像差是光學領域中重要的研究方向之一。目前已有許多發展成熟的像差理論，如:賽德像差理論、Zernike多項式等。本研究室過去已將波前像差函數以泰勒級數展開的形式呈現，並以此為基礎，成功計算出賽德像差係數，且與Zemax軟體數值相當接近，僅有些微誤差。

本研究旨在將先前以泰勒級數所展開的波前像差多項式轉換成Zernike多項式，並利用本研究室的FORTRAN程式，計算出在軸對稱光學系統中Zernike多項式係數數值。此外，本文也利用Zernike多項式本身的正交特性，搭配數值積分方法、FORTRAN程式與光線追蹤，來計算Zernike多項式的係數數值，並探討分割方法對數值積分的影響，將兩種方法的數值結果與Zemax軟體進行比較，以驗證本文方法之正確性。

根據最後的結果，本研究兩個方法的數值都與Zemax軟體非常相近，代表本研究的兩個方法都能成功計算出Zernike多項式的係數數值。

Aberration is one of the important research interests in optical field. So far, there are many well-developed theories for aberration, like Seidel aberration, Zernike polynomials etc. In the past, research in my laboratory has presented the wavefront aberration function in the form of Taylor series expansion. And based on this, we have successfully calculated the coefficient of Seidel aberration which is quite close to the value in Zemax, and only with slight error. The purpose of this study is to convert the wavefront aberration polynomials developed by my laboratory in the past into Zernike polynomials, and use the FORTRAN program to calculate the coefficient of Zernike polynomials in axis-symmetrical optical systems. Furthermore, we also use the orthogonality of Zernike polynomials, with numerical integration methods, FORTRAN program and ray tracing program, to calculate the coefficient of the Zernike polynomials, and discuss the influence of segmentation on numerical integration methods. In the end, we compare the numerical values of this two methods with Zemax. The numerical values of both are close to Zemax, which means that the methods in this study is feasible.

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