The fundamental solution of unidirectional pulse propagation equation

Abstract

In the article the fundamental solution of a variant of the three-dimensional wave equation known as unidirectional pulse propagation equation (UPPE) and its paraxial approximation is obtained. It is shown that the fundamental solution can be presented as a projection of a fundamental solution of the wave equation to some functional subspace. We discuss the degree of equivalence of the UPPE and the wave equation in this respect. In particular, we show that the UPPE, in contrast to the widespread belief, describes the wave propagation in both directions simultaneously, and remark non-causal character of its solutions