Research On Topological Phenomena In Photonic Crystals With Dispersive Background

With the discovery of topological materials,the characteristics of unidirectional edge states and robustness to perturbations have caused much research interests,and topological electronics has also become a hot field for studying this new state.Meanwhile,in the fields of optics,acoustics and circuits,analogizing to the fermion system,people have also realized a variety of topological phenomena,which developed into a new research direction in optical field,i.e.,topological photonics.Compared with electronic system,the matter in optical system is not necessarily quantum or condensed states,which makes it physically easier to understand and implement.Moreover,the band structure in optical system can be flexibly designed through the lattice structure and materials without the limitation of Fermi level,which greatly increases the tunability and controllability,and may also help to discover the phenomena that are hard to be observed in the electronic system.In this thesis,we mainly discuss the realization of several topological states and the related theories in two-dimensional dispersive photonic crystal,from the study of first-order photonic topological insulators and edge states to the higher-order photonic topological insulators and corner states.We compare the numerical results with the tight-binding calculations and verify their correctness,thus discovering interesting topological phenomena in optical systems.The thesis mainly includes topics as follows:Utilizing the coupled air annuluses in a honeycomb structure,we realize the high transmission phenomenon at the Dirac-like point.Considering a two-dimensional bulk Drude metal material,we drill air annuluses in it to form a honeycomb lattice with C6v symmetry.Then,in the band structure,we observe the bonding and anti-bonding states,and an optical tight-binding approximation can be formed by the coupling of evanescent waves between adjacent modes.When we further tune the ratios of inner and outer radii,band inversion can be achieved at the ? point.Meanwhile,the accidental degeneracy of the fundamental mode and the dipole mode forms a triple degeneracy point,i.e.,Dirac-like point.By numerical calculation,we find that the electromagnetic wave can pass through the whole system at the frequency of Dirac-like point.Therefore,this honeycomb system achieves high transmission in the dispersive metal materials,even with losses of metal,the PC structure is still helpful in enhancing the propagation of light when compared to a purely metallic system.Second,using a gear metamolecular structure,we design a triangle topological insulator with C6 symmetry.In this system,by rotating the gear structure,the symmetry will be reduced and we can observe the bandgap closing and re-opening between two higher-order compound modes.This is the band inversion mechanism.After that,a new kind of metamolecular crystals with topological nontrivial phases is formed.At the interface between two topologically distinct metamolecular crystals,we finally observe the edge state transport in a metallic system.We study the changes of localized corner state of a two-dimensional breathing Kagome lattice under different truncated boundaries.First,we analyze the topological properties of Kagome lattice under the two transformations of stretching and compressing.Then,for a finite Kagome lattice with C3 and mirror symmetries:when the boundaries have flat truncations,there are three localized zero-energy modes under the stretching lattice,i.e.,the second-order corner states,which are protected by topological character.When we change the cutting boundaries,which do not cross the Wannier center,a breathing Kagome lattice is thus obtained with three bearded truncations.In this supercell,we find the bonding and anti-bonding corner states in the compressing trivial structure,which are protected by geometry symmetry,while the bound states in the continuum are observed in the stretching topological structure.Using the couplings between multi orbital modes,we realize a plasmon-polaritonic quadrupole topological insulators.The key ingredient is the sign-reversal mechanism for the couplings between the plasmon-polaritonic cavities from the pole-pole orientation to the node-node orientation,which satisfies the conditions of three positive couplings and one negative coupling in each unit cell.Besides,we also study the evolution of the Wannier bands and eigenmode spectra,and observe the localized corner states.These topological states exist in the lattice structure where the topological invaritant is nonzero—polarization is 1/2,and the quadrupole moment is protected by mirror symmetry.At last,we further verify the edge states under different structural parameters,including quadrupole edge state and those in weak topological phases.We construct an optical Lieb lattice by using Drude metal as background,and study its band structure of monopole,dipolar and quadrupole modes.The compact localized states(CLSs)as well as its destructive wave interference of flat bands are also analyzed here.Comparing with the tight-binding method,we demonstrate the validity of designed photonic model to calculate the band structure of s,p and multi orbital modes under nearestneighbor couplings.Interestingly,in such a system,we numerically obtain the noncompact photonic zero-energy modes for both monopole and dipolar states in a finite Lieb lattice with flat truncations,which belong to the missing states that result from the incompleteness of the CLSs at flat band.Moreover,the photonic zero-energy modes under chiral symmetry-breaking are also discussed.