Efficient semivectorial mode solvers

A method, based on a semivectorial finite difference scheme, is described to construct modal fields for any two-dimensional refractive-index profile which is constant except at abrupt interfaces. The modal fields correspond to eigenvectors of the matrix equation to be solved. In order to find the eigenvectors and their corresponding eigenvalues, the matrix equation is formulated according to the inverse iteration method (IIM). Two versions of the IIM are compared. Further, two matrix equations are compared: one is based on the propagation equation, following from the three-dimensional paraxial wave equation, and the other is the Fresnel equation, leading to the standard eigenvalue equation. A new solution method for the matrix equation is presented. It is a refinement of the alternating direction implicit (ADI) method. This refined ADI method is compared to the standard conjugate gradient (CG) method. Both methods are tested for waveguides having a rectangular core cross section. The refined ADI method is found to be computationally more efficient than the unpreconditioned CG method.