點擴散函數（point spread function, PSF）反映了一光學系統對於點光源之解析能力。目前商用計算光學透鏡點擴散函數一般採用幾何光學或傅氏光學。在多種光學方法中，幾何光學雖然容易計算，但其中隱含光波波長無限小之假設；傅氏光學採用波動光學理論並含括光學繞射現象，然傅氏光學採近光軸假設與菲涅耳繞射（Fresnel diffraction）近似，若光學系統稍複雜些即難得到精確的解。由計算角度來看，幾何光學須計算多點才能得到精確的統計資訊，而波動光學雖然嚴謹正確，但計算點擴散函數時因光學系統尺寸相對波長過大、計算方法複雜且耗用大量資源，無法適用於商用計算光學透鏡點擴散函數。嚴謹波動光學使用純量近似的米氏解（scalar Mie solution），可用於計算球透鏡點擴散函數，但要計算很多項式其解才能收斂。
　　本論文採用大角度幾何光學加上相位補償的方式計算球透鏡點擴散函數，同時比較使用點擴散函數米氏解（scalar Mie solution）。若此差異能到達可接受之誤差範圍，那麼我們將能使用相位補償幾何光學的方法計算一些更複雜光學透鏡系統之點擴散函數，得到比幾何光學更為精確、比波動光學更快速且被廣泛應用的計算光學成像有效數值方法。
　　相位補償幾何光學採用了局部平面波 (local plane wave) 假設以趨近波動光學解，但不論是否將此局部平面波侷限於特定範圍，其結果與波動光學所預測光點尺寸皆有40% - 50% 之正偏差；然而，此正偏差在不同條件下仍相當一致。考量計算時間並修正此誤差比例，相位補償幾何光學仍不失為是一個方便的光學近似方法。

Point spread function (PSF) reflects the ability of an optical system to resolve an image point. Nowadays, commercial computational software for analyzing optical systems are mainly based on methods of geometrical optics (GO) and (rarely) Fourier optics (FO). Computation of the PSF of an optical lens by GO is relatively simple, flexible, and it can be tailored to any complex optical system. GO implies that light possesses infinitely small wavelengths. To obtain accurate statistical information of the point spread by GO, tracing up to several million light rays are needed. FO which is based on the theory of wave optics uses the paraxial approximation and Fresnel diffraction. That limits the accuracy of FO in analyzing complex optical systems. Wave optics (WO) -based methods for obtaining highly accurate PSFs require the use of complicated numerical schemes that consume tremendous computing resource. In practice, WO-PSF method is only applicable to the simplest case of a small spherical lens (ball lens) in which case modified Mie solution provides exact solutions due to a point source excitation.
　　In this thesis, we use large-angle Geometric Optics with phase compensation (PCGO) to calculate the PSF of a ball lens. In PCGO, reflection is neglected but the transmitted angles are computed by WO-based Fresnel equation. At the imaging plane, contribution of each optical ray is computed by a (truncated) constant-amplitude local plane wave weighted with the complex phase term. The phase of each ray is the cumulative sum of the product of local refractive index and the local ray propagation distance as the optical ray passes through each optical section. We then compute the effective optical spot size at the focus point. The results are compared with those computed by WO-based PSF. We find that PCGO spot sizes are in general 40%-50% larger than those predicted by modified Mie solutions.
　　We conclude that PSFs computed by PCGO are more accurate than the standard GO results. It is also computational less intensive due to the spreading of the ray by the local plane with the specific complex phase modulation. Hence, PCGO uses less number of light rays to obtain an accurate PSF.

論文審定書 i
誌謝 ii
中文摘要 iii
英文摘要 iv
目錄 v
圖次 vi
表次 vii
第一章　序論 1
　1-1　研究背景 1
　1-2　研究動機及目的 3
第二章　點擴散函數 4
　2-1　光點尺寸 4
　2-2　以幾何光學對球透鏡分析之點擴散函數 5
　2-3　以波動光學對球透鏡分析之點擴散函數 12
　2-4　以傅氏光學對球透鏡分析之點擴散函數 18
第三章　相位補償幾何光學對球透鏡之分析 23
　3-1　相位補償 23
　3-2　球透鏡之點擴散函數與光點尺寸 28
　3-3　計算參數與波動光學解之比較 39
第四章　結論 46
附錄 47
參考文獻 50

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