Roulette caustics in transformation optics of structured light beams

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.