Space-time exchanged symplectic integrator approach to temporal-mode selective optical three-wave mixing frequency conversion

Abstract

A space-time exchanged symplectic integrator (SI) method is developed to solve the three-wave resonance interaction equations commonly used in describing a number of fundamental nonlinear optical phenomena, such as second harmonic generation, optical parametric amplification, etc. Exchanging the roles of the space and time coordinates allows us to solve these nonlinearly coupled equations with hard-wired boundary conditions using a highly accurate SI scheme instead of standard finite difference methods. The developed method is then applied to the optical three-wave mixing frequency conversion processes in which an input signal light is converted into an idler light guided by a strong pump pulse. The temporal-mode selectivity in the frequency conversion processes is examined by systematically varying system parameters and the results are summarized as contour maps of selectivity on the ‘group-slownesses’, defined by the inverse of the group velocities, of the signal pulse and of the idler pulse relative to that of the pump pulse. The areas showing the highest peaks on these undergo significant and characteristic changes in their positions as well as the heights for different values of the coupling strength γ. The observed trends are rationalized on the basis of the Schmidt modes obtained by decomposing the two-time transfer function that fully characterizes the dynamics of the signal–idler wave conversion.