Fig. 1. Geometry, band structures, and topological phase transition of LC circuit. (a) Schematic diagram of the LC circuit for the lumped element of the triangular lattice structure; the unit cell is marked by the red dashed lines, and two unit vectors are indicated by black arrows. (b) The lumped element circuit of the hexagonal unit cells, where the cell uses six nodes to describe LC circuits with equal capacitance on the nodes and unequal inductance on the link inductance within the cell (denoted in red and green) and between the cells (denoted in dark blue). (c)–(e) Band structures and topological phase transition in the LC circuit. The left panel shows the band diagram of the unit cell with L1=6.87nH, L2=3.24nH, and L3=3.13nH. The middle panel shows the band diagram of the unit cell with L1=L2=L3=4.1nH. The right panel shows the band diagram of the unit cell with L1=3.13nH, L2=3.24nH, and L3=6.87nH. The on-node capacitance is taken as C=7.27pF for all three cases. (f) Eigenmode distribution at Γ point (purple and red dots) and (g) phase distribution at K point (yellow and green dots) in (c), (e).

Fig. 2. Design principle of a planar microstrip topological LC circuit and the LPWSs and LVWSs transportation. (a) Schematic view of the proposed planar microstrip topological LC circuit supporting the coexistence of LPWSs and LVWSs. (b) Schematic view of the proposed heterostructure composed of three honeycomb lattice domains. Black dashed hexagon displays the unit cells of the three domains. Right panels: unit cell with honeycomb microstrip structure; unit cell in domain A of the system; the metallic strips of intra/inter hexagonal unit cells have widths of 0.5, 2.5, and 3.2 mm, whereas in domain C, they are 3.2, 2.5, and 0.5 mm, respectively, and in domain B the width of metallic strips is taken as 2 mm. The length of all metallic strip segments is 10.9 mm and a lumped capacitor of C=5.6pF is loaded on the nodes. The whole microstrip system is fabricated on an F4B dielectric film with thickness of 1.6 mm and relative permittivity of 2.2. (c) Calculated frequency band structure for LPWSs, where the blue and red lines denote non-topological waveguide mode and topological waveguide mode, respectively. The blue regions represent the projection of the bulk bands. (d) The width of the topological frequency window as a function of n, where the small gap between the two branches of topological waveguide modes is marked by the light orange region. (e), (f) Distributions of the out-of-plane electric field |Ez| of LPWSs by the full-wave simulations at f=1.47GHz (e) and experimental measurements at f=1.45GHz (f). (g) Calculated frequency band structure for LVWSs, where the blue and red lines denote non-topological waveguide mode and topological waveguide mode, respectively. The blue regions represent the projection of the bulk bands. (h) The width of the topological frequency window as a function of the number of layers n in domain B. (i), (j) Distributions of the out-of-plane electric field |Ez| of LVWSs by the full-wave simulations at f=0.941GHz (i) and experimental measurements at f=0.937GHz (j).

Fig. 3. (a) Schematic view of topological-photonic-waveguide-based channel intersection supporting the coexistence of pseudospin- and valley-locked properties. (b), (c) Distributions of the out-of-plane electric field |Ez| of LPWSs by the full-wave simulations at f=1.48GHz (b) and experimental measurements at f=1.46GHz (c). (d), (e) Same as (b), (c) but the results of LVWSs obtained by the full-wave simulations at f=0.951GHz (d) and experimental measurements at f=0.947GHz (e).

Fig. 4. (a) Schematic view of topological photonic waveguide for beam modulator. (b), (c) Distributions of the out-of-plane electric field |Ez| of LPWSs by the full-wave simulations at f=1.50GHz (a) and experimental measurements at f=1.48GHz (b). The number of layers in domain B abruptly changes from one layer to five layers. (d), (e) Distributions of the out-of-plane electric field |Ez| of LVWSs by the full-wave simulations at f=0.945GHz (d) and experimental measurements at f=0.941GHz (e).

Fig. 5. Spatial demultiplexer by coupling the large-area waveguide states to the background space. (a) Schematic of the demultiplexer to route topological photonic waveguide states into different channels. Unit structures of the D domain are enlarged on the right. The width of the metallic strips along the x and y directions of a tetragonal lattice circuit based on microstrip lines is w=1.5mm and the lattice constant is a1=9.4mm. (b), (c) K-space analysis of the out-coupling of LPWSs (b) and LVWSs (c). Black dashed lines represent the termination; red, cyan, and red dashed lines represent Γ, K, and K′ conserving lines. (d) Distributions of the out-of-plane electric field |Ez| of LPWSs by the full-wave simulations at f=1.49GHz. (e) Distributions of the out-of-plane electric field |Ez| of LVWSs by the full-wave simulations at f=0.938GHz.

Fig. 6. The selection of supercells in heterostructure of topological microstrip-based LC circuit.

Fig. 7. Strong robustness against large defects of the LPWSs and LVWSs. (a), (b) Schematic views of straight topological large-area waveguide modes with different defects. (c), (d) Distributions of simulated electric field intensity of LPWSs at f=1.46GHz corresponding to (a), (b), respectively. (e), (f) Distributions of simulated electric field intensity of LVWSs at f=0.94GHz corresponding to (a), (b), respectively. The white circle depicts the location of the defect.

Fig. 8. Selective excitation of topological waveguide modes with pseudospin- and valley-momentum-locking unidirectional propagation. (a), (b) Schematic views of topological large-area waveguide mode propagation excited by different circularly polarized chiral sources. (c), (d) Propagation of LPWSs with a circularly polarized source at f=1.50GHz corresponding to (a), (b), respectively. (e), (f) Propagation of LVPWSs with a circularly polarized source at f=0.935GHz corresponding to (a), (b), respectively.

Fig. 9. Photo and diagram of the experimental setup. (a) Schematic of the experiment. The sample is excited by an electric dipole antenna, and the near field is scanned by an electric field probe. (b) Positioning of the probe scanning the electric field component perpendicular to the structure. (c) Photo of the fabricated experimental sample. The inset shows the zoomed-in view of the sample around the center, where the source is marked by red dots.