Discrete transparent boundary conditions for the numerical solution of Fresnelʹs equation

The paper presents a construction scheme of deriving transparent, i.e., reflection-free, boundary conditions for the numerical solution of Fresnelʹs equation (being formally equivalent to Schrödingerʹs equation). In contrast to previous suggestions, the method advocated here treasts the discrete problem after discretization of the time-like variable, i.e., in a Rothe method, which leads to a sequence of coupled boundary value problems. The thus obtained boundary conditions appear to be of a nonlocal Cauchy type. As it turns out, each kind of linear implicit discretization induces its own discrete transparent boundary conditions. Numerical experiments on technologically relevant examples from integrated optics are included.