Fig. 2: Dispersion relations and symmetries of TESC modes.
a Optical images of topological waveguide with a straight zigzag interface (brown line) and TESC with the zigzag interface forming a triangular closed path. The lattice constant a = 220 μm and ΔL = 0.3a. The side length of the TESC is L = 8a. b Calculated band diagram of TESC modes, presenting as topological edge states with discrete wavevectors. The wavevector of TESC modes is determined at the kx with the maximum intensity of the projected field in the momentum space, which is analyzed through spatial Fourier transformation of the electric field of eigen TESC modes (inset of b). Here ΔL = 0.7a and L = 16a are designed to show the 0th-order TESC mode at Γ point. δk = 2π/Reff represents the wavevector spacing between TESC modes, which is governed by the effective round-trip length of the cavity, Reff. c Simulated magnetic field Hz distributions of the eigen TESC modes. The TESC modes are noted as \({\text{TE}}_{P,I}^{m}\), where TE denotes transverse electric modes, m = 0, 1, 2, 3, … represents the order of modes, P = C3v, C1h corresponds to the point group, and I = A, B or A1, A2 represent the irreducible representations of the point groups C1h and C3v, respectively. The dashed black lines indicate the reference lines where the mirror operation is applied
