光子晶體利用完全相適吸收層之模擬分析

Abstract

本篇論文提出一個全新的方程式，以依照一個包含完全相適吸收層系統的形式，處理具有週期介電性結構中受到區域性微擾的共振模組的外殼方程。模擬的結果顯示了施子型態的缺陷模組解，並用以討論及最佳化完全相適吸收層的特徵參數。 本論文中系統的結構包含了一個二維光子晶體平板波導，其上具有三角晶格排列的空氣孔。垂直方向的波導則利用有效折射率方法來模擬，即將整個系統視為準二維。本論文只討論其中的類橫向電波模組，並將焦點放在傳導帶邊緣，觀察具有施子型態的缺陷模組。 依循O. Painter研究群的成果，本篇論文由一個解耦合和簡化的瓦尼爾外殼方程起始。微擾的位能則被概略地假設為一個拋物線型的位能井。 其後，引入一個型態相似於完全相適吸收層的延伸係數之特徵係數，此修訂後的瓦尼爾外殼方程可用以檢視所有非束縛的洩漏模組解。 模擬結果則利用12 組不同設定來加以比較並使得達到最佳化。經由計算導引模組和洩漏模組的外殼方程。本論文首先展示出了將完全相適吸收層放入一個以瓦尼爾方程描述的系統是可行的；此系統所有的洩漏模組的外殼方程及其對應的本徵值皆可有效獲得。在最後的部分，本篇論文展示了計算得到的部分施子型態缺陷模組。

This thesis proposes a novel formulation, in a form with the concept from the perfectly matched layer, to treat the envelope functions of resonant modes of localized perturbations within periodic dielectric structures. Some simulation results of the envelope of the donor type defect modes are demonstrated to discuss and optimize the performance of the parameters for the PML. The structure of the system in this thesis consists of a 2D photonic crystal slab waveguide with a triangular array of air holes. By using the effective index method for vertical guiding, whole system is viewed as quasi-2D. Only the TE-like modes are here under discussion. And putting emphasis around the conduction band edge, several donor-type defect modes are found. Following the work of the O. Painter’s group, a decoupled and simplified Wannier envelope function is adopted. The perturbed potential is approximately designed as a parabolic potential well. Then, introducing parameters which is a form like the stretched parameters in the PML into the Wannier envelope functions makes investigating the unbounded leaky modes become practical. The simulation results with 12 sets of different settings are presented for comparison and optimization. From the results for calculating the envelope of guided modes and leaky modes, it firstly shows that the PML is suitably used to be coupled with the Wannier envelope function. The envelope function for the leaky modes and their eigenvalues are obtained easily and efficiently. By comparing the results, it’s found that the desire envelope functions and precise eigenvalues are obtained with appropriate total length of the system and thickness of the PML. Finally, some donor type defect modes are calculated and shown.