Linear and quadratic exponential modulation of the solutions of the paraxial waveequation

Abstract

A review of well-known transformations, which allow us to pass from one solution of theparaxial wave equation (PWE) (in one transverse space variable) to another, is presented.Such transformations are framed within the unifying context of the Lie algebra formalism,being related indeed to symmetries of the PWE. Due to the closure property of thesymmetry group of the PWE we are led to consider as not trivial only the linear and thequadratic exponential modulation (accordingly, accompanied by a suitable shift orscaling of the space variables) of the original solutions of the PWE, which areseen to be just conveyed by a linear and a quadratic exponential modulation ofthe relevant ‘source’ functions. We will see that recently introduced solutions ofthe 1D PWE in both rectangular and polar coordinates can be deduced fromalready known solutions through the resulting symmetry transformation relatedschemes.