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We combine two different fields, topological physics and graded metamaterialsto design a topological metasurface to control and redirect elastic waves. Westrategically design a two-dimensional crystalline perforated elastic plate,using a square lattice, %consisting of bore holes, that hosts symmetry-inducedtopological edge states. By concurrently allowing the elastic substrate tospatially vary in depth, we are able to convert the incident slow wave into aseries of robust modes, with differing envelope modulations. This adiabatictransition localises the incoming energy into a concentrated region where itcan then be damped or extracted. For larger transitions, different behaviour isobserved; the incoming energy propagates along the interface before beingpartitioned into two disparate chiral beams. This "topological rainbow" effectleverages two main concepts, namely the quantum valley-Hall effect and therainbow effect usually associated with electromagnetic metamaterials. Thetopological rainbow effect transcends specific physical systems, hence, thephenomena we describe can be transposed to other wave physics. Due to thedirectional tunability of the elastic energy by geometry our results havefar-reaching implications for applications such as switches, filters andenergy-harvesters.

Localising elastic edge waves via the topological rainbow effect