The performance of the recently developed time-domain beam-propagation methods (TD-BPMs) is compared with that of the finite-difference time-domain (FDTD) method. For the TD-BPMs, we investigate full-band (FB), wide-band (WB), and narrow-band (NB) methods based on the implicit finite-difference (FD) schemes. Owing to the use of the slowly varying envelope, a time step of the TD-BPM can be chosen to be larger than that of the FDTD. Although the numerical results of a waveguide grating obtained from the FB-and WB-TD-BPMs agree well with that from the FDTD, the CPU times are longer than that of the FDTD due to the solution of broadly banded matrices. Introducing the alternating-direction implicit method (ADIM) into the WB-and NB-TD-BPMs contributes to a reduction in the CPU time. To make the methods more efficient, a fourth-order accurate FD formula is applied to the ADIM-based WB-and NB-TD-BPMs, leading to reduced CPU times to 40% and 6% of that of the FDTD, respectively.
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