中文摘要：

应用李变换方法研究了超环面光栅的成像及其像差，介绍了李变换基本原理．光栅成像过程可分成五个部分，分别对应五个李变换；其中最重要的变换就是超环面光栅的衍射李变换．利用这五个李变换推导出了超环面光栅的成像公式，并应用光线追迹，对研究结果进行了验证，说明了李变换的方法能准确地描述超环面光栅的成像．该方法可以处理平面光源，能计算远离子午焦平面处的像差．李变换方法导出的成像公式描述了像平面坐标与物平面坐标、方向余弦之间的函数关系，体现了物空间变量与像空间对应变量之间映射关系．

英文摘要：

The Lie method to characterize imaging of a toroidal grating was presented. A brief introduction of Lie method was given. There are the five transformations to describe the imaging processes of a toroidal grating. The formulae for calculation of the coordinate and direction of light ray were obtained with the above Lie transformation, and its numerical results were compared with those by raytracing program of Shadow. An advantage of the Lie theory of optical imaging was that the pupil （aperture） coordinates did not appear in the Lie theory. The phase space variables x and y in the Lie theory provided for the extent of the object. The image equations show that the direction relationship between variables of the object space coordinate system and those of the image space coordinate system.