Fig. 1. Topological surface states with on-demand dispersion. (a) Schematic of the topological surface states with on-demand dispersion. The dispersion of the topological surface states can be effectively manipulated by modifying the structural parameter ds. (b) The unit cell of the 2D kagome PhC supporting topological surface states. The lattice constant is a, and the 2D unit cell is formed by three dielectric cylinders with a radius r=0.15a. The relative permittivity εr is 9.6. (c) Band diagrams of kagome PhCs with (blue, d=0.5a) and without (red, d=0.56a) inversion symmetry. The valley gap generated at the K point by breaking the inverse symmetry of the unit cell is indicated by the gray region. (d) Phase distributions of Ez at the K point for the first and second bands. (e) Berry curvature for the first band of the unit cell without inversion symmetry shown in (c). (f) The super cell of the 2D PhC supporting topological surface states. The radius of the boundary cylinder is re, and the distance between the boundary cylinder and the PhC is ds. (g) The eigenmode dispersion of the super cell shown in (f) with different structural parameters, where a typical electric field distribution Re(Ez) of the surface state is also provided in the inset. (h), (i) Schematics for the dynamical spatial trapping of the EM pulse and backward beam routing mediated by topological surface states with on-demand dispersion, respectively.

Fig. 2. Eigen- and excitation solutions of topological surface states in a PhC slab. (a) The band diagrams of the kagome PhC with (blue, d=0.5a) and without (red, d=0.6a) inversion symmetry, where the 3D unit cell is shown by the inset. The height of the dielectric cylinders is h=1.3483a. The light and dark gray areas indicate the light cone and the bandgap, respectively. (b) The eigenmode dispersion of the super cell for different structural parameters ds. (c) The corresponding calculated group indices of the topological surface states for different structural parameters shown in (b). (d), (e) Real parts of Ez, Poynting vectors, and phase distributions of Ez for the cases of ds=0.3a and ds=1.0a, respectively. (f) A finite PhC slab consists of 23 periods along the propagation direction. The red star indicates the location of the CPMD sources, where the energy fluxes PL and PR are calculated at two planes indicated by two blue dashed boxes (1.34a×3.75a) labeled by L and R. Both planes are 3.5a away from the boundary of the waveguide along the Y-axis, where parts of them are inside the PhC (2.2a in length) to take the whole surface state into account. The background material is air. (g)–(j) The calculated normalized transmission spectra for different structural parameters ds, which are obtained by integrating the power in the blue dashed boxes (see Appendix A). The value “1” (“−1”) represents the energy flux exclusively to the left (right) side. The red and blue curves represent the transmission spectra under the excitation of pseudospin-down and pseudospin-up CPMD sources, respectively. (k) The corresponding distributions of electric field |E| under the excitation of the same CPMD source (pseudospin-down) shown in (g)–(j), where the red stars indicate the locations of the CPMD sources.

Fig. 3. Steady results of microwave experiments. (a) The experimental setup and fabricated dielectric PhC slab to demonstrate the topological surface states with on-demand dispersion. (b)–(e) Experimentally acquired |S21| (red points) and calculated normalized electric field amplitude |Ez| (blue solid lines) for different structural parameters ds acquired at the blue star marked in (a) under the excitation of a loop antenna. (f)–(i) The distributions of |S21| for different ds measured by a monopole antenna at the red dashed box outlined in (a). (j)–(m) Experimentally retrieved band structures calculated by the Fourier transform of the steady states and their corresponding simulation results (white dotted line).

Fig. 4. Spatiotemporal results of microwave experiments. (a)–(c) Evolutions of experimentally retrieved |Re(S21)| in spatial and time domains, which are acquired along the blue line (Fig. 13 in Appendix G), and the corresponding simulation results are also provided. The red arrows marked in (b) indicate the dynamical positions of the Gaussian pulse. (d)–(f) Fourier transform of experimental results at every moment. The zoom-in views for I, II, and III marked by the white dashed boxes are provided. The simulation results are also provided.

Fig. 5. Snapshots of spatiotemporal results in experiment. (a)–(c) The near-field distributions of |S21| for different ds at four typical moments.

Fig. 6. Experimental results of backward beam routing. (a) The experimental setup and fabricated dielectric PhC slab to demonstrate the beam routing effect. (b), (c) Experimentally acquired |S21| (red points) and simulated electric field amplitude |Ez| (blue solid lines) for different structural parameters ds obtained at the positions marked by the blue and red stars in (a), respectively, where the excitation of the loop antenna is placed on the dielectric waveguide. (d), (e) The acquired distribution of |S21| for ds=0.3a and ds=1.0a measured by a monopole antenna. Red arrows and blue areas indicate the direction of transmission and the location of the PhC slab, respectively. (f), (g) The typical near-field distributions of |S21| at a specified moment (t=114.009  ns). The central frequencies of the reconstructed pulses are set to 4.362 GHz and 4.507 GHz, respectively. And the corresponding pulse width and offset are set to 40 ns and 100 ns, respectively.

