Topological Edge States And Waveguide Propagation Of The Two-dimensional Stampfli-triangle Photonic Crystals

Topological insulators have been widely studied because of their internal insulation,external conductivity and unidirectional properties.Their unidirectional lossless,scattering-immune bending-resistant transmission and other characteristics have been introduced into optics,providing a solution for optical communication to achieve low loss transmission,which has a great application prospect in optical switching,optical communication and computing fields.The research of photonic crystals and metamaterials has formed a complete research idea include of band theory,structural design and characterization methods,which provides an effective method for constructing artificial band gap materials to achieve optical topological states.The research of topological photonics initially focused on the quantum Hall effect,which breaks time reversal symmetry,but its applicable frequency is limited to the microwave frequency band.The quantum spin Hall effect（QSHE）,protected by time reversal symmetry has been realized in optical systems several years ago.Subsequently,topological states based on the different symmetry protection without introducing strong spin-orbit coupling have been proposed successively,such as the QSHE protected by C₆ symmetry,and the quantum valley Hall effect（QVHE）protected by C₃symmetry.Topological states based on the symmetry protection can be realized used all-dielectric materials,the lattices arrangement only needs to meet the corresponding symmetry,and the applicable frequency band can be adjusted by the lattice constant.Since photonic crystals only support 1,2,3,4 and 6 rotational symmetries,the QSHE based on photonic crystals can only achieve single-frequency and single-mode topological transmission,and the QVHE can only realize the single-frequency and multi-mode or multi-frequency and single-mode topological transmission,to some extent,limits the development of the topological edge state in the optical communication device with the low loss and high capacity.Photonic quasicrystals have higher symmetry and richer band gaps than photonic crystals,and have been proved to be able to achieve topological edge states.However,photonic quasicrystals cannot obtain their bloch band structure due to lack of translational symmetry.In this paper,two-dimensional（2D）Stampfli-triangle photonic crystals,composed of periodic distribution of the basic structural units of Stampfli type photonic quasicrystals,were used to realize the QSHE and QVHE in the optical system based on symmetry only,and the influence of the interface formed by the combination of topological state protected by C₃ symmetry and C₆ symmetry on the topological edge states was also discussed.The main contents and innovation value of this paper are as follows.（1）2D Stampfli-triangle photonic crystals was proposed,to realize the QSHE and its topological edge states.2D Stampfli-Triangle photonic crystals take more complex quasicrystalline basic structural units as the primary cell,which increases the tunability of topological structure the diversity of the methods for constructing topological edge states.Different from the previous work,which the topological edge states were constructed by expanding or shrinking the honeycomb lattice primitive cell or changing the inner and outer diameters of the annular scatterers.In this paper,QSHE and its topological edge states and unidirectional propagation protected by C₆ symmetry were realized by only changing the inner and outer ring scatterers diameters of the primary cell.The frequency of the topological edge state and the propagation path and direction of the unidirectional waveguide can be changed by adjusting the diameter of scatterers in the inner and outer ring.The topological edge state can be widely used in optical integrated circuits devices such as logic gate.（2）After removing the central scatterers of the primitive cell for 2D Stampfli-triangle photonic crystal,it is found that its structure can be obtained by using six scatterers as the primitive cell and arranging in a triangular lattice.By adjusting the scatterer diameter,the valley photonic crystal（VPC）which can realize the QVHE can be obtained.By changing the scatterers in valley photonic crystals,its degeneracies in the low-frequency and high-frequency bands were broken simultaneously,and the QVHE protected by C₃ symmetry with large valley Chern number was realized simultaneously in two frequency bands.The multifrequency and multimode topological transmission was realized through the U-shaped waveguide constructed with two VPCs with opposite valley Chern numbers.According to the bulk-edge correspondence principle,the Chern number is equal to the number of topological edge states or topological waveguide modes.Therefore,we can determine the valley Chern number of the VPC by the number of topological edge states or topological waveguide modes,further determine the realization of large valley Chern number.These results provide new ideas for high-efficiency and high-capacity optical transmission and communication devices and their integration,and broaden the application range of topological edge states.（3）The photonic crystals for topological trivial and nontrivial states which can realize the QSHE and valley photonic crystals which can realize the QVHE were selected as the research objects.The Wilson-loop distribution of the photonic crystals for topological trivial and nontrivial states and the Berry curvature distribution of the valley photonic crystal are calculated respectively,and the corresponding relationship between the topological invariants and the number of topological edge states was analyzed.Two topological edge states were obtained for the two interfaces respectively constructed by the combination of the photonic crystal for topological nontrivial state and the two groups of valley photonic crystals,while one topological edge state was obtained for the two interfaces respectively constructed by the combination by the photonic crystal for topological trivial state and two groups of valley photonic crystals.The realization of topological edge states was verified by constructing waveguides with the two groups of valley photonic crystals and photonic crystals which correspond to topological trivial and nontrivial states.By analyzing the parity of topological edge states with bandgaps in the four interfaces,it is found that some of the interfaces can realize topological corner states.Therefore,the box structures were constructed to verify the relationship between the interface-dependent topological corner states and topological edge states.The results are useful for the study of the interactions of topological states with different symmetry protection.