Fig. 1. Topological properties of exceptional points and rings of non-Hermitian optical systems. (a) Multiple exceptional points of high-order non-Hermitian system^[20]; (b) non-Hermitian Fermi arc^[21]; (c) non-Hermitian exceptional lines realized by planar photonic crystals^[22]; (d) Weyl exceptional ring realized by helical waveguide array^[23]

Fig. 2. Edge states of non-Hermitian topological systems. (a) Non-Hermitian dimer chain realized by loss waveguide array^[26]; (b) non-Hermitian dimer chain composed of loss dielectric cylinders^[27]; (c) non-Hermitian skin effect realized by asymmetric coupling^[31]; (d) high-order non-Hermitian coupled ring topological system^[33]

Fig. 3. Schematic of topological dimer chain composed of composite coil resonators^[65]. (a) Schematic of dimer chain. In this tight-binding model, unit cell is composed of two resonators which are marked in a dashed rhombus; (b) resonance coil on top layer of composite coil resonator. Resonant frequency of coil is 5.62 MHz; (c) non-resonance coil with a LED lamp on bottom layer of composite coil resonator; (d) band structure and winding number of topological trivial dimer chain; (e) band structure and winding number of topological phase transition phase; (f) band structure and winding number of topological non-trivial dimer chain

Fig. 4. Measured DOS and LDOS spectra of topological dimer chains with and without perturbations^[65]. (a) Measured DOS spectrum of topological dimer chain without perturbation. Bulk state (5.43 MHz) and edge state (5.62 MHz) in bandgap are marked by two arrows; (b) similar to (a), but for case with disorder perturbation. Structure disorder is realized by randomly moving central ten resonators by 5 mm; (c) measured normalized LDOS distribution of bulk state in topological dimer chain without perturbation; (d) similar to (c), but for case with disorder perturbation; (e) measured normalized LDOS distribution of edge state without perturbation; (f) similar to (e), but for case with disorder perturbation

Fig. 5. Comparison of transmission efficiency between dimerized topological non-trivial chain and trivial chain^[65]. (a) Schematic of a multicoil WPT system based on second-order PT symmetry of topological edge states (TEMs). Energy is input and output by source coil and receiving coil, respectively; (b) calculated transmittance spectra as a function of frequency for topological non-trivial chain (solid line) and topological trivial chain (dashed line); (c) ratio of transmission efficiency of topological non-trivial chain to that of topological trivial chain. At working frequency of 5.62 MHz, efficiency ratio is 44.63; (d) experimental demonstration of lighting two 0.5 W LED lamps. Non-resonant source coil is placed in center of chain. All resonators are added with LED lamps except for two resonators near source coil

Fig. 6. Three coupled topological modes in composite topological dimer chain^[65]. (a) Schematic of multicoil WPT system based on third-order PT symmetry in composite topological dimer chain (κ₁ < κ₂). Effective third-order PT symmetry is formed by interaction of three topological modes, including two TEMs at two ends of chain and one topological interface state (TIM) at center of chain; (b) eigenvalues spectra of topological dimer chains with third-order PT symmetry (dots) and second-orderPT symmetry (circles). Discrete topological modes are identified at bandgap (marked by gray region) of chain; (c)-(e) normalized intensity distributions of three topological modes in composite topological dimer chain with third-order PT symmetry

Fig. 7. Comparison of transmission efficiency between topological dimer chains with third-order PT symmetry and with second-order PT symmetry^[65]. (a) Real and (b) imaginary eigenfrequencies of topological WPT system with second-order PT symmetry. Coupling coefficient between two TEMs is κee= 0.1 MHz; (c), (d) similar to (a), (b) but for topological WPT system with third-order PTsymmetry; (e) calculated transmittance spectra of topological chain with third-order PT symmetry (solid line) and second-order PT symmetry (dashed line); (f) ratio of transmission efficiency of topological chain with third-order PTsymmetry to that of chain with second-order PT symmetry

Fig. 8. Experimental demonstration of WPT with small standby power loss in topological dimer chain with third-order PT symmetry^[65]. (a) Measured reflection spectrum of topological dimer chain with effective third-order PT symmetry in working state; (b) experimental demonstration of topological WPT system with effective third-order PT symmetry by lighting two LED lamps at two ends of chain; (c) measured reflection spectrum of topological dimer chain with effective third-order PT symmetry in standby state; (d) similar to (b), but one resonator at right end of chain is removed, which corresponds to standby state. Standby power loss in topological dimer chain with effective third-order PT symmetry under standby state is small because LED lamps remain dark; (e)-(h) similar to (a)-(d), but for topological dimer chain with effective second-order PT symmetry under working and standby states. LED lamp at left end of chain is illuminated even if system is in standby state, which means standby power loss is large

Fig. 9. Quasiperiodic 1D topological Harper chain^[80]. (a) Schematic of 1D topological Harper chain constructed by resonance coils; (b) energy band structure of finite-size Harper chain; (c) measured DOS spectrum of 1D topological Harper chain

Fig. 10. LDOS distribution of asymmetric edge states in Harper chain^[80]. (a) Calculated left edge state; (b) calculated right edge state; (c) measured left edge state; (d) measured right edge state

Fig. 11. [in Chinese]

Fig. 12. Actively controlled directional long-range WPT based on asymmetric topological state of Harper chain^[80]. (a) Photo of resonant unit with active control of external voltage; (b) equivalent circuit diagram of active resonant coil; (c) resonant frequency of active coil resonator as a function of bias voltage. Dotted line and solid line are experimental data and fitted line, respectively; (d) relationship between coupling strength of active coil resonators and distance. Measured coupling strengths of U=0 and U=4 V are marked with dots and lines, respectively; (e) photo of experimental setup for measurement of actively controlled directional WPT; (f) transmission spectrum when external voltage is U=4 V; (g) transmission spectrum without external voltage

Fig. 13. Composite resonator with tunable gain and loss^[112]. (a) Equivalent circuit diagram of composite resonator, composed of a simple LC resonator, a negative resistance convertor (NIC) component, and a tunable resistor; (b) structure diagram of composite resonator, where left and right images indicate front and back of structure, respectively; (c) equivalent circuit diagram of NIC component; (d) structure diagram of NIC component; (e) reflection spectra measured with different resistances; (f) reflection spectra measured with different external voltages

Fig. 14. Topological properties of 1D non-Hermitian topological dimer chain^[112]. (a) Schematic of a non-Hermitian topological dimer chain with 10 resonators; (b) real eigenfrequencies for different parameters gL; (c) phase diagram of edge states in non-Hermitian topological dimer chain; (d) wave functions of high frequency edge state ω+; (e) wave functions of low frequency edge state ω-

Fig. 15. Sensitivity of edge states in non-Hermitian topological dimer chain^[112]. (a) Photo of non-Hermitian topological dimer chain; (b) reflection spectrum of non-Hermitian topological dimer chain; (c) sensitivity of non-Hermitian topological dimer chain varying with frequency detuning; (d) sensitivity of non-Hermitian topological dimer chain varying with coupling strength disturbance