本論文改良自蔡雅倫博士先前之研究，將原本的環形中空圓柱波導結構之截面由圓形改為正方形，排列方式為四方晶格排列結構，以利於車床或半導體製程技術製作樣品。並選用折射率為 3.3 的材料，形成中空波導。模擬方法採用圓柱座標的有限時域差分法，分析光在四方晶格排列結構中傳輸及其出射行為。本研究透過掃描正方形截面的寬度與入射光之波長，尋找能產生較低擴散角、較高能量傳遞效率的環形中空方柱波導結構。其中排列方式除了四方晶格之外，我們也將結構的排列方式依照Bessel函數的最低值來排列(以下簡稱Bessel 型排列)。研究輸出光束的能量傳遞效率及擴散角。

模擬結果顯示，在週期型排列中，最佳能量傳遞效率可達27.33%，、最小擴散角為 1.38°。相較之下，Bessel 型排列在能量傳遞效率與擴散角之表現上皆不如週期型排列。此外，與本實驗室歐奕辰同學所進行的六角晶格排列環形中空方柱波導結構研究相比，在週期型排列下，四方晶格排列在能量傳遞效率與擴散角兩方面皆優於六角晶格排列；而於Bessel 型排列中，四方晶格排列之能量傳遞效率及擴散角表現也都稍優於六角晶格排列。整體而言，無論是四方晶格排列或是六角晶格排列中，週期型排列結構在能量傳遞效率與擴散角表現上皆優於 Bessel 型排列結構。

根據本論文中的研究結果中，在週期型排列結構下，正方形寬度為0.442a、波長為0.794a時有最佳能量傳遞效率，將其換算成美國海軍研究實驗室的Scope-M 計畫中所使用的頻率9.7 GHz，得到a為3.895公分。本論文之結構為一個15a x 20a的環形中空方柱波導結構，其體積直徑為113 cm，體積高度為77.9 cm，比美國海軍研究實驗室所使用的直徑為5.4 m的圓盤發射器來的小非常多。

Based on the previous research by Dr. Ya-Lun Tsai, the circular cross-section of the torus hollow waveguide structure was modified to be square. The periodical squares are arranged in a square lattice to facilitate the fabrication process. The refractive index of the tori was 3.3. We use the Cylindrical Coordinates Finite-Difference Time-Domain method to analyze the light propagation passing through the torus waveguide. By scanning the width of the square and the wavelength of the input light, we aimed to obtain the structure parameters of the torus waveguide to minimize the divergence angle of the output light and to obtain a higher energy transmission efficiency of the light. We also modified the arrangement of the position of the squares according to the minima of the Bessel function. The divergence angle of the output light and the energy transmission efficiency of the light were analyzed.

The simulation results show that for the periodic arrangement, the best energy transmission efficiency reached 27.33%, and the smallest divergence angle was 1.38°. In contrast, the Bessel-type arrangement performed worse than the periodic one in both energy transmission efficiency and divergence angle. Additionally, compared with the hexagonal lattice waveguide structure studied by I-Chen Ou of our lab, the structure in square lattice performed better in both energy transmission efficiency and divergence angle in the periodic arrangement. In the Bessel-type arrangement, the square lattice structure also exhibited slightly better performance than the hexagonal lattice in terms of both energy transmission efficiency and divergence angle. Overall, in both square and hexagonal lattice structures, the periodic arrangement exhibited better performance in energy transmission efficiency and divergence angle compared to the Bessel-type arrangement.

According to the optimal research results presented in our research, the best energy transmission efficiency was achieved when the square width was 0.442a and the wavelength was 0.794a. Converting this to the frequency of 9.7 GHz used in the U.S. Naval Research Laboratory's Scope-M program, we obtain a=3.895 cm. The structure proposed in our research was a torus hollow waveguide structure with dimensions of 15a × 20a with diameter of 113 cm and height of 77.9 cm, which is significantly smaller than the transmitter used by the U.S. Naval Research Laboratory, which has a diameter of 5.4 m.

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