臺灣博碩士論文加值系統

因為一般光學系統之成像通常都會伴隨著像差之產生，所以像差在幾何光學中是很重要的理論，其中最著名也最廣為使用的即為Seidel-aberration、Buchdahl-aberration之研究。然而其係數之計算往往包含著複雜的迭代運算。因此本研究提供一個更簡便之方法，利用習知的泰勒級數展開，分別以波前與光線之觀點，來計算出四階波前像差與五階光線像差。
四階波前像差是以物高h與入射瞳之直角座標為系統的獨立變數，將光程對獨立變數進行泰勒展開；五階光線像差則是將像點對獨立變數做泰勒級數展開，再利用光線的對稱性質將展開式進行簡化，就可以分別得到另一種形式之四階波前像差與五階光線像差函數。此外本論文配合本研究室自行開發的軟體，計算出一個軸對稱光學系統的四階波前像差與五階光線像差數值，並與光學軟體Zemax模擬得到的像差值做比較，以驗證本論文理論之正確性。

The published work of calculating the aberration coefficients showing lots of iterative operations are quit complicated. Therefore, this study based on the well-known Taylor series expansions of a skew ray with respect to the object height and the Cartesian coordinates of entrance pupil provides another simpler methodology to investigate Seidel-aberrations and Buchdahl-aberrations from the viewpoint of wavefronts and rays respectively, namely the fourth-order wavefront aberrations and the fifth-order ray aberrations.
Regarding the object height and the Cartesian coordinates of entrance pupil as the independent variables, we expand the OPL function by Taylor series expansion with respect to independent variables. Then, the polynomial of the fourth-order wavefront aberration can be obtained by the relation between OPD and the wavefront aberration. Similarly, the polynomial of the fifth order ray aberration can be attained by using Taylor series expansion of incident point on image plane with respect to independent variables. As we simplify both polynomials determined by utilizing the symmetric properties of the axis-symmetrical optical systems, an alternative aberration function would be derived.
Finally, substituting the value of aberration coefficients calculated by the numerical software of our laboratory into the aberration functions, we can solve the magnitude of variety aberrations as well as compare the results with which from optical software to verify the correctness of this theory. It is found that the obtained values of this thesis agree remarkably with those attained by Zemax simulation.

中文摘要 i
ABSTRACT ii
誌謝 vi
目錄 vii
表目錄 x
圖目錄 xi
符號表 xiii
第一章 緒論 1
1.1前言 1
1.2光線像差 2
1.3波前像差 7
1.3.1波前 8
1.3.2光程與光程差 9
1.4文獻回顧 11
1.4.1光線像差 11
1.4.2波前像差 14
1.5本文架構 16
第二章 軸對稱光學系統之光程函數 18
2.1建立軸對稱光學系統 18
2.2軸對稱系統光程函數之對稱性 21
2.2.1軸上物點 23
2.2.2離軸物點 23
2.3軸對稱系統波前像差函數之對稱性 27
2.3.1軸上物點 27
2.3.2離軸物點 28
2.4建立光程多項式 30
2.5出射瞳至參考波前之光程 33
第三章 四階波前像差函數 36
3.1有限差分法 36
3.2光程函數之係數數值 39
3.3建立波前像差函數 43
3.4數值計算結果 47
3.5本章結論 49
第四章 軸對稱光學系統之光線像差多項式 50
4.1像差函數 50
4.2建立光線像差多項式 52
4.3軸對稱系統光線像差之對稱性 54
4.4對稱規則下像差函數之泰勒展開式 58
第五章 五階光線像差函數 61
5.1五階像差係數表示式 61
5.2偏微分係數數值結果 68
5.3五階像差係數數值計算結果與討論 73
5.3.1五階球差 74
5.3.2線性彗差 75
5.3.3斜球差 77
5.3.4橢圓彗差 79
5.3.5五階像散、場曲 80
5.3.6五階畸變 81
5.4本章結論 81
第六章 結論與展望 83
6.1本文結論 83
6.2未來展望 84
參考文獻 86
附錄 89

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