We theoretically propose a Coupled-Corner-State Waveguide (CCSW), which is composed of a zigzag edge-like structure based on C-4 symmetrical lattice. CCSW mode is achieved by weak coupling between a sequence of higher order topological corner state (TCS). Based on the tight-binding (TB) approximation, the flat dispersion relation of CCSW mode is obtained, and suitable for slowing down light. The characteristics of slow light, including the group index and group velocity dispersion, are discussed in detail. At the eigenfrequency of individual TCS, the group velocity dispersion of CCSW mode is zero. Importantly, the CCSW mode shows strong robustness when introducing disorders, compared with the conventional Coupled-Resonator-Optical Waveguide (CCROW) based on photonic crystal defect cavities. Our findings may find topological slow light applications such as optical buffers, the processing of optical signals, optical delay lines and so on.

In this work, a photonic valley-locked heterostructure is proposed, which is composed of a Dirac photonic crystal (DPC) and two valley photonic crystals (VPC) with opposite valley Chern numbers. By modifying the size of rods nearest to the domain walls, the topological valley waveguide state (TVWS) with large group index (ng=100) can be found, which is called topological slow-light waveguide mode (TSWM). The simulation results based on finite element method demonstrate that the TSWM supports high energy capacity in the proposed heterostructure waveguide, which is suitable for integrating with the existing waveguides. Besides, TSWM is also valley-locked and immune to backscattering. Our findings pave a way of topological slow light, enrich the research of VPCs, and have new applications in optical communication devices.