Fig. 1. Geometric structure, bulk bands, Zak phase, and the interface states. (a) Non-Hermitian 2D PC structure with square lattices of dielectric column (n = 2.4 + n_Ii) embedded in the air (n = 1), in which different rows of the dielectric materials have different loss factors n_I. a is the lattice constant, and the radius r is 0.27a. (b), (c) Real and imaginary parts of dispersion bands of the 2D PC in TM mode, respectively, and n_I is 0.3. The inset is the corresponding unit cell. (d) Calculated values of the Zak phase for the translational lattice at the right of the structure shown in (a), which shows the topological properties of our proposed structure; (e) evolution of the Zak phase for the first bulk band as a function of loss factor n_I; (f) dispersion curves of different super unit cells, as shown in the inset. Each color curve corresponds to different loss factors n_I.

Fig. 2. Non-Hermitian topological rainbow in the interface of two kinds of PCs. (a) Group velocity contour as a function of n_I and frequency; (b) magnitude of electric field distributions of all interface states; Y represents the distance of any positions along the y direction, and the fluctuation of the curve shows the magnitude of E. (c) |E| field distribution of the topological rainbow trapping phenomenon of nine topological states shown in (b); (d) Poynting vector power flow |S| distribution of the topological rainbow trapping phenomenon.

Fig. 3. Samples and experimental setup; calculated and measured topological rainbow for square lattices in non-Hermitian systems. (a), (b) Proposed experimental structure consists of a square lattice PC array of wrapped alumina cylinders with a relative permittivity of 7. The thicknesses of lossy materials (sponges, the measured permittivity is 1.1 + 0.4i) for each group increase gradually from bottom to top. (c), (d) Simulated and experimentally measured |E| of topological rainbow in non-Hermitian systems.

Fig. 4. Robustness of topological rainbow. (a) Schematic of the PC for square lattices with defects; (b) corresponding measured electric field amplitude distributions |E|; the appearance of the topological rainbow has been slightly affected by disorders, which shows the robustness of our structures.

Fig. 5. Tuning topological rainbow. (a) Calculated electric field |E| distributions with different wrapping thickness of lossy materials at frequency 8.13 GHz; (b) experimentally measured electric field amplitude |E| distributions at frequency 8.13 GHz. Under the different loss distributions, the interface state stops at different spatial positions.