六角形共振腔依靠全反射可以將電磁波侷限在結構內，形成具有高品質因子的迴廊模態與準迴廊模態的共振。但是因為在模擬完整六角形共振腔的時候，我們很難去分辨同波長激發下兼併態的模態，因此我們根據共振模態的對稱性將結構與模擬空間切成四分之一，不僅可以由特殊的對稱性分開處理，並輕易地分辨同波長下的兼併態，還可以節省四分之一的模擬時間。之後我們又發現，在使用一般方形格子點模擬的時候，在六角形的斜邊會有stair case的情況產生，當模擬空間的解析度不夠的時候，會因為結構邊緣的平整度不好而造成模態無法有效的激發出來，為了解決stair case帶來的模擬誤差，我們特別的去開發正三角形的格子點，使得格子點的幾何結構能夠貼合材料本身的結構。三角形格子點的設置比正方形的網格相對複雜，本論將會有詳細的設置流程，當三角形格子點的設置經過模擬的測試沒問題後，我們就拿來模擬六角形氧化鋅晶柱以及六角形石墨烯薄片的結構。六角形氧化鋅晶柱可以視為一二維介電質的模擬，所以我們使用二維的三角形格子點成功的模擬5~12階的模態，並與迴廊模態的理論的計算進行比較。而六角形石墨稀薄片我們使用三維的三角形格子點來模擬各模態，在過去使用方形格子點時只能免強的看到第10階的模態左右，但是三角形格子點可以相對容易的模擬出10階以上的模態，這也代表成功的解決了stair case所造成的問題，不過在石墨烯的例子中Fabry-Perot mode卻因為stair case的解決變的更明顯，甚至有主導共振腔模態的現象。總結來說三角形格子點的開發與模擬結果是可以接受的，他成功的模擬出了六邊形結構共振腔的模態，也解決了stair case的問題，並利用對稱性分辨出同波長下的兼併態，也有效的節省的模擬時間，不過因為stair case被解決後衍生出來而外的問題就有待後續想其他的辦法解決。

The hexagonal cavity confines the electromagnetic field inside the structure by total reflection, and has high Q-factor caused by the whispering gallery modes or Quasi- whispering gallery modes. However, it is difficult to discriminate the degenerated modes with the same wavelength when the whole hexagonal structure is used in the simulation. Hence, according to the symmetry of the hexagonal cavity, only a quarter of the hexagonal cavity with x-y axis symmetry can be used. It can discriminate the degenerated modes and reduce the computational time. Another issue found in the simulation is that the “stair case” will appear on the edge of the quarter hexagonal structure when square mesh is used. The stair case will not correctly excitate the modes in low resolution. In order to fix the problem, a “triangular mesh” FDTD method is developed in this work. The triangular mesh is used to simulate the hexagonal cavity of ZnO rod and graphene flakes. ZnO rod is a dielectric 2D hexagonal structure. The first 5~12th modes of the ZnO hexagonal cavity are successfully simulated and compared with previous results using square meshes. The graphene flake’s hexagonal cavity modes are simulated using three dimensional triangular mesh. In Cartesian mesh, it is difficult to find the mode pattern with mode number higher than 10. The error caused by stair case can be easily solved in the triangular mesh.

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