Polynomial Gaussian beams

Abstract

Polynomial Gaussian beams, which are laser beams with Gaussian envelops and complex bivariate polynomial prefactors, provide us with a tractable means to investigate the evolution of optical vortices. A formalism for the propagation of such beams allows one to determine how the coefficients of the polynomial transform during propagation. This formalism is used to proof that global topological charge is conserved, provided that the Gaussian envelope of the beam is rotationally symmetric. For astigmatic Gaussian beams the global topological charge is not conserved and can change during propagation in steps of 2 when one of the optical vortices undergoes topological charge inversion. The global topological charge is bounded by the order of the polynomial prefactor. One can also investigate the behavior of vortices in random vortex fields by modelling them as polynomial Gaussian beams. The phase functions that exist in the vicinity of the annihilation of a vortex dipole are similar, regardless of the type of beam in which the vortices exist. One can therefore use polynomial Gaussian beams to find a way to force vortices in random vortex fields to annihilate. The number of optical vortices that can exist in a polynomial Gaussian beam depends on the reducibility of the polynomial prefactor. During propagation the reducibility of the prefactor is generally destroyed. However, if the morphologies of the vortices of a fully reducible prefactor are all the same, the reducibility is maintained during propagation. The results obtained from the analyses of polynomial Gaussian beams are confirmed by numerical simulations.