A short note on the asymptotic stability of an oscillation-free eikonal splitting method

Different beam propagation methods (BPMs) have been fundamental in modern electromagnetical wave simulations. Challenges of the numerical strategy include the computational efficiency and stability, in particular when highly oscillatory optical waves are present. This paper concerns an eikonal splitting BPM scheme for two-dimensional paraxial Helmholtz equations together with transparent boundary conditions in slowly varying envelope approximations of active laser beams. It is shown that the finite difference method investigated is not only oscillation-free, but also asymptotically stable. This ensures the high efficiency and applicability in highly oscillatory wave applications.