Fig. 1. Transmission/reflection spectra and band structures in transmission line network: (a) The transmission line network with input and output (open boundary); (b) one-dimensional (1D) periodic transmission line network; (c) two-dimensional (2D) periodic transmission line network; (d) three-dimensional (3D) periodic transmission line network; (e)−(g) correspond to the band structures of the networks shown in (b)−(d) respectively.

Fig. 2. Anderson localization in 1D and 2D transmission line networks^[9]: (a) 1D random transmission line network, the randomness is introduced by changing the coordination numbers and cable lengths; (b) 2D random transmission line network, the randomness is introduced by removing the cables randomly in the square lattice network.

Fig. 3. Experimental observation of Anderson localization in 3D transmission line network^[10]: (a) The structure of the sample (it is the AB alternatively stacked structure); (b) the left panel is the transmission spectrum measured for the sample in (a), the circles and solid curves are the measured and calculated results respectively; the right panel is the transmission spectrum after introducing a defect, the circles and solid curves are the measured and calculated results respectively, the dotted curve is the calculated result without dissipation; (c) the transmission spectrum after introducing randomness, where the circles and solid curves are the measured and calculated results, and the dotted curve is the calculated result without considering dissipation; (d) inverse participation ratios (IPRs) for different sample scales; (e) the field intensity pattern of the scattered wave function at 46.66 MHz; (f) and (g) are measured and calculated field intensity pattern of localized states respectively, (g) also shows the connections of the random network sample.

Fig. 4. Transmission spectra and band structures of Sierpinski fractal transmission line networks^[11]: (a) A 4^th-generation Sierpinski network with three input/output cables on its three vertices; (b) the 1^st-generation Sierpinski network with input/output cables, the upper panel corresponds to the single-exist case while the lower one to the double-exist case; (c) the eigen-frequencies of an isolated first-generation Sierpinski network, the vertical axis corresponds to the absolute value of the determinant of matrix M, the horizontal axis corresponds to frequency. When the absolute value of determinant equals to zero, the corresponding frequency is an eigen-frequency; (d)−(f) correspond to the transmission spectra of the 1^st, 2^nd, 3^rd Sierpinski networks respectively, the upper rows correspond to the single exit channel case and the lower rows correspond to the double exit channel case.

Fig. 5. Influence of the fundamental loop on the band gaps of the periodic and quasiperiodic networks^[12]: (a) The transmission spectra of the triangle (solid curve), square (dotted curve), hexagon (dash curve) loop networks; (b) the band structure (left panel) and the transmission spectrum (right panel) of the square lattice; (c) the same as (b), but for the lattice network which add cables on the diagonals of the square lattice (see the inset at the right lower corner).

Fig. 6. Transmission line network possessing topological properties^[16]: (a) A hexagonal ring formed by the meta-atoms which possess angular momentum; (b) discretizing the meta-atom into three nodes; (c) shows how to introduce the coupling between the angular momentum and wave vector to the model in (b). Here we only show a hexagon of the honeycomb lattice; (d), (e) calculated band structures of the model in (c) for the m = 0 and m = 1 sectors respectively; (f), (g) projected band structures along x and y directions for m = 1 modes, the red and blue curves represent the edge states at the opposite boundaries respectively.

Fig. 7. Robustness of topological edge states and its experimental observation^[16]: (a) The sample configuration to verify the robustness of edge states in simulations; (b) the simulated transmission spectrum of m = 1 modes (purple curve) for the sample in (a), compared with the one without defect (black curve); (c) the experimental configuration used to observe the edge states; (d) the photo of experimental sample, red dots and lines are used to highlight its underlying honeycomb structure; (e), (f) the experimental and simulated field intensity patterns of the edge state for m = 1 modes at 34.5 MHz.

Fig. 8. Calculation of local Chern number in the finite samples^[16]: (a) The computation domain for local Chern number calculations; (b) from left to right: energy level, the relation between local Chern number and cutoff frequency, local Chern number patterns. The size of sample can be seen in the right panel; (c) the same as (b), but the sample size equals to the size in experiments.