Validity of the Fresnel approximation in the near field

By direct numerical-integration comparisons, it is established that the Fresnel approximation for collimated propagation is quite good (within about 2% in amplitude and 0.02 rad in phase) in every case, including that with the limit of a high Fresnel number. Moreover, the Fresnel approximation begins to break down in phase for spherical-wave propagation for beams faster than about f/12. It has been discovered, however, that if one also invokes the paraxial approximation, that is, replaces the spherical wave by a quadratic phase front, then the Fresnel approximation becomes valid for expanding (or diverging) beams as well. This result is substantiated through the use of stationary-phase arguments.