Fig. 1. Setup for LPWSs. (a) Schematic (left) of the three-layer heterostructure A|Bx|C and the corresponding band structures (right) of the domains A, B, and C. Here, x labels the width (number of unit cells along y) of the middle domain B, and unit cells of the three domains are highlighted as insets. (b) Projected band structure along kx for A|B5|C, where the black, blue, and red dotted lines denote bulk, nontopological waveguide, and topological waveguide modes, respectively. The amplitude and phase distributions of the topological waveguide modes (EZ) marked by red and green dots are shown on the right. (c) Width of the topological frequency window (marked by the light shadow in (b) as a function of x, where the light orange region marks the small gap between the two branches of topological waveguide modes. (d) Simulated |EZ| distributions of the pseudo-spin-momentum-locking unidirectional propagation at frequency 0.5601c/a in A|B5|C, where the chiral source for excitation is marked by the green star. (e) Simulated |EZ| distribution of the topological propagation against four sharp bends in an Ω-shaped waveguide at frequency 0.5601c/a using A|B5|C. (f) Simulated |EZ| distributions of the topological waveguide modes at frequency 0.5627c/a for x=1, 5, and 9, where the green dashed lines highlight the boundaries of domain B. (g) Simulated transmission spectrum versus frequency in a straight waveguide with x=5, where the light and orange regions denote the topological frequency window and topological mode gap, respectively.

Fig. 2. Experimental observations of LPWSs. (a) Photograph of an experimental sample from the top view, where the red dashed lines denote interfaces A|B and B|C, respectively, and all alumina cylinders are surrounded by microwave-absorbing foams (blue regions). The chiral source is composed of three antennas with phase winding 2π/3 along the clockwise direction, which is placed parallel to the alumina cylinders for launching the EM wave. (b) Measured amplitude of the EZ field at frequency of 7.92 GHz with source location indicated by the green star. (c) and (d) Measured amplitudes of the EZ field at frequency of 8.02 GHz for two different structural imperfections—disorder in (c) and a void cavity in (d), where the green dashed rectangles highlight the positions of the imperfections in experiments (see the top-right for details).

Fig. 3. Strong robustness against large cavity defects of the LPWSs. (a)–(d) Simulated propagation of the topological waveguide modes with cavity defects along the propagation path in A|B5|C waveguide at frequency 0.5601c/a. The size of the cavity defects increases along the transverse direction from (a)–(d) (see schematics of the insets). The white dashed lines denote the boundaries of the domain B, and the green dashed rectangles highlight the location of the air cavities. (e) and (h) Similar to (a)–(d), but for cavity defects with size increasing along the longitudinal direction (see the schematics of the insets). (i) Transmission spectra versus frequency for the waveguides with different cavity defects in (a)–(d). (j) Transmission spectra versus frequency for the waveguides with different cavity defects in (e)–(h).

Fig. 4. Applications of LPWSs. (a) and (b) Simulated amplitudes of the EZ field in topological channel intersections using LPWSs, where the chiral source is placed at port 1 and port 3 in (a) and (b), respectively. (c) Simulated amplitude of the EZ field in topological energy concentrator using LPWSs. (d) and (e) Transmission spectra corresponding to (a) and (b). (f) Normalized |EZ| profiles along three dashed green lines in (c).