和对称式光学系统比较,含有非球面透镜的光学系统能够实现一些特殊要求的变换。采用傍轴近似,用几何光学的方法分析了锥形透镜,得到了椭圆锥透镜透射函数的解析表达式。在此基础上进一步推广,得出了圆锥透镜和柱透镜的透射函数。利用以上结论构建了双柱透镜系统,用于实现变形分数变换。分数变换的可加性使得这一光学系统成为构建变形分数系统的基本单元,利用标量衍射理论,得到了该结构输出函数的解析式,并详细分析了结构参数对实现各种分数变换所需要满足的条件。

Compared with symmetrical optical system,the optical system which consists of non-spherical lens can implement some special mathematical transforms. Based on the paraxial approximation,the tapered lens are investigated by geometrical optics,and the transmission function is derived. Furthermore,the transmission functions of cone and cylinder lens are also derived. With the help of these conclusions,the optical system consisting of double cylinder lens is constructed. This setup is the basic unit to implement anamorphic fractional Fourier transform. Finally,on the basis of scalar diffraction theory,the analytical solution is obtained. Furthermore,the system parameters are studied in detail in order to satisfy the requirements for various fractional Fourier transforms.