An analytical closed-form equation to design a freeform element that replicates an axicon is presented. In this paper, we get the mentioned analytical closed-form equation step by step. The equation comes from applying both the Fermat principle and the Snell's law at the first refractive surface. This Freeform axicon with azimuthal variation can be applied to produce annular conical beams with the possibility to receive wavefronts not necessarily collimated. In the paper, we test the equation in several scenarios to test the design of the Freeform axicon and the results of the performance of the Freeform axicon were as expected.

中文翻译：

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具有方位角变化的自由轴锥

提出了一个分析封闭形式方程来设计一个复制轴锥的自由形式元素。在本文中，我们一步一步地得到了所提到的解析闭式方程。该方程来自在第一折射表面应用费马原理和斯涅尔定律。这种具有方位角变化的自由曲面棱镜可用于产生环形锥形光束，并有可能接收不一定准直的波前。在论文中，我们在多个场景中对方程进行了测试，以测试 Freeform axicon 的设计，Freeform axicon 的性能结果符合预期。