Performance evaluation of several implicit FDTD methods for optical waveguide analyses

The performance of the recently developed implicit finite-difference time-domain methods (FDTDs) is compared with that of the traditional explicit FDTD. For the implicit methods, the alternating-direction implicit (ADI) FDTD and the envelope ADI-FDTD are investigated. In addition, the locally one-dimensional (LOD) scheme is newly introduced into implicit FDTDs, i.e., LOD- and envelope LOD-FDTDs are developed for simple implementation of the algorithm and reduced CPU time. Numerical dispersion analysis is performed, demonstrating the improved dispersion properties of the envelope FDTD. Numerical results of a waveguide grating reveal that the wavelength responses obtained from the ADI/LOD-FDTD gradually shift toward a longer wavelength as the time step (Δt) is increased. For the envelope ADI/LOD-FDTD with Δt=8Δt_CFL, in which Δt_CFL is determined by the stability criterion, the responses are in good agreement with the response of the explicit FDTD, showing the comparable CPU time to that of the explicit FDTD. Further calculations of a waveguide with high-reflection coatings reveal that the CPU time of the envelope LOD-FDTD with Δt=32Δt_CFL is reduced to 25% of that of the explicit FDTD.