在過去幾年本研究室已使用泰勒級數展開法，探討第五階光線像差與第四階波前像差，並得到很好的結果。過去的文獻都使用出射瞳座標系來作該兩像差轉換，且所列的方程式沒闡明各階光線像差的分解能力，也無轉換的誤差分析。為了進一步了解此二者之間的關聯，本論文第二章利用兩種像差的幾何關係、光程函數、與漢密頓點特徵函數(Hamilton’s Point Characteristic Function)，來探討波前像差轉換至光線像差的問題。本論文第三章用一個Petzval光學系統為例，用文獻方法由各階波前像差轉換至各階光線像差，並計算其轉換差異(difference)。第四章再用連鎖律(chain’s rule)於入射瞳座標系，探討波前像差與光線像差的轉換條件。本論文轉換所得的光線像差，都與本研究室的資料及Zemax作比對，檢視其準確性。
上述的轉換都只能求得高斯像平面的光線像差，本論文第五章修改轉換方程式，可計算出非高斯像平面的光線像差，與文獻的方法比較，本論文所得的光線像差數值更精確。

Over the past few years, our research laboratory has derived wavefront aberration functions and ray aberration functions using Taylor series expansion from low to high orders. To further understand the relationship between these two, we investigate how wavefront aberration is transformed into ray aberration by utilizing the geometric relationship, optical path function, and Hamilton's Point Characteristic Function. Additionally, unlike previous literature that used the exit pupil coordinate system for aberration function transformation, this paper adopts the entrance pupil coordinate system as the form of wavefront aberration function and achieves the condition of using transformation equations by converting the coordinate systems between entrance and exit pupils. Finally, we will use a simulated Petzval optical system to calculate the ray aberration obtained through the transformation equations. A comparison will be made with the data from our previous research and Zemax to examine possible errors in the transformation equations and validate their accuracy.
Furthermore, this research goes beyond the conventional framework of calculating ray aberrations using the Gaussian image plane. By modifying the transformation equations, we derive new transformation equations for calculating changes in ray aberration after moving the image plane. This method improves upon some limitations found in previous literature.

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