Analysis of the gradual transition from the near to the far field in single-slit diffraction

In Optics it is common to split up the formal analysis of diffraction according to two convenient approximations, in the near and far fields (also known as the Fresnel and Fraunhofer regimes, respectively). Within this scenario, geometrical optics, the optics describing the light phenomena observable in our everyday life, is introduced as the short-wavelength limit of near-field phenomena, assuming that the typical size of the aperture (or obstacle) that light is incident on is much larger than the light wavelength. With the purpose to provide an alternative view on how geometrical optics fits within the context of the diffraction theory, particularly how it emerges, the transition from the near to the far field is revisited here both analytically and numerically. Accordingly, first this transition is investigated in the case of Gaussian beam diffraction, since its full analyticity paves the way for a better understanding of the paradigmatic (and typical) case of diffraction by sharp-edged single slits. This latter case is then tackled both analytically, by means of some insightful approximations and guesses, and numerically. As it is shown, this analysis makes explicit the influence of the various parameters involved in diffraction processes, such as the typical size of the input (diffracted) wave or its wavelength, or the distance between the input and output planes. Moreover, analytical expressions have been determined for the critical turnover value of the slit width that separates typical Fraunhofer diffraction regimes from the behaviors eventually leading to the geometrical optics limit, finding a good agreement with both numerically simulated results and experimental data extracted from the literature.