Validity of the Fresnel approximation in the diffraction of collimated beams

Most studies of the validity of the Fresnel approximation have relied principally on numerical results, because cancellation due to the oscillatory integrands suggests that the resulting field errors are difficult to analyze. A simple analysis is shown here, however, to give an excellent prediction of the associated errors in modeling the diffraction of a collimated beam. Further, the error estimates are presented as a universal contour map where only the contour labels depend on the aperture-size-to-wavelength ratio. The inaccuracy of the Kirchhoff boundary conditions effectively sets error bounds that are essential in deriving this universal map.