Many LED lighting applications involve the design of multiple optical surfaces. A prime example is a single lens with two refractive surfaces. In this paper, we consider an LED light source approximated as a point and a far-field target intensity. Using Hamilton’s characteristic functions, the design problem is converted into two generalized Monge–Ampère equations by deriving a generating function for each optical surface. The generating function is a generalization of the cost function in optimal transport theory. The generalized Monge–Ampère equations are solved using an iterative least-squares algorithm. To compute the first optical surface, we choose an intermediate far-field target intensity. By choosing different intermediate target intensities based on the source and target intensity, we develop a “knob” to distribute the refractive power over both surfaces of the lens. We apply the algorithm on two example problems and show it is capable of producing complicated target distributions.

中文翻译：

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双重透镜自由曲面设计的生成函数方法

许多LED照明应用涉及多个光学表面的设计。一个典型的例子是具有两个折射面的单透镜。在本文中，我们考虑将LED光源近似为一个点和远场目标强度。利用汉密尔顿的特征函数，通过推导每个光学表面的生成函数，将设计问题转换为两个广义的蒙格-安培方程。生成函数是最优运输理论中成本函数的概括。使用迭代最小二乘算法求解广义的Monge–Ampère方程。为了计算第一个光学表面，我们选择一个中间远场目标强度。通过根据源强度和目标强度选择不同的中间目标强度，我们开发了一个“旋钮”以将屈光力分布在镜片的两个表面上。我们将算法应用于两个示例问题，并证明了该算法能够产生复杂的目标分布。