在LED照明应用中为实现矩形均匀照度分布要求，提出了一种基于一阶线性偏微分方程的自由曲面反射器设计方法。基于微分几何理论和折射定律描述了光线与自由曲面的相互作用。根据LED光源特性建立了朗伯光源与矩形被照面之间的能量拓扑关系，推导了自由曲面反射器的一阶线性偏微分方程和边界条件。分别使用Runge-Kutta法和有限差分法对边界条件和偏微分方程进行数值计算，并对计算结果进行光线追迹仿真。仿真结果表明自由曲面反射器光通利用率达到了94%，矩形被照面横向照度均匀度达到了0.9，纵向照度均匀度达到了0.8。程序计算时间少于1 s。

A freeform reflector design method, which is mainly based on a first-order linear partial differential equation, is proposed for uniform rectangular illuminance distribution in the field of LED illumination. The interaction between the freeform surface and the light beam is depicted based on theory of the differential geometry and Snell′s law. The energy topological relation between the Lambertian luminaire and the illuminated rectangular surface is established according to the LED luminous intensity distribution. The method deducts a first-order linear partial differential equation with some boundary conditions to represent the freeform reflector. The boundary conditions and the partial differential equation are solved by the Runge-Kutta method and finite difference method, respectively. The numerical results are validated in the form of raytracing, which reveal that the luminous flux efficiency is about 94%, the transverse uniformity of illuminance on the target surface is 0.9 and the longitudinal uniformity of illuminance on the target surface is 0.8. The numerical computation time is less than 1 s.