Theoretical Analysis And Numerical Calculation Of Parameter Dependence Of Bound States In Continuum

The continuous spectral bound state is a special kind of square integrable characteristic solution of time harmonic Maxwell equations,and its frequency lies in the continuous spectrum.Generally,the square integrable characteristic solutions of time harmonic Maxwell equations exist outside the continuous spectrum.However,for some special structures,the square integrable characteristic solution can also exist in the continuous spectrum.The continuous spectrum bound state has very special properties,such as Fano resonance near it,total reflection and transmission,resonance mode with arbitrary large quality factor,etc.it can be applied to hypersensitive sensors,low threshold lasers and nonlinear optics.Symmetry plays an important role in the existence of continuous spectrum bound states.Continuous spectrum bound states can be divided into symmetric protected continuous spectrum bound states and asymmetric protected continuous spectrum bound states.The bound state of symmetric continuous spectrum exists in symmetric structure,and its symmetry is incompatible with the symmetry of radiation wave,so it cannot be coupled.When the symmetry of the structure is destroyed,the bound state of symmetry protection continuous spectrum will disappear.At present,the bound state of asymmetric protection continuous spectrum only exists in symmetric structure.The parameter dependence of continuous bound states on arbitrary structural disturbances is of great significance in practical application,and it is related to its topological properties.The continuous spectral bound state has a topological charge,which is defined as the winding number in the polarization direction.When the symmetry of the structure is destroyed,a continuous spectrum bound state with nonzero topological charge is split into several pairs of circularly polarized states with semi-integer topological charge,and the total topological charge is conserved.Recently,it has been found that some scalar continuous spectral bound states can be found by adjusting a single structural parameter for a two-dimensional structure with a periodic direction and a symmetrical direction.In this paper,we analyze the vector continuous spectrum bound states in two-dimensional structures,and prove that by adjusting two structural parameters,we can find the typical vector continuous spectrum bound states with non-zero wavenumber in both constant direction and periodic direction,while other vector continuous spectrum bound states,including scalar continuous spectrum bound states we studied before,can be found by adjusting one structural parameter.Numerical examples verify our theory.By using the fullorder perturbation method,it is proved that this kind of vector continuous spectrum bound state exists in the form of curve in three-dimensional space with three parameters.The numerical results also show that when the structural parameters change,if the continuous spectrum bound state and circular polarization state are considered at the same time,the topological charge is conserved.Our research reveals the basic properties of continuous spectrum bound states in periodic structures,and provides a systematic method for finding continuous spectrum bound states in asymmetric structures.