  Spatially accelerating beams that are solutions to the Maxwell equations may
propagate along incomplete circular trajectories, after which diffraction
broadening takes over and the beams spread out. Taking these truncated Bessel
wave fields to the paraxial limit, some authors sustained that it is recovered
the known Airy beams (AiBs). Based on the angular spectrum representation of
optical fields, we demonstrated that the paraxial approximation rigorously
leads to off-axis focused beams instead of finite-energy AiBs. The latter will
arise under the umbrella of a nonparaxial approach following elliptical
trajectories in place of parabolas. Deviations from full-wave simulations
appear more severely in beam positioning rather than its local profile.
