a–c, Pressure distributions on the sample surfaces XZ1, YZ1 and XZ2 (Fig. 2d–f), respectively, scanned step by step at 5.75 kHz. The white stars indicate the positions of the point-like sound source. The propagation directions of the beams, as predicted from most of the SAWs hosted on the corresponding facets, are closely related to the positive and negative refractions observed in Fig. 3. The data are used to obtain the EFCs (with kz > 0) in Fig. 2l–n through a Fourier transform. Similar data can be collected to obtain the frequency-dependent surface band dispersions at a given kz (Fig. 2i–k). d–f, Decay signatures identified for the surface states in a–c. The data are measured along the normal directions of the sample surfaces, the in-plane coordinates of which are marked by the black circles (A–G) in a–c. The pressure magnitudes do not exhibit precise (oscillatory) exponential decay, mainly because the surface beam consists of many surface arc states with different out-of-plane decay lengths. The inset shows the kz dependence of the decay length l simulated for each surface.