Responsive image
博碩士論文 etd-0924112-225553 詳細資訊
Title page for etd-0924112-225553
論文名稱
Title
球透鏡光學點擴散函數的計算
Computation of the Optical Point Spread Function of a Ball Lens
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
110
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2012-08-27
繳交日期
Date of Submission
2012-09-24
關鍵字
Keywords
球透鏡、點擴散函數、光點尺寸、幾何光學、富氏光學、電磁波光學、波動光學
ball lens, point spread function, spot size, geometric optics, Fourier optics, electromagnetic optics, wave optics
統計
Statistics
本論文已被瀏覽 5703 次,被下載 1677
The thesis/dissertation has been browsed 5703 times, has been downloaded 1677 times.
中文摘要
在本篇論文中,我們將對最簡單的光學成像系統「球透鏡」進行分析。欲對光學成像系統進行分析,傳統幾何光學可以作為初階的定性分析工具。然而,幾何光學並未將光的波動性與極化特性納入考量,欲在定量上獲得較精確的結果,應該採用波動光學做為分析的方式。相較於大部分的複雜光學系統,球透鏡具有球對稱性,因此我們得以使用精確的分析手段對其成像特性進行研究。
在本篇論文中,我們將採用幾何光學、富式光學、純量波動光學、電磁波光學等四種方法計算球透鏡的點擴散函數 (point spread function, PSF),並比較各理論所預測的光點尺寸 (spot size)。以下簡單說明:
1. 幾何光學 (geometric optics)
假設點光源為均向性光源,利用 Snell’s law 對所有通過球透鏡的光束進行解析光束追蹤法 (analytic ray tracing method),預測各光束在像平面上的落點,並根據平面上的落點密度計算出能量密度分佈函數 (點擴散函數)。
2. 富氏光學 (Fourier optics)
富氏光學考量了光的波動性,將光場視為平面波函數的疊加,在近軸近似下,運算中的積分核心 (integral kernel) 可近似成 Fresnel integral kernel,因此最終可以獲得球透鏡的點擴散函數的解析解。相較於幾何光學,它所預測的結果可視為考量波動性後更好的定性解。但在此階段我們並未考慮透鏡的反射效應。
3. 純量波動光學 (scalar wave optics)
相較於富式光學的計算,我們直接對光場函數所應滿足的非齊性赫姆霍茲方程式進行求解。在光的純量波理論中,光場函數並沒有具體的定義,故我們合理要求光場函數在介面上應滿足函數連續及一次導函數連續。
4. 電磁波光學 (electromagnetic optics)
考量光源的極化特性,由馬克士威爾方程式推導出滿足球形介面的兩種極化的赫茲向量所應滿足的方程式。相較於純量波動光學,此處我們確切地了解方程式中波函數所代表的是電場或是磁場,故我們能在介面上給予確切的數學條件。最後將兩種極化所預測的結果取平均值作為最終的點擴散函數。此方法最完整地考量了光的波動性、反射、折射、極化等性質。
Abstract
In this thesis, we analyze the simplest optical imaging system: a ball lens. The traditional method of using a geometric optics analysis on an optical system only gives the roughest qualitative solution due to the lack of consideration of wave properties. Therefore, for accurate quantitative results, we need to analyze said system with a complete wave theory approach. The reason that we chose a ball lens as the focus of this research is due to its spherical symmetry properties which allows us to rigorously investigate it with different analytic methods. We will apply geometric optics, Fourier optics, scalar wave optics, and electromagnetic optics methods to compute the point spread functions (PSF) of a ball lens under the assumption that the point source is isotropic. We will follow up by predicting the spot sizes that correspond to each mean.
First, with geometric optics (GO), we apply the analytic ray tracing method to correlate the origins of light rays passing through the ball lens to their respective positions on the receptive end. We can then evaluate the energy distribution function by gathering the density of rays on image plane. Second, in the theory of Fourier optics (FO), to obtain the analytic formula of the point spread function, the integral kernel can be approximated as the Fresnel integral kernel by means of paraxial approximation. Compared to GO, the results from FO are superior due to the inclusion of wave characteristics. Furthermore, we consider scalar wave optics by directly solving the inhomogeneous Helmholtz equation which the scalar light field should satisfy. However, the light field is not assigned to an exact physical meaning in the theory of scalar wave optics, so we reasonably require boundary conditions where the light field function and its first derivative are continuous everywhere on the surface of ball lens. Finally, in the theory of electromagnetic optics (EMO), we consider the polarization of the point source, and the two kinds of Hertz vectors and , both of which satisfy inhomogeneous Helmholtz equation, and are derived from Maxwell’s equations in spherical structures. In contrast with the scalar wave optics, the two Hertz vectors are defined concretely thus allowing us to assign exact boundary conditions on the interface. Then the fields corresponding to and are averaged as the final point spread function.
