Abstract

Theoretical treatments of three and four-wave interactions are typically based on the transformation $q_j=\sqrt{{\rho }_j}\exp \left(i{\varphi }_j\right)$. Using the Manley-Rowe relations and the Hamiltonian, these systems are integrated in terms of elliptic functions [1]. A new approach, which ultimately leads to the same elliptic function solutions and is also based on the Hamiltonian structure, is introduced. It yields a useful geometric picture of wave interactions and simplifies the understanding and analysis of these processes. In doing so, it facilitates the development and optimization of control strategies for wave interactions such as quasi-phase-matching [1–3].

© 1998 Optical Society of America