非球面檢測之迭代相移干涉與子孔徑相位接合演算法開發

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姓名

林柏至(Po-Chih Lin) 查詢紙本館藏

畢業系所

光電科學與工程學系

論文名稱

非球面檢測之迭代相移干涉與子孔徑相位接合演算法開發
(Development of an Iterative Phase-Shifting Algorithm and a Subaperture Phase-Stitching Algorithm for Aspheric Testing)

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摘要(中)

光學元件檢測中，量測非球面透鏡是一項極具挑戰的研究工作，從干涉儀校準與控制、雷射光源穩定性、重建非球面演算法與量測平台的設計，每一個主題，都需要不同領域的專家，精心的設計與研究來完成。本研究將重心放在非球面量測資料擷取完成後的計算處理，將開發一系列非球面相位重建演算法。
本研究利用子孔徑接合的方式，完整量測非球面透鏡，而欲使用這樣的方法，必須完成兩個主要的目標，第一：單一子孔徑相位重建演算法，本研究以五步相移干涉術與最小平方誤差法為基礎，利用迭代的非線性擬合方式來開發高誤差容忍度的相移干涉術，並搭配自行開發、以Zernike多項式擬合法為基礎的相位解纏繞演算法。第二：子孔徑相位接合演算法，以最小平方法為基礎，找出重疊區域的誤差最小值，作環狀相位接合，最後再以中央孔徑為基準，利用線性擬合的方式，接合不同半徑的環狀區域，完整重建非球面鏡。
演算法模擬實驗結果，當所有的子孔徑干涉條紋內含振動誤差時，經過一系列非球面重建演算法之後，所得到的誤差殘餘值為0.006 waves，符合一般干涉儀的精度標準。此演算法搭配研究團隊所開發的非球面干涉儀，將可處理1000 個波長的非球面度，相較於現有儀器的可量測範圍，提高了20倍之多，將可透過精密與高動態範圍的非球面檢測能力，提升非球面光學元件的產品層次。

摘要(英)

In the field of optical testing, measuring an aspheric lens surface is a challenging task. It involves multiple topics such as the interferometer control, optical alignment, laser-source stability, phase reconstruction algorithm and mechanical stage design. Completing the work in each topic requires specific expertise, careful design and research in different fields. The main focus of this thesis was on developing a series of phase reconstruction algorithms.
Subaperture stitching interferometry was adopted for measuring aspheric surfaces in this research. The corresponding algorithm could be divided into two parts. The first one was the phase reconstruction algorithm for single subaperture. An iterative phase-shifting algorithm highly tolerant to phase-shift errors was developed based on the Hariharan five-step algorithm and nonlinear least-squares fitting. The phase map was then unwrapped by a Zernike-polynomial-based phase unwrapping process. The second one was the subaperture stitching algorithm. All subapertures in one annulus were stitched simultaneously in least-squares sense. By eliminating the relative piston and tilt between adjacent subapertures, the sum of squared errors in the overlapped regions was minimized. The phase stitching between annuli also utilized the least-squares method in the overlapped region.
Simulation studies were carried out to demonstrate the effectiveness of the proposed algorithm. Random phase shifts were introduced into the subaperture interferograms. The resulting rms phase residue after the phase-shifting, phase-unwrapping and phase-stitching processes was 0.006 waves, which met the precision requirement of common interferometers. The algorithm in conjunction with the aspheric interferometer developed in the research group will be capable of measuring aspheres with 1000-wave departure. The dynamic range is extended by 20 times compared with that of typical non-stitching optical-testing instruments.

關鍵字(中)

★ 非球面檢測
★ 子孔徑接合
★ 相移干涉術

關鍵字(英)

★ Interferometry
★ Subaperture Stitching
★ Aspheric Testing

論文目次

中文摘要 I
Abstract II
致謝 III
目錄 IV
圖目錄 VII
表目錄 XII
第一章緒論 1
1-1 前言 1
1-2 研究動機與目的 3
1-3 文獻回顧與研究方法 4
1-4 論文架構 6
第二章光學基礎理論與原理 8
2-1 基本干涉原理 8
2-2 基本軸對稱像差 10
2-3 相移干涉術-五步法 14
2-4 相移干涉術-最小平方差演算法 16
2-5 Standard Zernike 多項式 19
第三章非球面透鏡之干涉檢測方法 22
3-1 Fizeau干涉儀基本介紹 23
3-2 量測平台簡介 24
第四章相位重建演算法 26
4-1 消除傾斜誤差之迭代相移干涉術 26
4-2 二維相位解纏繞 33
第五章子孔徑相位接合演算法 35
5-1 最小平方差之環狀接合演算法 35
5-2 環狀波前接合演算法 42
第六章結果與討論 44
6-1 消除傾斜波之迭代相移干涉術 44
6-1-1 模擬參數設定 44
6-1-2 模擬結果 48
6-1-3 非線性擬合之光強均勻化的誤差分析 52
6-1-4 二維相位解纏繞模擬結果 55
6-1-5 平板標準鏡之量測結果 58
6-2 最小平方差之子孔徑相位接合演算法 62
6-2-1 無雜訊之子孔徑相位的模擬結果 63
6-2-2 加入相移干涉術之模擬結果 66
6-2-3 增加重疊區域之模擬結果 68
6-2-4 球面透鏡之環狀量測結果 70
6-3 環狀波前接合演算法 79
6-3-1 無雜訊之環狀相位模擬結果 79
6-3-2 加入相移干涉術之環狀相位模擬結果 83
6-3-3 全域球面透鏡之量測結果 84
第七章結論與未來工作 89
參考文獻 91

參考文獻

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指導教授

陳怡君(Yi-Chun Chen)

審核日期

2011-8-24

推文