本論文研究將歐拉-拉格朗日方程式(Euler-Lagrange Equation)的物理特性，運用到多模態椎狀波導(multimode waveguide taper)設計上，並用數值模擬進行分析。首先，我們會介紹歐拉-拉格朗日方程式的理論，說明如何將此理論運用到設計椎狀波導上。再來利用已知理論基礎，透過數值模擬方法，時域有限差分法(finite difference time domain, FDTD)和有限差分本徵模態方法(finite difference eigenmdoe, FDE)來進行椎狀波導設計的工作。最後透過時域有限差分法驗證設計出來的多模態固定損耗椎狀波導(multimode constant loss taper, MMCLT)的特性，寬度0.5-10μm的多模態固定損耗椎狀波導，在長度220μm時，具有99%以上的穿透率、偏振不相關性(polarization independence)和容忍度(robustness)。相較於其他常見的椎狀波導，長度縮短70μm，所以此椎狀波導將可利於縮小未來光纖耦合元件的尺寸。

In this thesis, we devoted our effort to design a new waveguide taper which is called the multimode constant loss taper(MMCLT). According to Euler-Lagrange equation, if we can fix the step loss of a taper at a constant, we’ll be able to get a taper has lowest loss. This taper was designed to reduce the loss which is caused by mode conversion to high order modes or radiation modes. The width of the taper is from 0.5μm to 10μm. The substrate’s width is 20μm. The model is based the on silicon-on-insulator(SOI) material. We use the two dimensional finite difference time domain (2D FDTD) method and the finite difference eigenmode (FDM) to find the width of each propagation step to make the step loss constant. We also use the 2D FDTD method to demonstrate the performance of the MMCLT. We find that the MMCLT can achieve a transmission of 99% at a shorter length than the other tapers, such as linear, parabolic and exponential tapers. And as the length of the MMCLT reaches 220μm, it is polarization-independent and robustness.

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