Distant small spot presentation in midair haptics using polyhedral reflector

In midair haptics, ultrasound phased arrays are mainly used due to their high spatiotemporal controllability. The constraint on the presentation distance of phased arrays to form a focus can be mitigated by utilizing concave reflectors. This paper numerically examines the convergence of a surface approximated by multiple planes serving as a reflector, substituted for an ideal concave surface. A mirrored phased array produced by the planar segments forms a focus and concurrently creates interference among imaginary sources. A single-point convergence condition is derived that constrains the accuracy of the approximated reflector and the phased array size. As long as it satisfies the convergence condition, the approximated reflector can form a single focal point. Numerical simulations were conducted to confirm the validity of the convergence equation and the 5% tolerance of the segment size for the reflector deformation. An experimental campaign was also conducted and confirmed that the polyhedral reflector was able to form a single small focus by controlling the phase shift of the sound source.