Appell transformation and symmetry transformations for the paraxial wave equation

The analysis presented in this paper is the natural continuation of that developed in a previous paper, where the Appell transformation, well known in the theory of the heat equation, has been interpreted in relation to the paraxial (free) propagation under both a rectangular and a circular cylindrical symmetry as connecting solutions of the pertinent paraxial wave equation, which are generated by Fourier or Hankel pairs of functions. Indeed, here we will reformulate in optical terms the result proved by Leutwiler relative to the n-dimensional heat equation. Accordingly, we will show that the optical Appell transformation is essentially-in the sense clarified in the text-the only symmetry transformation for the paraxial wave equation.