Counterpropagating quasi-phase matching: a generalized analysis

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.