Engineering structured light with optical vortices

In this work, we demonstrate the possibility of generating and controlling any given kind of structured radially symmetric intensity profile with an embedded optical vortex. This is achieved with the use of Sturm–Liouville theory on a circular domain with Bessel, Laguerre–Gauss, Zernike, and Fourier bases. We show that the core intensity profile can be constructed independently of the topological charge of the vortex.