Fig. 1. (a) Geometry of the edge waveguide consisting of a magnetically biased ferrite half-space and a metal half-space separated by an air gap with thickness d. (b) Band diagram (TE waves) and gap Chern number of a lossless biased ferrite material with ωl/(2π)=4.43GHz and ωg/(2π)=5.18GHz (green solid lines). The bandgap is delimited by the horizontal-dashed black lines. The figure also depicts the dispersion of the edge modes for the air gap thicknesses: d=0.1mm (brown solid curve); d=0.75mm (red solid curve); d=1.5mm (black solid curve); and d=5mm (blue solid curve). (c) Dispersion of the edge modes when the response of the ferrite is regularized with a spatial-frequency cutoff kmax (dashed curves) superimposed on the dispersion of the edge waveguide of panel (a) for an air gap thickness d=1/kmax (solid curves). The color code is as in (b).

Fig. 2. Geometrical illustration of the concept of an ill-defined topology. (a) A torus with a vanishingly small inner radius has a cross section consisting of two kissing circles and thus it is topologically ill-defined. The topology can be regularized with an arbitrarily weak perturbation. For example, in panel (b) by increasing slightly the inner radius of the object one obtains a geometrical shape topologically equivalent to a torus (g=1), (c) whereas by separating the top and bottom caps by an infinitesimal distance at the singular point the object becomes topologically equivalent to a sphere (g=0).

Fig. 3. (a) Prototype of the unidirectional edge waveguide. The structure is formed by a hollow aluminum waveguide partially filled with a magnetically biased ferrite block with dimensions Lx=150mm, Ly=50mm, and Lz=10mm. The ferrite is biased by a permanent magnet placed at the back of the waveguide (not shown in the figure). The ferrite block is separated from the bottom lateral metallic wall by an air gap with thickness d=5mm. Absorbent blocks are placed at the left- and right-hand sides of the ferrite block. The structure is excited by small monopole antennas fed through ports 1 and 2. (b) Measured amplitude of the scattering parameters S12 and S21. The transmission level from port 1 to port 2 is larger due to the excitation of an edge mode at the ferrite–air–metal interface. (c) Edge mode measurement apparatus. The top cover of the waveguide is replaced by a metallic plate drilled with small holes. The holes have a 2 mm diameter and are separated from each other by 1.5 mm in each direction. The electric field is measured at 1 mm from the surface of the hole grid using a small metallic probe that is moved by a robotic arm and is connected to a vector network analyzer (R&S ZVB20).

Fig. 4. Measured electric field distributions |Ez/E0|2 near the perforated top metallic plate. (a) and (b) Port 1 excitation (propagation from the right to the left) for an oscillation frequency f1=7GHz and f2=7.25GHz, respectively. (c) and (d) are similar to (a) and (b) but for a port 2 excitation (propagation from the left to the right). In all panels, the small green dot represents the position of the excited monopole antenna. The lateral walls are absorbers.

Fig. 5. (a) Time snapshot of the measured normalized electric field distribution Re{Ez/E0} above the surface of the perforated metal cover of the waveguide for a port 1 excitation and for f2=7.25 GHz. (b) Similar to (a) but for the full-wave simulation.⁵⁸ (c) Similar to (b) but the electric field is monitored inside the waveguide. In all panels, the small blue dot on the bottom right corner represents the position of the monopole antenna.

Fig. 6. Normalized electric field distribution |Ez/E0|2 measured above the surface of the perforated metallic plate for a port 2 excitation at the oscillation frequency (a) f1=7GHz and (b) f2=7.25GHz. The small green dot represents the position of the monopole antenna.