On the selection rules for three-wave interactions along ray trajectories

Abstract

Three-wave interactions may occur in media with quadratic nonlinearities, which allow for a flow of power between linear waves. The interactions require selection rules similar to conservation of energy and momentum to be satisfied. Equations are presented to solve these selection rules efficiently along the ray trajectory of a pump wave via integration similar to how ray trajectories are determined numerically. This is convenient when dealing with large amplitude beams which may interact with waves along its trajectory. Reformulating the selection rules as a system of ODEs means that the selection rules may be solved using dispersion relations for the three waves, even if the dispersion relations cannot be solved for frequency or wavevector, which would otherwise be needed. In numerical implementations, root-finding algorithms, which may be unstable for complicated dispersion relations, can be avoided. A simple set of equations valid in one-dimensional are presented first. The corresponding equations in arbitrary dimension, including 2D and 3D, are then derived. A set of equations are also derived to find different solutions to the selection rules at a fixed point. Examples with the derived equations applied to plasma physics are presented.