The generation, manipulation, and detection of beams carrying orbital angular momentum (OAM) have acquired paramount importance in a wide range of applications, from imaging to information and communication technology. To this aim, optical conformal mappings have been designed and implemented to transform light beams in an efficient and controlled way. In this paper, we analyze for the first time the caustics generated by conformal transformations of structured light beams, focusing in particular on log-pol mapping and circular-sector transformations, exploited for OAM multiplexing/sorting and multiplication/division, respectively. Using analytical models in the stationary phase approximation and numerical simulations, we show how the generated caustics find the correct description in the framework of the roulette curves described by a circle rotating around the corresponding transform. In addition, we consider and analyze the effect of the input OAM, showing how the twisted wavefront introduces a distortion on the caustics pattern which can be correlated to the amount of the input value of OAM.

携带轨道角动量（OAM）的光束的产生，操纵和检测在从成像到信息和通信技术的广泛应用中已变得至关重要。为了这个目的，已经设计并实现了光学共形映射，以有效且受控的方式来变换光束。在本文中，我们首次分析了结构光束的保形变换产生的焦散，特别是对数极化映射和循环扇区转换，分别用于OAM复用/排序和乘法/除法。使用固定相近似中的解析模型和数值模拟，我们显示了生成的焦散如何在轮盘曲线的框架中找到正确的描述，该轮盘曲线由围绕相应变换旋转的圆描述。此外，我们考虑并分析了输入OAM的影响，显示了扭曲的波前如何在苛性图形上引入失真，该失真可以与OAM的输入值量相关。