Comparison between Staggered and Unstaggered Finite-Difference Time-Domain Grids for Few-Cycle Temporal Optical Soliton Propagation

Abstract

The auxiliary-differential-equation formulation of the finite-difference time-domain method has become a powerful tool for modeling electromagnetic wave propagation in linear and nonlinear dispersive media. In the first part of this paper, we compare the stability and accuracy of second- and fourth-order-accurate spatial central discretizations on staggered grids with a third-order-accurate spatial discretization on an unstaggered grid, combined with a second-order leapfrog time integration scheme for modeling linear dispersive phenomena in a one-dimensional single-resonance Lorentz medium. We use on the unstaggered grid the NS2 scheme introduced by Y. Liu, which combines forward and backward differencing for the spatial derivatives. Phase and attenuation errors are determined analytically across the entire frequency band of the low-loss single-resonance Lorentz model. In the second part of the paper, we compare the use of one-dimensional staggered and unstaggered grids for modeling the transient evolution of few-cycle optical localized pulses in a dispersive Lorentz medium with a delayed third-order nonlinearity. The focus of the study is on the decay of higher-order solitons under the combined action of dispersion, self-phase modulation, self-steepening, and stimulated Raman scattering. Numerical results from the simulations on the unstaggered and staggered grids are in excellent agreement. This study demonstrates the accuracy of the unstaggered scheme augmented with a linear/nonlinear dispersive media formulation for temporal soliton propagation.