A stable complex Jacobi iterative solution of 3D semivectorial wide-angle beam propagation using the iterated Crank–Nicholson method

Abstract

An extension of the recently proposed three-dimensional (3D) wide-angle (WA) beam propagation method (BPM) using complex Jacobi iteration (CJI) taken into account polarization effects is presented. The resulting iterative BPM is faster than BPMs based on the traditional direct matrix inversion for waveguides with unchanging refractive index profiles during propagation direction. However, for varying refractive index waveguides the iterative method suffered from the fact that the iteration count between two successive cross-sections increases dramatically during the propagation direction. To overcome this problem, we propose the utility of the iterated Crank-Nicholson method. At each propagation step, the propagation equation is divided in multiple stages by the iterated Crank-Nicholson method and then each stage is recast in terms of a Helmholtz equation with source term, which is solved effectively by the complex Jacobi iterative method. The resulting approach is stable and well-suited for large structures with long propagation paths.