The unitarity of split-operator finite difference and finite-element methods: Applications to longitudinally varying semiconductor rib waveguides

Split-operator finite-difference and finite-element techniques are applied to the calculation of losses in semiconductor rib waveguide Y junctions. It is shown that, unlike the finite-difference procedure, which is unitary for both uniform and nonuniform grid-point spacings, considerable care must be exercised in formulating the split-operator finite-element method in order to preserve the power in the propagating electric field. The calculations are performed in the context of the Fresnel approximation to the scalar Helmholtz equation, which yields accurate solutions for transverse electric (TE)-polarized electric fields in rib waveguides far from cutoff