Fig. 1. (a) Schematic diagram of the proposed non-Hermitian 2D PC geometry with a triangular lattice of air holes embedded in a dielectric substrate. (b) Dispersion bands of 2D PC with a complete photonic bandgap (gray stripe) at na=1 and n=3.45+nI×i (nI=0.5), where the inset shows the unit cell with high symmetry points. (c) Chern number distribution as a function of nI, which shows phase transition from topology to triviality with the increasement of loss. (d) Zak phase evolution for nI=0.5, and ξ is the displacement parameter along the a2 direction.

Fig. 2. Topological rainbow effect at the interface of two kinds of PCs with different topological properties in non-Hermitian system. (a) Schematic diagram of the proposed 2D topological PC. The solid black line illustrates the interface between trivial PC (normal lattice) and topological PC (twisted lattice). (b) Map of the group velocity distribution as a function of ξ and frequency. All dashed lines denote zero group velocity. (c) Normalized electric field distributions of |E|2 along the y direction.

Fig. 3. (a) Position and electric intensity distribution for the photonic state of frequency 0.5156c/a, which is controlled by loss. Loss can tune the operating frequency of interface states. (b) Normalized electric field distributions of |E|2 of the certain state in (a) under different losses.

Fig. 4. (a) Dispersion bands of 2D non-Hermitian PC of triangular lattice with a complete photonic bandgap (gray stripe). Blue and orange solid lines represent TM and TE mode bulk states, respectively. (b) Normalized electric field distributions of |E|2 of topological rainbow phenomenon in TM mode.

Fig. 5. (a) Structure of proposed non-Hermitian 2D PC with fewer disorders. (b) Normalized electric field distributions of |E|2 of topological rainbow phenomenon with fewer disorders in TE mode.

Fig. 6. (a) Structure of proposed non-Hermitian 2D PC with bigger disorders. (b) Normalized electric field distributions of |E|2 of topological rainbow effect with bigger disorders in TE mode, which are affected by disturbances.

Fig. 7. (a) Calculated energy bands of unit cells for different radii, which show apparent changes of bandgaps. (b) Normalized electric field distributions of |E|2 of topological rainbow phenomena in cases of r=0.46a and r=0.5a. The performances of rainbow devices are affected by the increase of radius.

Fig. 8. (a) Schematic diagram of the proposed 2D topological rainbow structure with different angles. (b) Normalized electric field distributions of |E|2 along the y direction when the rotated angle is 12°.