Streamlines of modified vector Mathieu–Gauss beams

We characterize the streamlines and polarization structure of the modified vector Mathieu-Gauss (mvMG) beams, a class of vector solutions to the paraxial Maxwell equations in elliptic cylindrical coordinates. We derive analytical expressions for the TE and TM modes and demonstrate that their streamlines can be obtained either exactly or via asymptotic and trigonometric approximations. Furthermore, we explore superpositions of the TE and TM modes, including circularly polarized helical beams, and provide closed-form expressions for the corresponding streamlines and Stokes parameters. Our results reveal the topological richness of mvMG beams and offer new tools for their description, with potential applications in structured light, optical singularities, and beam shaping in non-Cartesian coordinate systems.