High numerical aperture focusing of singular beams

Rigorous vector analysis of high numerical aperture optical systems encounters severe difficulties. While existing analytic methods, based on the Richards-Wolf approach, allow focusing of nearly planar incident wavefronts, these methods break down for beams possessing considerable phase jumps, such as beams containing phase singularities. This work was motivated by the need to analyze a recently introduced metrological application of singular beams that demonstrated an experimental sensitivity of 20nm under a moderate numerical aperture of 0.4. One of the possibilities to obtain even better sensitivity is by increasing the numerical aperture of the optical system. In this work we address the issue of high numerical aperture focusing of the involved singular beams. Our solution exploits the superposition principle to evaluate the three dimensional focal distribution of the electromagnetic field provided the illuminating wavefront can be described as having piecewise quasi constant phase. A brief overview of singular beam microscopy is followed by deeper discussion of the involved high numerical aperture focusing issue. Further, a few examples of different singular beam focal field distributions are presented.