Finite difference expressions for arbitrarily positioned dielectric steps in waveguide structures

The finite difference expressions for arbitrary positions of the dielectric steps in waveguide structures are derived and inserted into the beam propagation algorithm based on the method of lines (MoL). The accuracy of these expressions is tested by calculating the effective indexes of slab waveguides and comparing the results with analytic solutions, Tapered waveguides are analyzed as an application of this new approach, The propagation behavior of the field is presented and the modal power loss is compared with other methods.