Optimal focusing of a beam in a ring vortex

Conventional light focusing, i.e. concentration of an extended optical field within a small area around a point, is a frequently used process in Optics. An important extension to conventional focusing is the generation of the annular focal field of an optical beam. We discuss a simple optical setup that achieves this kind of focusing employing a phase plate as unique optical component. It is assumed that the annular focal field is modulated by an azimuthal phase of integer order q that converts the field in a ring vortex. We first establish the class of beams that being transmitted through the phase plate can be focused into a ring vortex. Then, for each beam in this class we determine the plate transmittance that generates the vortex with the maximum possible intensity, which is referred to as optimal ring vortex.