## 1 Introduction  
Regardless of the immediate environment of a receiving or radiating aerial, the space surrounding an antenna is usually subdivided into two regions: the near-field and the far-field regions [1], also known as the Fresnel and the Fraunhofer regions, respectively. While there is no clear boundary between these regions, a common and acceptable criterion is that \(2D^{2}/\lambda\) represents a safe far-zone distance, where \(D\) is the maximum dimension of an antenna aperture and \(\lambda\) is the operating wavelength [2]. At such a Fraunhofer distance, the maximum path-length difference between the contribution from the edge of the aperture and that from the center corresponds to the fraction \(\lambda/16\) or to a phase \(\pi/8\) radians, as very well illustrated by Figure 1.13 in [3]. However, as suggested by Equation (9) in [4], this criterion might not be sufficiently strict for aperture synthesis imaging applications. This is what is observed with the study reported in this note, which aims at comparing the well-known far-field conditions to the near-field ones in detail and to estimate if this criterion, which has been historically established for elementary antennas outside any imaging context, has to be revisited, or not, for antenna arrays when they are used for synthesis imaging with interferometric data. Such a comparison between near-field and far-field imaging has already been addressed in synthetic aperture imaging radiometry. However, among the differences that can be found between these two situations, one or the other, or several particularities, are very often neglected. This is not the case in this study, where near-field imaging is considered with spherical waves and without neglecting any term in the modeling equation, whatever the price to pay for taking them all into account in the numerical processing conducted with the aid of a computer. On the other hand, far-field imaging approximation is considered with plane waves according to the Van Citter-Zernike theorem [5; 6]. The question of the boundary between these two regions arises in synthetic aperture interferometry when complex visibilities of a scene observed at a given distance from the antenna array are inverted in order to retrieve the brightness temperature distribution of the synthesized field of view. Indeed, as soon as these complex visibilities have been obtained in the near-field region, they have to be inverted with the appropriate resolving matrix, namely the inverse of the modeling matrix of the instrument in near-field imaging conditions, not that in the far-field approximation; otherwise, errors might be expected in the retrieved map. Regarding synthesis imaging operations with an array of elementary antennas, it is often suggested in the literature to convert near-field visibilities to far-field ones and then to invert the latter with a standard far-field resolving matrix, thus making the calculation and the inversion of the near-field imaging matrix unnecessary. However, the conditions of validity and reliability of this kind of pre-processing are not always mentioned. Although it obviously has many advantages in terms of computational effort, it is well suited only for unresolved point sources, but it is definitively not for extended ones. As argued in this note, these approaches are therefore useless in many cases. This provides clear support for the approach adopted in this study, which makes no approximation in the modeling and no compromise in the processing as far as near-field conditions are concerned. It aims at inverting near-field complex visibilities with the appropriate resolving matrix obtained from the near-field modeling matrix without neglecting any term in the corresponding equation and whatever the computational effort to calculate it and to invert it.