Finite-difference derivation of the reflectivity at output facets of dielectric waveguides with a highly diverging output beam

Brute force numerical computations are considered for the reflectivity of the output facet of a dielectric waveguide with or without an antireflection coating. Only scalar fields are considered, mainly in two-dimensional systems, through both exact equations and a series expansion, discretized with finite differences. A fair approximation of the series can be obtained with fast Fourier transforms (FFT's) only, with neither matrix inversion nor diagonalization. Junctions with a highly diverging output beam require a very large computational domain that can be coped with FFT methods only. This is emphasized through the detailed analysis of a three-dimensional example. (C) 1998 Optical Society of America.