Characteristics And Applications Research Of Topological Photonic Crystal Edge States And Corner States

The quantum Hall effect and quantum spin Hall effect demonstrate the electron motion properties of the one-way and anti-backscattering at the interfaces,resulting in the emergence of topological interface states with the global and stable characteristics.Hence the concept of topological insulators is proposed.Compared to electrons limited by the Fermi level,photons have faster transmission speed,lower energy consumption,larger bandwidth,and stronger anti-interference capability as information carriers.Researchers have found that optical topological insulators analogous to electronic topological insulators can be realized in the photonic crystal systems.The photonic crystal is artificial structure composed of different materials arranged periodically in space,which has properties such as photonic bandgaps and localization.Topological interface states based on photonic crystals have attracted extensive attentions of researchers and have been intensively researched.Topological interface states provide a new idea and method for signal transmission and modulation.For example,one-dimensional topological edge states can transmit signal with robustness,and zero-dimensional topological corner states can achieve strong localization of signals.Based on topological band theory and topological polarization theory,we analyze the forming conditions of topological interface states,construct different combination structures with trivial and nontrivial photonic crystal,realize different types of topological edge states and topological corner states,and study the formation mechanisms,transmission properties,localization properties,modulation methods,and application prospects of one-dimensional edge states and zero-dimensional corner states.The main research contents are as follows:（1）Investigation on the formation mechanisms and control methods of one-dimensional topological edge states.A square photonic crystal with C₄ symmetry is constructed by circular and semi-circular dielectric columns.Trivial and nontrivial photonic crystals can be obtained when the band inversion and the topological phase transition are achieved by adjusting the positions of the dielectric columns of the square lattice photonic crystal.One-dimensional topological edge states are realized by combining trivial and nontrivial photonic crystal structures,and the strong robustness of topologically protected one-dimensional edge states is studied in the presence of defects such as disorder and impurities.The working frequency range of one-dimensional topological edge states is controlled by fine-tuning the size of the dielectric columns at the interface of the photonic crystal combined structure.Additionally,a kagome photonic crystal structure with C₃ symmetry is constructed by triangular dielectric columns,and the trivial and nontrivial photonic crystals can be obtained when the band inversion and the topological phase transition are achieved by expanding or shrunking the photonic crystal lattice,resulting in the open of the bandgap at the high symmetry points in the first Brillouin zone.One-dimensional topological edge states are realized by combining trivial and nontrivial photonic crystal structures in their common photonic bandgap.We find that two different topological edge states can be generated by exchanging the positions of trivial and nontrivial photonic crystals.（2）Research on the formation mechanisms and control methods of zero-dimensional topological corner states.Square photonic crystals with C₄ symmetry are constructed by circular and semicircular dielectric pillars,as well as circular and 90°sector-shaped dielectric pillars.The formation conditions of topological corner states are analyzed using the edge polarization theory.Zero-dimensional topological corner states can be achieved in a square box structure composed of different topological photonic crystals and the robustness of these states to defects such as impurities and disorder is studied.The frequencies of zero-dimensional topological corner states can be tuned by introducing point defects or adjusting the dimensions of the corner pillars.In addition,a triangular lattice structure photonic crystal is constructed to analyze the role of nearest-neighbor coupling and next-nearest-neighbor coupling in the formation of zero-dimensional topological corner states.Various types of topological corner states are achieved in the triangular and parallelogram box structures consist of two basic topological insulator structures with the kagome structure photonic crystal,and the characteristics of different types of corner states are studied.Finally,the energy intensity of localized zero-dimensional topological corner states is significantly improved by the nearest-neighbor coupling and the superimposition of corner states in a hexagonal combined structure.（3）Application research on topological edge states and topological corner states.Based on a square lattice photonic crystal,a one-dimensional topological photonic crystal edge state waveguide is constructed,and the strong topological protection robustness of edge state waveguides to defects is studied.By adjusting the size of the dielectric pillars at the interface between the trivial and nontrivial photonic crystals,the overlapping frequency band between one-dimensional topological edge states and zero-dimensional topological corner states is achieved.A resonant system supporting both topological edge state waveguides and topological corner state cavities is constructed and we further optimize the resonant system by introducing point defects and reducing the number of corner states to eliminate the coupling split,narrow the resonant bandwidth,and increase its quality factor to 9900 approximately.In addition,one-dimensional topological edge state waveguides with different edge structures are constructed by two types of kagome lattice photonic crystals.And a single-channel photonic crystal waveguide that supports dual frequency bands is realized in a photonic crystal system with sandwich structure.Finally,a topological frequency divider on the photonic crystal platform is achieved when a Y-shaped branching structure is constructed at the end of the waveguide to output signals with different frequency bands from different ports.The research results can be applied to integrated photonic devices such as photonic waveguides,wavelength division multiplexers,resonators,and lasers.