高品質因子(high-quality factor)有強度提高及光譜線寬變窄的優點，因此對於光學相關的研究而言，如何在系統中達到 high- Q 是非常重要的，其中有許多不同的方法能夠達到這樣的效應，在此選擇連續頻譜內的束縛態(Bound states in the continuum ,BIC)，此法顛覆傳統對於波的想法，雖然是束縛態，但存在於輻射波的連續頻譜中。而本篇論文主要是在一維波導陣列中以添加缺陷，利用調整系統的對稱性以及在不同的位置添加缺陷的方式，使其達到各種不同的效應，這邊模擬了三種不同的現象進行分析與探討。

首先是法諾共振下的光學漩渦(Optical vortices at Fano resonances)，從最初的三角形環狀結構開始，會發現在這樣的系統下能觀察到隨著頻率的改變，拓樸荷也產生了變化，不過因為激發點對稱性的關係缺少指向性，因此加入陣列結構並從左方激發，便成功觀察到了光學漩渦的現象，在不同拓樸荷的情況下，能量流會呈現順時鐘及逆時鐘的旋轉方向。但在能量流為零也就是法諾共振發生的頻率位置，拓樸荷±1彼此強度相同能量相互抵銷，使中間能量旋轉的地方場強為零，而其他地方仍有洩漏產生，表示他只是一般的侷限態。

為了將洩漏給抵消，將缺陷其中一側變為半無限的陣列結構，另一側則為特定個數的波導陣列，利用這樣的不對稱性就可以同時達到法諾及法布立－佩羅共振，能量會因為法布立－佩羅共振被侷限在特定個數的陣列波導中，半無限陣列結構的能量則會因破壞性法諾共振而沒有能量產生，進而達到法諾共振下的BIC。

接著在陣列結構的中央波導上下都加入缺陷，想利用這樣的對稱性來完成BIC，也就是被對稱保護的 BIC。此方法是利用模態的對稱性將陣列結構的能量給抵銷，使能量被侷限在兩個缺陷之中，而單一缺陷的三角環狀結構無法增加垂直方向的對稱性，因此無法在那樣的系統下達到被對稱保護的BIC，最後將材料改成石墨烯，期望在這樣的片狀結構也能達到BIC的效應。

關鍵字:連續頻譜內的束縛態、光學漩渦、法諾共振、有限差分時域法

We theoretically and numerically analyzed the bound states in a one-dimensional array with side-coupled waveguides. When one side-coupled system passes through the Fano resonance dip, its topological characteristics changes and the phenomenon of optical vortex occurs. However, the system only achieves a bound state with finite linewidth. By changing into a semi-infinite array with one side-coupled waveguide above a finite number in the array end, Fano and Fabry-Perot resonances can be achieved simultaneously in this system. The finite waveguide array end produces total reflection and the semi-infinite waveguide array produces destructive Fano resonance. We demonstrate that all the energy is restricted in a few waveguides and no field penetration beyond the side-coupled waveguide position and achieve bound states in the continuum (BIC). We further analyzed the BIC protected by symmetry by adding two up-down coupled waveguides to the array structure. The odd-symmetry mode of the vertical coupled structure can be decoupled from the even-symmetry mode of the array structure with the energy is confined in these two defects. This BIC is achieved by symmetry different from the above single defect case. Finally, the waveguide material is changed from dielectric to graphene to observe whether the BIC can be achieved in the 2D plasmonic material.

Keywords: bound states in the continuum, optical vortex, Fano resonance, FDTD

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