The aim of this work is threefold. First, following Luneburg and using our own notation, we review the Cartesian ovals. Second, we obtain analytical expressions for the reflecting and refracting surfaces that transform a prescribed smooth two-dimensional wavefront into a spherical one. These results are applied to show that the reflecting surface that connects a plane wavefront to a spherical one is a parabolical surface, and we design a lens, with two freeform surfaces, that transforms a spherical wavefront into another spherical one. These examples show that our equations provide the well-known solution for these problems, which is given by the Cartesian ovals method. Third, we present a procedure to obtain exact expressions for the refracting and reflecting surfaces that connect two given arbitrary wavefronts; that is, by assuming that the optical path length between two points on the prescribed wavefronts is given by the designer the refracting and reflecting surfaces we are looking for are determined by a set of two algebraic equations, which in the general case have to be solved in a numerical way. These general results are applied to compute the analytical expressions for the reflecting and refracting surfaces that transform a parabolical initial wavefront into a plane one.

中文翻译：

---

折射和反射界面将给定的波前转换为另一个波前

这项工作的目的是三重的。首先，按照 Luneburg 并使用我们自己的符号，我们回顾笛卡尔椭圆。其次，我们获得了反射面和折射面的解析表达式，将指定的光滑二维波前转换为球面波前。这些结果用于表明连接平面波阵面和球面波阵面的反射面是抛物面，我们设计了一个具有两个自由曲面的透镜，将球面波阵面转换为另一个球面波阵面。这些例子表明我们的方程为这些问题提供了众所周知的解决方案，这是由笛卡尔椭圆方法给出的。第三，我们提出了一个程序来获得连接两个给定任意波前的折射和反射表面的精确表达式；那是，假设指定波前两点之间的光程长度由设计者给出，我们正在寻找的折射和反射表面由一组两个代数方程确定，在一般情况下，必须以数值求解道路。这些一般结果用于计算将抛物线初始波前转换为平面波前的反射和折射表面的解析表达式。