Fig. 5. Demonstrations of rainbow edge waves in chirped valley phononic crystals.

a Configuration of chirped valley phononic crystals (PCs) for transporting edge modes of surface acoustic waves (SAWs) along a straight interface. The blue and brown circles in the m^th column represent shallow and deep cavities with depths of h_sl,m and h_dp,m. The cavity depths gradually increase with the increase of column index m following the relations h_sl,m = h_sl,1 + (m − 1)h_δ and h_dp,m = h_dp,1 + (m − 1)h_δ. The depths of the shallow and deep cavities in the first column are h_sl,1 = 8.58 mm and h_dp,1 = 10.58 mm, respectively. The step size h_δ is 0.08 mm. b Wavenumber-frequency dispersion curves of edge modes for type A–B supercells with H_δ = 0 mm in the 1st column (left) and H_δ = 1.36 mm in the 18th column (right). The depth change H_δ equals to (m − 1)h_δ. c Dispersion curves of edge mode group velocities c_g for type A–B supercells with different H_δ from 0.16 to 1.76 mm. The group velocity is zero at the cutoff frequency f_cut. d–f Theoretical group velocities versus location x along the interface of the chirped valley PCs at frequencies of 5.13, 5.02, and 4.96 kHz. g–i Experimental energy fields of edge waves in the chirped valley PCs at frequencies of 5.13, 5.02, and 4.96 kHz. The energy fields show that SAWs at different frequencies stop propagating forward at different locations in the interface. Accumulation of edge mode energy in the low group velocity region can be observed in g, h. j Two-dimensional distribution of the edge mode energy with respect to location x along the interface and frequency. The locations where edge waves stop propagating forward agree well with the theoretically predicted locations with zero group velocity.