理想的光學系統，因為假設光學邊界沒有厚度且將正弦函數 以角度 簡化，其方程式稱為近軸光線追蹤方程式，該方程式的成像是完美成像。但是實際的幾何光學方程式是司乃爾方程式，其成像會產生像差，物點 發出的光線是5個變數的向量 的函數，無法在像平面聚焦於一點 。若想要有清晰的成像，便需要讓像差小。計算主要像差係數(稱為B係數)的像差理論是Buchdahl所發展的，其須要使用複雜又困難的迭代計算。本研究室曾於去年發展一種全新的賽德像差係數的計算方法，該理論使用泰勒級數展開，將像平面的光線 展開至第三階(即 )，但本研究室過去只有第二階的微分 ，必須用差分法求第三階微分。所以本論文旨在發展球面與平面邊界的第三階微分 ，使得光線主要像差B係數能有解析解。本文也探討軸對稱系統的個別邊界，對B係數的貢獻，並將分析結果與Zemax結果進行比較，證明了本文方法在完整系統與個別邊界的分析結果皆精準且正確。

An ideal optical system assumes that the optical boundary has no thickness and the sine function is simplified to its first-order polynomials, which is called the paraxial ray tracing equation, and the image of this equation is perfect. However, the actual geometrical optics equation is formed by the Snell equation, and its image will occur aberrations. The light emitted by the object point is a function of five variables and cannot focus on a single point. If you want to create a clear image, you need to minimize the aberrations. The aberration theory for calculating the main aberration coefficients (known as the B coefficients) was developed by Buchdahl, and it requires complicated and difficult iterative calculations. Last year, our laboratory developed a new calculation method to calculate the aberration coefficients. The method expands the light variables of the image plane to the third order of Taylor series, but we only had the second-order differential equations for the boundaries then. Therefore, this paper aims to develop the third-order differential equations of the spherical and plane boundaries, so that the B coefficients of the main aberration of light can have an analytical solution. This paper also discusses the contribution of the individual boundaries of the axisymmetric system to the B coefficients and compares the analysis results with the Zemax results, which proves that the results of the method in both the complete system and individual boundaries are accurate and correct.

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