Optical vortex trajectories in an astigmatic and elliptical Gaussian   beam
  An optical vortex, produced at one point in an optical beam, would propagate through an optical system to another point where the vortex can be used for some purpose. However, asymmetrical optical elements in such a system can cause astigmatism or at least distroy the rotational symmetry of the beam, which may affect the propagation of the vortex in an undesirable way. While an optical vortex in a rotationally symmetric, stigmatic Gaussian beam retains its initial morphology for as far as it propagates, the morphology of an optical vortex in an asymmetric or astigmatic Gaussian beam changes. The vortex can even be replaced by another with the opposite topological charge. We consider the behavior of single noncanonical vortices propagating in Gaussian beams that are asymmetric and/or astigmatic. General expressions for the vortex trajectories are provided. The locations of the flip planes and the evolution of the anisotropy of the vortex are considered for different non-ideal situations. 
Filippus S. Roux