In this paper, we present sampling conditions for fast-Fourier-transform-based field propagations. The input field and the propagation kernel are analyzed in a combined manner to derive sampling criteria that guarantee accurate calculation results in the output plane. These sampling criteria are also applicable to the propagation of general fields. For focal field calculations, geometrical optics is used to obtain a priori knowledge about the input and output fields. This a priori knowledge is used to determine an optimum balance between computational load and calculation accuracy. In a numerical example, correct results are obtained even though both the input field and the propagation kernel are sampled below the Nyquist rate. Finally, we show how chirp z-transform-based zoom-algorithms may be analyzed using the same techniques.