Fig. 1. Illustrations of tunable topological negative refraction. (a) Schematic illustration. Through tailoring the Bloch wavevector k of incident one-way topological states, the variation of in-plane wavevector k∥ can turn the refracted wave from negative to positive refraction, according to Snell’s law k∥=k0sinθ. The gray dashed line represents the normal line of refraction. (b) Band structure of the topological photonic crystal that supports a one-way state. As the frequency increases, the Bloch wavevector of the one-way state moves along the high-symmetry lines K′−M−K. (c) Tailoring the Bloch wavevector of topological states in two-dimensional k-space. (d) Designed gyromagnetic photonic crystal with oppositely magnetized sublattices along the z-axis. Tailoring the Bloch wavevectors of topological states in k-space drives a transition from negative (θ1<0°) to critical (θ2=0°) to positive (θ3>0°) refraction.

Fig. 2. Topological refraction via tunable Bloch wavevectors of one-way waveguide states. (a) Experimental sample. The constant of the photonic crystals along the x direction (zigzag edge) is a=10mm and the width of the channel is w=w0+0.2a, where w0=a/3. Red and blue stars are two dipole sources with opposite phases. Blue arrow represents the transport of the waveguide state, while green arrow indicates the refracted beam at the interface (white dotted line). The supercell (white dotted rectangle) is used to obtain the dispersions in simulation. (b) Calculated and measured dispersions. The colored lines and background maps denote the simulated values and measured data, respectively. The blue curve indicates the one-way waveguide state. (c) Eigenmodal fields of the waveguide states. (d)–(h) Simulated Fourier spectra of the waveguide states at different frequencies: (d) f=9.51GHz, (e) f=9.62GHz, (f) f=9.80GHz, (g) f=10.00GHz, (h) f=10.11GHz.

Fig. 3. Demonstration of topological refraction. (a) Negative refraction with θ=−32° at f=9.68GHz. (b) Critical refraction with θ=−5° at f=9.80GHz. (c) Positive refraction with θ=+14° at f=9.92GHz. (a1), (b1), (c1) Simulated and measured electric field distribution in real space through near-field mapping. (a2), (b2), (c2) Fourier spectra of the simulated electric field (background) and theoretical k-space analysis (dots, lines, and arrows) of refraction process. The blue dots are k-components of incident waveguide states obtained from projected bands, the hexagon and semicircle represent the first Brillouin zone of the photonic crystal and the light cone of air, respectively, and the green arrows are theoretically calculated refraction angles. (d), (e) Immunity to metallic obstacles. (d) Simulated electric field distribution while inducing metallic obstacles in the channel at 9.68 GHz, together with the partial view of experimental sample. (e1), (e2), (e3) Experimental measurements of field distributions in the air side at (e1) f=9.68GHz, (e2) f=9.80GHz, and (e3) f=9.92GHz. The metallic obstacles in the transport channel do not affect the final emergence and refraction angles of the negative refraction waves.

Fig. 4. Active control of topological refraction. (a) Experimental sample. (b) Partially enlarged view of (a) to show the interlayer structures. (c)–(e) Physical mechanism. (f) Measured refraction angles θ varying with the gap width g at 9.9, 10.0, and 11.1 GHz. (g)–(n) Active control of topological refraction realized by tuning the gap width g at 10 GHz. (g), (k) g=0mm, μ0H0=1850G, θ=−38°. (h), (l) g=0.5mm, μ0H0=1650G, θ=−18°. (i), (m) g=1.0mm, μ0H0=1550G, θ=−5°. (j), (n) g=2.0mm, μ0H0=1475G, θ=+12°. (g)–(j) Simulated results. (k)–(n) Measured results.

Fig. 5. Sample fabrication and experiment measurement. (a) Exploded view of the experimental sample. (b) Structural details of a unit cell of antichiral gyromagnetic photonic crystal (PC). (c) Measurement of internal electromagnetic wave in sample. (d) 180° 3 dB bridge for creating a pair of dipole sources of opposite phases. (e) Schematic for measurement system. (f) Transmission spectra of one-way topological state.

Fig. 6. Eigenmodes in gyromagnetic photonic crystal. (a) Simulation setup for energy bands and eigenmodal field calculation. (b), (c) Projected energy bands and eigenmodal field distributions for the photonic crystal excitation analysis, respectively.

Fig. 7. Projected band structures with varying distances Δw. (a) Schematic diagram. (b) Δw=0.1a. (c) Δw=0.2a. (d) Δw=0.3a.

Fig. 8. Topological negative refraction excited by distinct types of sources. (a) A pair of odd-symmetric dipole sources. (b) Single dipole source. (c) Electric field distributions at the blue dashed line positions in (a) and (b).

Fig. 9. Scattering-based evaluation of topological properties in an antichiral photonic crystal. (a) Schematic of the simulation setup. Twisted boundary conditions are applied along the lateral edges, mapping the right boundary field to the left with an added twisting phase term eiΦx. An incident wave is launched through a waveguide formed by perfect magnetic conductors (PMCs). (b) Reflection phase φr as a function of the twisting angle Φx, demonstrating a full 2π winding at 9.8 GHz. This phase winding confirms the nontrivial topological nature of the antichiral edge state.

Fig. 10. Fourier transform of the eigenmodal field of the topological state. (a) Evolution of the topological state’s eigenmodal field along the x direction. (b) Periodic extension of the topological state’s eigenmodal field to obtain complete wave packets. (c) Fourier spectra of the topological state at kx=(5/12) (2π/a). (d)–(h) Simulated Fourier spectra under point-source excitation in finite-sized photonic crystals, used to verify consistency with the eigenfield-based method.

Fig. 11. Theoretical calculation of refraction angle through wavevectors matching. (a) Mapping relationship between two-dimensional k-space and projected bands. (b) Wavevectors matching between incident waveguide states and refraction beam on the interface. (c) Theoretically calculated refracted angle under various magnetic biases.

Fig. 12. Simulated Fourier spectra (background) and theoretical wavevector projections (symbols and arrows) at 10 GHz under (a) μ0H0=1900G, (b) μ0H0=1700G, and (c) μ0H0=1500G.

Fig. 13. Negative refraction in valley photonic crystals. (a) Schematic diagram of valley photonic crystals. The channel is composed of an interface between two photonic crystals with opposite valley Chern numbers. (b), (c) Projected band structures and eigenmodal fields. The channel supports valley-dependent topological states, locked in the K′ and K valleys. The right edge of the photonic crystals supports trivial edge states. (d)–(g) Negative refraction excited by an RCP source at varying frequencies. Due to a small Δk, the refracted beam covers only a narrow angular range. The presence of edge states significantly reduces refraction efficiency, resulting in substantial energy leakage.

Fig. 14. Efficiency of topological refraction in valley and antichiral photonic crystals. (a) Simulation setup for evaluating the transmission efficiency of topological refraction. (b) Electromagnetic wave transmission channels in photonic crystals. (c)–(f) Transmission and efficiency in (c), (d) antichiral photonic crystals and (e), (f) valley-Hall photonic crystals.

Fig. 15. Refraction in a chiral system. (a) Schematic diagram of the chiral system. (b), (c) Projected band structures and eigenmodal fields. The channel supports odd and even modes locked between the K′ and K valleys, respectively. The interface of the system also supports one-way edge states. (d)–(g) Negative refraction under the excitation of a pair of odd-symmetric dipole sources. (h)–(k) Positive refraction under the excitation of a pair of even-symmetric dipole sources. The refraction angle shows only a small range of variation, and the efficiency is significantly reduced due to the presence of edge states at the interface.