Stability And Numerical Calculation Of Symmetry-protected Bound States In The Continuum On A Three-dimensional Photonic Crystal Plates

The bound states in the continuum are a kind of squared-integrable feature solutions of the system of Maxwell equations in the open structure.The resonant mode with an infinite quality factor exists only at a number of discrete frequencies.The special properties of resonant modes with arbitrarily large quality factors near the bound states in the continuum make them applicable in broad fields of optics and photonics.The continuum bound states in symmetric structures can be divided into two categories:symmetry-protected bound states in the continuum and asymmetryprotected bound states in the continuum.The existence of symmetry-protected bound states in the continuum has been proved mathematically.This thesis mainly studies symmetry-protected bound states in the continuum in rotationally symmetric structures.When the structure is slightly disturbed,the bound states in the continuum may disappear into a resonant mode,or evolve into another bound states in the continuum.It is very important to study the stability of the bound states in the continuum for further understanding their properties and practical applications.Symmetry-protected bound states in the continuum can be found in photonic crystal plates with C2,C3,C4,C6 rotationally symmetric properties.There are some numerical results on the dependence of symmetry-protected bound states in the continuum on the rotational symmetry of C2 and C3,but there is a lack of rigorous theoretical analysis.In this thesis,the dependence of symmetry-protected bound states in the continuum on the rotational symmetry of C4 and C6 is studied.Firstly,the problem whether a characteristic solution is a bound state in the continuum is transformed into the problem whether the eigenvalues of the rotation matrix are the same as the solutions of a simple algebraic equation by analyzing the properties of characteristic solutions of the system of Maxwell equations with rotational symmetry.Secondly,the dependence theory of symmetry-protected bound states in the continuum on different rotational symmetries is established.Then,the evolution modes of the bound states with different symmetries are discussed when C4 rotational symmetry changes to C2 rotational symmetry and C6 rotational symmetry changes to C3 or C2 rotational symmetry.The conclusions are as follows:(1)For the symmetrically protected continuous spectrum bound state in the C4 rotationally symmetric structure,the continuous spectral bound state still exists when C2 is kept by breaking C4.(2)For the non-degenerate continuous spectrum bound states in C6 structure,① Breaking C6 rotation symmetry and keeping C3 rotation symmetry,bound states in the continuum still exists;② When the C6 rotation symmetry is broken and the C2 rotation symmetry is maintained,bound states in the continuum corresponding to the irreducible representation still exists,but bound states in the continuum corresponding to the irreducible representation does not exist,and becomes a resonant mode.(3)For the double-degenerate symmetryprotected bound states in the continuum in C6 structure,①Destroys C6 rotation symmetry and maintains C3 rotation symmetry,so that the double-degenerate symmetry-protected bound states becomes a pair of non-degenerate resonant modes;When C6 rotation symmetry is broken and C2 rotation symmetry is maintained,the double degenerate symmetry-protected bound states in the continuum are transformed into a pair of nondegenerate symmetry-protected bound states in the continuum.Finally,a large number of numerical experiments are carried out by using the finite element method through FreeFEM programming software to verify the correctness of the above theory.