General expression for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves

General expressions are derived for the intensities of plane waves interacting in a second-order nonlinear process in which there are only two incident waves. The analysis covers, in a unified way, sum-frequency mixing and difference-frequency mixing and includes parametric amplification. The unified solutions are compared with several approximations in common use in order to establish regions in which such approximations can be used with reasonable accuracy.>