全真映像存在於近軸光區，一旦系統有了定值光瞳孔徑或視場，像差便產生。若用幾何光學的觀念，當光線和光軸的角度大到近軸光學理論無法精確描述，成像缺陷便應運而生，一般來說入射光線角度越大，像差越顯著。
本論文主要是探討波前像差多項式與光線像差多項式的數學理論，分析入射光線兩個角度α0與β0，對軸對稱光學系統的兩種像差多項式的展開式做討論。上述兩種像差多項式，傳統上是以出光瞳極座標系為冪級數展開式。本論文不用出光瞳極座標系，而是以入射光線的角度α0與β0，推導波前像差多項式與光線像差多項式。此法雖然對主光線的描述較複雜，角度α0與β0也不同於出光瞳極座標系具有方便性，但卻能真實的呈現入射光線的角度和波前像差多項式與光線像差的關係。希望對於光學系統設計者，可直接從入射光線的角度α0與β0，評估各種類型的像差，進而設計更完善的光學系統。

Ideal image formation is located at paraxial region, once the system has pupil diameter value or field of view, aberration will immediately appear. If uses the concept of geometrical optics, when the angle between light and optical axis is too big to be described accurately, it will cause the imaging defects. For general speaking the larger the angle, the greater the difference of aberration will be appear.
This paper will place its focus on discussing mathematical theories of polynomials of wave-front and ray aberrations. To discuss these two kind of aberration polynomial expansions in a axisymmetric optical system, We analyze the incident light angles α0 and β0. Traditionally, these two aberration polynomials are based on the pupil polar coordinates for the power series expansion. In this paper, we derived wave-front aberration polynomials and ray aberration polynomials by the angles α0 and β0 of incident light instead of using the pupil polar coordinates. Although it is complex to describe the chief ray, and the angles α0 and β0 of incident light are not convenience as the pupil polar coordinates, but it can be truly presented the relationship between the angles of incident light and wave-front aberration polynomials or ray aberration polynomials. We hope optics system designers can evaluate the various types of aberrations by tracing the angles α0 and β0 of incident light, and then to design better optics systems.

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