Interpretation of the $w$-term

The term $w\sqrt{1-l^2-m^2}$ is often referred to as the $w$-term in the literature. The origin of this term is purely geometrical and arises due to the fact that fringe rotation effectively phases the array for a point in the sky referred to as the phase center direction. A wave front originating for this direction will then be received by all antennas and the signals will be multiplied in-phase at the correlator (effectively phasing the array). The locus of all points in 3D space, for which the array will remain phased is a sphere, referred to as the celestial sphere. A wave front from a point away from the phase tracking center but on the surface of such a sphere, will carry an extra phase, not due to the geometry of the array but because of its separation from the phase center. In that sense, the phase of the wavefront measured by a properly phased array in fact carries the information about the source structure and the $w$-term is the extra phase due to the spherical geometry of the problem. The sky can be approximated by a 2D plane close to the phase tracking center and the $w$-term can be ignored, which is another way of saying that a 2D approximation can be made for a small field of view. However sufficiently far away from the phase center, the phase due to the curvature of the celestial sphere, the $w$-term, must be take into account, and to continue to approximate the sky as a 2D plane, we will have to rotate the visibility by the $w$-term. This will be equivalent to shifting the phase centre and corresponds to a shift of the equivalent point in the image plane. Since the $w$-term is a function of the image co-ordinates, this shift is different for different parts of the image. Shifting the phase centre to any one of the points in the sky, will allow a 2D approximation only around that direction and not for the entire image. Hence the errors arising due to ignoring the $w$-term cannot be removed by a constant phase rotation of all the visibilities. This is another way of understanding that, in the strict sense, the sky brightness is not a Fourier transform of the visibilities.