Research On Line States And Topological Insulator In Photonic Lattice

The photonic lattice has received extensive attention from many condensed matter physics and optics scholars because of its periodic structure and unique characteristics.Reasonably adjusting the photonic lattice parameters,setting the propagation path and predicting the transmission result have important research significance and application value in the fields of optical imaging.It has become one of the research hot-spots in recent years to achieve the goal of optical localization by using optical lattices.The idea of using optical lattices to realize light wave regulation comes from the regulation of electronic wave functions by semiconductor lattices,because the mathematical equations describing the two are the same in form.In the photonic lattice,photonic localization can be achieved by applying disorder,magnetic field,designing defects,and using the nonlinear effect of photonic crystals.However,these methods all modulate the photonic crystal,which increases the difficulty of experiment and application.In recent years,researchers have proposed a method that does not require additional modulation,and only uses the topology of the lattice band structure to offset the discrete diffraction of light waves during propagation,thereby realizing photon positioning.Such a system is called a flat-band system.Flat-band systems only exist in photonic lattices with specific structures,such as Lieb lattices,Kagome lattices,and super honeycomb lattices.The flat-band system has a perfect compact local state that can be linearly superimposed and has been widely studied.In recent years,studies have found that there are new types of “flat-band line states” in the flat-band system of truncated two-dimensional photonic lattices(Lieb and Super-honeycomb lattices),which still have strong optical local characteristics,showing straight or zigzag line.Edge states in photonic topological insulators have broad application prospects in the fields of optical transmission and optical quantum computing.Because there are boundary states that are topologically protected and suppress backscatter in the photonic topological insulator.Therefore,the study of photonic topological insulators has become a research hot-spot in the field of photonics.The combination of the concept of the topological phase of condensed matter and the optical system has given birth to a series of novel physical phenomena,such as the optical integer quantum Hall effect,the optical quantum spin Hall effect,and the optical Floquet topological insulator.Studies have shown that in Lieb and honeycomb lattices,the tight-binding method is used to introduce spin-orbit coupling to produce the quantum spin Hall Effect,and topologically non-trivial energy gap is opened,thereby obtaining edge states.In this paper,the “Flat-band line states” and “edge states” in the Kagome lattice are introduced respectively,and the system of photon localization and edge states are improved based on the above two points.The main contents are as follows:(1)First,we analyzed the band structure and flat band eigenvectors of the Kagome photonic lattice through the tight-binding method,and the “Flat-band line states” mode was theoretically analyzed;Then based on the paraxial Schr(?)dinger equation,we numerically simulated the strong local characteristics of the “Flat-band line states” in the continuous Kagome lattice model by using the split-step Fourier algorithm.(2)Secondly,we constructed Kagome photonic lattices with three semi-infinite boundary structures(namely,mixed boundary,polyline boundary and rocking chair boundary).And the spin-orbit coupling is introduced into the tight-binding model of the three photonic lattices,the energy band is deformed,and the topological edge state is generated.It needs to be pointed out that adjusting the spin-orbit coupling coefficient can produce two topological edge states with different frequencies based on the mixed boundary Kagome lattice.This work lays the foundation for the realization of topological lasers based on the Kagome lattice introducing spin-orbit coupling.