Perfect optical vortices (POVs) provide a solution to address the challenge induced by strong dependence of classical optical vortices on their carried topological charges. However, traditional POVs are all shaped into bright rings with a single main lobe along the radial direction. Here we propose a method for enhanced control on the ring profile (the radial intensity profile of circular rings) of POVs based on modulated circular sine/cosine radial functions, which is realized by a circular Dammann grating embedded with a spiral phase. Specifically, a type of “absolute” dark POVs surrounded by two bright lobe rings in each side is presented, which provides a perfect annular potential well along those dark impulse rings for trapping low-index particles, cells, or quantum gases. In addition, several POVs with different ring profiles, including conventional POVs with bright rings, the dark POVs mentioned above, and also POVs with tunable ring profiles, are demonstrated. This work opens up new possibilities to controllably tune the ring profile of perfect vortices, and this type of generalized POVs will enrich the content of singular optics and expand the application scope of perfect vortices in a range of areas including optical manipulation, both quantum and classical optical communications, enhanced optical imaging, and also novel structured pumping lasers.