Asymptotic behavior of the boundary value problem of four-wave mixing in photorefractive media

Solutions to the four-wave mixing problem asymptotically approach a spatial steady state over large interaction distances. In this limit, intensities of the two-point boundary value problem may be found from solutions to quadratic equations. The different limiting behavior exhibited for positive and negative gain has important implications for applications in which linearity of system response is important. In the regime studied, positive gain leads quickly to maximally depleted pumps with increasing input signal. For negative gain, such depletion does not occur until higher signal levels are reached.