目次 Table of Contents
誌謝 i
中文摘要 ii
英文摘要 iii
目錄 iv
圖目錄 vi
表目錄 ix
第一章 序論 1
1-1研究動機及背景 1
1-2研究目的及方法 3
第二章 幾何光學對球透鏡之分析 6
2-1 幾何光學簡述 6
2-2 光的折射定律和反射定律 6
2-3 球透鏡之點擴散函數推導與分析 8
2-3-1 無球透鏡時的PSF與光點尺寸計算 8
2-3-2 球透鏡的PSF與光點尺寸計算 11
2-4球透鏡之PSF與光點尺寸 20
第三章 富氏光學對球透鏡之分析 26
3-1 富氏光學理論 26
3-2 繞射 26
3-3 夫琅禾費繞射 (Fraunhofer diffraction) 28
3-4 應用富氏光學計算球透鏡PSF 31
第四章 純量波動光學對球透鏡之分析 50
4-1 波動光學理論 50
4-2 格林函數(點擴散函數)基本理論 50
4-3 以波動光學解球透鏡之格林函數 52
4-4 球透鏡的PSF與光點尺寸 (波動光學解) 61
第五章 電磁波光學理論對球透鏡分析與計算 70
5-1 球型結構中的波動方程式 70
5-2 應用電磁波光學計算球透鏡PSF 78
5-3 球透鏡的PSF與光點尺寸(電磁波光學解) 86
5-4 結論 88
附錄 92
附錄A 重要特殊函數 92
附錄B 折射率、物距與像距 92
附錄C 無窮級數有效加總項數決定 94
參考文獻 97
參考文獻 References
[1] Dean G. Duffy, Green’s Functions with Applications, Chapman and Hall/CRC, Boca Raton, London, New York, Washington D. C., pp.282, 2001,
[2] Abramowitz and Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972.
[3] Eugene Hecht, Optics, 4th Ed., Addison Wesley, New York, 2002.
[4] Akira Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering, Prentice Hall, Englewood Cliffs, New Jersey, 1991.
[5] T. D. Visser and S. H. Wiersma, ‘‘Spherical abbberation and the elcectromagnetic field in high-aperture systems,’’ J. Opt. Soc. Am. A, Vol. 8, No. 9, September 1991.
[6] Max J. Riedl, Optical Design Fundamentals for Infrared Systems, Second Edition, SPIE Press Book, 2001.
[7] Joseph W. Goodman, Introduction to Fourier Optics, Third Edition, Roberts & Company Publishers, United States of America, 2005.
[8] Stratton, Electromagnetic Theory, IEEE Press, United States of America, 2007.
[9] 葉玉堂,饒建珍,肖峻,幾何光學 (Principle of Optics),五南圖書出版股份有限公司,台中市,2008.
[10] 黃衍介,近代實驗光學 (Modern Experimental Optics),東華書局,台北市,2005.
[11] H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, New Jersey, 1984.
[12] Arfken and Weber, Mathematical Methods for Physicists, Sixth Edition, Elsevier Academic Press, United States of America, 2005.
[13] Robert D. Guenther, Modern Optics, John Wiley & Sons, Inc., Canada, 1990.
[14] David Voelz, Computational Fourier Optics-A MATLAB Tutorial, SPIE Press, Bellingham, Washington, 2011.
[15] Okan K. Ersoy, Diffraction, Fourier Optics and Imaging, John Wiely & Sons, Inc., Hoboken, New Jersey, 2007.
[16] Donald C. Stinson, Intermediate Mathematics of Electromagnetics, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
電子全文 Fulltext
本電子全文僅授權使用者為學術研究之目的,進行個人非營利性質之檢索、閱讀、列印。請遵守中華民國著作權法之相關規定,切勿任意重製、散佈、改作、轉貼、播送,以免觸法。
論文使用權限 Thesis access permission:自定論文開放時間 user define
開放時間 Available:
校內 Campus: 已公開 available
校外 Off-campus: 已公開 available


紙本論文 Printed copies
紙本論文的公開資訊在102學年度以後相對較為完整。如果需要查詢101學年度以前的紙本論文公開資訊,請聯繫圖資處紙本論文服務櫃台。如有不便之處敬請見諒。
開放時間 available 已公開 available

QR Code