  Perfect optical vortex (POV) is a type of vortex beam with an infinite thin
ring and a fixed radius independent of its topological charge. Here we propose
the concept of generalized perfect optical vortex along arbitrary curves beyond
the regular shapes of circle and ellipse. Generalized perfect optical vortices
also share the similar properties as POVs, such as defined only along infinite
thin curves and owning topological charges independent of scales. Notably, they
naturally degenerate to the POVs and elliptic POVs along circles and ellipses,
respectively. We also experimentally generated the generalized perfect optical
vortices through a digital micromirror device (DMD) and measured the phase
distributions by interferometry, exhibiting good agreements with the
simulations. Moreover, we derive a proper modified formula to yield the
generalized perfect optical vortices with uniform intensity distribution along
predesigned curves. The generalized perfect optical vortices might find the
potential applications in optical tweezers and communication.
