The beam propagation method (BPM) is well known as the most widely used propagation technique for modelling optoelectronic and photonic devices. The finite difference beam propagation method (FD-BPM) is the most commonly employed numerical technique for simulating field propagation in optical components. FD-BPM still offers computational advantages over time domain numerical techniques such as Finite Difference Time Domain (FDTD) method.
FD-BPM is usually implemented by using implicit schemes such as Crank-Nicolson scheme (CN) due to its stability. However, in the case of modelling three-dimensional (3D) photonic structures the CN scheme uses iterative matrix solvers and thus requires huge computational resources and long run-times. The way out might be the implementation of explicit Du-Fort Frankel (DFF) finite difference schemes. DFF is three-level explicit algorithm, but providing better stability condition than simple explicit schemes and very attractive computational efficiency for modelling realistic waveguide based 3D photonic devices.
Some examples of FD-BPM field simulation using the DFF scheme are given in this paper. The computational efficiency and stability of DFF FD-BPM formulation and inherent downsides of the method (such as spurious or “ghost” solutions) are compared against standard implicit CN FD-BPM schemes.