Noniterative vectorial beam propagation method with a smoothing digital filter

The alternating direction implicit (ADI) technique is used in the numerical solution scheme for each longitudinal step, in the well-known vectorial beam propagation method (VBPM). It is used to reduce the linear-system matrix into a set of tridiagonal ones, which can then be solved by Thomas algorithm. Solutions using standard (Crank-Nicolson) VBPM for step-index waveguides contain small oscillations, which can significantly reduce the accuracy of the solutions. We show that this “noise” can be removed by a properly tailored low-pass digital filter in the spatial frequency domain. Results become as accurate as those obtained by iterative techniques. Examples deal first with the scaler LP₀₁ mode and a Gaussian field in a single-mode fiber converging to the vector HE₁₁ mode. Second, with an optimized Ti:LiNbO₃ bent waveguide using MgO. Finally, with a rib waveguide where advantages related to the use of the digital filter become very evident