Mathematical theory for topological photonic materials in one dimension
Mathematical theory for topological photonic materials in one dimension
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DOI：
10.1088/1751-8121/aca9a5
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发表时间：
2021-01
期刊：
影响因子：
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通讯作者：
Junshan Lin;Hai Zhang
中科院分区：
文献类型：
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作者：
Junshan Lin;Hai Zhang
这一工作为一维拓扑光子材料提供了一个严格的理论。主要的焦点是由体结构的拓扑性质引起的界面模式的存在。对于具有时间反演对称性的一维光子晶体结构，我们研究了由狄拉克点引起的界面模的存在性。具体来说，我们建立的扰动，保证周围的狄拉克点的带隙的开放和存在的接口模式的条件。对于具有时间反演和反转对称的周期性光子结构，Zak相位被量化，仅取两个值0，π。我们表明，Zak相位是由奇偶校验（偶或奇）的布洛赫模式在带边。对于一个由两个半无限系统组成的光子结构，在一个具有不同拓扑指数的界面两侧，我们证明了在公共带隙内存在一个界面模。在扰动下的模式的稳定性也进行了研究。最后，我们研究了有限拓扑结构的共振。我们的结果是基于转移矩阵方法和Sturm-Liouville算子的振动理论。所得方法和结果可推广到一般的一维拓扑Sturm-Liouville系统。
This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence of interface modes that are induced by topological properties of the bulk structure. For a general 1D photonic structure with time-reversal symmetry, we investigate the existence of an interface mode that is induced by a Dirac point upon perturbation. Specifically, we establish conditions on the perturbation which guarantee the opening of a band gap around the Dirac point and the existence of an interface mode. For a periodic photonic structure with both time-reversal and inversion symmetry, the Zak phase is quantized, taking only two values 0,π . We show that the Zak phase is determined by the parity (even or odd) of the Bloch modes at the band edges. For a photonic structure consisting of two semi-infinite systems on the two sides of an interface with distinct topological indices, we show the existence of an interface mode inside the common gap. The stability of the mode under perturbations is also investigated. Finally, we study resonances for finite topological structures. Our results are based on the transfer matrix method and the oscillation theory for Sturm–Liouville operators. The methods and results can be extended to general topological Sturm–Liouville systems in one dimension.