光系統的聚焦品質一直是光學發展上需要改善的對象，以往傳統在分析聚焦品質大部分是使用光學設計軟體，藉由參數的設定進行最佳化，由調頻轉換函數(MTF)圖決定聚焦的好壞。本論文提出不同的方法，分析當平行光軸光線經過非球面鏡若聚焦於一點時，這面透鏡會滿足兩條常微分方程式，並使用數值分析的Runge Kutta 四階法求數值解，造出完美透鏡。接著對傾斜角度光線入射透鏡後其折射線與光軸的夾角進行分析，我們知道聚焦位置與入射高度有關，因此這個角度可用 的多項式表示，將已知的聚焦位置與入射高度寫成線性方程組，利用最小平方法找出此多項式各項係數，而一面透鏡的特性將由這些係數唯一決定，便可利用這些係數進行複合透鏡的聚焦分析，最後提出複合透鏡折射線與光軸夾角的多項式如何由這些係數決定，因此即使不使用軟體分析兩面透鏡複合後的聚焦品質，只要知道該面透鏡的係數，便可知道聚焦的好壞。

In optics, researchers devote to improve focusing quality in lens system.
Traditionally, the optical program is used to analysis focusing quality. Researchers usually optimize the quality by changing the parameters used in the program, like refracted angles and thickness of lens, and they determined which sets of parameters are better by frequency modulation transfer function (MTF).
This thesis proposes a different method to analysis. The lens will satisfy two Ordinary Differential Equations (ODE) when parallels lights pass through aspherical lens and focus in a spot. Our purpose is to find out the numerical solutions with Runge Kutta fourth-order method and use these solutions to establish perfect lens.
Then, we pay attention to angles between refracted ray and optical axis when the ray is not parallel to the axis. Because focusing position is related to the incident height, these angles could be represented by polynomials which variables are the heights. Using focusing positions and the incident heights to set linear equations and using the least squares method to find out each coefficient. The specialty of the lens will be determined only by these coefficients. So we use these coefficients to analysis focusing quality of compound lens in advance.
Last part in this thesis, we concentrate on how to use these coefficients to determine polynomials of angles in compound lens system when refracted ray is not parallel to the axis. Therefore, as long as we get the coefficients the lens, we could realize the focusing quality of compound lens by these coefficients even though we do not use optical program.

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