Fig. 1. Topological states in momentum space. (a) Robust topological edge state based on integer quantum Hall effect^[15]; (b) topological photonic crystals based on quantum spin Hall effect^[102]; (c) topological photonic crystals and robust transmission states based on valley Hall effect^[18]

Fig. 2. Topological states in synthetic dimension space. (a)‒(c) Synthetic dimensions constructed by introducing intrinsic degrees of freedom, including angular momentum, frequency, and spatial modes^{[128, 131, 134]}; (d)‒(g) synthetic dimensions constructed by introducing system parameters, including inter-waveguide separation, dielectric layer thickness, and lattice translation parameters^{[71, 74, 138, 140]}

Fig. 3. Schematic diagram of Brillouin zone discretization^[153]. (a) Schematic diagram of discretized Brillouin zone in a square lattice; (b) schematic diagram of discretized Brillouin zone in a hexagonal lattice

Fig. 4. Topological rainbow based on gradient magnetic fields^[23]. (a) Schematic diagram of strongly coupled topological states in a magneto-optical photonic crystal waveguide; (b) projected band structure of the magneto-optical photonic crystal waveguide; (c) energy band distribution at different magnetic field strengths; (d) group velocities distribution at different magnetic field strengths; (e) spatial distribution of electromagnetic waves at different frequencies; (f) schematic diagram of broadband slow-light rainbow trapping in a magneto-optical photonic crystal waveguide with gradient magnetic fields

Fig. 5. Topological rainbow based on structural height gradients^[25-27]. (a)‒(c) Regulation of photonic crystal dispersion in honeycomb lattice by water column height and topological rainbow trapping based on surface acoustic waves; (d)‍(e) topological rainbow trapping based on the graded-SSH‒metawedge; (f)(g) periodically perforated soil plate varying in depth and topological rainbow in the elastic wave system

Fig. 6. Topological rainbow based on contraction-expansion lattices^[33-34]. (a)‍(b) Schematic diagram of contraction and expansion of lattices and topological rainbow in the interface; (c)(d) schematic diagram of gradual sandwiched structure and double modes of topological rainbow

Fig. 7. Topological rainbow based on rotation angle gradients. (a)‒(e) Topological rainbow based on rotation angle gradients in phononic crystals^[44]:^{(a) two-dimensional triangular lattice phononic crystal with rotation angle θ of blue region relative to y-axis; (b) dispersion relations under different rotation angles; (c) gradient phononic crystal plate with rotation angle variation; (d) dispersion relations of edge states under different rotation angles; (e) normalized intensity distribution along the lower boundary of Fig. 7(c). (f)‒(h) Topological rainbow based on gradient rotation angle of valley photonic crystals[45]: (f) schematic of valley photonic crystal unit cell and band structure for a given rotation angle; (g) valley photonic crystal plate with rotation angle variation; (h) rainbow distribution at different frequencies}

Fig. 8. Topological rainbow based on dielectric filling factor gradients. (a)‒(c) Topological rainbow based on one-dimensional chirped topological photonic crystal^[48]: (a) schematic diagram of one-dimensional photonic crystal heterostructure; (b) electric field intensity distribution of unidirectional rainbow trapping at different wavelengths; (c) electric field intensity distribution of bidirectional rainbow trapping at different wavelengths. (d) Topological rainbow based on a hybrid gyromagnetic photonic crystal waveguide composed of different cylinder radii and waveguide widths^[158]

Fig. 9. Topological rainbow based on the assembly of different higher-order topological states. (a)‍(b) Symmetry-breaking mechanisms in Kagome photonic crystal lattices and four common lattice configurations with corresponding band structures^[160]; (c)‒(e) microwave photonic crystal structures and experimentally observed higher-order topological rainbow distributions^[160]; (f)‒(h) experimental structures in elastic wave systems and observed higher-order topological rainbow distributions^[159]

Fig. 10. Topological rainbow based on material loss gradients^[69-70]. (a) Square lattice dielectric pillar structure composed of a normal lattice on the left and a shifted lattice on the right, with a shift of half a lattice period and a loss gradient along y-direction; (b) group velocity distribution as a function of loss and frequency, with the yellow dashed line indicating zero group velocity; (c) experimental structure; (d) experimentally observed non-Hermitian topological rainbow phenomenon; (e)(f) rotational lattice and topological rainbow achieved by tuning the material loss of photonic crystal dielectric plates

Fig. 11. Topological rainbow based on synthetic dimensions^[69,71]. (a) Schematic of photonic crystal structure, where the blue region represents dielectric column and the gray region represents air; (b) band structure of the unit cell in TM mode, with the orange region indicating the bandgap; (c) evolution of the Zak phase with translation; (d) group velocity distribution in synthetic dimension space; (e) electric field distribution in TM mode, where topological photonic states are localized at different positions along the interface at various frequencies, leading to the topological rainbow phenomenon; (f) two-dimensional photonic crystal structure formed by stitching two triangular lattices with different rotation angles; (g) electric field distribution in TE mode

Fig. 12. On-chip topological rainbow based on synthetic dimensions^[72-73]. (a)‍(b) Schematic and SEM image of geometric structure of topological rainbow device; (c) evolution of group velocity of topological edge states with translation ξ and wavelength; (d) surface morphology of topological rainbow device and experimentally observed rainbow phenomenon; (e) phononic crystal structure formed by stitching non-translated and translated lattices; (f) experimentally measured surface acoustic wave displacement field intensity at different frequencies

Fig. 13. Higher-order topological rainbow based on synthetic dimensions. (a)(b) Band diagram and electric field distribution of gapless topological corner modes^[74]; (c) acoustic higher-order topological rainbow sound intensity distribution^[62]; (d)(e) photonic crystal structures and electric field intensity distributions in higher-order topological rainbows^[75]

Fig. 14. Topological rainbow based on pseudo-electromagnetic fields^[83]. (a) Landau rainbow phenomenon under synergistic effects of pseudo-magnetic and pseudo-electric fields, where the introduction of pseudo-magnetic field generates a series of flat Landau levels, and the pseudo-electric field breaks the degeneracy of these Landau levels; (b)(c) implementation of pseudo-magnetic and pseudo-electric fields in photonic crystal systems; (d) schematic diagram of experimental sample for microwave near-field measurements; (e)(f) experimentally observed and numerically simulated Landau rainbow phenomenon

Fig. 15. Topological rainbow based on continuous Landau modes^[84-85]. (a) One-dimensional lattice with non-reciprocal nearest-neighbor coupling; (b) corresponding complex energy spectra, and the color of each dot indicates the eigenstate's position along x direction; (c) site-dependent amplitudes under steady state excitation; (d)(e) circuit sample schematics and measured steady-state voltage response