The term
is often referred to as the -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
-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 -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 -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 -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 -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 -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.