利用条纹泽尼克多项式来表征自由曲面的光学元件，并将多项式中表示初级球差、彗差、像散项转换为矢量形式。利用矢量波像差理论,研究了自由曲面光学元件校正光学系统初级像差的特性。通过分析可知，自由曲面在光学系统中不同位置时所校正的像差特性不同。当自由曲面位于光学系统的孔径光阑（入瞳或出瞳）上可以校正光学系统全视场内为常数的初级像差；当自由曲面远离孔径光阑时，由于轴外视场成像光束口径的缩放与偏移，自由曲面可以校正非对称初级像差，且不同初级像差与视场依据关系不同。

The optical element of free-form surface can be expressed by the fringe Zernike polynomial and the terms of initial spherical sag, coma and astigmatism are transformed into the vector forms. Based on the vector wave aberration theory, the free-form optical element characteristic of optical system initial aberration correction is analyzed. Through the analysis, the free-form surfaces in optical system have different aberration emendation specialties at different positions. The free-form optical element can be corrected in full field of view for constant initial aberration as a stop aperture (entrance pupil or exit pupil) in optical system. When the free-form surface is far from the stop aperture of optical system, due to the scaling and offset of the imaging beam aperture for an off-axis field point, the free-form surface can correcte the asymmetric initial aberration, and the relationship of different initial aberration and field of view is different.