矽光子學是光電科學領域中的重點研究發展方向之一，基於矽光子學在積體光路中的諸多優點，例如高速並低損耗的信號傳輸能力，使矽光子學還在光譜學以及光子晶片應用上，扮演重要的角色。更重要的是，矽光子元件的製作與半導體製造中時常用到的CMOS製程上有具高度的兼容性。使得矽光子學的發展可以立基成熟的半導體製程技術，從元件快速進步到系統的層級，大幅提升積體光路效能以及複雜度。
本論文中，我們利用圓柱或圓孔為建構單元，以非均勻二維方式排列形成光子晶體(Phc)結構，計算可以實現高效率耦合至目標波導中基本模態的介面結構，減少積體光路中傳輸導光模態的不匹配。為了進行分析與設計，我們開發一種結合平面波展開法與多重散射法的計算工具。透過此工具，我們研究以光子晶體結構將分別20μm與2μm的雷射光束耦合到寬度為0.7μm的波導上。我們首先考慮多層半圓弧排列的光子晶體結構，在不考慮光子晶體傳播的損耗上，可以將耦合效率從0.55提高到0.87。為了更進一步提高耦合效率，我們也考慮了不同孔徑尺寸的光子晶體陣列作為等效透鏡以及等效錐形波導，即使考慮光子晶體結構中具有傳播的損耗，仍可以高達0.87的耦合效率。

Silicon photonics is one of the focused topics in photonics and optics society because of its potential in the integrated photonics circuit (IPC) for high speed, low loss signal processing and transmission. It also features promising applications in spectroscopy on photonics chips. To provide efficient light transmission in the IPC photonic structures facilitate one of the good alternatives because it is highly compatible to CMOS fabrication process.
In this dissertation, we design nonuniform photonic crystal (PhC) structures based on the two dimensional arrangement of either cylindrical posts or holes to shape the modes to couple into the fundamental mode of a target waveguide which reduces the mismatch at the connecting interface for guided modes. A method combining plane wave decomposition and multiple scattering is developed. Using this tool, we study PhC structure that focuses laser beam of 20 μm and 2 μm in radius, respectively to a waveguide of 0.7μm in width. Adopting parameters that cab be processed under current fabrication technology, we first consider PhC array arranged in multiple layers of curved lines and it can improve the coupling efficiency from 0.55 to 0.87 without considering the PhC transmission loss. Furthermore, we consider PhC arrays of varying holes sizes to serve as an effective lens and also taping PhC array structure. Up to 0.87 coupling efficiency is attained when the transmission loss of the PhC structure is involved.

中文論文審定書 i
英文論文審定書 ii
摘要 iii
Abstract iv
目錄 v
圖目錄 vii
表目錄 xii
第一章、 緒論 1
1-1矽光子學簡介 1
1-2光子晶體簡介 1
1-3光子晶體應用 6
1-4研究動機 8
1-5論文架構 10
第二章、多重散射法的推論 11
2-1簡介 11
2-2波動方程式 11
2-2-1馬克斯威爾方程式(Maxwell’s equations) 11
2-2-2齊次向量漢姆霍茲方程式(homogeneous vector Helmholtz equations) 12
2-3場在圓柱的傅立葉-貝索展開(Fourier-Bessel expansion) 14
2-4散射矩陣 19
2-5場在圓柱外無窮遠處的傅立葉-貝索展開 21
2-6推論總結 22
2-7模擬測試 23
2-7-1平面波入射 23
2-7-2高斯光入射 32
第三章、週期性排列光子晶體以平面波入射聚焦現象與暗區之研究 36
3-1簡介 36
3-2模擬分析與討論 37
3-2-1聚焦現象 37
3-2-2暗區現象 40
第四章、週期性排列光子晶體以高斯光入射聚焦現象與光波導耦合之研究 45
4-1簡介 45
4-2模擬分析與討論 46
4-2-1多層半圓弧排列結構聚焦現象 46
4-2-2多層半圓弧排列結構耦光效率計算(coupling efficiency) 52
4-2-3透鏡結構聚焦現象 54
4-2-4透鏡結構耦光效率計算(coupling efficiency) 61
4-2-5透鏡結構層數對焦距影響 65
第五章、結論與未來展望 67
5-1結論 67
5-2未來展望 67
參考文獻 69

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