Fresnel number

For an electromagnetic wave passing through an aperture and hitting a screen, the Fresnel number F is defined as where Conceptually, it is the number of half-period zones in the wavefront amplitude, counted from the center to the edge of the aperture, as seen from the observation point (the center of the imaging screen), where a half-period zone is defined so that the wavefront phase changes by

Another criterion called Gaussian pilot beam allowing to define far and near field conditions, consists to measure the actual wavefront surface curvature for an unaberrated system.

Lawrence[3] and now adopted in propagation codes like PROPER,[4] allows one to determine the realm of application of near and far field approximations taking into account the actual wavefront surface shape at the observation point, to sample its phase without aliasing.

This criterion is named Gaussian pilot beam and fixes the best propagation method (among angular spectrum, Fresnel and Fraunhofer diffraction) by looking at the behavior of a Gaussian beam piloted from the aperture position and the observation position.

Near/far field approximations are fixed by the analytical calculation of the Gaussian beam Rayleigh length and by its comparison with the input/output propagation distance.

the surface wavefront maintains itself nearly flat along its path, which means that no sampling rescaling is requested for the phase measurement.

In this case the beam is said to be near field at the observation point and angular spectrum method is adopted for the propagation.

On the contrary, once the ratio between input/output propagation distance and Gaussian pilot beam Rayleigh range yields

Fraunhofer diffraction returns then to be an asymptotic case that applies only when the input/output propagation distance is large enough to consider the quadratic phase term, within the Fresnel diffraction integral, negligible irrespectively to the actual curvature of the wavefront at the observation point.