The propagation of light in homogeneous media is a crucial technology in optical modeling and design as it constitutes a part of the vast majority of optical systems. Any improvements in accuracy and speed are therefore helpful. The far-field integral is one of the most widely used tools to calculate diffraction patterns. As a general rule, this approximate method requires the observation plane located in the far-field region, i.e., a very considerable propagation distance. Only in the well-designed (namely aberration-free) optical system does the far-field integral not suffer from the limitation of the large distance. Otherwise, the far-field integral cannot provide accurate results. In the present work, we generalize the far-field integral to a more general concept with a much more flexible application scope, which allows for the inclusion of aberrations as well. Finally, as an essential part of this generalization, the propagation to arbitrarily oriented planes is also taken into account.