Non-paraxial beam propagation methods

Exact analytical results are employed in the testing of split-step and finite difference approaches to the numerical solution of the non-paraxial non-linear Schrödinger equation. It is shown that conventional split-step schemes can lead to spurious oscillations in the solution and that fully finite difference descriptions may require prohibitive discretisation densities. Two new non-paraxial beam propagation methods, that overcome these difficulties, are reported. A modified split-step method and a difference-differential equation method are described and their predictions are validated using dispersion relations, an energy flow conservation relation and exact solutions. To conclude, results concerning 2D (transverse) beam self-focusing, for which no exact analytical solutions exist, are presented.