为在特定平面上实现所需照明，从而满足各类LED照明系统的要求，通过透镜的第一面球面和第二面自由曲面的两次折射实现LED的配光设计.根据给出的LED辐射特性和所需实现的照明面上的能量分布，并预先定义透镜球面的结构参量，可得到一组一阶偏微分方程，数值差分求解直接求得透镜的自由曲面.光源采用发光面积1×1 mm2朗伯体发光的LED，视角180°，由自由曲面透镜得到的各种形状照明面的模拟均匀性皆接近90%，且能量利用率高于98%，编程解一阶偏微分方程组的计算时间少于20 s.

Secondary optical design of LED was realized through refractions on the first spherical surface and the second freeform surface of a freeform lens.Thereby the desired illumination was achieved and the demand of various LED illumination system was met.The numerical results of first-order partial differential equations sets were investigated to obtain a freeform surface and these equations could be obtained on knowing the characters of LED and the desired illumination,with a pro-defining of the first spherical surface of the lens.LED with 1×1mm2 Lambertian light emitting surface and viewing angle of 180°was applied,and the simulated illumination of various shapes all have a uniformity near to 90% and an energy efficiency higher than 98%.It costs less than 20 seconds to calculate the first-order partial differential equations.Once this method was applied in the secondary optical design of LED,the designing efficiency and the performance of the illumination system would surely be improved.