An iterative finite difference beam propagation method based on the Crank-Nicholson scheme is presented to simulate continuous wave (CW) second-order nonlinear effects in optical waveguides with the depletion of the pump wave taken into account. This method is an extension of the linear finite-difference beam propagation method and preserves the same order of accuracy. Comparisons with the previously published explicit finite difference beam propagation method and the rectangular approximation method are presented. Quasi-phase matched difference frequency generation in AlGaAs and quasi-phase-matched second harmonic generation in LiNbO3 are considered in the evaluation, showing that one iteration for the IFD-BPM is sufficient for the simulation with good accuracy and without increasing much computation time.
Hsu-Feng Chou, Ching-Fuh Lin, and Gin-Chung Wang, "An Iterative Finite Difference Beam Propagation Method for Modeling Second-Order Nonlinear Effects in Optical Waveguides," J. Lightwave Technol. 16, 1686- (1998)