A stable DuFort-Frankel beam-propagation method for lossy structures and those with perfectly matched layers

The computationally efficient DuFort-Frankel beam-propagation method (BPM) is ideally suited to parallel computing. Although the scheme is conditionally stable for structures with lossless materials, in the presence of material loss it can become unstable. It is shown that the use of perfectly matched layer (PML) boundary conditions can also cause instability, especially in the three-dimensional (3-D) case. These instabilities are characterized and a stabilized DuFort-Frankel scheme is presented that extends the scope of this powerful method to cover these practically important scenarios.