目前的光學系統所使用的光學邊界面，大部份都是球邊界面的形式，但球邊界面會有像差的存在，會造成像的品質降低。非球面的光學邊界面，在製作上較困難，價格也較昂貴，但在許多幾何光學系統中，非球邊界面仍然佔有極重要的地位，深值得吾人對它做深入的研究與探討。
在幾何光學中，近軸光線追蹤法僅適合初步的分析。光學系統的最後性能，仍須利用歪斜光線追蹤法，才能精確地評估出。利用本實驗室所發展的歪斜光線追蹤及系統靈敏度分析，即能在光學儀器的設計與分析上，做最有效的發揮。
本文利用齊次轉換矩陣和幾何光學的Snell’s定理，推導出歪斜光線行經二次曲面(拋物面、橢圓面及雙曲面)邊界時，其折(反)射光線方向及入射點位置，並推導出光線行經此二次曲面的靈敏度方程式。且以FORTRAN電腦程式，作數值模擬分析、並與球面的入射點位置、折射線方向及靈敏度作比較，以驗証方程式的正確性。

The majority of optical systems are based on spherical components because they are easy to manufacture and low in costs. But the aberration form a spherical boundary is inevitable and may degrade the quality of image. Aspherical boundary surfaces are more difficult and expensive to make. However, there are cases where components with aspherical boundary surfaces (one example is paraboloids) have a significant advantage over spherical ones. Thus the aspherical boundary surfaces are worthful for study.
One of the most popular mathematical tools in fields of robotics, mechanisms and computer graphics is the 4x4 homogeneous transformation matrix. In previous work we applied this matrix to the optical domains of planar and spherical surfaces for: (1) skew ray tracing to determine the paths of skew rays being reflected/refracted; (2) sensitivity analysis to determine by direct mathematical analysis the differential change of incident point and reflected/refracted vector with respect to change in incident light source or orientation of optical boundary; 3) a sensitivity analysis-based merit function derived directly from mathematical expression of system components. The present work extends our previous work to include the cases of parabaloidal, eliiptical, and hyperbolical boundary surfaces.

摘要 Ⅰ
英文摘要 Ⅱ
誌謝 Ⅲ
目錄 Ⅳ
圖目錄 Ⅵ
表目錄 Ⅹ
符號表 ⅩⅠ
第一章 緒論 1
1.1 前言 1
1.2 幾何光學原理 2
1.3 光線追蹤簡介 6
1.4 文獻回顧 9
1.5 齊次座標表示法 12
1.6 內容概要 17
第二章 各種曲面的光線追蹤 18
2.1軸對稱曲面 18
2.2軸對稱曲面的法線向量 20
2.3拋物面邊界的光線追蹤 23
2.4拋物面邊界的靈敏度分析 31
2.5光源在焦點的反射線追蹤 37
2.6數值模擬 40
第三章 橢圓邊界的光線追蹤 45
3.1橢圓邊界的光線追蹤 45
3.2橢圓面邊界的靈敏度分析 53
3.3數值模擬 58
第四章 雙曲面邊界的光線追蹤 63
4.1雙曲面邊界的光線追蹤 64
4.2雙曲面邊界的靈敏度分析 72
4.3數值模擬 77
第五章 結論 82
考文獻84

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