Is the Rayleigh-Sommerfeld diffraction always an exact reference for high speed diffraction algorithms?

In several areas of optics and photonics the behavior of the electromagnetic waves has to be calculated with the scalar theory of diffraction by computational methods. Many of these high-speed diffraction algorithms based on a fast-Fourier-transformation are approximations of the Rayleigh-Sommerfeld-diffraction (RSD) theory. In this article a novel sampling condition for the well-sampling of the Riemann integral of the RSD is demonstrated, the fundamental restrictions due to this condition are discussed, it will be demonstrated that the restrictions are completely removed by a sampling below the Abbe resolution limit and a very general unified approach for applying the RSD outside its sampling domain is given.