Mathieu’s modes: historical path of a “new beam”

In this article we show a comprehensive and didactically detailed analysis of elliptical optical beams, covering theoretical and experimental aspects. It begins with an introduction to the importance of studying light phenomena, highlighting research on technological and scientific advances provided by light beams. Our work highlights the scarcity of publicity about Mathieu’s optical beams, and the importance of producing technical teaching material for the training of Physics students in Brazil. We start from Maxwell’s equations to derive the Helmholtz equation, then look for a solution for the Helmholtz equation in cylindrical elliptical coordinates, detailing how to arrive at the Mathieu beams. The relationship between Mathieu beams and Bessel beams is explored, as well as their non-diffractive propagation. Furthermore, the importance of phase singularity in determining the order or topological charge of beams and the practical limitation of the propagation distance of non-diffractive beams are highlighted. Finally, the experimental generation of Mathieu beams is mentioned through rustic experimental setups such as holograms.

Keywords:
Non-diffractive beams; Mathieu beams; Modulated light; Elliptical beams