A full-vector finite-element beam propagation method (VFE-BPM) in terms of all the components of slowly varying electric fields is described for the analysis of three-dimensional (3-D) nonlinear optical waveguides. Electric fields obtained with this approach can be directly utilized for evaluating nonlinear refractive index distributions. To eliminate nonphysical, spurious solutions, hybrid edge/nodal elements are introduced. Furthermore, to avoid spurious reflections from the computational window edges, anisotropic perfectly matched layer boundary conditions are implemented, and to reduce computational effort for the nonlinear optical waveguide analysis, an iterative algorithm is also introduced. The effectiveness of the present approach is verified by way of numerical examples: nonlinear directional couplers, spatial soliton emission phenomena, and soliton couplers.
Takeshi Fujisawa and Masanori Koshiba, "Full-Vector Finite-Element Beam Propagation Method for Three-Dimensional Nonlinear Optical Waveguides," J. Lightwave Technol. 20, 1876- (2002)