Fig. 1. Coexistence of 0D Weyl points and 1D nodal lines in chiral photonic metamaterials. (a) 3D band structures (kx=0), Weyl points, and nodal lines for the chiral metamaterials. (b) and (c) Density distributions of the photonic Weyl points and nodal lines in 2D momentum space, respectively. (d) and (e) 2D dispersion lines ω(ky)[ω(kz)] at fixed kz (ky) of the low-frequency negative (high-frequency positive) chirality Weyl points in panel (a), respectively. (f) and (g) 2D equifrequency lines, nonzero Berry curvatures, and the Berry phase of the negative (positive) Weyl points in panel (a). The corresponding electromagnetic parameters of the chiral metamaterials are εx=1, εt=3, α=1/3, ω0=0.3, ωp=1, and μx=μz=1, respectively.

Fig. 2. Comprehensive topological phase diagrams, 3D equifrequency surfaces, and multiple topological transitions. (a) Evolution phase diagram of electromagnetic parameters εx(y,z), μx(y,z), and γ varying with angular frequency ω. Roman numerals I–IV represent four different phase regions. (b) The medium exhibits high anisotropy as the angular frequency approaches 0. (d), (f), and (h) The 3D equifrequency surfaces at the resonance frequency and low-frequency (high-frequency) photonic Weyl frequency ω=0.3 and ω=0.549242 (ω=1.0), respectively. (c), (e), (g), and (i) The multiple 3D equifrequency surfaces caused by the evolution of parameters ω. They correspond to different phases I–IV regions in panel (a), respectively. The other electromagnetic parameters of the chiral metamaterials are set to the same as in Fig. 1.

Fig. 3. Complete topological band gaps and critical points of multiple phase transitions. (a)–(i) The 2D cross-section views of the 3D equifrequency surfaces for different phase regions in Fig. 2(a) by varying the angular frequency ω. Here, the cyan, orange, and blue lines represent the 2D equifrequency lines in the ky−kz, kx−kz, and kx−ky planes, respectively. The inset of panel (e) shows the existence of a self-intersection point in kx−kz plane. The resonance frequency ω0, low-frequency, and high-frequency Weyl points serve as critical points for phase transitions of these different topological phases. The other parameters of the chiral metamaterials are set to the same as in Fig. 1.

Fig. 4. Topological phase diagrams and frequency chirality-dependent Fermi arc surface states. (a) Topological phase diagram of the kz band gap. Here, Roman numerals I–IV are four different topological phase regions. Only regions I and III have kz band gaps. (b) and (c) 2D equifrequency lines, Berry fluxes, and Chern numbers of the gapped phase regions I and III in panel (a), respectively. REP (LEP) is right (left) elliptical polarization. (d) and (h) Gap Chern numbers and frequency chirality-dependent Fermi arc surface states of the vacuum-metamaterial systems, corresponding to phases I and III, respectively. (e) and (i) Skin depths y/λ of the common gap surface states in panels (d) and (h), respectively. (f), (g), (j), and (k) Mode profiles |E| and numerical simulation of frequency chirality-dependent surface waves of the gap modes A and B in panels (e) and (i), respectively. The other parameters of the chiral metamaterials are set to the same as in Fig. 1.

Fig. 5. Topologically protected properties of Fermi arc surface states. (a)–(h) 2D equifrequency lines of the bulk states and Fermi arc surface states as a function of angle frequency ω on the ky−kz plane. The low-frequency and high-frequency photonic Weyl points are highlighted by the blue and red dots, respectively. (i) The band gap sizes of the Fermi arc surface states are accompanied by the evolution of the angular frequency ω, corresponding to the cases in panels (b)–(g), respectively. The other parameters of the chiral metamaterials are set to the same as in Fig. 1.

Fig. 6. Multichannel and directional topological photonic routings. (a) and (b) The two-channel wave vector dependence for photonic routing that consists of the vacuum–chiral metamaterial systems, corresponding to the points D and E in Fig. 5(e), respectively. (c) and (d) 1D electric field distributions at lines L1/L2/L3 in panels (a) and (b), respectively. (e)–(g) Harpoon-like multichannel topological photonic routings. (h) and (i) Electric field intensity |E| of the multichannel topological photonic routings, corresponding to the cases 1–3 in panels (e)–(g). (j) and (k) Numerical simulations of transmission and electric field distributions of the cross multichannel topological photonic routings (+ type). The other electromagnetic parameters of the chiral metamaterials are set to the same as in Fig. 1.