A Stable and Accurate Preconditioner for Bidirectional Beam Propagation Method

A stable and accurate preconditioner based on the Denman-Beavers iteration (DBI) for the square-root operator to construct an efficient bidirectional beam propagation method (Bi-BPM) is proposed. Utilizing a branch-cut rotation technique, numerical instability from the evanescent wave in classical DBI is nearly removed. The biconjugate gradient stabilized method is applied to solve the preconditioned formulations due to its fast convergence rate. Moreover, the present preconditioner is shown to be highly efficient for dealing with the optical devices with multiple longitudinal dielectric interfaces, especially for those with aperiodic structures since it can be simultaneously obtained during the calculation of the square root of the characteristic matrix. To validate the present approach, a typical single facet waveguide structure and two kinds of multilayer coating structures are considered as the numerical examples. Results show that the present Bi-BPM has the advantages of excellent convergence, and high accuracy and stability, superior to the widely used Padé approximation.