Topological Edge Modes in Non-Hermitian Photonic Aharonov-Bohm Cages

We investigate the topological edge modes in a rhombic waveguide array with imaginary coupling, which is realized by incorporating auxiliary waveguide between the coupled waveguides. By suitably tuning both real and imaginary couplings to generate an effective π flux, a non-Hermitian analog of Aharonov-Bohm (AB) cage is formed as the band structures become flat and coalesce into third-order exceptional points (EPs). We show the array can support gapped topological edge modes, which can be explained by mapping the system Hamiltonian into the square root of an anti-parity-time (PT)-symmetric waveguide array. In contrast to the linear power increase of bulk modes, the propagation of edge modes can be conserved or exponentially amplified depending on which termination is initially excited. Our study provides a promising way to realizing robust light propagation.