In this work, the propagation of vortex beams is treated using a catastrophe theory approach. Analytic expressions are deduced to describe caustic surfaces produced by vortex lenses and vortex axicons. The obtained analytics allow us to explain the formation of the shadow region along the optical axis for vortex beams using geometric optics (previously, the zero axial intensity was explained just by diffraction effects). Thus, the presence of a vortex eikonal leads to a fundamental change in the type of axial caustic. Another important distinction of the caustics produced by vortex beams from those produced by nonvortex radial beams has been shown to consist in wavelength-dependence. The results of numerical simulation show that the propagation operator defined using a geometrical optics approximation agrees well with the numerical simulation results obtained using a nonparaxial diffraction operator based on the conical wave expansion.

中文翻译：

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涡旋透镜和涡旋轴锥产生的涡旋光束的焦散。

在这项工作中，使用突变理论方法来处理涡流束的传播。推导了解析表达式来描述由涡旋透镜和涡旋轴锥产生的苛性表面。所获得的分析结果使我们能够使用几何光学来解释涡旋光束沿光轴的阴影区域的形成（以前，零轴强度仅通过衍射效应来解释）。因此，涡旋电子波的存在导致轴向苛性碱类型的根本变化。涡流束与非涡流径向束所产生的碱的另一个重要区别已显示出与波长有关。