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研究生:吳惟
研究生(外文):WeiWu
論文名稱:微分計算在光學系統設計的應用
論文名稱(外文):Applications of Differential Computation Methods on Optical System Design
指導教授:林昌進
指導教授(外文):Psang-Dain Lin
學位類別:博士
校院名稱:國立成功大學
系所名稱:機械工程學系碩博士班
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
語文別:中文
論文頁數:143
中文關鍵詞:幾何光學齊次座標轉換矩陣調變轉移函數波前像差
外文關鍵詞:Geometric opticsHomogeneous coordinate transform matrixModulation transfer function(MTF)Wavefront aberration
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光線追蹤數據,可以使光學系統設計者準確地評估設計結果,並做正確的判斷與修正。光學系統若以近軸理論進行光線追蹤,成像將接近完美;然而對大部分實際光線而言,因大視角、光源離軸等因素,已超出近軸理論,使成像產生像差,因此必須有正確的光線追蹤理論,才足以建立嚴謹的光學系統模型。
傳統幾何光學以向量方法推導光線追蹤方程式,向量方法只能考慮位置變化,無法處理座標系轉換,因此對非軸對稱光學系統的建模極為困難。本研究以齊次座標轉換矩陣,建構一種新的歪斜光線追蹤方法,將光學系統的數學模型以矩陣形式表現,不但可以應用於軸對稱的光學系統,也可應用於非軸對稱光學系統中,使得幾何光學的研究能更深入有效。
因為光線追蹤方程式是多層次的複合函數,若光學系統有n個邊界,則光線追蹤方程式會是n層的複合函數,其第一階微分甚難求出,因此甚少文獻探討幾何光學的第一階微分。所以目前光學軟體常使用有限差分法求解,且必須追蹤多條光線才能求得近似解。本研究已發展出微分所須的齊次座標轉換矩陣,可使光學微分運算程式化。與有限差分法比較,本法更精確,可做為光學系統設計上的有力工具。
本研究所發展的計算,可應用於調變轉移函數、波前像差分析和光學系統的最佳化。整體而言,本方法具有下列兩大優點:(1) 可深入探討非軸對稱光學系統;(2)將微分計算可程式化,未來可做幾何光學的高階微分。

Raytracing data of optical systems allows the designer to precisely evaluate the design results, make correct judgments and amendments. For paraxial ray tracing theory of an optical system, the light rays near axis region are assumed, so close to the perfect image. However, for most of the actual light rays, its horizon beyond the many near-axis region, and the imaging will produce aberrations, it is necessary have the correct ray tracing to establish a strict model of the optical system.
The traditional geometrical optics uses vector method to derive the raytracing equation, vector method only to be able to consider the position variation, and unable to process coordinate system's transformation. Consequently, it is really difficult to the non-axial symmetry optical system's modeling. In this study, the homogeneous coordinate transfer matrix is used for constructs a new skew raytracing method. The optical system's mathematical model presented by matrix notation, not only apply in the axial symmetry optical system, but also apply in the non-axial symmetry optical system, and enables the geometrical optics research more effective.
Due to the tracing equation is the multi-level composite functions, that is a function which operates in turn on other functions. If the optical system has n boundary, then the tracing equation can be n layers composite function, it is really difficult to extract its first order differential quantity. Therefore, the present commercial optics software often uses the finite difference method, and must trace many rays to obtain the approximate solution. The proposed study has developed the homogeneous coordinate transfer matrix for the goal, and making the differential computation of optics parameters being programmable. Compares with the finite difference method, proposed method is more accuracy, and will be a powerful tool for optical system design.
The proposed method already applied in investigating of MTF, wavefront aberration and optimization of an optical system. In summary, the proposed methodology provides two advantages: 1) to penetrate discussion non-axial symmetry optical system. 2) a potential basis for the future development of a numerical technique for computing the high-order derivatives of the optical quantities of an optical system.

摘 要 III
ABSTRACT IV
誌謝 VI
目錄 VII
表目錄 IX
圖目錄 X
符號說明 XV
NOMENCLATURE XVI
第一章 概論 1
1.1 幾何光學簡介 1
1.1.1 基本原理 2
1.1.2 光學元件 7
1.1.3 像差 19
1.2 文獻回顧 25
1.3 本文研究動機 28
1.4 本文架構 29
第二章 歪斜光線追蹤 30
2.1 齊次座標轉換 30
2.2 球面邊界的光線追蹤 33
2.3 平面邊界的光線追蹤 42
2.4 本章小結 48
第三章 光學系統的數學模型 49
3.1 透鏡的建模 49
3.2 軸對稱光學系統的建模 60
3.3 非對稱光軸光學系統的建模 65
3.4 光線追蹤數據的功用 76
3.5 本章小結 86
第四章 光學系統參數的微分 87
4.1 邊界變數對系統變數的一階微分矩陣 89
4.2 波前像差對系統變數的一階微分矩陣 101
4.3 以波前像差為基礎的評估函數 107
4.4 本章小結 108
第五章 微分在幾何光學MTF的應用 109
5.1 軸光源的MTF 計算 111
5.2 離軸光源的 MTF 計算 113
5.3 實例 124
5.4 本章小結 130
第六章 結論與未來展望 132
6.1結論 132
6.2未來展望 133
參考文獻 134
LIST OF PUBLICATIONS 141
VITA 143



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