Efficient interface conditions (EICs) are derived for the propagation equation using the slowly varying envelope approximation for the dominant electric field component. At the interface between two different media, the two lateral second derivatives in the discretized propagation equation are adapted such that the discretized modal field equation is correct up to second order in the lateral grid spacing. Since the error term is then of the order of the lateral grid spacing, our EICs are first-order EICs. These interface conditions are compared with well-known zero-order EICs derived by Stern and Kim and Ramaswamy. It is shown that the first-order EICs yield faster convergence to the exact effective index value as the lateral grid spacing is decreased than do the zero-order EICs. It turns out that our EICs are very much like those derived by Vassallo. Using essentially the same method, he derived EICs of second and first order for the field component respectively parallel and perpendicular, to the interface. Hence the accuracy of his EICs is one order higher for the field component parallel to the interface, although it introduces an extra asymmetry in the propagation matrix.
Wijnands, F., Wijnands, F. H. G. M., Rasmussen, T., Hoekstra, H., Povlsen, J. H., Bjarklev, A., & de Ridder, R. M. (1995). Efficient interface conditions for the semi-vectorial finite difference beam propagation method. Optical and quantum electronics, 27(10), 961-975. https://doi.org/10.1007/BF00558488