes of the travelling-wave type in periodic crystal layers and localized vibration in layered crystal resonators. A unique feature of this type of focusing is the absence of a fixed focusing axis in the uniform layered structures, that is, local and relatively long-lived vibration might be excited at an arbitrary point on the surface of planar crystal resonators. Therefore, such localized resonances could be used both as virtual keys for keyless keyboards and as movable pixels for imaging in touchscreen panels and graphics sensors.