The generalized Douglas scheme is applied to the time-domain finite-difference beam propagation methods (TD-BPM's) in rectangular and cylindrical coordinates. High accuracy and efficiency are demonstrated through the analysis of optical pulse propagation in slab and circularly symmetric waveguides. As an example of a reflection problem, the TD-BPM in cylindrical coordinates is applied to the analysis of a fiber Bragg grating with a sinusoidal index change. Effectiveness of the present scheme is discussed in comparison with the conventional TD-BPM and the finite-difference time-domain method.
Jun Shibayama, Tomokazu Takahashi, Junji Yamauchi, and Hisamatsu Nakano, "Efficient Time-Domain Finite-Difference Beam Propagation Methods for the Analysis of Slab and Circularly Symmetric Waveguides," J. Lightwave Technol. 18, 437- (2000)