A Comparison of Transparent Boundary Conditions for the Fresnel Equation

We consider two numerical transparent boundary conditions that have been previously introduced in the literature. The first condition (BPP) was proposed by Baskakov and Popov (1991, Wave Motion14, 121–128) and Papadakis et al. (1992, J. Acoust. Soc. Am.92, 2030–2038) while the second (SDY) is that of Schmidt and Deuflhard (1995, Comput. Math. Appl.29, 53–76) and Schmidt and Yevick (1997, J. Comput. Phys.134, 96–107). The latter procedure is explicitly tailored to the form of the underlying numerical propagation scheme and is therefore unconditionally stable and highly precise. Here we present a new derivation of the SDY approach. As a result of this analysis, we obtain a simple modification of the BPP method that guarantees accuracy and stability for long propagation step lengths.