Optimized method of lines for non-linear waveguide problems

The Method of Lines is a semi-analytical versatile tool for the solution of partial differential equations. For the analysis of spatial complex linear waveguide structures, this method is combined with impedance/admittance and field transformation, as well as with finite differences. This paper extends this approach to the treatment of structures with non-linear dielectric materials. The non-linear generalized transmission line equations are derived. An iterative algorithm based on the impedance/admittance transformation with the field transformation obtains efficient and self-consistent solutions. For demonstration, a non-linear stripe waveguide is considered. The Kerr non-linearity was investigated, though the general case is treatable. As a criterion for the correctness of the algorithm, a second harmonic generation as well as a bidirectional, spatially, and temporally energy exchange between the harmonics was examined. The specific limiting factors for the algorithm were explored. The approach can be used for any spatial structure, including, for instance, photonic crystal waveguides and metamaterials.