The system of four differential equations governing counterpropagating quasi-phase matching are recast in a Hamiltonian form that leads to immediate insight into the nonlinear mixing process by inspection for all possible boundary conditions. A reduced Hamiltonian is found using conservation relations that are dependent on only two normalized field efficiencies and two aggregate phases. Hamiltonian contours are plotted in a series of phase-space cross sections to provide insight into the generalized behavior. The nonlinear eigenmodes are found, and their stability is examined. Finally, two specific counterpropagating quasi-phase-matching configurations, i.e., mirrored and mirrorless, are analyzed using this general Hamiltonian with the appropriate boundary conditions.
© 2004 Optical Society of America
(190.0190) Nonlinear optics : Nonlinear optics
(190.1450) Nonlinear optics : Bistability
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.3100) Nonlinear optics : Instabilities and chaos
Gary D. Landry and Theresa A. Maldonado, "Counterpropagating quasi-phase matching: a generalized analysis," J. Opt. Soc. Am. B 21, 1509-1521 (2004)