Fig. 7. Excitation solutions of topological surface states in the 2D PhC structure. (a) Schematic of the 2D PhC structure for topological surface states. The location of CPMD sources is marked by the red star. (b), (c) The calculated normalized transmission spectra for topological surface states with different structural parameters ds, which can be obtained by integrating the power along the gray dashed lines (L and R) marked in (a) (see Appendix A). The red and blue curves represent the transmission spectra under the excitation of pseudospin-down and pseudospin-up CPMD sources, respectively. (d), (e) The corresponding distributions of electric field |E| under the excitation of CPMD sources with opposite pseudospin states.

Fig. 8. Eigen-solutions of topological surface states in a 2D PhC structure. (a) The eigenmode dispersion of the super cell shown in Fig. 1(f) with different structural parameters ds. (b) The corresponding distributions of Re(Ez) and Arg(Ez) for the topological surface states at kx=1.1π/a shown in (a). (c) The eigenmode dispersion of the super cell shown in Fig. 1(f) with different structural parameters re. Here, ds=0.5a (a is the lattice constant). (d) The corresponding distributions of Re(Ez) and Arg(Ez) for the topological surface states at kx=1.1π/a shown in (c).

Fig. 9. Topological surface states in a PhC slab with finite height. (a) The 3D unit cell of the kagome PhC supporting topological surface states. The lattice constant is a, and the 3D unit cell is formed by dielectric cylinders with a radius r=0.15a and a height h=1.3483a. The relative permittivity εr is 9.6. (b) The band diagrams of the kagome PhC with (blue, d=0.5a) and without (red, d=0.6a) inversion symmetry. The light and dark gray areas indicate the light cone and the bandgap, respectively. (c), (d) Distributions of Re(Ez) and Arg(Ez) of the gapless PhC at the K point for the first and second bands. (e), (f) Distributions of Re(Ez) and Arg(Ez) of the non-trivial PhC with a valley gap at the K point for the first and second bands. (g) The 3D super cell of the PhC slab supporting topological surface states. The corresponding distributions of Re(Ez) and Arg(Ez) at kx=1.1π/a for different structural parameters ds are also provided.

Fig. 10. Excitation solutions of topological surface states in the PhC slab. (a)–(d) The distributions of electric field |E| under the excitation of the same CPMD source (pseudospin-up) for the cases of ds=0.3a, ds=0.5a, ds=0.8038a, and ds=1.0a.

Fig. 11. Topological surface states under the excitation of the magnetic dipole source. (a)–(d) The near-field distributions of electric field amplitude (|Ez|) for the cases of ds=0.3a, ds=0.5a, ds=0.8038a, and ds=1.0a. The red star represents the location of the magnetic dipole source.

Fig. 12. Benchmarks of the retrieved propagation dynamics using simulated data in the frequency domain. (a) A finite PhC slab supporting topological surface states. The location of the magnetic dipole is marked by the red star. The distributions of Re(Ez) in the frequency and time domains are extracted by the 3D FDTD method at the red dashed box. (b)–(d) Retrieved time dependent Re(Ez) using the frequency domain data obtained by the 3D FDTD method and the direct output results of the 3D FDTD method at the blue star marked in (a). The retrieved near-field distributions [Re(Ez)] at the red dashed box and the direct output results of the 3D FDTD method at typical moments are also provided. The central frequencies of the Gaussian pulse are set to 4.361 GHz, 4.465 GHz, and 4.511 GHz for the cases of ds=0.3a, ds=0.8038a, and ds=1.0a, respectively, while the width and offset of all pulses are 40 ns and 100 ns, respectively.

Fig. 13. Propagation dynamics of topological surface states. (a) Schematic of the finite PhC slab supporting topological surface states. The location of the magnetic dipole is marked by the red star. (b), (c) Evolutions of experimentally retrieved Re(S21) in time domain for the cases of ds=0.5a and ds=0.7a, which are both acquired along the blue line shown in (a). The corresponding 3D FDTD results are also provided. The central frequencies of the Gaussian pulses are set to 4.384 GHz and 4.427 GHz for the cases of ds=0.5a and ds=0.7a, respectively, while the width and offset of the two pulses are 40 ns and 100 ns, respectively.

Fig. 14. Snapshots of time domain experimental results. (a), (b) The near-field distribution of |S21| for the cases of different ds at four typical moments.

Fig. 15. The PhC slab structure for beam routing. The top view of the PhC structure and the straight coupling waveguide shown in Fig. 6.