Evolution of optical vortices in gradient media and curved spaces.

Light propagation in gradient media and curved spaces induce intriguing phenomena, such as focusing and self-imaging, thus delivering a wide range of applications. However, these systems are limited to excitations without orbital angular momentum, which may produce unforeseen results. Here, we demonstrate the reconstructions (or called imaging to some extent) of optical vortices (OVs) in two-dimensional (2D) gradient media and three-dimensional (3D) curved spaces. We present the evolution of OVs in two types of generalized Maxwell fisheye (GMFE) lenses from the perspective of geometrical and wave optics, and use coherent perfect absorbers (CPAs) to better recover the OVs in the converging position. Furthermore, we also demonstrate such phenomena in two types of 3D compact closed manifolds-sphere and spindle-which are also called geodesic lenses. Surprisingly, the results we obtained in 3D curved spaces can be seen as a strong verification of the Poincaré-Hopf theorem. Our work provides a new, to the best of our knowledge, platform to investigate the evolution of OVs on curved surfaces.