A general conformal framework for regular cusp beams

Abstract

In the last decades, structured optics unveiled a vast diversity of spatial modes and configurations to customize optical beams, in combination with a rich portfolio of techniques to control the degrees of freedom of light. Among all, accelerating cusp beams have known increasing attention, while conformal mappings have become popular as efficient and scalable methods for beam shaping and manipulation. In this work, we bridge the gap between these two fields, presenting a conformal framework to describe the generation and propagation of the so-called regular polygon beams. In particular, we show how their peculiar regular structure of cusps can be described as the envelope of hypocycloid caustics generated by a circular-sector transformation acting on a standard input beam. The good agreement between theory, numerical simulations, and experimental data confirms the validity of the model developed in the stationary phase approximation, providing an elegant analytical tool to describe the generation and control of regular cusp beams.