C4旋转对称光子晶体平板中的对称保护连续谱束缚态

在光子晶体平板中,连续谱束缚态关于C2和C6旋转对称的依赖性已经在数值上进行了广泛研究,但是缺少严格的理论分析过程,此外还缺少对C4旋转对称的研究,鉴于此,构建了系统分析连续谱束缚态关于所有旋转对称的依赖性的理论,并且重点研究了C4旋转对称的情况;首先,通过分析具有旋转对称的结构中麦克斯韦方程组特征解的性质,将连续谱束缚态的存在性问题转变为旋转矩阵的特征值是否与一个简单代数方程的解相同的问题;其次,给出了C4旋转对称的结构中连续谱束缚态存在时所对应的条件;然后,证明了破坏C4旋转对称保持C2旋转对称时,连续谱束缚态依然存在;最后,利用有限元软件FreeFEM进行了大量的数值验证;上述理论可适用于所有旋转对称的情况,深入揭示了旋转对称对连续谱束缚态存在的重要性,深入揭示了高阶旋转对称性与低阶旋转对称性之间的依赖关系,为连续谱束缚态的实际应用提供了理论指导。

Symmetry-protected Bound States in the Continuum in C4 Rotationally Symmetric Photonic Crystal Plates

The dependence of bound states in the continuum(BICs) on C2 and C6 rotational symmetry in photonic crystal plates has been extensively studied numerically. However, a rigorous theoretical analysis process is lacking, and there is a lack of studies on C4 rotational symmetries. In view of this, a theory of systematic analysis of the dependence of BICs on all rotational symmetries was constructed, and the case of C4 rotational symmetry was mainly studied. Firstly, by analyzing the characteristic solutions of Maxwell''s equations with rotationally symmetric structure, the problem of the existence of BICs was transformed into the question of whether the eigenvalue of the rotation matrix was the same as the solution of a simple algebraic equation. Secondly, the conditions for the existence of BICs in C4 rotationally symmetric structures were given. Then, it was proved that the BICs still existed when C4 rotation symmetry was destroyed and C2 rotatory symmetry was maintained. Finally, the finite element software, FreeFEM, was used to do a lot of numerical verifications. The above theory can be applied to all cases of rotational symmetries, revealing the importance of rotational symmetry for the existence of BICs. The dependence between high-order and low-order rotational symmetries was revealed providing theoretical guidance for applying BICs.