Reflection and refraction of waves occur at the interface between two different media. These two fundamental interfacial wave phenomena form the basis of fabricating various wave components, such as optical lenses. Classical refraction—now referred to as positive refraction—causes the transmitted wave to appear on the opposite side of the interface normal compared to the incident wave. By contrast, negative refraction results in the transmitted wave emerging on the same side of the interface normal. It has been observed in artificial materials1,2,3,4,5, following its theoretical prediction6, and has stimulated many applications including super-resolution imaging7. In general, reflection is inevitable during the refraction process, but this is often undesirable in designing wave functional devices. Here we report negative refraction of topological surface waves hosted by a Weyl phononic crystal—an acoustic analogue of the recently discovered Weyl semimetals8,9,10,11,12. The interfaces at which this topological negative refraction occurs are one-dimensional edges separating different facets of the crystal. By tailoring the surface terminations of the Weyl phononic crystal, constant-frequency contours of surface acoustic waves can be designed to produce negative refraction at certain interfaces, while positive refraction is realized at different interfaces within the same sample. In contrast to the more familiar behaviour of waves at interfaces, unwanted reflection can be prevented in our crystal, owing to the open nature of the constant-frequency contours, which is a hallmark of the topologically protected surface states in Weyl crystals8,9,10,11,12.
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We thank M. Xiao for discussions. This work is supported by the National Basic Research Program of China (grant number 2015CB755500), the National Natural Science Foundation of China (grant numbers 11774275, 11674250, 11534013 and 11747310) and the Natural Science Foundation of Hubei Province (grant number 2017CFA042). F.Z. was supported by UT Dallas research enhancement funds.
Nature thanks A. Grushin, B. Zhang and the other anonymous reviewer(s) for their contribution to the peer review of this work.
The authors declare no competing interests.
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Extended data figures and tables
a–c, The data are evaluated at kz = 0.5π/h for the three side surfaces XZ1, YZ1 and XZ2, respectively, the terminations of which are specified in the main text. Two gapless surface bands (green lines) traverse the two gaps between the lowest three projected bulk bands (grey regions). We focus on the lowest surface band in the main text.
a, Berry flux contributions to the lowest two bulk bands, derived from the numerically calculated Weyl charge distributions (see Fig. 2h). Here |C| labels the amplitude of the Weyl charge, and the red and black spheres indicate sources and sinks of Berry fluxes, respectively. b, Chern numbers for the lowest two bulk gaps opened at kz > 0, labelled according to the Berry flux distributions in a. (Here, the band structure is simulated at kz = 0.5π/h.) The Chern numbers Cgap of both gaps are −1, consistent with the topologically non-trivial surface spectra presented in Extended Data Fig. 1.
Extended Data Fig. 3 Different EFC properties attained by engineering the surface terminations of the Weyl phononic crystal.
a, Surface EFCs evaluated for the XZ surfaces at the Weyl frequency of 5.75 kHz, where the parameter dy defined at the top of the column characterizes the surface truncation (black dashed line). The grey regions display the projections of bulk bands, the blue spheres display the projections of the Weyl points K and K′ at 5.75 kHz, and the arrows indicate the directions of the group velocities for the surface arc states. b, Similar to a, but for the YZ surfaces specified by dx. The evolution of the EFCs for different surface terminations shows various possibilities of manipulating the surface states according to their group velocities. We focus on the cases in a2, a4 and b4 in the main text to attain the desired SAW properties. Throughout, we use 5.75 kHz to shrink the momentum regions projected by the bulk bands, which is favourable for the experimental observation of the surface arc states and the associated interfacial phenomena.
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.
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He, H., Qiu, C., Ye, L. et al. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. Nature 560, 61–64 (2018). https://doi.org/10.1038/s41586-018-0367-9
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