  We present a general, Gaussian spatial mode propagation formalism for
describing the generation of higher order multi-spatial-mode beams generated
during nonlinear interactions. Furthermore, to implement the theory, we
simulate optical angular momentum transfer interactions, and show how one can
optimize the interaction to reduce the undesired modes. Past theoretical
treatments of this problem have often been phenomenological, at best. Here we
present an exact solution for the single-pass no-cavity regime, in which the
the nonlinear interaction is not overly strong. We apply our theory to two
experiments, with very good agreement, and give examples of several more
configurations, easily tested in the laboratory.
